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+{"text":"\\section{Introduction}\n\nMuch of our knowledge of gas giant exoplanets comes from those transiting in front of bright stars on close-in orbits. These planets are easier to detect and are more favorable to follow-up observations such as radial velocity measurements. Their atmospheric characterization through transit spectroscopy is also facilitated by the large amount of flux received from bright stars. The vast majority of these planets have been discovered by dedicated ground-based photometric surveys, the most prolific being WASP \\citep{Pollacco2006, Collier2007} and HAT \\citep{Bakos2002, Bakos2004}. These two projects detected \\simi80 hot Jupiters around stars of magnitude $V \\leq 11$. The \\textit{CoRoT}\\xspace and \\textit{Kepler}\\xspace missions provided little addition to this sample because they target fainter stars and much smaller sky areas.\nThe XO project \\citep{McCullough2005} aims at detecting transiting exoplanets around bright stars from the ground using small telescopes. The project started in 2003 and discovered five close-in gas giant planets, XO-1b to XO-5b \\citep{McCullough2006, Burke2007, JohnsKrull2008, McCullough2008, Burke2008}. A second version of XO was deployed in 2011 and 2012 and operated nominally from 2012 to 2014. This yielded the discovery of the hot Jupiter XO-6b orbiting a bright, hot, and fast rotating star \\citep{Crouzet2017}, and of other planet candidates.\nIn this chapter, we provide an overview of this second version of XO. First, we present the goals of the project, the instrumental setup, and the observation strategy. Then, we detail the various steps of the data reduction pipeline, and we review the instrumental performances. Finally, we present the search for periodic signals in the lightcurves, the identification and follow-up observations of transit candidates, and we compare the detection yield to the number of detected planets.\n\n\n\n\\section{Goals of the project}\n\\label{sec: Goals}\n\nThe XO project aims at detecting transiting exoplanets around bright stars ($V < 11$) including some with long orbital periods ($P > 10$ days). Bright host stars ensure that follow-up observations can be achieved to confirm the planet candidates, and make their atmospheric characterization possible. Besides, whereas close-in transiting giant planets have been discovered and studied by dozens, only a few transiting ones with long orbital periods and bright host stars are known. The long period severely decreases the transit probability making such detections challenging. As a result, constraints on the physical properties of long period giant exoplanets have been obtained only recently from a statistical study of \\textit{Kepler}\\xspace objects, and their atmospheric properties are largely unknown \\citep[see][and the chapter entitled \\underline{Hot Jupiter Populations from Transit and RV Surveys} of this book]{Santerne2016}. \n\n\n\n\n\\section{Instrumental setup}\n\\label{sec: Instrumental setup}\n\nXO is composed of three identical units installed at Vermillion Cliffs Observatory, Kanab, Utah, at Observatorio del Teide, Tenerife, Canary Islands, and at Observatori Astron\\`omic del Montsec, near \\`Ager, Spain. Each unit is composed of two 10 cm diameter and 200 mm focal length Canon telephoto lenses equipped with Apogee E6 1024$\\times$1024 pixels CCD cameras, mounted on a German-Equatorial Paramount ME mount, and protected by a shelter with a computer-controlled roof (Figure \\ref{fig: XO units}). \nThe basic hardware is described in \\citet{McCullough2005} but was changed a bit in the transition from the first version of XO at a single observatory at Haleakala, Hawaii, to the three observatories in the small roll-off roof enclosures. The main change was from back-illuminated SITe CCDs with parallel port interface to front-side illuminated Kodak CCDs with Ethernet interface, as a cost compromise (the manufacturer SITe stopped making the inexpensive back-side illuminated CCDs).\nAll six lenses and cameras operate in a network configuration: they point toward the same fields of view over the night at the three locations. The CCDs are read in time delayed integration (TDI): pixels are read continuously while stars move along columns on the detector. The scan and read rates are adjusted to maintain round PSFs (Point Spread Functions). This setup results in strips of $43^{\\circ}.2\\times7^{\\circ}.2$ on the sky instead of square images. This technique maximizes the number of observed bright stars by enlarging the effective field of view. The exposure time to acquire a full strip is 5.3 minutes, which corresponds to 53 seconds on each $7^{\\circ}.2\\times7^{\\circ}.2$ square subimage. The nominal PSF FWHM (Full Width Half Maximum) is 1.2 pixels and varies in practice between 1 and 2.5 pixels (see Section \\textit{\\nameref{sec: FWHM variations}}). The units can be controlled remotely and operate robotically. Data from a weather sensor are recorded and analyzed in real time; they are used to send instructions such as opening or closing the roof, running the observations, etc. A webcam sensitive in the optical and infrared is mounted inside each enclosure and can be accessed from a web interface. A computer running on Linux is installed inside each unit to control the operations and store the data. An IDL program commands the operations of the unit and runs the observations. This program as well as other daily tasks are launched using crontab. \n\n\n\n\\begin{figure}\n\\includegraphics[width=5cm]{crouzet_fig1.png}\n\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig2.png}\n\\caption{XO units at Vermillion Cliffs Observatory, Kanab, Utah (left) and at the Observatorio del Teide, Canary Islands (right). A third unit is installed at the Observatori Astron\\`omic del Montsec near \\`Ager, mainland Spain.}\n\\label{fig: XO units}\n\\end{figure}\n\n\n\\newpage\n\n\\section{Observations}\n\\label{sec: Observations}\n\nThis section presents the observation strategy that we adopted for XO and gives an overview of the operations.\n\n\n\\subsection{Observation strategy}\n\\label{sec: Observation strategy}\n\nWe observed two strips with respective centers at $RA = 90^{\\circ}$ and $RA = 270^{\\circ}$ and $Dec$ extending from $+90^{\\circ}$ to $+54^{\\circ}$. This corresponds to an effective sky area of $520^{\\circ 2}$. We observed these strips for twice nine months between 2012 and 2014, one strip at a time depending on their observability. An example of one night observations with XO is shown in Figure \\ref{fig: XO observations}. Darks and flats are taken at the beginning and end of the night, science strips are taken the rest of the night. The scan direction is either North or South with a switch occurring around the meridian crossing. Under nominal conditions, we collect from 60 to 100 TDI strips each night depending on the season. \n\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig3.png}\n\\caption{Example of observing night with XO. These data were taken by one camera of the unit located at the Observatorio del Teide, Canary Islands, during the night of June 8, 2013. Only a sample of the data is shown. The TDI mode yields long strips instead of square images. The operations are summarized at the bottom from dusk (left) to dawn (right). The blue triangle near the meridian crossing indicates parts of the strips that are contaminated by one of the enclosure's walls (see Section \\textit{\\nameref{sec: Data selection}}).}\n\\label{fig: XO observations}\n\\end{figure}\n\n\n\n\\subsection{Data quality check}\n\nWe developed a data quality check software program that analyzes the observations after each night. All science and dark files are listed and their size is checked. Files with a very small size are corrupted (usually they contain a header with no data) and are removed from the list.\nThen, for each science image (strip), we measure its size in pixel units, the corresponding number of 1024$\\times$1024 sub-images, the median intensity, the sky background, and the number of point sources. We also compute the FWHM in the $x$ and $y$ directions using 100 bright, non-saturated stars. The median intensity and the number of point sources are also computed for each 1024$\\times$1024 sub-image. This program is ran for both cameras at each site.\nIn addition, we summarize the weather data and the CPU temperature over the night. We also report the status of the network power strip (NPS) and of the enclosure roof, the last commands that were issued at the end of the observations, and the available disk space.\nThe results are sent via email (Figure \\ref{fig: diagnostics}) with a separate alert email if the NPS is not in a nominal state. This program runs every day at each site.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig4.png}\n\\caption{Example of nightly diagnostics file for the night of March 2, 2013 for the XO unit at Observatori Astron\\`omic del Montsec. ``Michael'' and ``Walt'' are the nicknames of the two cameras. Each camera recorded 101 TDI strips; the diagnostics information for only three strip is shown for clarity (as indicated by `...'). The weather sensor information, NPS state, enclosure's roof state, CPU temperature, and Sun altitude are also reported.}\n\\label{fig: diagnostics}\n\\end{figure}\n\n\n\n\\subsection{Data management}\n\nThe data are stored on the computers at each site and are transferred periodically on servers at Space Telescope Science Institute (STScI) through internet using a $rsync$ command. A 8 TB storage space was allocated to the project. The full analysis of the data is performed on the STScI servers.\n\n\n\n\n\\section{Data reduction}\n\\label{sec: Data reduction}\n\nThe data reduction pipeline is divided in two phases. The first phase consists of reducing the data of each night. The second phase consists of gathering the data taken on the same field over all nights and to build the lightcurves. Data from the six cameras are reduced independently and the lightcurves are merged at the very end of the data reduction.\nThe pipeline is composed of \\textit{IDL} routines that are launched with \\textit{Bash} programs. The \\textit{Bash} commands iterate on each night and then on each field. The main advantage is that if \\textit{IDL} crashes, the \\textit{Bash} program just moves on to the next night or field.\n\n\n\n\\subsection{Carving the strips}\n\n\nFor each night, the strips are carved into $7^{\\circ}.2\\times7^{\\circ}.2$ square images. A World Coordinate System (WCS) solution is found using the astrometry.net software program \\citep{Lang2010} plus a six parameter astrometric solution. We use a two step process: first, each square image is centered on the coordinates of a predefined field (listed in Table \\ref{tab: xo fields}), then a precise astrometric solution is found. Observing in TDI mode allows us to observe these square fields more efficiently than simply cycling through them using standard telescope pointing and CCD readout, as done by other transit surveys.\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Central coordinates of the fields of view observed by XO. Because we observe in TDI mode, fields with the same $RA$ are observed within one scan. Only a small amount of data is obtained for fields 10 to 12.}\n\\label{tab: xo fields}\n\\vspace{3mm}\n\\begin{tabular}{cccccccccccccc}\n\\hline\n\\hline\nField & 00 & 01 & 02 & 03 & 04 & 05 & 06 & 07 & 08 & 09 & 10 & 11 & 12  \\\\\n\\hline\n$RA \\rm \\; [^{\\circ}]$ & 270 & 270 & 270 & 270 & 270 & 90 & 90 & 90 & 90 & 270  & 270 & 90 & 90  \\\\\n$Dec \\rm \\; [^{\\circ}]$ & +90 & +83 & +76 & +69 & +62 & +83 & +76 & +69 & +62 & +55 & +48 & +55 & +48  \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Image calibration}\n\n\nAbout five dark frames and five flat frames are taken for each camera at the beginning and end of the night. They come as strips of the same size as the science images, but because we use the TDI mode, the final darks and flats are 1-D vectors with 1024 elements corresponding to the columns.\nFor each dark frame, an outlier resistant mean is computed for each column after excluding the first 1024 rows. The resulting 1-D vectors that have a mean value between 500 and 2000 ADU (Analog to Digital Unit) are averaged to create a final dark. Those with higher or lower median values are not used, as they are contaminated by parasitic light for example. One dark is created for each night. In the absence of dark frames, we use the dark of the previous or the following night.\n\n\n\nOne flat is built for each camera. We use the TDI images recorded during twilight and calibrate them with the dark. We eliminate pixels that are close to saturation ($>$~50,000 ADU) as well as a region of $3\\times3$ pixels around them.\nWe eliminate stars in the following way. We build a median-filtered image where each pixel is replaced by the median of the $15\\times15$ surrounding pixels, and subtract it to the initial image. We compute the outlier-resistant standard deviation of the residual image. Pixels that are above three times this standard deviation are flagged and are removed from the initial image. This yields an image equivalent to the initial image but with the stars removed.\nThen, we divide each row by its median in order to give the same weight to all rows. We compute an outlier-resistant mean of the image along the columns (starting at row 1024) and normalize the resulting vector by its median. This yields a flat vector. We average the flat vectors obtained from each twilight image to build a flat vector for each day (Figure \\ref{fig: flat-fields}), and we average the daily flat vectors obtained from the first months of observations. \nPermanent warm columns are identified using the darks and are replaced by the average of the 10 neighboring valid columns (here a ``column'' refers to one vector element). We suppress high frequency variations by replacing each element by the average of its 10 neighbors. We normalize this vector by the median of the 50 central elements. This yields a masterflat that we use for all the observations. This procedure is repeated for all six camera.\nFinally, the science image calibration is performed by subtracting the dark vector to each row and dividing it by the flat vector.\n\n\n\nThe strips are affected by warm columns. There are typically 10 warm columns per image except for one CCD that produces about 100 warm columns. About a third are permanent and two thirds vary from image to image. We correct for them in each image in the following way. We compute a smoothed version of each row using a 11-pixel running median and subtract it to the original row. Then, we compute the median of each column in the residual image. Those exceeding a threshold are considered as warm; we use a threshold of 15 ADU for the six CCDs after a visual inspection. For these columns, this median is subtracted to the same column in the original image in order to remove the excess counts. We record the number of corrected columns, their indices, and their original values in the image header.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig5.pdf}\n\\caption{Example of flats built from twilight images taken during March, 2013 with one camera, zoomed in between columns 400 and 560. Because we use the TDI mode, flats are one-dimensional vectors with 1024 elements. We normalize and average these flats to built a masterflat for each camera and use it for all the observations.}\n\\label{fig: flat-fields}\n\\end{figure}\n\n\n\n\\subsection{Flux extraction}\n\n\nWe select around 6000 target stars in each field for the photometry using an image taken under excellent conditions (the ``reference image'', Figure \\ref{fig: reference image}). Stellar fluxes are extracted by circular aperture photometry using the \\textit{Stellar Photometry Software} program \\citep[\\textit{SPS,}][]{Janes1993}. We compute the sky background for each star using an annulus around the aperture and subtract it. To measure the flux, \\textit{SPS} does not simply sum the pixels inside the aperture: instead, it computes an intensity profile by interpolating between pixels, and measures the flux by integrating this profile over the aperture size. In the original version of the XO pipeline, only one aperture of 3 pixel radius was used for all the stars. We improved this by optimizing the aperture for each star, as described next. For this process, we use only high quality data that were selected after a first flagging procedure and visual inspection of the star -- epoch magnitude arrays (see Sections \\textit{\\nameref{sec: Building the lightcurves}} and \\textit{\\nameref{sec: Data selection}}). \nFor each star, we build lightcurves for nine apertures with radii ranging from 2 to 10 pixels, calculate their RMS, and select the aperture that leads to the smallest RMS. We use the point-to-point RMS rather than the standard RMS so the aperture optimization is not affected by long term trends. This is justified because point sources are identified in each image and the apertures are placed at their centroids.\nThe fields of view are crowded so the best aperture for a given star may be affected by other sources, and it shows large fluctuations for stars of similar magnitudes. To suppress these fluctuations, we build a function that sets the optimal aperture for each stellar magnitude. We divide the magnitude range [6, 15] into 100 bins, calculate the mean best aperture in each bin, and fit this distribution by a fourth order polynomial (Figure \\ref{fig: aperture function}). The last step is an iterative process to have the magnitude of each star correspond to that measured in its optimal aperture. \nWe run this process for two representative fields (00 and 02) and average them to obtain a final aperture function. We build an aperture function for each camera. Differences between cameras are due to different focus or stability, for example. Finally, the optimal aperture for each star is given by the aperture function rounded to the closest integer (because only a small number of apertures can be used, due to a limitation of \\textit{SPS}).\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig6.png}\n\\caption{Reference image for field 06. Point sources identified in all the images of this field are matched against this image. The image size is 1024$\\times$1024 pixels and $7^{\\circ}.2\\times7^{\\circ}.2$ on the sky. The gray scale at the bottom indicates the intensity in ADU (most bright pixels are beyond the scale limit). The star hosting the transiting hot Jupiter XO-6b is indicated by a green circle.}\n\\label{fig: reference image}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=5.7cm]{crouzet_fig7.png}\n\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig8.png}\n\\caption{Best photometric aperture as a function of stellar $R$ magnitude for two cameras, at Vermillion Cliffs Observatory (left) and at Observatori Astron\\`omic del Montsec (right). Each star is represented by a black dot. A mean is performed in each magnitude interval (red points) and a fourth order polynomial is fitted to define an aperture function (blue line). We use this function to define an aperture for each star.}\n\\label{fig: aperture function}\n\\end{figure}\n\n\n\n\\subsection{Building the lightcurves}\n\\label{sec: Building the lightcurves}\n\nThe strip carving, astrometry, image calibration, and flux extraction are performed for each night. We obtain a star coordinate -- magnitude file for each image. Because the \\textit{SPS} software identifies point sources in each image before doing the photometry (it does not handle input coordinates), we must correlate the sources found in each image to build the lightcurves. This correlation is made against the reference image. We include an image distortion correction performed in $x, y$ rather than in $RA, Dec$ because the observed fields are close to the celestial pole which yields degeneracies in $RA$. Then, we build a star -- epoch array that gathers the magnitudes recorded for all stars at all epochs. Similar star -- epoch arrays are built for the magnitude error, $RA$, $Dec$, $x$, $y$, sky background, airmass, and PSF cross-correlation parameter. We obtain one set of such arrays for each night, field, and camera.\n\n\nThe second phase of the pipeline is ran for each camera and each field. We combine all the nights together: we correlate the star coordinates from night to night and build one set of star -- epoch arrays covering all the observations. Examples of such arrays are shown in Figure \\ref{fig: cal arrays}. Because we are interested in bright stars, we truncate these arrays and keep only the 2000 brightest stars in each field, which corresponds to a limit $R$ magnitude around 12.5 and a photometric precision of 2-3\\%. We use only these stars in the rest of the pipeline.\n\n\n\n\n\\subsection{Photometric calibrations}\n\nThe next steps are photometric calibrations that are essential to improve the lightcurves from the raw photometry. These calibrations are applied independently for each scan direction (the strips are observed scanning North or South which yield different systematic effects). First, we compute the mean magnitude of each lightcurve and subtract it. From this point, the lightcurves consist of residual magnitudes rather than absolute magnitudes and we work with the residual magnitude arrays. We select reference stars in the following way. For a given star $s$, we subtract the photometric time series of $s$ from those of all the stars and evaluate the mean absolute deviation (MAD) of the resulting time series. Stars are sorted by increasing MAD; the first index is excluded because it corresponds to star $s$, and the following $N$ stars are kept as reference stars. Then, we build a reference lightcurve using an outlier-resistant mean of the $N$ reference stars, and subtract it to the lightcurve of star $s$. This yields a calibrated lightcurve for star $s$. We use $N = 10$. Lightcurves of stars with a bright nearby reference star are usually of better quality. For example, XO-6 has a reference star of magnitude $R = 9.6$ located at 5.6 arcmin (14 pixels) separation. This reference star makes XO-6b a good target for atmospheric characterization by differential spectrophotometry from the ground, if both stars can be placed on the detector. \nThen, for each epoch, we remove a 3rd order polynomial corresponding to low-frequency variations of the magnitude residuals in the CCD's \\textit{x,y} plane, also called ``L-flats\". We also remove a linear dependence of each lightcurve with airmass. At the end of the process, the mean magnitude of each lightcurve is subtracted again.\n\n\n\\subsection{Data selection}\n\\label{sec: Data selection}\n\n\nParts of the data are affected by poor weather, high sky background, instrumental malfunctions, etc., and must be removed. A lot of effort is put into inspecting the magnitude arrays, identifying the low quality parts, searching for the causes, defining criteria and procedures to flag them, and adjusting the flagging parameters. We summarize below the main causes of low quality data and the data selection procedures. Figure \\ref{fig: cal arrays} shows an example of star -- epoch magnitude arrays before and after the flagging.\n\n\n\\begin{itemize}\n\n \\item The main cause of bad data is poor weather. To quantify this, we calculate the standard deviation of the magnitude residuals at each epoch, $\\sigma_{ep}$, using stars in the magnitude range [8.3, 10]. We flag epochs with $\\sigma_{ep}$ above a given threshold that we set manually for each camera and each field (Figure ~\\ref{fig: histo sigep}). These thresholds are between 1.9\\% and 4.5\\% with an average at 2.3\\%.\n \n \\item We flag data with a sky background larger than 10,000 ADU and those with an airmass larger than 3.\n \n \\item We flag epochs where one of the enclosure's wall is visible in the images, which occurs when the observed field is at low altitude. Although part of the image could be used, some reference stars may be missing so we discard the whole image. These images show a large difference of sky background between the part pointing at the sky and that pointing at the wall. We identify them by calculating the standard deviation of the sky background across the image normalized to its mean. Those with a value larger than a given threshold are flagged.\n \n \\item We flag epochs where less than half of the stars have a valid magnitude measurement. \n\n \\item We flag epochs where the cross-correlation parameter averaged over all stars is lower than a threshold value, ranging from 0.61 to 0.74 depending on the camera and field. This parameter is a crude measurement of the PSF shape. In particular, it identifies images that are affected by imperfect tracking, which have a low correlation parameter (see Section \\textit{\\nameref{sec: Mount stability}}).\n\n\\end{itemize}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig9.png}\n\\caption{Star -- epoch data arrays generated by the XO pipeline. Stars and epochs are sorted on the $x$ and $y$ axis by increasing magnitude and time, respectively. The full arrays contain 2000 stars and around 12,000 epochs. A subset of 475 stars $\\times$ 1640 epochs is displayed here, and corresponds to field 02 observed with one camera at the Observatorio del Teide between March 12 and April 25, 2013. From left to right: magnitude residuals before flagging, magnitude residuals after flagging, sky background, airmass, cross-correlation function. In the magnitude residual arrays, stars that have been discarded appear as white columns, variable stars appear as alternatively dark and bright columns, and epochs with an increased dispersion $\\sigma_{ep}$ appear as black and white regions (before they are flagged).}\n\\label{fig: cal arrays}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig10.pdf}\n\\caption{Histogram of the standard deviation of the magnitude residuals at each epoch, $\\sigma_{ep}$, for field 00 observed with one camera at Vermillion Cliffs Observatory over the whole duration of the observations (black line). The tail of the histogram corresponds to data taken under poor conditions, generally due to bad weather, that yield larger values of $\\sigma_{ep}$. A threshold (red line) is set manually for each field and each camera to flag these data. A similar approach is used for other flagging parameters.}\n\\label{fig: histo sigep}\n\\end{figure}\n\n\n\n\n\\subsection{Correction of systematic effects and final lightcurves}\n\nWe correct the remaining systematic effects in the cleaned residual magnitude arrays using the \\textit{SysRem} algorithm \\citep[\\textit{Systematic Removal},][]{Tamuz2005}. We correct for 10 eigenfunctions using 20 iterations, still per camera and field independently. Finally, we combine the lightcurves and other parameters obtained from the six cameras, interleaving the epochs and sorting them by ascending Julian date. This yields one set of star -- epoch arrays for each field.\n\n\n\n\n\\section{Instrumental performances}\n\\label{sec: Instrumental performances}\n\nIn this section, we review some technical issues that we encountered along the observations and present the photometric performances of the instruments.\n\n\\subsection{FWHM variations}\n\\label{sec: FWHM variations}\n\nAfter analyzing the first months of observations, we found significant variations of the PSF FWHM that are well correlated with the ambient temperature (Figure \\ref{fig: fwhm variations}). This effect is common and due to the lenses. Such PSF variations affect the flux measurements. To minimize this effect, the lenses were surrounded by a thermal regulation system made of flat and soft resistor strips for heating, a thermal probe, and a synthetic isolator, but this system was not efficient enough. After the first nine months, we increased the heating power and the isolation layer which reduced these FWHM fluctuations.\n\n\n\\begin{figure}\n\\includegraphics[width=5.7cm]{crouzet_fig11.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig12.png}\n\\includegraphics[width=5.7cm]{crouzet_fig13.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig14.png}\n\\includegraphics[width=5.7cm]{crouzet_fig15.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig16.png}\n\\caption{Variations of the PSF FWHM as a function of the ambient temperature during the first months of observations for the six cameras. Top: Vermillion Cliffs Observatory; middle: Observatorio del Teide; bottom: Observatori Astron\\`omic del Montsec. These variations were minimized afterwards by improving the temperature control system.}\n\\label{fig: fwhm variations}\n\\end{figure}\n\n\n\n\\subsection{Mount stability}\n\\label{sec: Mount stability}\n\n\nSome series of images show elongated PSFs or even two point sources separated by a few pixels instead of a single star (Figure \\ref{fig: elongated fwhm}). The cause is a mechanical defect in the mount resulting in an imperfect tracking. This behavior is seen only for a small fraction of the data taken at Observatorio del Teide and was corrected after the first nine months of observations. The affected images are discarded following a procedure described in Section \\textit{\\nameref{sec: Data selection}}. \n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig17.png}\n\\caption{Zoom on a standard image (left) and on one with elongated PSFs due to a mount defect (right). This behavior is seen only for a small fraction of the data taken at Observatorio del Teide.}\n\\label{fig: elongated fwhm}\n\\end{figure}\n\n\n\n\\subsection{Interferences}\n\n\nAnother defect is the presence of electrical interferences that create divisions within a strip, each with a specific background level, usually separated by sharp horizontal boundaries, and which vary from one strip to another (Figure \\ref{fig: interferences}). This effect is seen for a small fraction of the data taken at Observatorio del Teide and for a very small fraction of the data taken at Vermillion Cliffs Observatory. We solved this effect by improving the electrical isolation of the systems. No clear consequence is identified on the photometry probably because the excess counts are removed while subtracting the sky background, so we keep these data as they are.\n\n\n\\begin{figure}\n\\includegraphics[width=6.15cm]{crouzet_fig18.png}\\hfill\n\\includegraphics[width=5.4cm]{crouzet_fig19.png}\n\\caption{Example of images affected by electrical interferences at the Observatorio del Teide. Left: effect on the strips taken during the night of February 25, 2013. Some strips are affected at the beginning and end of the night. The scale in ADU is indicated at the bottom. Right: effect on a square image (obtained after carving the parent strip) and projection along the $y$ direction downwards.}\n\\label{fig: interferences}\n\\end{figure}\n\n\n\n\\subsection{Photometric precision}\n\\label{sec: Photometric precision}\n\n\nTo evaluate the photometric precision of the instruments, we calculate the standard deviation of the lightcurves over all the observations and report them in a RMS -- magnitude diagram. An example is shown in Figure \\ref{fig: RMS diagram} for one camera and one field. We obtain a precision between 1\\% and 3\\% for stars of magnitude 9 to 12.5 considering all the exposures, on a timescale of 6.3 minutes. This precision is twice larger than the theoretical prediction, on average. We also calculate the point-to-point RMS which does not consider correlated noise. The point-to-point RMS agrees well with the theoretical predictions for stars fainter than magnitude 11 and is slightly larger for brighter stars. The difference between the true and point-to-point RMS indicates that lightcurves are dominated by correlated noise, which is often the case for ground-based time-series photometric observations. For XO, the large amount of collected data by six different cameras from three different sites on the same fields averages these variations and reduces the RMS drastically when folding the lightcurves at given periods, as shown in Figures \\ref{fig: xo poster}, \\ref{fig: lightcurves}, and \\ref{fig: lightcurves long}. For example, after folding the lightcurve of the hot Jupiter XO-6b at the orbital period of the planet, the RMS is 1.3 mmag on a 30 min timescale (the ingress and egress duration) and 0.8 mmag on a 1.5 hour timescale (half the transit duration). \n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig20.png}\n\\caption{RMS -- magnitude diagram for the stars of field 00 obtained from one camera at Vermillion Cliffs Observatory over the full duration of the observations (twice nine months). Each point represents the standard deviation of the lightcurve as a function of the stellar $R$ magnitude. The true RMS (black dots) and the point-to-point RMS (red dots) are calculated from the full unbinned lightcurves with one point taken every 6.3 minutes on average. We also show the point-to-point RMS on a one-hour timescale obtained after binning the lightcurves with a 10-point boxcar (gray dots). Several noise components computed theoretically for the unbinned lightcurves are indicated: Poisson noise (dashed line), sky background noise (dash-dot line), read-out noise (dotted line), a systematic noise floor arbitrarily set at 0.4\\% (dash-dot-dot line), and total noise (plain line).}\n\\label{fig: RMS diagram}\n\\end{figure}\n\n\n\n\n\\section{Planet search}\n\\label{sec: Planet search}\n\nIn this section, we present the search for transiting exoplanets from the extracted lightcurves. Several steps are necessary: searching for transit signals, selecting viable planet candidates, and confirming or rejecting them through follow-up observations. Then, we compare the number of detected planets with the expected yield. \n\n\n\\subsection{Search for periodic signals}\n\\label{sec: Search for periodic signals}\n\nWe search for periodic signals in the lightcurves using the BLS algorithm \\citep[Box Least Square,][]{Kovacs2002}. This program searches for square shapes in the lightcurves representing transit-like events. Because we are interested in short and long period planets, we perform the search over a wide period range: $0.4 < P < 100$ days. The frequency comb is set by several parameters: the expected transit duration $\\tau$, the total extent of the observations $t_{obs}$ (two years), and the maximum number of lightcurve bins $N_{bin}$ that BLS can handle. The transit duration depends on $P$ but also on the stellar density $\\rho_\\star$. We use the known exoplanet population discovered by the ground-based transit surveys WASP and HAT to bracket the range of stellar densities and adopt a range $0.15 < \\rho_\\star < 5 \\rm \\; g\\,cm^{-3}$. This bracketing is necessary to limit the search to plausible systems.\n\nThe transit duration $\\tau$ varies widely within the period search range so a unique frequency comb is not appropriate (again it would yield too many unphysical detections). We divide the period range into 15 intervals with sizes following approximately a logarithmic scale. For each interval, we calculate the limits of the plausible transit duty cycles $q$, where $q\\, = \\, \\tau \/ P$, and use them to build a specific frequency comb (Figure \\ref{fig: transit duty cycle}).\nEach frequency comb has a constant spacing in $\\Delta f \/ f$, which we set to keep a maximum timing error over $t_{obs}$ as one fourth of the transit duration. We also set the number of lightcurve bins to keep a minimum of four in-transit bins. This ensures that the transit events overlap well with each other when folding the lightcurve. We run the BLS search in each interval separately. During this search, we refine the frequency comb with nine times more frequencies which we apply locally, first around the forty best frequencies, then around the best frequency. For each lightcurve, BLS returns the period that yields the largest power. In addition, we calculate several parameters corresponding to that signal: $q$, $\\varphi_{min}$ and $\\varphi_{max}$ (the minimum and maximum in-transit phases), the transit depth $\\delta$, and $\\alpha$. This latter parameter is equivalent to the signal to noise ratio of the transit: $\\alpha = \\delta \/ \\sigma \\times \\sqrt{N \\times q}$, where $\\sigma$ is the standard deviation of the residual lightcurve after subtracting the transit signal, $N$ is the number of data points, and $N \\times q$ is the number of in-transit data points (assuming that the points are equally distributed).\n \nWe run BLS twice. After the first run, the histogram of identified periods shows several peaks that are mostly centered around integer values below 15 days, which indicate aliases of the day-night cycle. Another peak around 29 days is present and is probably due to the Moon cycle. We define filters manually to exclude periods around these peaks and perform a second run.\n\nFinally, we combine the 15 intervals into 3 period ranges: 0.4 -- 1 day, 1 -- 10 days, and 10 -- 100 days. For each lightcurve, we compare the results obtained for the intervals within each range and keep the period that yields the largest power in the BLS spectrum.\nAll the lightcurves end up with a best period but most of them do not contain a valid signal. We keep only those with $\\alpha > 15$ and sort them by decreasing $\\alpha$ (lightcurves with the most significant signals are ranked first). This yields around 300 lightcurves out of 2000 for each field. We developed an interactive program in $IDL$ to inspect these selected lightcurves and to identify transit candidates, as displayed in Figure \\ref{fig: xo poster}. From this search, we identified hundreds of variable stars and a few tens of transiting planet candidates. Examples of lightcurves of variable objects are shown in Figures~\\ref{fig: lightcurves} and ~\\ref{fig: lightcurves long}.\n\n\nTo evaluate the transit search efficiency, we inject fake transits in the lightcurves. We choose 20 lightcurves of stars of various magnitudes and assign them different periods as well as transit parameters randomly chosen within physically plausible ranges. We also inject the transits of the planets discovered by the WASP and HAT surveys in the lightcurves of stars of similar magnitudes (these planets have periods generally below 10 days). This yields a total of 167 lightcurves with injected transits that we concatenate at the end of the star -- epoch magnitude array, and that we analyze in the same way as regular lightcurves. More than 90\\% of the injected planets are recovered after the BLS search and the transit candidate selection.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig21.pdf}\n\\caption{Transit duty cycle $q$ as a function of orbital period $P$ for different stellar densities $\\rho_\\star$. Values of $\\rho_\\star$ from top to bottom are 0.15, 0.3, 1.41 (Sun), 3, and 5 $\\rm g\\,cm^{-3}$ and are represented by a red dashed, red dotted, green plain, blue dotted, and blue dashed line respectively. We split the period search range into 15 intervals and define the allowed duty cycles in each of them using the extremum values obtained for the stellar densities 0.15 and 5 $\\rm g\\,cm^{-3}$, as indicated by the boxes. A specific frequency comb is built for each interval.}\n\\label{fig: transit duty cycle}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{crouzet_fig22.png}\n\\caption{Image of the interactive interface that is used to analyze lightcurves and identify transit candidates. This example is that of the hot Jupiter XO-6b \\citep{Crouzet2017}. The interface shows the lightcurve folded at the best period (top left) with individual data points in black and a binning in yellow, the BLS spectrum (top right), the full lightcurve (middle left) where the transits are indicated in green, a zoom on the transit (middle right), the centroid variations, a visible and an infrared image of the star's neighborhood, the results of a transit fit, and a list of parameters (BLS outputs, transit parameters, magnitudes, color, spectral type, Simbad information, etc.).}\n\\label{fig: xo poster}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=5.7cm]{crouzet_fig23.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig24.png}\n\\includegraphics[width=5.7cm]{crouzet_fig25.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig26.png}\n\\includegraphics[width=5.7cm]{crouzet_fig27.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig28.png}\n\\includegraphics[width=5.7cm]{crouzet_fig29.png}\\hfill       \n\\includegraphics[width=5.7cm]{crouzet_fig30.png}\n\\caption{Lightcurves of eclipsing objects and variable stars with short periods obtained with XO. Individual data points obtained with a time interval of 6.3 minutes are shown in black and a binning is shown in red. The period in days and the $R$ magnitude are indicated on each plot. From left to right and top to bottom: a transiting planet candidate with a 12 day period, a probable eclipsing binary, a semi-detached eclipsing binary, two variable stars, three variable stars with beats (with the lowest period indicated).}\n\\label{fig: lightcurves}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=5.7cm]{crouzet_fig31.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig32.png}\n\\includegraphics[width=5.7cm]{crouzet_fig33.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig34.png}\n\\includegraphics[width=5.7cm]{crouzet_fig35.png}\\hfill\n\\includegraphics[width=5.7cm]{crouzet_fig36.png}       \n\\caption{Same as Figure \\ref{fig: lightcurves} for long period objects. From left to right and top to bottom: an eccentric eclipsing binary with a 88 day period, a heartbeat eccentric eclipsing binary with a 20 day period, a 24 day period variable star with amplitude variations, and three variable stars with 115 day, 31 day, and 20 day periods.}\n\\label{fig: lightcurves long}\n\\end{figure}\n\n\n\n\n\\subsection{Follow-up observations}\n\\label{sec: Follow-up observations}\n\nFollow-up observations are necessary to confirm or reject transit candidates. We built a large follow-up team of amateur and professional astronomers who run these observations using facilities reported in Table \\ref{tab: follow-up facilities}. We perform photometry at the predicted transit ephemerides to check the reality of the signals; most candidates are not detected and are rejected. Then, we perform multi-color photometry: we observe transits in different bandpasses to identify eclipsing binaries, which have a color-dependent transit depth. We also compare the odd and even transit depths, which usually differ for eclipsing binaries, and search for secondary eclipses that may be unseen in the XO lightcurves. Then, valid candidates are sent for radial velocity observations with the SOPHIE spectrograph at the Observatoire de Haute-Provence, France \\citep{Bouchy2009}. In practice, follow-up observations are very time consuming, so the candidates are ordered by priority and the type of follow-up observation is carefully chosen. For example, a few candidates show a very clear signal in the XO data and have all the properties of a transiting planet; these are sent directly for radial velocities. Others are sent to multi-color photometry: one transit observed in two colors can be enough to identify an eclipsing binary. Finally, some objects require specific observations. This was the case for the hot Jupiter XO-6b which orbits a fast rotating F5 star: the radial velocity precision was limited to about 70 $\\rm m \\,s^{-1}$ due to the stellar rotation, which was too large to confirm the presence of the planet. We performed Rossiter-McLaughlin observations with SOPHIE and detected the planet's signature in Doppler tomography, which confirmed its planetary nature \\citep{Crouzet2017}. Follow-up observations of other transit candidates are underway.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Observatories and telescopes used for follow-up observations of XO transit candidates. The diameters of the primary mirrors are given in inches and cm.}\n\\label{tab: follow-up facilities}\n\\begin{tabular}{lll}\n\\hline\n\\hline\nObservatory & Telescope & Purpose  \\\\\n\\hline\nObservatoire de Haute-Provence, France & 76\" (193 cm), SOPHIE spectrograph  &  Radial velocities \\\\\nHereford Arizona Observatory, Arizona, USA & Celestron 11\" (28 cm), Meade 14\" (36 cm)  & Photometry \\\\\nActon Sky Portal, Massachusetts, USA & 11\" (28 cm)  &  Photometry  \\\\\nObservatori Astron\\`omic del Montsec, Catalonia, Spain & Joan Or\\'o Telescope 31\" (80 cm)  & Photometry  \\\\\nObservatoire de Nice, France & Schaumasse 16\" (40 cm)  &  Photometry  \\\\\nVermillion Cliffs Observatory, Kanab, Utah, USA & 24'' (60 cm)  &  Photometry \\\\\nElgin Observatory, Elgin, Oregon, USA & 12\" (30 cm)  &  Photometry \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Planet detection yield and discoveries}\n\\label{sec: Yield and planet detection}\n\nWe estimate the yield of exoplanet discovery with XO and compare it to the number of detected planets. The aim is to inform us on the observation and data analysis efficiency and not to perform an accurate completeness study. Thus, we compute only simple estimates.\nWe use the yield simulations developed by \\citet{Sullivan2015} for the NASA \\textit{TESS}\\xspace mission \\citep[\\textit{Transiting Exoplanet Survey Satellite,}][]{Ricker2014}, which in turn are based off results from the \\textit{Kepler}\\xspace mission. We consider only planets with orbital periods $P < 20$ days, because they are easier to detect due to the decreasing transit probability for longer orbital periods, as evidenced by the WASP \\& HAT planet distributions peaking around 3-4 days.\nThe \\textit{TESS}\\xspace simulations cover 95\\% of the entire sky, \\textit{i.e.}\\xspace 38,000\\ensuremath{^{\\circ2}}\\xspace. The number of transiting planets with $P < 20$ days and radii greater than four times that of Earth ($R_p > 0.4 \\;R_{Jup}$) orbiting stars with Cousins $I$ band magnitude $I_c < 10$ over this area is expected to be \\simi100, and \\simi300 for $I_c < 11$ \\citep[][Figures 11 \\& 22]{Sullivan2015}. The XO units concentrate on 520\\ensuremath{^{\\circ2}}\\xspace (10 fields of 52\\ensuremath{^{\\circ2}}\\xspace each). Thus, the yield of such planets is 1.4 for $I_c < 10$ and 4.1 for $I_c < 11$.\nIn this calculation we assume ideal monitoring, \\textit{i.e.}\\xspace no limitation due to the observing window or instrumental precision. For \\textit{TESS}\\xspace, this is well justified as the monitoring will be continuous for at least 30 days with a precision that is much better than necessary for the detection of transiting giant exoplanets (in other words, \\textit{TESS}\\xspace will detect nearly every planet with those characteristics, at least around the selected target stars). For XO, the observing window per field extends to approximately the nighttime of eighteen months (twice nine months, two strips, two longitudes considering that Observatorio del Teide and Observatori Astron\\`omic del Montsec have a similar longitude), and the precision of the full lightcurves folded at short periods and binned on timescales that are relevant to transits is at the millimagnitude level (see Section \\textit{\\nameref{sec: Photometric precision}} and Figures \\ref{fig: xo poster} and \\ref{fig: lightcurves}). Thus, transiting close-in giant planets should be detected, if present. \nWe found two such planets with the XO units so far, XO-6b \\citep{Crouzet2017} and XO-7b \\citep{Crouzet-inprep}, which host stars have $I_c$ magnitudes of 10.12 and 10.27 respectively. This number is in line with the approximate yield. Thus, the observations and data analysis are as efficient as one could expect. A re-analysis of the XO data to find new transiting close-in gas giant planets would be of low gain and we will simply pursue the follow-up observations of the candidates we have identified. Although transiting planets with longer periods are more challenging to detect, we found a few candidates with $P > 20$ days which are also under follow-up observations.\n\n\n\n\n\\section{Conclusion}\n\\label{sec: Conclusion}\n\nIn this chapter, we presented an overview of the second version of the XO project: instrumental setup, operations, data reduction, instrumental performances, search for transit signals, planet yield and discoveries. We observed two strips covering an effective sky area of $520^{\\circ2}$ for twice nine months using the CCDs in time-delayed integration, and we extracted the lightcurves of \\simi20,000 bright stars up to magnitude $R \\approx 12.5$. The precision is at the millimagnitude level when folding the lightcurves on timescales that are relevant to transits. In addition, this setup allows us to detect long period signals, up to $P \\approx 100$ days. We identified several hundreds of variable stars and a few tens of planet candidates. The transiting hot Jupiter \\mbox{XO-6b} orbiting a fast rotating star has been discovered from this work \\citep{Crouzet2017}. Another planet, XO-7b, has been confirmed \\citep{Crouzet-inprep} and other transit candidates are under follow-up observations. The XO observations have been discontinued in anticipation of the NASA \\textit{TESS}\\xspace mission and the work on XO is now dedicated to the follow-up and study of individual objects.\n\n\n\n\n\\section{Acknowledgments}\nN.C. gratefully acknowledges Peter R. McCullough as the founder and Principal Investigator of the XO project. The XO project was supported by NASA grant NNX10AG30G.\nThis research has made use of the Extrasolar Planets Encyclopaedia (exoplanet.eu) and the Simbad database (simbad.u-strasbg.fr\/simbad\/).\nSoftware: astrometry.net \\citep{Lang2010}, Stellar Photometry Software \\citep{Janes1993}.\n\n\n\n\\bibliographystyle{spbasicHBexo}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor real values of $\\,s$, $s>1$, the Riemann zeta function is defined as $\\,\\zeta(s) := \\sum_{n=1}^\\infty{{\\,1\/n^s}}$.\\footnote{In this domain, this series converges according to the integral test. For $\\,s=1$, one has the harmonic series $\\:\\sum_{n=1}^\\infty{1\/n}$, which diverges to infinity.} For $\\,s=2k$, $\\,k \\in \\mathbb{Z}$, $k>0$, Euler (1740) did find that~\\cite{Euler} \n\\begin{equation}\n\\zeta(2k) = \\frac{2^{2k-1} \\, \\left|B_{2k}\\right|}{(2k)!} \\, \\: \\pi^{2k} \\, ,\n\\label{eq:Euler}\n\\end{equation}\nwhere $\\,B_{k}\\,$ is the $k$-th Bernoulli number.\\footnote{The (rational) numbers $B_{k}$ are the coefficients of ${\\,z^k\/k!}\\,$ in the Taylor series expansion of ${\\,z\/(e^z-1)}$, $|z| < 2\\,\\pi$.}\n\nAs a consequence, since $\\,B_2 = 1\/6\\,$ one has $\\,\\zeta(2) = \\pi^2\/6$, which is the Euler solution to the Basel problem (see Ref.~\\cite{Euler0} and references therein).\n\nBy noting that the series expansion approach introduced by Dancs and He (2006) on seeking for an Euler-type formula for $\\,\\zeta{(2k+1)}$, see Ref.~\\cite{Dancs}, could be modified in a manner to furnish similar formulas for $\\,\\zeta(2k)$, here in this note I show that the substitution of $\\,\\sin{(n \\pi)}\\,$ by $\\,\\cos{(n \\pi)}\\,$ in the Dancs-He initial series in fact yields a series expansion which can be reduced to a finite sum involving only even zeta values.  From the first few terms of this sum, I have found an elementary proof of $\\,\\zeta{(2)} = {\\,\\pi^2\/6}\\,$ and a recurrence formula for $\\zeta{(2k)}$.  The proofs are elementary in the sense they do not involve complex analysis, Fourier series, or multiple integrals.\\footnote{For non-elementary proofs, see, e.g., Refs.~\\cite{Kalman,Apostol} and references therein.}\n\n\n\\section{Elementary evaluation of $\\,\\zeta{(2)}$}\n\nFor any real $\\epsilon > 0$ and $u \\in [1,1+\\epsilon]$, we begin by taking into account the following Taylor series expansion considered by Dancs and He in Ref.~\\cite{Dancs}:\n\\begin{equation}\n\\frac{2 \\, e^t}{e^t + u} = \\sum_{m=0}^{\\infty}{\\phi_m(u) ~ \\frac{t^m}{\\,m!}} \\, ,\n\\label{eq:phiserie}\n\\end{equation}\nwhich converges absolutely for $\\,|t| < \\pi$.\n\nFrom the generating function for the Euler polynomial $\\,E_m(x)$, namely ${\\,2 \\, e^{x\\,t}\/(e^t + 1)} = \\sum_{m=0}^{\\infty}{E_m(x)\\, \\frac{t^m}{m!}}\\,$, it is clear that $\\,\\phi_m(1) = E_m(1)$, for all nonnegative integer values of $\\,m$.  For $u>1$, we have\n\\begin{equation}\n\\phi_m(u) = -2 \\, \\sum_{n=1}^{\\infty}{\\frac{n^{\\,m}}{(-u)^{\\,n}}} \\, .\n\\label{eq:phi}\n\\end{equation}\nLet us take this series as our definition of $\\,\\phi_{-m}(u)$, $m$ being a positive integer. Therefore\n\\begin{equation}\n\\phi_{-m}(1) = -2 \\, \\sum_{n=1}^\\infty{\\frac{(-1)^n}{n^m}} = -2 \\: \\zeta^{*}{(m)} = 2\\,(1-2^{1-m})\\,\\zeta{(m)}\n\\label{eq:phinegative}\n\\end{equation}\nfor all integer $\\,m>1$.\n\nNow, let\n\\begin{equation*}\nf(u) := \\sum_{n=1}^{\\infty}{\\frac{\\left({\\,1\/u}\\right)^n}{n^2}}\n\\end{equation*}\nbe an auxiliary function, with $u$ belonging to the same domain as above.  Since $\\,\\cos{(n \\pi)} = (-1)^n$, then $\\,f(u)\\,$ can be written in the form\n\\begin{equation*}\nf(u) = \\sum_{n=1}^{\\infty}{(-1)^n \\, \\frac{\\cos(n\\,\\pi)}{u^n\\,n^2}} \\, .\n\\end{equation*}\nOn expanding $\\,\\cos{(n \\pi)}\\,$ in a Taylor series, one has\n\\begin{equation*}\nf(u) = \\sum_{n=1}^{\\infty}{\\left[\\frac{(-1)^n}{u^n\\,n^2} \\cdot \\sum_{j=0}^{\\infty}{(-1)^j \\, \\frac{(n \\pi)^{2 j}}{(2 j)!}}\\right]} = \\sum_{j=0}^{\\infty}{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, \\sum_{n=1}^{\\infty}{(-1)^n \\, \\frac{n^{2 j}}{u^n\\,n^2}}} ,\n\\end{equation*}\nin which the change of sums justifies by Fubini's theorem.  By writing the last series in terms of $\\,\\phi_m(u)$, one has\n\\begin{eqnarray}\nf(u) = \\sum_{j=0}^{\\infty}{(-1)^j \\, \\frac{\\pi^{2 j}}{(2 j)!} \\, \\frac{\\phi_{2j-2}(u)}{(-2)}} \\, .\n\\label{eq:marcia}\n\\end{eqnarray}\n\nThis is sufficient for proofing our first result.\n\\newline\n\n\\begin{teo}[Short evaluation of $\\,\\zeta{(2)}\\,$]\n\\label{teo:z2}\n\\begin{equation*}\n\\sum_{n=1}^\\infty{\\frac{1}{\\,n^2}} = \\frac{\\pi^2}{6} \\, .\n\\end{equation*}\n\\end{teo}\n\n\\begin{prova}\n\\; By taking the limit as $u \\rightarrow 1^{+}$ on both sides of Eq.~\\eqref{eq:marcia}, one has\n\\begin{eqnarray}\n\\lim_{u \\rightarrow 1^{+}} \\sum_{n=1}^{\\infty}{\\frac{1}{u^n \\, n^2}} = -\\frac12 \\, \\phi_{-2}(1) \\, + \\frac12 \\, \\frac{\\pi^2}{2!} \\, \\phi_0{(1)} -\\frac12 \\, \\sum_{j=2}^{\\infty}{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, \\phi_{2j-2}(1)} \\, ,\n\\end{eqnarray}\nwhich, in face of the value of $\\phi_{-2}(1)$ stated in Eq.~\\eqref{eq:phinegative}, implies that\n\\begin{equation}\n\\sum_{n=1}^{\\infty}{\\frac{1}{n^2}} = -\\frac12 \\, \\left[ 2 \\left(1-2^{1-2}\\right) \\zeta{(2)} \\right] \\, +\\frac{\\pi^2}{4} \\, E_0{(1)} -\\frac12 \\, \\sum_{j=2}^{\\infty}{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, E_{2j-2}(1)} \\, .\n\\end{equation}\nSince $E_0(1)=1$ and $E_m(1)=0$ for all $m>0$, the right-hand side of this equation reduces to $\\,-\\frac12 \\, \\zeta{(2)} +{\\,\\pi^2 \/ 4}$, which implies that\n\\begin{equation*}\n\\zeta{(2)} = -\\frac12 \\, \\zeta{(2)} +\\frac{\\pi^2}{4} \\, ,\n\\end{equation*}\nand then $\\: \\dfrac{3}{2} \\: \\zeta{(2)} = \\dfrac{\\pi^2}{4}\\,$. \n\\begin{flushright} $\\Box$ \\end{flushright}\n\\end{prova}\n\n\n\\section{Recurrence formula for $\\zeta{(2 k)}$}\n\nInterestingly, our approach can be easily adapted to treat higher even zeta values by changing the exponent of $\\,n\\,$ from $\\,2\\,$ to $\\,2 k$. The result is the following recurrence formula for even zeta values.\n\n\\begin{teo}[Recurrence for $\\,\\zeta{(2 k)}\\,$]\n\\label{teo:z2k}\n\\;  For any positive integer $\\,k$,\n\\begin{equation*}\n\\left( 4 -\\frac{4}{2^{2k}} \\right) \\zeta{(2k)} =  \\sum_{m=1}^{k-1}{ \\frac{(-1)^{k-m+1}}{(2k-2m)!} \\left( 2 -\\frac{4}{2^{2m}} \\right) \\pi^{2k-2m} \\, \\zeta{(2m)}} \\,-(-1)^k \\frac{\\pi^{2k}}{(2k)!} \\, .\n\\label{eq:recorrencia}\n\\end{equation*}\n\\end{teo}\n\n\\begin{prova}\n\\; We begin by defining $\\,f_k{(u)} :=  \\sum_{n=1}^{\\infty}{\\left({\\,1\/u}\\right)^n \/ n^{2 k}}\\,$.  Again, since $\\,\\cos{(n \\pi)} = (-1)^n$, we may write\n\\begin{eqnarray}\nf_k{(u)} &=& \\sum_{n=1}^{\\infty}{(-1)^n \\, \\frac{\\cos(n\\,\\pi)}{u^n\\,n^{2k}}} = \\sum_{n=1}^{\\infty}{\\frac{(-1)^n}{u^n\\,n^{2k}} \\, \\sum_{j=0}^{\\infty}{(-1)^j \\, \\frac{(n \\pi)^{2 j}}{(2 j)!}}} \\nonumber \\\\\n&=& \\sum_{j=0}^{\\infty}{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, \\sum_{n=1}^{\\infty}{(-1)^n \\, \\frac{n^{2 j}}{u^n\\,n^{2k}}}}\\, .\n\\end{eqnarray}\nOn rewriting the last series in terms of $\\phi_m(u)$, one finds\n\\begin{equation*}\nf_k{(u)} = \\sum_{j=0}^{\\infty}{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, \\frac{\\phi_{2j-2k}(u)}{(-2)}}  \\, -\\frac12 \\, \\sum_{j=0}^{k-1}{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, \\phi_{2j-2k}(u)} \\,- \\frac12 \\, \\sum_{j=k}^\\infty{(-1)^j \\frac{\\pi^{2 j}}{(2 j)!} \\, \\phi_{2j-2k}(u)} \\, .\n\\end{equation*}\nNow, on substituting $\\,m = j -k\\,$ in the above series, one has\n\\begin{eqnarray}\nf_k{(u)} = -\\frac12 \\, \\sum_{m=-k}^{-1}{(-1)^{m+k} \\, \\frac{\\pi^{2m+2k}}{(2m+2k)!} \\, \\phi_{2m}(u)} -\\frac12 \\, \\sum_{m=0}^\\infty{(-1)^{m+k} \\, \\frac{\\pi^{2m+2k}}{(2m+2k)!} \\, \\phi_{2m}(u)} \\nonumber \\\\\n=  -\\frac12 \\, (-1)^k \\left[ \\, \\sum_{\\widetilde{m}=1}^{k}{ \\frac{(-1)^{\\widetilde{m}}\\,\\pi^{2k-2\\widetilde{m}}}{(2k-2\\widetilde{m})!} \\, \\phi_{-2\\widetilde{m}}(u)}\n+ \\sum_{m=0}^\\infty{\\frac{(-1)^{m} \\, \\pi^{2m+2k}}{(2m+2k)!} \\, \\phi_{2m}(u)} \\right] \\! . \\;\n\\label{eq:2seriesM}\n\\end{eqnarray}\nThe limit as $u \\rightarrow 1^{+}$, taken on both sides of Eq.~\\eqref{eq:2seriesM}, yields\n\\begin{equation}\n\\lim_{u \\rightarrow 1^{+}} \\sum_{n=1}^{\\infty}{\\frac{1}{u^n \\, n^{2k}}} = -\\frac12 \\, (-1)^k \\left[ \\, \\sum_{m=1}^k{ \\frac{(-1)^m \\, \\pi^{2k-2m}}{(2k-2m)!} \\, \\phi_{-2m}(1)}\n+\\sum_{m=0}^\\infty{\\frac{(-1)^m \\, \\pi^{2m+2k}}{(2m+2k)!} \\, \\phi_{2m}(1)} \\right]\\!.\n\\label{eq:aux}\n\\end{equation}\nFrom Eq.~\\eqref{eq:phinegative}, one knows that $\\phi_{-2m}(1) = 2 \\left( 1 -2^{1-2m} \\right) \\zeta{(2m)}$. For nonnegative values of $m$, one has $\\phi_{2m}(1) = E_{2m}(1) = 0$, the only exception being $\\,\\phi_0(1) = E_0(1)=1$. This reduces Eq.~\\eqref{eq:aux} to\n\\begin{equation*}\n\\sum_{n=1}^{\\infty}{\\frac{1}{n^{2k}}} = -(-1)^{k} \\sum_{m=1}^k{\\frac{(-1)^m \\, \\pi^{2k-2m}}{(2k-2m)!} \\, \\left( 1 -2^{1-2m} \\right) \\zeta{(2m)}} -(-1)^k\\, \\frac{\\pi^{2k}}{2\\,(2k)!} \\, .\n\\end{equation*}\nBy extracting the last term of the sum and isolating $\\,\\zeta{(2k)}$, one finds\n\\begin{equation*}\n\\left( 2 -\\frac{2}{2^{2k}} \\right) \\, \\zeta{(2k)} =  (-1)^{k+1} \\sum_{m=1}^{k-1}{\\frac{(-1)^m \\, \\pi^{2k-2m}}{(2k-2m)!} \\, \\left( 1 -2^{1-2m} \\right) \\zeta{(2m)}} -(-1)^k\\, \\frac{\\pi^{2k}}{2\\,(2k)!} \\, .\n\\end{equation*}\nA multiplication by $2$ on both sides yields the desired result.\n\\begin{flushright} $\\Box$ \\end{flushright}\n\\end{prova}\n\n\nThe first few even zeta values can be readily obtained from this recurrence formula. For $\\,k=1$, the sum in the right-hand side is null and one has\n\\begin{equation*}\n3 \\,\\zeta(2) = -(-1)\\,\\frac{\\pi^2}{2} \\, ,\n\\end{equation*}\nwhich simplifies to $\\,\\zeta(2) = \\pi^2\/6$, in agreement to Theorem~\\ref{teo:z2}.\n\nFor $k=2$, one has\n\\begin{equation*}\n\\frac{15}{4} \\: \\zeta(4) = \\frac{\\pi^2}{2!} \\: \\zeta(2) - \\frac{\\pi^4}{ 4!} \\, .\n\\end{equation*}\nBy substituting the value of $\\,\\zeta(2)$, above, and multiplying both sides by $4$, one finds\n\\begin{equation}\n15 \\: \\zeta(4) = \\frac{\\pi^4}{3} -\\frac{\\pi^4}{6} = \\frac{\\pi^4}{6} \\, ,\n\\label{eq:z4}\n\\end{equation}\nwhich implies that $\\,\\zeta(4) = \\pi^4 \/ 90$.\n\nNote that, by writing the recurrence formula in Theorem~\\ref{teo:z2k} in the form\n\\begin{equation}\n\\left( 1 -\\frac{1}{2^{2k}} \\right) \\, \\frac{\\zeta{(2k)}}{\\pi^{2k}} = \\sum_{m=1}^{k-1}{ \\, \\frac{(-1)^{k-m+1}}{(2k-2m)!} \\left( \\frac12 -\\frac{1}{2^{2m}} \\right) \\frac{\\zeta{(2m)}}{\\pi^{2m}}} -\\frac{(-1)^k}{4\\,(2k)!} \\, ,\n\\end{equation}\nit is straightforward to show, by induction on $k$, that the ratio ${\\, \\zeta{(2k)} \/ \\pi^{2k}}\\,$ is a rational number for every positive integer $\\,k$, without making use of Euler's formula for $\\zeta{(2k)}$, see Eq.~\\eqref{eq:Euler}, and Bernoulli numbers. In fact, this was the original motivation that has led the author to study the properties of the Dancs-He series expansions. The proofs developed here could well be modified to cover other special functions of interest in analytic number theory.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGoal of this paper is to provide a\none-dimensional symmetry result for a phase transition\nequation in a genuinely nonlocal regime in three spatial dimensions.\nThat is, we consider a fractional Allen-Cahn equation of the type\n\\begin{equation}\\label{ALLEN}\n(-\\Delta)^s u=u-u^3\\end{equation} in~$\\mathbb{R}^3$, with\n$s\\in\\left(0,\\frac12\\right)$, and, under boundedness and monotonicity\nassumptions, we prove that~$u$ depends only on one variable, up to a rotation.\n\nIn this setting,\nas customary, for~$s\\in(0,1)$, we consider the fractional Laplace operator\ndefined by\n$$ (-\\Delta)^s u(x):=c_{n,s}\\int_{\\mathbb{R}^n}\\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\\,dy,$$\nwith\n\\begin{equation}\\label{cns} c_{n,s}:=\\frac{2^{2s-1}\\,s\\,\\Gamma\\left( \\frac{n}2+s\\right)}{\n\\pi^{\\frac{n}2}\\,\\Gamma(1-s)},\\end{equation}\nbeing~$\\Gamma$ the Euler's Gamma Function.\n\nMoreover, we say that~$u$ is $1$D if there exist~$u_o:\\mathbb{R}\\to\\mathbb{R}$ and~$\\omega_o\\in\nS^{n-1}$ such that~$u(x)=u_o(\\omega_o\\cdot x)$ for any~$x\\in\\mathbb{R}^n$.\nThen, our main result in this paper is the following:\n\n\\begin{theorem}\\label{MAIN}\nLet~$n\\le3$, $s\\in\\left(0,\\frac12\\right)$ and~$u\\in C^2(\\mathbb{R}^n,[-1,1])$\nbe a solution of~$(-\\Delta)^s u=u-u^3$ in~$\\mathbb{R}^n$, \nwith~$\\partial_{x_n}u>0$ in~$\\mathbb{R}^n$.\nThen, $u$ is $1$D.\n\\end{theorem}\n\nRecently, a result similar to that in Theorem~\\ref{MAIN} has been established\nin Theorem~1.4 of~\\cite{DSVarxiv}, under the additional assumption that\n\\begin{equation} \\label{LIMIT}\n\\lim_{x_n\\to-\\infty} u(x',x_n)=-1\\quad{\\mbox{ and }}\\quad\n\\lim_{x_n\\to+\\infty} u(x',x_n)=1.\\end{equation}\nTherefore, Theorem~\\ref{MAIN} here\nis the extension of Theorem~1.4 of~\\cite{DSVarxiv}\nin which it is not necessary to assume the limit condition~\\eqref{LIMIT}.\\medskip\n\nWe recall that equation~\\eqref{ALLEN}\nrepresents a phase transition subject to long-range interactions, see e.g.\nChapter~5 in~\\cite{bucur} for a detailed description of the model.\nIn particular, the states~$u=-1$ and~$u=1$ would correspond to the\n``pure phases'' and equation~\\eqref{ALLEN} models\nthe coexistence between intermediate phases and studies\nthe separation between them.  \n\nAt a large scale, the separation between phases\nis governed by the minimization of a limit interface, which can be either\nof local or nonlocal type,\nin dependence of the fractional parameter~$s\\in(0,1)$,\nwith a precise bifurcation occurring at\nthe threshold~$s=\\frac12$. \nMore precisely, as proved in~\\cite{MR2948285, MR3133422},\nif~$u$ is a local energy minimizer for equation~\\eqref{ALLEN}\nand~$u_\\varepsilon(x):=u(x\/\\varepsilon)$, as~$\\varepsilon\\searrow0$\nwe have that~$u_\\varepsilon$ approaches a ``pure phase''\nstep function with values in~$\\{-1,1\\}$. That is, we can write, up to subsequences,\n$$ \\lim_{\\varepsilon\\searrow0} u_\\varepsilon = \\chi_E-\\chi_{\\mathbb{R}^n\\setminus E},$$\nand the set~$E$ possesses a minimal interface criterion, depending on~$s$.\nMore precisely, in the ``weakly nonlocal regime'' in which~$s\\in\\left[\\frac12,1\\right)$,\nthe set~$E$ turns out to be a local minimizer for the classical\nperimeter functional: in this sense, on a large scale, the\nweakly nonlocal regime is indistinguishable with respect to the classical case\nand, in spite of the fractional nature of equation~\\eqref{ALLEN},\nits limit interface behaves in a local fashion\nwhen~$s\\in\\left[\\frac12,1\\right)$.\n\nConversely, in the ``genuinely nonlocal regime''\nin which~$s\\in\\left( 0,\\frac12\\right)$,\nthe set~$E$ turns out to be a local minimizer for the nonlocal perimeter functional\nwhich was introduced in~\\cite{MR2675483}. That is, the interface\nof long-range phase transitions when~$s\\in\\left(0, \\frac12\\right)$\npreserves its nonlocal features at any arbitrarily large scale,\nand, as a matter of fact, the scaling properties of the associated\nenergy functional preserve this nonlocal character as well.\nNeedless to say, the persistence of the nonlocal\nproperties at any scale and the somehow unpleasant scaling of the associated\nenergies provide a number of difficulties in the analysis\nof long-range phase coexistence models.\\medskip\n\nIn particular, symmetry properties\nof the solutions of equation~\\eqref{ALLEN}\nhave been intensively studied, also in view of a celebrated\nconjecture by E. De Giorgi in the classical case, see~\\cite{DG}.\nThis classical conjecture asks whether or not bounded and monotone\nsolutions of phase transitions equations are necessarily $1$D.\nIn the fractional framework, a positive answer\nto this problem was known in dimension~$2$\n(see~\\cite{MR2177165} for the case~$s=\\frac12$\nand~\\cite{SV09, cabre-TAMS, MR3035063} for the full range~$s\\in(0,1)$).\nAlso, in dimension~$3$, a positive answer was known\nonly in the weakly nonlocal regimes~$s=\\frac12$\nand~$s\\in\\left(\\frac12,1\\right)$, see~\\cite{MR2644786,MR3148114}.\nSee also~\\cite{2016arXiv161009295S} for a very recent contribution\nabout symmetry results for equation~\\eqref{ALLEN}\nin the weakly nonlocal regime --\nas a matter of fact, the lack of ``good energy estimates''\nprevented the extension of the techniques of these articles\nto the strongly nonlocal regime~$s\\in\\left(0,\\frac12\\right)$.\nIn this sense, our Theorem~\\ref{MAIN} aims at overcoming\nthese difficulties, by relying on the very recent paper~\\cite{DSVarxiv},\nwhich has now taken into account the weakly nonlocal regime\nfor equation~\\eqref{ALLEN}.\\medskip\n\nAfter this work was completed we have \nalso received a preliminary version of the article \\cite{cabre}, \nin which symmetry results for fractional Allen-Cahn equations \nwill be obtained also in the setting of stable solutions.\n\\medskip\n\nThe rest of the paper is organized as follows.\nIn Section~\\ref{SEC:1}, we recall the notion of local minimizers\nand we introduce an equivalent minimization problem in an extended space:\nthis part is rather technical, but absolutely non-standard,\nsince the lack of decay of our solution and the strongly nonlocal condition~$s\\in\\left(0,\\frac12\\right)$\nmake the energy diverge, hence the standard extension methods\nare not available in our case and we will need to introduce a suitable energy\nrenormalization procedure.\n\nIn Section~\\ref{SEC:2}, we relate stable and minimal solutions in the one-dimensional\ncase, by relying also on some layer solution theory of~\\cite{cabre-sire-AIHP}.\n\nIn Section~\\ref{CLASS:A} we consider the profiles of the solution\nat infinity and we establish their minimality and symmetry properties.\n\nIn Section~\\ref{CLASS:B}, we discuss the minimization properties\nunder perturbation which do not overcome the limit profiles,\nand in Section~\\ref{CLASS:C} we recover the minimality of a solution from that\nof its limit profiles. The proof of Theorem~\\ref{MAIN}.\nis contained in Section~\\ref{CLASS:D}.\n\nFor completeness, in Section~\\ref{UL810} we also provide a variant of Theorem~\\ref{MAIN}\nthat gives minimality and symmetry results under the assumption that the limit profiles\nare two-dimensional.\n\\medskip\n\nIt is worth to point out that the setting in Sections~\\ref{SEC:1}--\\ref{CLASS:C}\nis very general and it applies to all the fractional powers~$s\\in(0,1)$, hence\nit can be seen as a useful tool to deal with a class of problems also in extended spaces,\nso to recover minimal properties of the solution from some knowledge of the limit profiles.\n\n\n\\section{Local minimizers in $\\mathbb{R}^n$ and extended local minimizers in~$\\mathbb{R}^{n+1}_+$}\\label{SEC:1}\n\nEquation~\\eqref{ALLEN} lies in the class of semilinear fractional equations of the type\n\\begin{equation}\\label{ALLEN-GEN} (-\\Delta)^s u=f(u).\\end{equation}\n{F}rom now on,\nwe will denote by~$f$ a bistable nonlinearity, namely,\nwe assume that~$f(-1)=f(1)=0$, and there exist~$\\kappa > 0$\nand~$c_\\kappa > 0$ such that~$ f'(t)<-c_\\kappa$ for\nany~$t\\in[-1,-1+\\kappa]\\cup[1-\\kappa,1]$. We also assume that\n\\begin{equation} \\label{BISTABLE}\n\\int_0^1 f(\\sigma)\\,d\\sigma>0>\\int_{-1}^0 f(\\sigma)\\,d\\sigma\n.\\end{equation}\nTo ensure that the solution is sufficiently regular\nin our computations, we assume that\n\\begin{equation}\\label{QUE}\nf\\in C^{1,\\alpha}_{\\rm loc}(\\mathbb{R})\\end{equation}\nwith~$\\alpha\\in(0,1)$\nand~$\\alpha>1-2s$.\nWe remark that, in view\nof~\\eqref{QUE} here and Lemma~4.4 in~\\cite{cabre-sire-AIHP}, bounded\nsolutions of~\\eqref{ALLEN-GEN} are automatically in~$C^2(\\mathbb{R}^n)$,\nwith bounded second derivatives.\n\nThe prototype for such bistable nonlinearity is, of course, the case in which~$f(t)=t-t^3$.\nAlso,\nto describe the energy framework of nonlocal phase transitions, given~$s\\in(0,1)$, $v:\\mathbb{R}^n\\to\\mathbb{R}$\nand~$\\omega\\subset\\mathbb{R}^n$, we consider the functional\n$$ {\\mathcal{F}}_\\omega(v):=\n\\frac{c_{n,s}}{2}\\iint_{Q_\\omega}\\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}}\\,dx\\,dy+\\int_\\omega\nF(v(x))\\,dx,$$\nwhere\n$$ Q_\\omega:= (\\omega\\times\\omega)\\cup\n(\\omega\\times\\omega^c)\\cup(\\omega^c\\times\\omega)$$\nand\n$$ F(t):= -\\int_{-1}^t f(\\tau)\\,d\\tau.$$\n\n\\begin{definition}\\label{D:L} We say that~$u$ is a local minimizer\nif, for any~$R>0$ and any~$\\varphi\\in C^\\infty_0(B_R)$, it holds that~$\n{\\mathcal{F}}_{B_R}(u)\\le{\\mathcal{F}}_{B_R}(u+\\varphi)$.\n\\end{definition}\n\nNow we describe an extended problem\nand relate its local minimization to the one in Definition~\\ref{D:L}\n(see~\\cite{caffasil}).\nFor this, we set~$a:=1-2s\\in(-1,1)$ and~$\\mathbb{R}^{n+1}_+:=\\mathbb{R}^n\\times(0,+\\infty)$.\nThen, given any~$V:\\mathbb{R}^{n+1}_+\\to\\mathbb{R}$ and~$\\Omega\\subset\\mathbb{R}^{n+1}$, we define\n$$ {\\mathcal{E}}_\\Omega(V):=\n\\frac{\\tilde c_{n,s}}{2}\\int_{\\Omega^+}z^a |\\nabla V(x,z)|^2\\,dx\\,dz+\\int_{\\Omega_0}\nF(V(x,0))\\,dx,$$\nwhere~$\\Omega^+:=\\Omega\\cap\\mathbb{R}^{n+1}_+$ and~$\\Omega_0:=\\Omega\\cap\\{z=0\\}$.\nHere, we used the notation~$\n(x,z)\\in\\mathbb{R}^n\\times(0,+\\infty)$ to denote the variables of~$\\mathbb{R}^{n+1}_+$ and~$\\tilde c_{n,s}>0$\nis a normalization constant.\nGiven~$R>0$, we also denote\n\\begin{eqnarray*}&&{\\mathcal{B}}_R:= B_R\\times(-R,R)=\\big\\{(x,z)\\in\n\\mathbb{R}^{n}\\times\\mathbb{R} {\\mbox{ s.t. }} x\\in B_R {\\mbox{ and }} |z|<R\\big\\}\\\\{\\mbox{and }}&&\n{\\mathcal{B}}_R^+:= {\\mathcal{B}}_R\\cap\\{z>0\\}=\nB_R\\times(0,R)=\\big\\{(x,z)\\in\n\\mathbb{R}^{n}\\times\\mathbb{R} {\\mbox{ s.t. }} x\\in B_R {\\mbox{ and }} z\\in(0,R)\\big\\}.\\end{eqnarray*}\nIn this setting, we have the following notation:\n\\begin{definition}\\label{D:L:2} We say that~$U$ is an extended local minimizer\nif, for any~$R>0$ and any~$\\Phi\\in C^\\infty_0({\\mathcal{B}}_R)$, it holds that~$\n{\\mathcal{E}}_{{\\mathcal{B}}_R}(U)\\le{\\mathcal{E}}_{{\\mathcal{B}}_R}(U+\\Phi)$.\n\\end{definition}\n\nThe reader can compare Definitions~\\ref{D:L} and~\\ref{D:L:2}. Also, given~$v\\in L^\\infty(\\mathbb{R}^n)$,\nwe consider the $a$-harmonic extension of~$v$ to~$\\mathbb{R}^{n+1}_+$ as the function~$E_v:\n\\mathbb{R}^{n+1}_+\\to\\mathbb{R}$\nobtained by convolution with the Poisson kernel of order~$s$. More explicitly, we set\n\\begin{equation}\\label{0djhicseodfeeyyeyeye} P(x,z):= \\bar c_{n,s} \\frac{z^{2s}}{(|x|^2+z^2)^{\\frac{n+2s}2}}.\\end{equation}\nIn this framework, $\\bar c_{n,s}$ is a positive normalization constant such that\n$$ \\int_{\\mathbb{R}^n} P(x,z)\\,dx=1,$$\nsee e.g.~\\cite{MR3461641}. Then we set\n$$ E_v(x,z):=\\int_{\\mathbb{R}^n} P(x-y,z)\\,v(y)\\,dy=\n\\int_{\\mathbb{R}^n} P(y,z)\\,v(x-y)\\,dy.$$\nWe remark that when~$v\\in C^\\infty_0(\\mathbb{R}^n)$, the function~$E_v$ can also be obtained\nby\nminimization\nof the associated Dirichlet energy, namely\n\\begin{equation} \\label{DIRmi}\n\\inf_{{V\\in C^\\infty_0(\\mathbb{R}^{n+1})}\\atop{V(x,0)=v(x)}}\n\\int_{\\mathbb{R}^{n+1}_+}z^a |\\nabla V(x,z)|^2\\,dx\\,dz\n=\n\\int_{\\mathbb{R}^{n+1}_+}z^a |\\nabla E_v(x,z)|^2\\,dx\\,dz,\\end{equation}\nsee Lemma 4.3.3 in~\\cite{bucur}. Nevertheless,\nwe want to consider here the more general framework in which~$v$\nis bounded, but not necessarily decaying at infinity, and this will produce a number\nof difficulties, also due to the lack of ``good'' functional settings.\n\nWe also remark that the setting in~\\eqref{DIRmi} and the normalization\nconstant~$\\tilde c_{n,s}$\nare compatible with the choice of the constant in~\\eqref{cns},\nsince, for any $v\\in C^\\infty_0(\\mathbb{R}^n)$,\n\\begin{equation}\\label{COMP:AK}\n\\frac{c_{n,s}}{2}\\iint_{\\mathbb{R}^{2n}}\\frac{|v(x)-v(y)|^2}{|x-y|^{n+2s}}\\,dx\\,dy=\n\\frac{\\tilde c_{n,s}}{2}\\int_{\\mathbb{R}^{n+1}_+}z^a |\\nabla E_v(x,z)|^2\\,dx\\,dz,\n\\end{equation}\nsee e.g. formula~(4.3.15) in~\\cite{bucur}.\n\nSince these normalization constants will not play any role in the following computations,\nwith a slight abuse of notation, for the sake of simplicity,\nwe just omit them in the sequel.\n\nIn our setting, for functions~$v$ with no decay at infinity,\nformula~\\eqref{COMP:AK} does not make sense, since both the terms could diverge.\nNevertheless, we will be able to overcome this difficulty by an energy\nrenormalization procedure, based on the formal substraction of the infinite energy.\nThe rigorous details of this procedure are discussed in the\nfollowing\\footnote{We observe that Proposition~\\ref{0OAPQO182:P}\nhere is also related to the extension method in Lemma~7.2\nin~\\cite{MR2675483}, where suitable trace and extended energies\nare compared in the unit ball:\nin a sense, since\nProposition~\\ref{0OAPQO182:P} here\ncompares energies defined in the whole of the space, it can be viewed\nas a ``global'', or ``renormalized'', version of\nLemma~7.2\nin~\\cite{MR2675483}. }\nresult:\n\n\\begin{proposition}\\label{0OAPQO182:P}\nLet~$n\\ge2$ and~$s\\in(0,1)$.\nFor any~$v\\in W^{2,\\infty}(\\mathbb{R}^n)$\nand any~$\\varphi\\in C^\\infty_0(\\mathbb{R}^n)$, it\nholds that\n\\begin{eqnarray*} +\\infty&>& \n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v+\\varphi)(x)-(v+\\varphi)(y)|^2-\n|v(x)-v(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\\\\&\n=& \\lim_{R\\to+\\infty} \\inf_{{\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n.\\end{eqnarray*}\n\\end{proposition}\n\n\\begin{proof} For concreteness, we\nsuppose that the support of~$\\varphi$\nlies in~$B_1$.\nWe take~$\\tau\\in C^\\infty_0(B_2,\\,[0,1])$,\nwith~$\\tau=1$ in~$B_1$, and we let\n\\begin{equation}\\label{v k} \\tau_k(x):=\\tau\\left(\\frac{x}{k}\\right)\n\\quad{\\mbox{ and }}\\quad\nv_k(x):=\\tau_k(x)\\,v(x).\\end{equation}\nWe also set~$w_k:= v-v_k$.\nIn this way, $w_k$ is bounded, uniformly\nLipschitz and\nvanishes in~$B_k$.\nIn particular,\n\\begin{equation}\\label{9:PAA}\n\\frac{\\big|w_k(x)-w_k(y)\\big|\\,\\big|\n\\varphi(x)-\\varphi(y)\\big|\n}{|x-y|^{n+2s}}\\le\n\\frac{C\\,\\min\\{1,|x-y|\\}\\,\\big|\n\\varphi(x)-\\varphi(y)\\big|\n}{|x-y|^{n+2s}}\\in L^1(\\mathbb{R}^{2n}).\\end{equation}\nFurthermore, we have that\n\\begin{eqnarray*}\n&& \\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v+\\varphi)(x)-(v+\\varphi)(y)|^2-\n|v(x)-v(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&&\\qquad\n-\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v_k+\\varphi)(x)-(v_k+\\varphi)(y)|^2-\n|v_k(x)-v_k(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\n\\\\ &=&2\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{\\big(v(x)-v(y)\\big)\\big(\n\\varphi(x)-\\varphi(y)\\big)\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&&\\qquad\n-2\\iint_{\\mathbb{R}^{2n}}\n\\frac{\\big(v_k(x)-v_k(y)\\big)\\big(\n\\varphi(x)-\\varphi(y)\\big)\n}{|x-y|^{n+2s}}\\,dx\\,dy\n\\\\ &=&\n2\\iint_{\\mathbb{R}^{2n}}\n\\frac{\\big(w_k(x)-w_k(y)\\big)\\big(\n\\varphi(x)-\\varphi(y)\\big)\n}{|x-y|^{n+2s}}\\,dx\\,dy\n.\\end{eqnarray*}\nThis, \\eqref{9:PAA}\nand the Dominated Convergence\nTheorem give that\n\\begin{equation}\\label{K9:00:01}\n\\begin{split}&\n\\lim_{k\\to+\\infty}\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v_k+\\varphi)(x)-(v_k+\\varphi)(y)|^2-\n|v_k(x)-v_k(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\n\\\\&\\qquad=\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v+\\varphi)(x)-(v+\\varphi)(y)|^2-\n|v(x)-v(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\n.\\end{split}\\end{equation}\nAlso,\n$$ \\frac{\\big|v(x)-v(y)\\big|\\,\\big|\n\\varphi(x)-\\varphi(y)\\big|\n}{|x-y|^{n+2s}}\\le\n\\frac{C\\,\\min\\{1,|x-y|\\}\\,\\big|\n\\varphi(x)-\\varphi(y)\\big|\n}{|x-y|^{n+2s}}\\in L^1(\\mathbb{R}^{2n}),$$\nand therefore\n\\begin{equation}\\label{K9:00:02}\n\\begin{split}&\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v+\\varphi)(x)-(v+\\varphi)(y)|^2-\n|v(x)-v(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&\\qquad=\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{\n\\big(v(x)-v(y)\\big)\\,\\big(\n\\varphi(x)-\\varphi(y)\\big)\n}{|x-y|^{n+2s}}\\,dx\\,dy+\n\\iint_{\\mathbb{R}^{2n}}\\frac{|\\varphi(x)-\\varphi(y)|^2}{|x-y|^{n+2s}}\\,dx\\,dy\n<+\\infty\n.\\end{split}\\end{equation}\nOn the other hand, recalling~\\eqref{0djhicseodfeeyyeyeye}, \n\\[\nz^a |\\nabla P(x,z)|\\le\n\\frac{Cz(|x|+z)}{(|x|^2+z^2)^{\\frac{n+2s+2}{2}}}\n+\\frac{C}{(|x|^2+z^2)^{\\frac{n+2s}{2}}}\n\\le\n\\frac{C}{(|x|^2+z^2)^{\\frac{n+2s}{2}}}\n.\\]\nThis implies that\n\\begin{eqnarray*}\n&& z^a |\\nabla E_\\varphi(x,z)|=\n\\left| z^a \\int_{\\mathbb{R}^n} \\nabla\nP(x-y,z)\\,\\varphi(y)\\,dy\n\\right|\\\\&&\\qquad\\le\nC \\int_{B_1} |\\nabla P(x-y,z)|\\,dy\n\\le C\\int_{B_1}\n\\frac{dy}{(|x-y|^2+z^2)^{\\frac{n+2s}{2}}}\n.\\end{eqnarray*}\nHence, for any~$x\\in\\mathbb{R}^n\\setminus B_2$,\n\\begin{equation}\\label{HENCE1}\nz^a |\\nabla E_\\varphi(x,z)|\n\\le C\\int_{B_1}\n\\frac{dy}{(|x|^2+z^2)^{\\frac{n+2s}{2}}}\n= \\frac{C}{(|x|^2+z^2)^{\\frac{n+2s}{2}}}\n\\end{equation}\nand, if~$x\\in B_2$,\n\\begin{equation}\\label{HENCE2}\nz^a |\\nabla E_\\varphi(x,z)|\\le\nC\\int_{B_1}\n\\frac{dy}{(0+z^2)^{\\frac{n+2s}{2}}}\n\\le\n\\frac{C}{z^{n+2s}}\n.\\end{equation}\nWe also notice that\n\\begin{equation}\\label{6bis}\\begin{split}& z^a\\,|\\nabla E_\\varphi(x,z)|\\le\nz^a\\,\\left| \\int_{B_1} P(y,z)\\,\\nabla\\varphi(x-y)\\,dy\\right|\n\\\\&\\qquad\\le C z^a\\int_{\\mathbb{R}^n} P(y,z)\\,dy\\le Cz^a\\in L^1(B_2\\times(0,2)).\\end{split}\\end{equation}\nIn addition, $\\| E_{v_k}\\|_{L^\\infty(\\mathbb{R}^{n+1}_+)}\\le\n\\|v_k\\|_{L^\\infty(\\mathbb{R}^n)}$;\nas a consequence\nof these observations, setting\n\\begin{equation}\\label{elle R}\nL_R:=\n(\\partial B_R)\\times (0,R)\\quad{\\mbox{\nand }}\\quad U_R:= B_R\\times\\{R\\}, \\end{equation}we have that\n\\begin{equation}\\label{889AO}\\begin{split}&\n\\lim_{R\\to+\\infty}\n\\int_{L_R\\cup U_R }\nz^a |E_{v_k}(x,z)|\\,|\n\\partial_\\nu \nE_{\\varphi}(x,z)|\\,d{\\mathcal{H}}^{n}(x,z)\n\\le\\lim_{R\\to+\\infty}\n\\int_{L_R\\cup U_R }\n\\frac{C\\,d{\\mathcal{H}}^{n}(x,z)}{\n(|x|^2+z^2)^{\\frac{n+2s}{2}}}\n\\\\ &\\qquad\\qquad\\le\\lim_{R\\to+\\infty}\n\\int_{L_R\\cup U_R }\n\\frac{C\\,d{\\mathcal{H}}^{n}(x,z)}{\nR^{n+2s}} \\le \\lim_{R\\to+\\infty}\n\\frac{CR^n}{R^{n+2s}}=0,\n\\end{split}\\end{equation}\nwhere~$\\partial_\\nu$ denotes the\nexternal normal derivative to the boundary\nof~${\\mathcal{B}}_R^+$.\n\nAccordingly,\n\\begin{eqnarray*}\n&& \\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v_k+\\varphi}(x,z)|^2\n-|\\nabla E_{v_k}(x,z)|^2\\big)\\,dx\\,dz\\\\\n&=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{\\varphi}(x,z)|^2\n+2\\nabla E_\\varphi(x,y)\\cdot\n\\nabla E_{v_k}(x,z)\\big)\\,dx\\,dz\n\\\\ &=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a |\\nabla E_{\\varphi}(x,z)|^2\n+2{\\rm div}\\,\\big(z^a\nE_{v_k}(x,y)\n\\nabla E_{\\varphi}(x,z)\\big) \\,dx\\,dz\\\\\n&=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a |\\nabla E_{\\varphi}(x,z)|^2\\,dx\\,dz\n+2\\int_{\\partial {\\mathcal{B}}_R^+ }\nz^a E_{v_k}(x,z)\n\\partial_\\nu \nE_{\\varphi}(x,z)\\,d{\\mathcal{H}}^{n}(x,z)\\\\\n&=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a |\\nabla E_{\\varphi}(x,z)|^2\\,dx\\,dz\n+2\\int_{B_R} v_k(x)\\,(-\\Delta)^s\\varphi(x)\\,dx\n\\\\ &=&\n\\int_{ \\mathbb{R}^{n+1}_+}\nz^a |\\nabla E_{\\varphi}(x,z)|^2\\,dx\\,dz\n+2\\int_{\\mathbb{R}^n} v_k(x)\n\\,(-\\Delta)^s\\varphi(x)\\,dx.\n\\end{eqnarray*}\nSimilarly,\n\\begin{eqnarray*}\n&& \\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_{v}(x,z)|^2\\big)\\,dx\\,dz\\\\\n\\\\&=&\n\\int_{ \\mathbb{R}^{n+1}_+}\nz^a |\\nabla E_{\\varphi}(x,z)|^2\\,dx\\,dz\n+2\\int_{\\mathbb{R}^n} v(x)\n\\,(-\\Delta)^s\\varphi(x)\\,dx.\\end{eqnarray*}\nSince~$|v_k (-\\Delta)^s\\varphi|\\le|v(-\\Delta)^s\\varphi|\\le\\frac{C}{1+|x|^{n+2s}}\\in L^1(\\mathbb{R}^n)$,\nwe thus obtain that\n\\begin{equation}\\label{89:0128esd}\n\\begin{split}\n&\\lim_{k\\to+\\infty}\n\\int_{\\mathbb{R}^{n+1}_+}\nz^a\\big(|\\nabla E_{v_k+\\varphi}(x,z)|^2\n-|\\nabla E_{v_k}(x,z)|^2\\big)\\,dx\\,dz\n\\\\=\\;&\\lim_{k\\to+\\infty}\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v_k+\\varphi}(x,z)|^2\n-|\\nabla E_{v_k}(x,z)|^2\\big)\\,dx\\,dz\n\\\\\n=\\;&\\lim_{k\\to+\\infty}\n\\left[\\int_{ \\mathbb{R}^{n+1}_+}\nz^a |\\nabla E_{\\varphi}(x,z)|^2\\,dx\\,dz\n+2\\int_{\\mathbb{R}^n} v_k(x)\n\\,(-\\Delta)^s\\varphi(x)\\,dx\n\\right]\\\\ =\\;&\n\\int_{ \\mathbb{R}^{n+1}_+}\nz^a |\\nabla E_{\\varphi}(x,z)|^2\\,dx\\,dz\n+2\\int_{\\mathbb{R}^n} v(x)\n\\,(-\\Delta)^s\\varphi(x)\\,dx\\\\\n=\\;&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_{v}(x,z)|^2\\big)\\,dx\\,dz.\n\\end{split}\\end{equation}\nMoreover, from~\\eqref{COMP:AK}, we know that\n\\begin{eqnarray*}\n&&  \n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v_k+\\varphi)(x)-(v_k+\\varphi)(y)|^2-\n|v_k(x)-v_k(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v_k+\\varphi}(x,z)|^2\n-|\\nabla E_{v_k}(x,z)|^2\\big)\\,dx\\,dz\n.\\end{eqnarray*}\nPutting together this with~\\eqref{K9:00:01},\n\\eqref{K9:00:02} and~\\eqref{89:0128esd}, we\nconclude that\n\\begin{eqnarray*} +\\infty&>& \n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v+\\varphi)(x)-(v+\\varphi)(y)|^2-\n|v(x)-v(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&=&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz.\\end{eqnarray*}\nTherefore, to complete the proof of the desired claim,\nit remains to show that\n\\begin{equation}\\label{0OAPQO182}\n\\begin{split}\n&\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n\\\\\n=\\;& \\lim_{R\\to+\\infty} \\inf_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n.\\end{split}\\end{equation}\nTo this end,\nwe take~$\\theta\\in C^\\infty_0({\\mathcal{B}}_1,\\,[0,1])$\nwith~$\\theta=1$ in~${\\mathcal{B}}_{1\/2}$\nand we set~$\\theta_R(x,z):=\\theta\\left(\\frac{x}{R},\\frac{z}{R}\\right)$\nand~$\\Phi_R:=E_\\varphi \\theta_R$.\nWe observe that\n$$ |E_\\varphi(x,z)|\\le \\int_{B_1}\n\\frac{C z^{2s}}{(|x-y|^2+z^2)^{\\frac{n+2s}2}}\\,dy$$\nand therefore, if~$|x|\\ge2$,\n\\begin{equation*} \n|E_\\varphi(x,z)|\\le \n\\frac{C z^{2s}}{(|x|^2+z^2)^{\\frac{n+2s}2}}\\le\n\\frac{C z^{2s}}{(1+|x|^2+z^2)^{\\frac{n+2s}2}}\n.\\end{equation*}\nSimilarly, if~$|x|\\le2$ and~$z\\ge1$,\n\\begin{equation*}\n|E_\\varphi(x,z)|\\le\\int_{ B_1}\n\\frac{C z^{2s}}{(0+z^2)^{\\frac{n+2s}2}}\\,dy\\le\n\\frac{C z^{2s}}{(1+|x|^2+z^2)^{\\frac{n+2s}2}},\n\\end{equation*}\nup to renaming~$C$.\nAlso, if~$|x|\\le2$ and~$z\\in(0,2)$,\n$$ |E_\\varphi(x,z)|\\le C\\int_{\\mathbb{R}^n}P(x-y,z)\\,dy=C.\n$$\nIn view of these estimates, we have that\n\\begin{equation}\\label{STIME:EPHI1}\n\\begin{split}& \\int_{\\mathbb{R}^{n+1}_+} z^a |E_\\varphi(x,z)|^2\\,dx\\,dz\n\\le\nC+\\int_{\\mathbb{R}^{n+1}_+\\setminus {\\mathcal{B}}_2} z^a |E_\\varphi(x,z)|^2\\,dx\\,dz\\\\\n&\\qquad\\le C+\\int_{\\mathbb{R}^{n+1}_+\\setminus {\\mathcal{B}}_2} \n\\frac{C z^{1+2s}}{(1+|x|^2+z^2)^{{n+2s}}}\n\\,dx\\,dz\\\\&\\qquad\n\\le\nC+\\int_{\\mathbb{R}^{n+1}_+\\setminus {\\mathcal{B}}_2} \n\\frac{C }{(1+|x|^2+z^2)^{{n+s-\\frac12}}}\n\\,dx\\,dz\\le C,\n\\end{split}\\end{equation}\nsince~$2n+2s-1>n+1$.\n\nSimilarly, recalling~\\eqref{HENCE1},\n\\eqref{HENCE2} and~\\eqref{6bis},\n\\begin{equation*}\n\\begin{split}& \\int_{\\mathbb{R}^{n+1}_+} z^a |\\nabla E_\\varphi(x,z)|^2\\,dx\\,dz\n\\\\ \\le\\;& C+\\int_{ \\{(x,z):\\, x\\in B_2,\\, z>1\\}}\nz^{-a} \\big(z^a |\\nabla E_\\varphi(x,z)|\\big)^2\\,dx\\,dz\n+\\int_{ \\{(x,z): \\, x\\in \\mathbb{R}^n\\setminus B_2,\\, z\\in(0,1]\\}}\nz^{-a} \\big(z^a |\\nabla E_\\varphi(x,z)|\\big)^2\\,dx\\,dz\n\\\\&\\qquad+\\int_{ \\{(x,z):\\, x\\in \\mathbb{R}^n\\setminus B_2,\\,z>1\\}}\nz^{-a} \\big(z^a |\\nabla E_\\varphi(x,z)|\\big)^2\\,dx\\,dz\n\\\\ \\le\\;& C+\n\\int_{ \\{(x,z):\\, x\\in B_2,\\, z>1\\}}\n\\frac{C}{z^{2n+1+2s}}\\,dx\\,dz\n+\\int_{ \\{(x,z):\\, x\\in \\mathbb{R}^n\\setminus B_2,\\, z\\in(0,1]\\}}\n\\frac{C}{z^{1-2s} (|x|^2+0)^{n+2s}}\\,dx\\,dz\\\\&\\qquad\n+\\int_{ \\{(x,z):\\, x\\in \\mathbb{R}^n\\setminus B_2,\\,z>1\\}}\n\\frac{C}{(|x|^2+z^2)^{n+2s}}\\,dx\\,dz\n\\\\ \\le\\;&C,\n\\end{split}\n\\end{equation*}\nand therefore\n\\begin{equation}\\label{STIME:EPHI2}\n\\int_{\\mathbb{R}^{n+1}_+\\setminus{\\mathcal{B}}_{R\/2}} z^a |\\nabla E_\\varphi(x,z)|^2\\,dx\\,dz\\le\\delta(R),\n\\end{equation}\nwith~$\\delta(R)$ infinitesimal as~$R\\to+\\infty$.\n\nIn addition,\n\\begin{eqnarray*} \\Big| |\\nabla \\Phi_R|^2-|\\nabla E_\\varphi|^2\\Big|\n&\\le& E_\\varphi^2|\\nabla\\theta_R|^2\n+|\\nabla E_\\varphi|^2 |1-\\theta_R^2|\n+2|E_\\varphi| |\\theta_R|\n|\\nabla E_\\varphi||\\nabla\\theta_R|\\\\\n&\\le& \\frac{C E_\\varphi^2}{R^2 }+\nC|\\nabla E_\\varphi|^2 \\chi_{\\mathbb{R}^{n+1}_+\\setminus {\\mathcal{B}}_{R\/2}}.\n\\end{eqnarray*}\nTherefore, in light of~\\eqref{STIME:EPHI1} and~\\eqref{STIME:EPHI2},\n\\begin{eqnarray*}&& \\left|\n\\int_{\\mathbb{R}^{n+1}_+} z^a |\\nabla \\Phi_R(x,z)|^2\\,dx\\,dz\n-\n\\int_{\\mathbb{R}^{n+1}_+} z^a |\\nabla E_\\varphi(x,z)|^2\\,dx\\,dz\\right|\\\\ &\\le&\n\\frac{C}{R^2}\\int_{\\mathbb{R}^{n+1}_+} z^a\n|E_\\varphi(x,z)|^2\\,dx\\,dz+\nC\\int_{\\mathbb{R}^{n+1}_+\\setminus {\\mathcal{B}}_{R\/2}} z^a\n|\\nabla E_\\varphi(x,z)|^2 \\,dx\\,dz \\\\&\\le&\n\\frac{C}{R^2}+\\delta(R).\n\\end{eqnarray*}\nConsequently, we find that\n\\begin{equation}\\label{PAal203948:PP}\n\\begin{split}\n& \\lim_{R\\to+\\infty} \\inf_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n\\\\ \\le\\;&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi_R)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n\\\\\n=\\;&\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Phi_R(x,z)|^2 +2\\nabla E_v(x,z)\\cdot\\nabla\\Phi_R(x,z)\\big)\\,dx\\,dz\n\\\\ \n\\le\\;&\n\\lim_{R\\to+\\infty} \\left[ \\frac{C}{R^2}+\\delta(R)+\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_\\varphi(x,z)|^2 +2\\nabla E_v(x,z)\\cdot\\nabla\nE_\\varphi(x,z)\\big)\\,dx\\,dz\\right.\\\\&\\qquad\n\\left.+2\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi_R(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right]\\\\\n\\le\\;&\n\\lim_{R\\to+\\infty} \\left[ \\frac{C}{R^2}+\\delta(R)+\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_\\varphi(x,z)|^2 +2\\nabla E_v(x,z)\\cdot\\nabla\nE_\\varphi(x,z)\\big)\\,dx\\,dz\\right.\\\\&\\qquad\\left.\n+2\\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right]\n.\\end{split}\\end{equation}\nNow we claim that\n\\begin{equation}\\label{PAal203948}\n\\lim_{R\\to+\\infty} \\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\left|\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right|\n=0.\n\\end{equation}\nTo check this, we recall the notation in~\\eqref{elle R}\nand observe that\n\\begin{equation}\\label{012owe2eudyf8i:00}\n\\begin{split}\n&\\lim_{R\\to+\\infty}\\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\left|\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right|\\\\\n=\\,&\n\\lim_{R\\to+\\infty}\\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\\left|\n\\int_{ {\\mathcal{B}}_R^+ } {\\rm div}\\,\\Big(\nz^a \\Phi(x,z) \\nabla E_v(x,z)\\Big)\\,dx\\,dz\n-\\int_{ {\\mathcal{B}}_R^+ } {\\rm div}\\,\\Big(\nz^a E_v(x,z)\\nabla E_\\varphi(x,z)\\Big)\\,dx\\,dz\\right|\\\\\n=\\,&\n\\lim_{R\\to+\\infty}\\left| \\int_{B_R} \\varphi(x)\\, (-\\Delta)^s v(x)\\,dx\n- \\int_{B_R} v(x)\\, (-\\Delta)^s \\varphi(x)\\,dx\\right.\\\\\n&\\quad\\qquad\\left.-\\int_{L_R\\cup U_R} \nz^a E_v(x,z)\\partial_\\nu E_\\varphi(x,z)\\,d{\\mathcal{H}}^n(x,z)\\right|\n.\\end{split}\n\\end{equation}\nAlso, as in~\\eqref{889AO}, we have that\n\\begin{equation*}\n\\lim_{R\\to+\\infty}\n\\int_{L_R\\cup U_R }\nz^a |E_{v}(x,z)|\\,|\n\\partial_\\nu \nE_{\\varphi}(x,z)|\\,d{\\mathcal{H}}^{n}(x,z)=0.\n\\end{equation*}\nHence, fixing~$k$ and\nrecalling the notation in~\\eqref{v k}, we derive from~\\eqref{012owe2eudyf8i:00}\nthat\n\\begin{eqnarray*}\n&&\\lim_{R\\to+\\infty}\\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\\left|\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right| \\\\\n&=& \\left|\\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s v(x)\\,dx\n- \\int_{\\mathbb{R}^n} v(x)\\, (-\\Delta)^s \\varphi(x)\\,dx\\right| \\\\\n&=& \\left|\\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s v_k(x)\\,dx\n- \\int_{\\mathbb{R}^n} v_k(x)\\, (-\\Delta)^s \\varphi(x)\\,dx\n\\right.\\\\\n&&\\qquad+\\left. \\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s w_k(x)\\,dx\n- \\int_{\\mathbb{R}^n} w_k(x)\\, (-\\Delta)^s \\varphi(x)\\,dx\\right|\n\\\\ &=&\\left|\n\\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s w_k(x)\\,dx\n- \\int_{\\mathbb{R}^n} w_k(x)\\, (-\\Delta)^s \\varphi(x)\\,dx\\right|.\n\\end{eqnarray*}\nSince~$|(-\\Delta)^s \\varphi(x)|\\le\\frac{C}{1+|x|^{n+2s}}\\in L^1(\\mathbb{R}^n)$,\nwe can take the limit as~$k\\to+\\infty$ and use the Dominated Convergence Theorem,\nto obtain that\n\\begin{equation}\\label{udofjvw9etgierufgh}\n\\begin{split}\n&\\lim_{R\\to+\\infty}\\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\left|\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right|\\\\\n=\\;&\\lim_{k\\to+\\infty}\\left| \\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s w_k(x)\\,dx\n- \\int_{\\mathbb{R}^n} w_k(x)\\, (-\\Delta)^s \\varphi(x)\\,dx\\right|\\\\\n=\\;&\\lim_{k\\to+\\infty}\\left| \\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s w_k(x)\\,dx\\right|\n\\\\ =\\;&\n\\lim_{k\\to+\\infty}\\left| \\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s v(x)\\,dx\n-\\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s (\\tau_k v)(x)\\,dx\\right|\n\\\\ =\\;& \\left|\n\\int_{\\mathbb{R}^n} \\varphi(x)\\, (-\\Delta)^s v(x)\\,dx\n-\n\\lim_{k\\to+\\infty}\\left[ \\int_{\\mathbb{R}^n} \\tau_k(x)\\,\\varphi(x)\\, (-\\Delta)^s v(x)\\,dx\\right.\\right.\\\\ &\n\\qquad\\left.\\left.+\n\\int_{\\mathbb{R}^n} v(x)\\,\\varphi(x)\\, (-\\Delta)^s \\tau_k(x)\\,dx+\n\\int_{\\mathbb{R}^n} \\varphi(x)\\, B(\\tau_k, v)(x)\\,dx\\right]\\right|\n\\\\ =\\;&\\lim_{k\\to+\\infty}\\left|\\frac{1}{k^{2s}}\n\\int_{B_1} v(x)\\,\\varphi(x)\\, (-\\Delta)^s \\tau\\left(\\frac{x}{k}\\right)\\,dx+\n\\int_{\\mathbb{R}^n} \\varphi(x)\\, B(\\tau_k, v)(x)\\,dx\\right|\\\\\n\\le\\;& \\lim_{k\\to+\\infty}\\left[ \\frac{C}{k^{2s}}\n\\int_{B_1} |v(x)|\\,|\\varphi(x)|\\,dx+\n\\int_{\\mathbb{R}^n} |\\varphi(x)|\\, |B(\\tau_k, v)(x)|\\,dx\\right]\n\\\\ =\\;&\\lim_{k\\to+\\infty}\\int_{B_1} |\\varphi(x)|\\, |B(\\tau_k, v)(x)|\\,dx,\n\\end{split}\\end{equation}\nwhere\n$$ B(f,g):= c\\, \\int_{\\mathbb{R}^n} \\frac{(f(x)-f(y))(g(x)-g(y))}{|x-y|^{n+2s}}\\,dy,$$\nfor some~$c>0$, see e.g. page~636 in~\\cite{MR3211862}\nfor such bilinear form.\n\nNotice now that\n\\begin{eqnarray*}\n|B(\\tau_k, v)(x)|&\\le&\nC\\int_{\\mathbb{R}^n} \\frac{\\min\\{ 1,\\frac{|x-y|}{k}\\}\\,\n\\min\\{ 1,|x-y|\\}}{|x-y|^{n+2s}}\\,dy\\\\\n&\\le&\n\\frac{C}{k} \\int_{B_1(x)} \\frac{|x-y|^2}{|x-y|^{n+2s}}\\,dy\n+\n\\frac{C}{k}\\int_{B_k(x)\\setminus B_1(x)} \\frac{|x-y|}{|x-y|^{n+2s}}\\,dy\n+\nC\\int_{\\mathbb{R}^n\\setminus B_k(x)} \\frac{dy}{|x-y|^{n+2s}}\\\\ &\\le&\\frac{C}{k}+\\frac{C}{k^{2s}}.\n\\end{eqnarray*}\nWe plug this information into~\\eqref{udofjvw9etgierufgh} and we obtain~\\eqref{PAal203948},\nas desired.\n\nNow, we insert~\\eqref{PAal203948} into~\\eqref{PAal203948:PP} and we conclude that\n\\begin{eqnarray*}&& \\lim_{R\\to+\\infty} \\inf_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n\\\\ &&\\qquad\\le \\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_\\varphi(x,z)|^2 +2\\nabla E_v(x,z)\\cdot\\nabla\nE_\\varphi(x,z)\\big)\\,dx\\,dz\\\\\n&&\\qquad=\n\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2 -|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz.\n\\end{eqnarray*}\nHence, to prove~\\eqref{0OAPQO182}, it remains to show that\n\\begin{equation}\\label{0OAPQO182:BIS}\n\\begin{split}\n&\\lim_{R\\to+\\infty}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n\\\\\n\\le \\;& \\lim_{R\\to+\\infty} \\inf_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\n.\\end{split}\\end{equation}\nFor this, we fix~$\\Psi\\in\nC^\\infty_0({\\mathcal{B}}_R)$ with~$\\Psi(x,0)=\\varphi(x)$ and we use again~\\eqref{PAal203948}\nto see that\n\\begin{equation} \\label{00:10023a:023948}\n\\begin{split}\n& \\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Psi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\\\\\n=\\;&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi(x,z)|^2+2z^a\\nabla\\Psi(x,z)\\cdot\\nabla E_v(x,z)\\big)\\,dx\\,dz\n\\\\\n=\\;&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi(x,z)|^2+2z^a\\nabla E_\\varphi(x,z)\\cdot\\nabla E_v(x,z)\\big)\\,dx\\,dz\n\\\\&\\qquad+2\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Psi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\n\\\\ \\ge\\;&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi(x,z)|^2+2z^a\\nabla E_\\varphi(x,z)\\cdot\\nabla E_v(x,z)\\big)\\,dx\\,dz\n\\\\&\\qquad-2\\sup_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\left|\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla E_v(x,z)\\cdot\\nabla\\big(\\Phi(x,z)-E_\\varphi(x,z)\\big)\\,dx\\,dz\\right|\n\\\\ =\\;&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi(x,z)|^2+2z^a\\nabla E_\\varphi(x,z)\\cdot\\nabla E_v(x,z)\\big)\\,dx\\,dz\n-\\mu(R),\n\\end{split}\\end{equation}\nwith~$\\mu(R)$ independent of~$\\Psi$ and infinitesimal as~$R\\to+\\infty$.\n\nNow we take~$\\Psi_*\\in C^\\infty_0({\\mathcal{B}}_R)$ such that~$\\Psi_*(x,0)=\\varphi(x)$\nthat minimizes the functional~$\\Psi\\mapsto\\int_{ {\\mathcal{B}}_R^+ }z^a|\\nabla\\Psi(x,z)|^2\\,dx\\,dz$\nin such class. Then, using the variational equation for minimizers\ninside~${\\mathcal{B}}_R^+$ and the fact that~$\\Psi_*(x,0)=E_\\varphi(x,0)$, we see that\n\\begin{eqnarray*}\n&& \\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi(x,z)|^2-|\\nabla E_\\varphi(x,z)|^2\\big)\\,dx\\,dz\\\\\n&\\ge&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi_*(x,z)|^2-|\\nabla E_\\varphi(x,z)|^2\\big)\\,dx\\,dz\\\\\n&=&\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\nabla\\big(\\Psi_*-E_\\varphi\\big)(x,z)\\cdot\n\\nabla\\big(\\Psi_*+E_\\varphi\\big)(x,z)\n\\,dx\\,dz\\\\\n&=& \\int_{ {\\mathcal{B}}_R^+ }{\\rm div}\\,\\Big(\nz^a\\big(\\Psi_*-E_\\varphi\\big)(x,z)\n\\nabla\\big(\\Psi_*+E_\\varphi\\big)(x,z)\\Big)\n\\,dx\\,dz\\\\\n&=& \\int_{ L_R\\cup U_R}\nz^a\\big(\\Psi_*-E_\\varphi\\big)(x,z)\n\\partial_\\nu\\big(\\Psi_*+E_\\varphi\\big)(x,z)\\,d{\\mathcal{H}}^n(x,z)\\\\\n&=& -\\int_{ L_R\\cup U_R}\nz^a E_\\varphi(x,z)\n\\partial_\\nu E_\\varphi(x,z)\\,d{\\mathcal{H}}^n(x,z)\n.\n\\end{eqnarray*}\nTherefore, recalling~\\eqref{HENCE1} and~\\eqref{HENCE2}, we conclude that\n\\[\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla \\Psi(x,z)|^2-|\\nabla E_\\varphi(x,z)|^2\\big)\\,dx\\,dz\\ge -\\int_{ L_R\\cup U_R}\n\\frac{C}{R^{n+2s}}\n\\,d{\\mathcal{H}}^n(x,z)\\ge- \\frac{C}{R^{2s}}.\n\\]\n{F}rom this and~\\eqref{00:10023a:023948} we thus obtain that\n\\begin{eqnarray*}\n&& \\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Psi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\\\\\n&\\ge& \n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_\\varphi(x,z)|^2+2z^a\\nabla E_\\varphi(x,z)\\cdot\\nabla E_v(x,z)\\big)\\,dx\\,dz\n-\\mu(R)- \\frac{C}{R^{2s}}\\\\\n&=&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{\\varphi+v}(x,z)|^2\n-|\\nabla E_{v}(x,z)|^2\n\\big)\\,dx\\,dz-\\mu(R)- \\frac{C}{R^{2s}}.\n\\end{eqnarray*}\nSince this is valid for any~$\\Psi\\in C^\\infty_0({\\mathcal{B}}_R)$ with~$\\Psi(x,0)=\\varphi(x)$,\nwe conclude that\n\\begin{eqnarray*}\n&&\\inf_{ {\\Phi\\in\nC^\\infty_0({\\mathcal{B}}_R)}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla (E_{v}+\\Phi)(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz\\\\\n&\\ge&\n\\int_{ {\\mathcal{B}}_R^+ }\nz^a\\big(|\\nabla E_{v+\\varphi}(x,z)|^2\n-|\\nabla E_v(x,z)|^2\\big)\\,dx\\,dz-\\mu(R)- \\frac{C}{R^{2s}}\n.\\end{eqnarray*}\nBy taking the limit as~$R\\to+\\infty$,\nwe obtain~\\eqref{0OAPQO182:BIS}, as desired,\nand so we have\ncompleted the proof of Proposition~\\ref{0OAPQO182:P}.\n\\end{proof}\n\nIn view of Proposition~\\ref{0OAPQO182:P}, \nwe can now\nrelate the original and the extended energy functionals,\naccording to the following result:\n\n\\begin{corollary}\\label{CPOR:EQ}\nLet~$n\\ge2$ and~$s\\in(0,1)$.\nFor any~$v\\in W^{2,\\infty}(\\mathbb{R}^n)$ it holds that\n\\begin{equation}\\label{0o0oP}\n\\inf_{{R>0}\\atop{\\Phi\\in C^\\infty_0(\\mathcal{B}_R)}}\n{\\mathcal{E}}_{\\mathcal{B}_R}(E_v+\\Phi)-{\\mathcal{E}}_{\\mathcal{B}_R}(E_v)\\;\n=\\;\n\\inf_{{R>0}\\atop{\\varphi\\in C^\\infty_0(B_R)}}\n{\\mathcal{F}}_{B_R}(v+\\varphi)-\n{\\mathcal{F}}_{B_R}(v).\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\n Notice that, given~$\\varphi\\in C^\\infty_0(\\mathbb{R}^n)$,\n\\begin{equation}\\label{IDEPHI}\n\\begin{split}\n&\\inf_{ {{R>0}\\atop{\\Phi\\in C^\\infty_0(\\mathcal{B}_R)}}\\atop{\\Phi(x,0)=\\varphi(x)}}\n{\\mathcal{E}}_{\\mathcal{B}_R}(E_v+\\Phi)-{\\mathcal{E}}_{\\mathcal{B}_R}(E_v)\\;\n\\\\=\\;& \n\\inf_{ {{R>0}\\atop{\\Phi\\in C^\\infty_0(\\mathcal{B}_R)}}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{\\mathbb{R}^{n+1}_+} z^a\\Big( |\\nabla (E_v+\\Phi)(x,z)|^2-|\\nabla E_v(x,z)|^2\n\\Big)\\,dx\\,dz\\\\ &\\qquad\\qquad\\qquad+\n\\int_{\\mathbb{R}^n}\\Big( F(v(x)+\\varphi(x))-F(v(x))\\Big)\\,dx\n\\\\ =\\;&\\lim_{R\\to+\\infty}\n\\inf_{ {{\\Phi\\in C^\\infty_0(\\mathcal{B}_R)}}\\atop{\\Phi(x,0)=\\varphi(x)}}\n\\int_{\\mathbb{R}^{n+1}_+} z^a\\Big( |\\nabla (E_v+\\Phi)(x,z)|^2-|\\nabla E_v(x,z)|^2\\Big)\n\\,dx\\,dz\\\\&\\qquad\\qquad\\qquad+\n\\int_{\\mathbb{R}^n}\\Big( F(v(x)+\\varphi(x))-F(v(x))\\Big)\\,dx\n\\\\=\\;&\n\\iint_{\\mathbb{R}^{2n}}\n\\frac{|(v+\\varphi)(x)-(v+\\varphi)(y)|^2-\n|v(x)-v(y)|^2\n}{|x-y|^{n+2s}}\\,dx\\,dy\\\\&\\qquad\\qquad\\qquad\n+\n\\int_{\\mathbb{R}^n}\\Big( F(v(x)+\\varphi(x))-F(v(x))\\Big)\\,dx\\\\\n=\\;&\n{\\mathcal{F}}_{B_R}(v+\\varphi)-\n{\\mathcal{F}}_{B_R}(v),\n\\end{split}\\end{equation}\nthanks to Proposition~\\ref{0OAPQO182:P}.\nSince the identity in~\\eqref{IDEPHI}\nis valid for any~$\\varphi\\in C^\\infty_0(\\mathbb{R}^n)$\n(i.e. for any~$R>0$ and any~$\\varphi\\in C^\\infty_0(B_R)$), taking the infimum in such\nclass we obtain~\\eqref{0o0oP}, as desired.\n\\end{proof}\n\nDue to Corollary~\\ref{CPOR:EQ},\nthe following equivalence result for minimizers holds true:\n\n\\begin{proposition}\\label{EQUI DL}\n$E_v$ is an extended local minimizer according to Definition~\\ref{D:L:2}\nif and only if~$v$ is a local minimizer according to Definition~\\ref{D:L}.\n\\end{proposition}\n\n\\begin{proof} We observe that~$E_v$ is an extended local minimizer according to Definition~\\ref{D:L:2}\nif and only if the first term in~\\eqref{0o0oP} is nonnegative;\non the other hand, $v$ is a local minimizer according to Definition~\\ref{D:L}\nif and only if the last term in~\\eqref{0o0oP} is nonnegative; since\nthe two terms in~\\eqref{0o0oP} are equal, the desired result is established.\n\\end{proof}\n\nIn view of Proposition~\\ref{EQUI DL} (see also Lemma 6.1 in~\\cite{cabre-TAMS}),\nit is natural to say that~$u$ is a stable solution of~\\eqref{ALLEN-GEN}\nif the second derivative of the associated energy functional is nonnegative,\naccording to the following setting:\n\n\\begin{definition}\nLet~$u$ be a solution of~\\eqref{ALLEN-GEN} in~$\\mathbb{R}^n$. We say that~$u$ is stable if\n$$ \\int_{\\mathbb{R}^{n+1}_+} z^a |\\nabla \\zeta(x,z)|^2\\,dx\\,dz+\\int_{\\mathbb{R}^n} F''(u(x))\\,\\zeta^2(x,0)\\,dx\\ge0$$\nfor any~$\\zeta\\in C^\\infty_0(\\mathbb{R}^{n+1})$.\n\\end{definition}\n\n\\section{Variational classification of $1$D solutions}\\label{SEC:2}\n\nThe goal of this section is to establish the following result:\n\n\\begin{lemma}\\label{CL:MO}\nLet~$s\\in(0,1)$ and\n$v\\in C^2(\\mathbb{R},[-1,1])$ be a stable solution of~$(-\\Delta)^s v=f(v)$ in~$\\mathbb{R}$.\nAssume also that~$\\dot v\\ge0$. Then~$v$ is a local minimizer.\n\\end{lemma}\n\n\\begin{proof} The monotonicity of~$v$ implies that the following limits exist:\n$$ \\underline\\ell:= \\lim_{t\\to-\\infty}v(t)\\le\\lim_{t\\to+\\infty}v(t) =:\\overline\\ell.$$\nWe also consider the sequence of functions~$v_k(t):=v(t+k)$.\nBy the Theorem of Ascoli, up to a subsequence we know that~$v_k$ converges to~$\\overline\\ell$\nin~$C^2_{\\rm loc}(\\mathbb{R})$, and so, passing the equation to the limit, we conclude that~$f(\\overline\\ell)=0$.\nSimilarly, one sees that~$f(\\underline\\ell)=0$.\nAs a consequence,\n\\begin{equation}\\label{889:p}\n\\underline\\ell,\\,\\overline\\ell\\in\\{-1,0,1\\}.\\end{equation}\nNow, we claim that\n\\begin{equation}\\label{NOT Z}\n{\\mbox{$v$ is not identically zero.}}\n\\end{equation}\nThe proof is by contradiction: if\n$v$ is identically zero, we take~$\\psi\\in C^\\infty_0(\\mathbb{R}^2)$\nand, for~$\\varepsilon>0$, we let~$\\psi_\\varepsilon(x,z):=\\psi(\\varepsilon x,\\varepsilon z)$. \nThe stability inequality for~$\\psi_\\varepsilon$ gives that\n\\begin{eqnarray*}\n0 &\\le& \\int_{\\mathbb{R}^2_+} z^a |\\nabla\\psi_\\varepsilon(x,z)|^2\\,dx\\,dz+\n\\int_{\\mathbb{R}} F''(v(x))\\,\\psi_\\varepsilon^2(x,0)\\,dx\\\\ &=&\n\\varepsilon^2 \\int_{\\mathbb{R}^2_+} z^a |\\nabla\\psi(\\varepsilon x,\\varepsilon z)|^2\\,dx\\,dz+\n\\int_{\\mathbb{R}} F''(v(x))\\,\\psi^2(\\varepsilon x,0)\\,dx\n\\\\ &=&\n\\varepsilon^{-a} \\int_{\\mathbb{R}^2_+} Z^a |\\nabla\\psi(X,Z)|^2\\,dX\\,dZ-\\varepsilon^{-1}\n\\int_{\\mathbb{R}} f'(0)\\,\\psi^2(X,0)\\,dX\n\\\\ &=& C_1\\varepsilon^{2s-1}-C_2\\varepsilon^{-1} f'(0),\n\\end{eqnarray*}\nfor some~$C_1$, $C_2>0$. {F}rom this, one obtains that\n$$ f'(0)\\le \\lim_{\\varepsilon\\searrow0} \\frac{C_1\\varepsilon^{2s}}{C_2}=0.$$\nThis is a contradiction, since~$f'(0)>0$ and thus~\\eqref{NOT Z} is proved.\n\nTo complete the proof of Lemma~\\ref{CL:MO},\nwe now distinguish two cases, either~$v$ is constant or not.\nIf~$v$ is constant, then it is either identically~$-1$ or identically~$1$,\ndue to~\\eqref{NOT Z}, and this implies the desired result.\n\nSo, we can now focus on the case in which $v$ is not constant.\nThen, $\\underline\\ell<\\overline\\ell$. So, from~\\eqref{889:p},\nwe have that~$v$ is a transition layer connecting:\n\\begin{enumerate}\n\\item either $-1$ to~$0$,\n\\item or $0$ to $1$,\n\\item or $-1$ to $1$.\n\\end{enumerate}\nIn view of Theorem 2.2(i) in~\\cite{cabre-sire-AIHP}, the first two cases cannot occur\nand therefore\n\\begin{equation}\\label{78s:10234}\n\\lim_{t\\to\\pm\\infty} v(t)=\\pm1.\n\\end{equation}\nSince the proof of this fact relies on the theory of layer solutions,\nwe provide the details of the argument that we used.\nWe argue for a contradiction and we suppose that\n\\begin{eqnarray}\n&& \\label{C:caso1}\n{\\mbox{ either }} \\lim_{t\\to-\\infty} v(t)=-1 {\\mbox{ and }} \\lim_{t\\to+\\infty} v(t)=0\\\\&&\\label{C:caso2}\n{\\mbox{ or }} \\lim_{t\\to-\\infty} v(t)=0 {\\mbox{ and }} \\lim_{t\\to+\\infty} v(t)=1.\n\\end{eqnarray}\nBy maximum principle, we know that~$\\dot v>0$.\nThen, if we set either~$\\tilde v(t):= 2v(t)+1$ (if the case in~\\eqref{C:caso1} holds true)\nor~$\\tilde v(t):= 2v(t)-1$ (if~\\eqref{C:caso2} holds true), we have that\nthe derivative of~$\\tilde v$ is strictly positive and\n\\begin{equation}\\label{78s:10234:BIS} \\lim_{t\\to\\pm\\infty}\\tilde v(t)=\\pm1.\\end{equation}\nIn addition,\n$$ (-\\Delta)^s \\tilde v(t)= 2(-\\Delta)^s v(t)= 2f(v(t))=2 \nf\\left( \\frac{\\tilde v(t)\\mp1}{2}\\right)=\n-G'(\\tilde v(t)),$$\nwith\n$$ G(r):= -2\\int_{0}^r f\\left( \\frac{\\tau\\mp1}{2}\\right)\\,d\\tau\n=\n-4\\int_{\\mp1\/2}^{(r\\mp1)\/2} f(\\sigma)\\,d\\sigma.$$\nThis and~\\eqref{78s:10234:BIS} give that we are in the setting of\nTheorem 2.2(i) in~\\cite{cabre-sire-AIHP}. In particular, from formula~(2.8)\nin~\\cite{cabre-sire-AIHP} we know that\n$$ 0=G(-1)-G(1)=\n4\\int_{\\mp1\/2}^{(1\\mp1)\/2} f(\\sigma)\\,d\\sigma\n-4\\int_{\\mp1\/2}^{(-1\\mp1)\/2} f(\\sigma)\\,d\\sigma=\n4\\int_{(-1\\mp1)\/2}^{(1\\mp1)\/2} f(\\sigma)\\,d\\sigma\n,$$\nhence\n$$ {\\mbox{ either }}\\quad\n\\int_{-1}^{0} f(\\sigma)\\,d\\sigma=0 \\quad{\\mbox{ or }}\\quad\n\\int_{0}^{1} f(\\sigma)\\,d\\sigma=0.$$\nThis is in contradiction with~\\eqref{BISTABLE}\nand so it proves~\\eqref{78s:10234}.\n\nHence, necessarily~$v$ is a transition layer connecting~$-1$ to~$1$\nand so it is minimal due to the sliding method (see e.g. the proof of Lemma~9.1\nin~\\cite{VSS}).\n\\end{proof}\n\n\\section{Classification of the profiles at infinity}\\label{CLASS:A}\n\nIn this section, we consider the two profiles \nof a given solution at infinity. Namely, if~$s\\in(0,1)$ and~$u\\in C^2(\\mathbb{R}^n,[-1,1])$\nis a solution of~$(-\\Delta)^s u=f(u)$ in~$\\mathbb{R}^n$, \nwith~$\\partial_{x_n}u>0$ in~$\\mathbb{R}^n$,\nwe set\n$$ \\underline u(x'):=\\lim_{x_n\\to-\\infty} u(x',x_n)\\quad{\\mbox{ and }}\\quad\n\\overline u(x'):=\\lim_{x_n\\to+\\infty} u(x',x_n).$$\nIn this setting, we have:\n\n\\begin{lemma} \\label{02738}\nAssume that~$n=3$. Then,\nboth~$\\underline u$ and~$\\overline u$ are $1$D and local minimizers.\n\\end{lemma}\n\n\\begin{proof} By passing the equation to the limit, we have that\n\\begin{equation}\\label{sta000}\n{\\mbox{both~$\\underline u$ and~$\\overline u$ are \nstable solutions in~$\\mathbb{R}^{n-1}=\\mathbb{R}^2$.}}\\end{equation}\nThe proof of~\\eqref{sta000} is based on a general\nargument (see e.g.~\\cite{MR2483642}), given in details\nhere for the sake of completeness.\nLet~$\\xi\\in C^\\infty_0(\\mathbb{R}^{n})$ and~$\\eta\\in C^\\infty_0((-1,1))$, with~$\\eta(0)=1$.\nGiven~$\\varepsilon>0$, we set~$\\xi_\\varepsilon(x,z)=\\xi_\\varepsilon(x',x_n,z):=\n\\xi(x',z)\\eta(\\varepsilon x_n)$. Notice that~$\\xi_\\varepsilon\\in C^\\infty_0(\\mathbb{R}^{n+1})$,\ntherefore by the stability of~$u$ and the translation invariance we\nhave that\n\\begin{eqnarray*}\n0 &\\le& \\lim_{t\\to+\\infty}\nC \\int_{\\mathbb{R}^{n+1}_+} z^a |\\nabla \\xi_\\varepsilon(x,z)|^2\\,dx\\,dz+\n\\int_{\\mathbb{R}^n} F''( u(x',x_n+t))\\,\\xi_\\varepsilon(x,0)\\,dx\\\\&=&\nC \\int_{\\mathbb{R}^{n+1}_+} z^a |\\nabla \\xi_\\varepsilon(x,z)|^2\\,dx\\,dz+\n\\int_{\\mathbb{R}^n} F''( \\overline u(x'))\\,\\xi_\\varepsilon^2(x,0)\\,dx\\\\\n&=&\nC \\int_{\\mathbb{R}^{n+1}_+} z^a \\Big(\n|\\nabla \\xi(x',z)|^2 \\eta^2(\\varepsilon x_n)+\n\\varepsilon^2 |\\nabla \\eta(\\varepsilon x_n)|^2 \\xi^2(x',z)+2\\varepsilon\n\\eta(\\varepsilon x_n)\\xi(x',z)\\nabla\\eta(\\varepsilon x_n)\\cdot\\nabla \\xi(x',z)\n\\Big)\\,dx\\,dz\\\\&&\\quad+\n\\int_{\\mathbb{R}^n} F''( \\overline u(x'))\\,\\xi^2(x',0)\\eta^2(\\varepsilon x_n)\\,dx.\n\\end{eqnarray*}\nHence, taking the limit as~$\\varepsilon\\to0$,\n$$ 0\\le\nC \\int_{\\mathbb{R}^{n+1}_+} z^a \n|\\nabla \\xi(x',z)|^2 \\,dx\\,dz+\n\\int_{\\mathbb{R}^n} F''( \\overline u(x'))\\,\\xi^2(x',0)\\,dx,$$\nand so~$\\overline u$ is stable in~$\\mathbb{R}^{n-1}$.\nThis proves~\\eqref{sta000} for $\\overline u$ (the case of~$\\underline u$\nis similar).\n\nAs a consequence of \\eqref{sta000} and of the classification results in the plane\n(see in particular Theorem 2.12 in~\\cite{cabre-TAMS}, or~\\cite{SV09}), we conclude that~$\n\\underline u$ and~$\\overline u$ are~$1$D and monotone.\nThen, the local minimality is a consequence of Lemma~\\ref{CL:MO}.\n\\end{proof}\n\n\\section{Local minimization by range constraint}\\label{CLASS:B}\n\nIn this section, we point out that perturbations which do not pointwise exceed\nthe limit profiles necessarily increase the energy. For the classical case,\nthis property has been exploited in Theorem~4.5 of~\\cite{AAC},\nTheorem~10.4 of~\\cite{danielli} and \nLemma~2.2 of~\\cite{FV11}. In the framework of this paper, the result that we need is the following:\n\n\\begin{lemma}\\label{INTRAP}\nLet~$s\\in(0,1)$ and~$u\\in C^2(\\mathbb{R}^n,[-1,1])$\nbe a solution of~$(-\\Delta)^s u=f(u)$ in~$\\mathbb{R}^n$, \nwith~$\\partial_{x_n}u>0$ in~$\\mathbb{R}^n$.\nLet\n$$ \\underline u(x'):=\\lim_{x_n\\to-\\infty} u(x',x_n)\\quad{\\mbox{ and }}\\quad\n\\overline u(x'):=\\lim_{x_n\\to+\\infty} u(x',x_n).$$\nLet also~$R>0$ and~$\\Phi\\in C^\\infty_0({\\mathcal{B}}_R)$ and suppose that\n$$ (E_u+\\Phi)(X)\\in \\big[ E_{\\underline u}(X),\\,E_{\\overline u}(X)\\big]\\qquad{\\mbox{for \nany }}X\\in\\mathbb{R}^{n+1}_+.$$\nThen~${\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_u)\\le{\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_u+\\Phi)$.\n\\end{lemma}\n\n\\begin{proof} The argument is by contradiction. \nWe suppose that there exist~$R>0$\nand a perturbation~$\\Phi\\in C^\\infty_0({\\mathcal{B}}_R)$ with\n$$ (E_u+\\Phi)(X)\\in \\big[ E_{\\underline u}(X),\\,E_{\\overline u}(X)\\big]\\qquad{\\mbox{for \nany }}X\\in\\mathbb{R}^{n+1}_+$$\nand such that\n$$ {\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_u+\\Phi) < {\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_u).$$\nThat is, letting~$\\varphi(x):=\\Phi(x,0)$, \nby Proposition~\\ref{0OAPQO182:P},\n\\begin{eqnarray*}\n0 &>& \n\\frac{1}{2}\\int_{\\mathbb{R}^{n+1}_+}z^a \\big(|\\nabla (E_u+\\Phi)(x,z)|^2-\n|\\nabla E_u(x,z)|^2\\big)\\,dx\\,dz\n+\\int_{B_R} \\big( F((u+\\varphi)(x))-F(u(x))\\big)\\,dx\n\\\\ &=&\n\\frac{1}{2}\\iint_{\\mathbb{R}^{2n}}\\frac{|(u+\\varphi)(x)-(u+\\varphi)(y)|^2\n-|u(x)-u(y)|^2}{|x-y|^{n+2s}}\\,dx\\,dy\n+\\int_{B_R} \\big( F((u+\\varphi)(x))-F(u(x))\\big)\\,dx\\\\&=&\n{\\mathcal{F}}_{B_R}(u+\\varphi)-{\\mathcal{F}}_{B_R}(u).\n\\end{eqnarray*}\nTherefore, there exists a perturbation~$w_o:=u+\\varphi_o$ of~$u$, with~$\\varphi_o\\in C^\\infty_0(B_R)$,\nsuch that\n\\begin{equation}\\label{TRA:1}\nw_o(x)\\in \\big[ {\\underline u}(x'),\\,{\\overline u}(x')\\big]\\qquad{\\mbox{for \nany }}x\\in\\mathbb{R}^n\\end{equation}\nand\n\\begin{equation}\\label{GHL:ocon} {\\mathcal{F}}_{B_R}(w_o)=\\inf_{{\\varphi\\in C^\\infty_0(B_R)}\\atop{\n(u+\\varphi)(x)\n\\in [ {\\underline u}(x'),\\,{\\overline u}(x')]}}\n{\\mathcal{F}}_{B_R}(u+\\varphi)<{\\mathcal{F}}_{B_R}(u).\\end{equation}\nAs a consequence, by taking energy perturbations,\nwe see that~$(-\\Delta)^s w_o=f'(w_o)$ inside~$\\{\nx\\in\\mathbb{R}^n {\\mbox{ s.t. }} w_o(x)\\in [ {\\underline u}(x'),\\,{\\overline u}(x')]\\}$.\nIn addition, if~$w(x_o)={\\overline u}(x_o)$, then~$(-\\Delta)^s w_o(x_o)\\le f'(w_o(x_o))$.\nSimilarly,\nif~$w(x_o)={\\underline u}(x_o)$, then~$(-\\Delta)^s w_o(x_o)\\ge f'(w_o(x_o))$.\n\nNow we claim that strict inequalities hold in~\\eqref{TRA:1}, namely\n\\begin{equation}\\label{TRA:2}\nw_o(x)\\in \\big( {\\underline u}(x'),\\,{\\overline u}(x')\\big)\\qquad{\\mbox{for \nany }}x\\in\\mathbb{R}^n.\\end{equation}\nTo check this, suppose by contradiction, for instance,\nthat there exists~$x_o\\in\\mathbb{R}^n$\nsuch that~$w_o(x_o)={\\overline u}(x_o')$. Then, the function~$\\zeta(x):=\n{\\overline u}(x')-w_o(x)$ has a minimum at~$x_o$. Accordingly,\n$$ 0\\ge(-\\Delta)^s \\zeta(x_o) \\ge f({\\overline u}(x_o'))- f'(w_o(x_o)) =0.$$\nThis gives that~$\\zeta$ must vanish identically, and thus that~$w_o$ is identically equal to~$\n{\\overline u}$. Then, taking~$\\bar x\\in\\mathbb{R}^n\\setminus B_R$, we have that\n$$ {\\overline u}(\\bar x')= w_o(\\bar x)=u(\\bar x)<\n{\\overline u}(\\bar x').$$\nThis is a contradiction, and therefore~\\eqref{TRA:2} is established.\n\n{F}rom~\\eqref{TRA:2}, it follows that\n$$ \\max_{\\overline{B_R}}( w_o-{\\overline u})<0.$$\nNow, we let~$w_k(x):=u(x+ke_n)$. We claim that there exists~$\\bar k\\in\\mathbb{N}$ such that\n\\begin{equation} \\label{df:29}\nw_{\\bar k}(x)>w_o(x) {\\mbox{ for any }} x\\in\\mathbb{R}^n.\\end{equation}\nTo prove this, we argue by contradiction and suppose that for any~$k\\in\\mathbb{N}$\nthere exists~$x_k\\in\\mathbb{R}^n$ such that~$\nw_{k}(x_k)\\le w_o(x_k)$.\nThen, $x_k\\in B_R$\n(otherwise~$w_o(x_k)=u(x_k)<u(x_k+ke_n)=w_k(x_k)$).\nAccordingly, up to a subsequence, we may suppose that~$x_k\\to\\bar x$,\nfor some~$\\bar x\\in\\overline{B_R}$ as~$k\\to+\\infty$.\nConsequently,\n$$ 0\\le\\lim_{k\\to+\\infty} w_o(x_k)-w_k(x_k)\n= w_o(\\bar x)-\n\\lim_{k\\to+\\infty} u(x_k-ke_n)=\nw_o(\\bar x)-\n\\lim_{k\\to+\\infty} u(\\bar x-ke_n)=\nw_o(\\bar x)-\\overline u(\\bar x').\n$$\nBut this inequality is in contradiction with~\\eqref{TRA:2}\nand thus we have proved~\\eqref{df:29}.\n\nNow, starting from~\\eqref{df:29}, we can reduce~$k$ till\na touching between~$w_k$ and~$w_o$ occurs.\nThat is, we define~$k_\\star:=\\inf\\{ k\\in [0,\\bar k]\n{\\mbox{ s.t. }}w_k>w_o\n\\}$. We claim that\n\\begin{equation}\\label{k0}\nk_\\star=0.\n\\end{equation}\nOnce again, suppose not. Then, the function~$v:=w^{k_\\star}-w_o$\nwould satisfy~$v\\ge0$ in~$\\mathbb{R}^n$, with~$v(\\tilde x)=0$, for some~$\\tilde x\\in\\overline{B_R}$.\nAs a consequence,\n$$ 0\\ge (-\\Delta)^s v(\\tilde x) = f(w^{k_\\star}(\\tilde x))-f(w(\\tilde x))=0,$$\nwhich implies that~$v$ vanishes identically. In particular, fixing~$x_\\star$\noutside~$B_R$, we would have that\n$$ 0=v(x_\\star)=\nw^{k_\\star}(x_\\star)-w_o(x_\\star)=\nu(x_\\star+k_\\star e_n)-u(x_\\star)>0.$$\nThis contradiction completes the proof of~\\eqref{k0}.\n\nNow, in view of~\\eqref{k0}, we obtain that, for any~$x\\in\\mathbb{R}^n$,\n$$ w_o(x)\\le\\lim_{k\\searrow0} w_k(x)=\n\\lim_{k\\searrow0} u(x+ke_n)=u(x).$$\nSimilarly, one can prove that~$w_o(x)\\ge u(x)$.\nTherefore, $w_o$ and~$u$ must coincide\nand so~${\\mathcal{F}}_{B_R}(w_o)={\\mathcal{F}}_{B_R}(u)$.\nBut this fact is in contradiction with~\\eqref{GHL:ocon}\nand so we have completed the proof of Lemma~\\ref{INTRAP}.\n\\end{proof}\n\n\\section{Local minimization properties inherited from those of the profiles at infinity}\\label{CLASS:C}\n\nIn this section, we show that if the profiles at infinity are local minimizers,\nthen so is the original solution. In the classical case of the Laplacian, this property\nwas discussed, for instance, in Proposition~2.3 of~\\cite{FV11}.\nIn our setting, the result that\nwe need is the following (and it uses\nthe pivotal definition of extended local minimizer in Definition~\\ref{D:L:2}):\n\n\\begin{lemma}\\label{0tyg27427412}\nLet~$s\\in(0,1)$ and~$u\\in C^2(\\mathbb{R}^n,[-1,1])$\nbe a solution of~$(-\\Delta)^s u=f(u)$ in~$\\mathbb{R}^n$, \nwith~$\\partial_{x_n}u>0$ in~$\\mathbb{R}^n$.\nLet\n$$ \\underline u(x'):=\\lim_{x_n\\to-\\infty} u(x',x_n)\\quad{\\mbox{ and }}\\quad\n\\overline u(x'):=\\lim_{x_n\\to+\\infty} u(x',x_n)$$\nand suppose that~$\\underline u$ and~$\\overline u$ are local minimizers (in~$\\mathbb{R}^{n-1}$).\nThen, $E_u$ is an extended local minimizer (in~$\\mathbb{R}^{n+1}_+$)\nand~$u$ is a local minimizer (in~$\\mathbb{R}^{n}$).\n\\end{lemma}\n\n\\begin{proof} Our goal is to show that $E_u$ is an extended local minimizer \nin the sense of Definition~\\ref{D:L:2} (from this, we also obtain that~$u$ is a local minimizer,\nthanks to Proposition~\\ref{EQUI DL}). To this aim, fixed~$x_n\\in\\mathbb{R}$,\nwe use the following ``slicing notation'' for a domain~$\\Omega\\subset\\mathbb{R}^{n+1}$:\nwe let\n$$ \\Omega^{x_n} := \\{ (x',z)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R} {\\mbox{ s.t. }}\n(x',x_n,z)\\in\\Omega\\}.$$\nAlso, the function~$\\overline u:\\mathbb{R}^{n-1}\\to\\mathbb{R}$ can be seen as a function on~$\\mathbb{R}^n$,\nby defining~$\\overline u_\\star(x',x_n):=\\overline u(x')$ and so we can consider its $a$-harmonic\nextension~$E_{\\overline u_\\star}$.\nGiven~$R>0$ and~$\\Phi\\in C^\\infty_0({\\mathcal{B}}_R)$, with~$\\varphi(x):=\\Phi(x,0)$,\nwe also define~$\\Phi_{x_n}(x',z):=\\Phi(x',x_n,z)$ and~$\\varphi_{x_n}(x'):=\\varphi(x',x_n)$.\nHence\nwe have that\n\\begin{equation}\\label{8920tr}\n\\begin{split}\n{\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_{\\overline u_\\star}+\\Phi)\\;\n&= \n\\frac{1}{2}\n\\int_{{\\mathcal{B}}_R^+}z^a |\\nabla (E_{\\overline u_\\star}+\\Phi)(x,z)|^2\\,dx\\,dz+\n\\int_{B_R} F(\\overline u(x')+\\varphi(x))\\,dx\n\\\\ &\\ge \\int_\\mathbb{R}\\left[\n\\frac{1}{2}\n\\int_{(B_R)_{x_n}\\times(0,R)}z^a |\\nabla_{(x',z)} E_{\\overline u}(x',z)+\\nabla_{(x',z)}\n\\Phi_{x_n} (x',z)|^2\\,dx'\\,dz\\right.\\\\ &\\qquad+\\left.\n\\int_{(B_R)_{x_n}} F(\\overline u(x')+\\varphi_{x_n}(x))\\,dx'\n\\right]\\,dx_n\n\\end{split}\\end{equation}\nAlso, since~$\\overline u$ is a local minimizer in~$\\mathbb{R}^{n-1}$, we have\nthat~$E_{\\overline u}$ is an extended local minimizer in~$\\mathbb{R}^{n-1}\\times(0,+\\infty)$,\nthanks to\nProposition~\\ref{EQUI DL}, and therefore\n\\begin{eqnarray*}&&\n\\frac{1}{2}\n\\int_{(B_R)_{x_n}\\times(0,R)}z^a |\\nabla_{(x',z)} E_{\\overline u}(x',z)+\\nabla_{(x',z)}\n\\Phi_{x_n} (x',z)|^2\\,dx'\\,dz+\n\\int_{(B_R)_{x_n}} F(\\overline u(x')+\\varphi_{x_n}(x))\\,dx'\n\\\\ &\\ge&\n\\frac{1}{2}\n\\int_{(B_R)_{x_n}\\times(0,R)}z^a |\\nabla_{(x',z)} E_{\\overline u}(x',z)|^2\\,dx'\\,dz\n+\\int_{(B_R)_{x_n}} F(\\overline u(x'))\\,dx'.\n\\end{eqnarray*}\nBy inserting this inequality into~\\eqref{8920tr}, we obtain that~$\n{\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_{\\overline u_\\star}+\\Phi)\\ge\n{\\mathcal{E}}_{ {\\mathcal{B}}_R }(E_{\\overline u_\\star})$.\nThat is, \n\\begin{equation}\\label{OVER}\n{\\mbox{$E_{\\overline u_\\star}$ is an extended local minimizer in~$\\mathbb{R}^{n+1}_+$.\n}}\\end{equation}\nSimilarly,\none can define~$\\underline u_\\star(x',x_n):=\\underline u(x')$\nand conclude that\n\\begin{equation}\\label{UNDER}\n{\\mbox{$E_{\\underline u_\\star}$ is an extended local minimizer in~$\\mathbb{R}^{n+1}_+$.\n}}\\end{equation}\nNow,\ngiven~$R>0$ and~$\\Psi\\in C^\\infty_0({\\mathcal{B}}_R)$, with~$\\psi(x):=\\Psi(x,0)$,\nwe consider the perturbation~$E_u+\\Psi$.\nWe define\n\\begin{eqnarray*}\n&&\\alpha(X):=\\left\\{\n\\begin{matrix}\nE_{\\overline u_\\star}(X) & {\\mbox{ if }} (E_u+\\Psi)(X)\\le E_{\\overline u_\\star}(X),\\\\\n(E_u+\\Psi)(X) & {\\mbox{ if }} (E_u+\\Psi)(X)> E_{\\overline u_\\star}(X),\n\\end{matrix}\n\\right. \\\\\n&&\\beta(X):=\\left\\{\n\\begin{matrix}\nE_{\\overline u_\\star}(X) & {\\mbox{ if }} (E_u+\\Psi)(X)> E_{\\overline u_\\star}(X),\\\\\nE_{\\underline u_\\star}(X) & {\\mbox{ if }} (E_u+\\Psi)(X)< E_{\\underline u_\\star}(X),\\\\\n(E_u+\\Psi)(X) & {\\mbox{ if }} (E_u+\\Psi)(X)\\in\\big[ E_{\\underline u_\\star}(X),\n\\,E_{\\overline u_\\star}(X)\\big],\n\\end{matrix}\n\\right. \\\\\n{\\mbox{and }}&&\\gamma(X):=\\left\\{\n\\begin{matrix}\nE_{\\underline u_\\star}(X) & {\\mbox{ if }} (E_u+\\Psi)(X)\\ge E_{\\underline u_\\star}(X),\\\\\n(E_u+\\Psi)(X) & {\\mbox{ if }} (E_u+\\Psi)(X)< E_{\\underline u_\\star}(X).\n\\end{matrix}\n\\right. \n\\end{eqnarray*}\nBy~\\eqref{OVER}, we have that\n\\begin{equation}\\label{P:X1}\n{\\mathcal{E}}_{ \\{ E_u+\\Psi>E_{\\overline u_\\star}\\} }(E_{\\overline u_\\star}) \\le \n{\\mathcal{E}}_{ \\{ E_u+\\Psi>E_{\\overline u_\\star}\\} }(\\alpha).\n\\end{equation}\nSimilarly,\nby~\\eqref{UNDER}, we have that\n\\begin{equation}\\label{P:X2}\n{\\mathcal{E}}_{ \\{ E_u+\\Psi<E_{\\underline u_\\star}\\} }(E_{\\underline u_\\star}) \\le \n{\\mathcal{E}}_{ \\{ E_u+\\Psi<E_{\\underline u_\\star}\\} }(\\gamma).\n\\end{equation}\nIn addition, from Lemma~\\ref{INTRAP},\n\\begin{equation}\\label{P:X3}\n{\\mathcal{E}}_{{\\mathcal{B}}_R }(E_u) \\le \n{\\mathcal{E}}_{{\\mathcal{B}}_R}(\\beta).\n\\end{equation}\nNotice also that~$ \\{ E_u+\\Psi>E_{\\overline u_\\star}\\}$ and~$\n\\{ E_u+\\Psi<E_{\\underline u_\\star}\\}$ are subset of~$ {{\\mathcal{B}}_R } $.\nThus, using~\\eqref{P:X1}, \\eqref{P:X2} and~\\eqref{P:X3},\n\\begin{eqnarray*}\n{\\mathcal{E}}_{{\\mathcal{B}}_R }(E_u+\\Psi)&=&\n{\\mathcal{E}}_{\\{ E_u+\\Psi>E_{\\overline u_\\star}\\} }(E_u+\\Psi)+\n{\\mathcal{E}}_{\\{ E_u+\\Psi<E_{\\underline u_\\star}\\} }(E_u+\\Psi)+\n{\\mathcal{E}}_{{\\mathcal{B}}_R\\cap\n\\{ E_{\\underline u_\\star}\\le E_u+\\Psi\\le E_{\\overline u_\\star}\\} }(E_u+\\Psi)\\\\\n&=&\n{\\mathcal{E}}_{\\{ E_u+\\Psi>E_{\\overline u_\\star}\\} }(\\alpha)+\n{\\mathcal{E}}_{\\{ E_u+\\Psi<E_{\\underline u_\\star}\\} }(\\gamma)+\n{\\mathcal{E}}_{{\\mathcal{B}}_R\\cap\n\\{ E_{\\underline u_\\star}\\le E_u+\\Psi\\le E_{\\overline u_\\star}\\} }(\\beta)\\\\\n&\\ge& \n{\\mathcal{E}}_{\\{ E_u+\\Psi>E_{\\overline u_\\star}\\} }(E_{\\overline u_\\star})+\n{\\mathcal{E}}_{\\{ E_u+\\Psi<E_{\\underline u_\\star}\\} }(E_{\\underline u_\\star})+\n{\\mathcal{E}}_{{\\mathcal{B}}_R\\cap\n\\{ E_{\\underline u_\\star}\\le E_u+\\Psi\\le E_{\\overline u_\\star}\\} }(\\beta)\\\\\n&=&\n{\\mathcal{E}}_{\\{ E_u+\\Psi>E_{\\overline u_\\star}\\} }(\\beta)+\n{\\mathcal{E}}_{\\{ E_u+\\Psi<E_{\\underline u_\\star}\\} }(\\beta)+\n{\\mathcal{E}}_{{\\mathcal{B}}_R\\cap\n\\{ E_{\\underline u_\\star}\\le E_u+\\Psi\\le E_{\\overline u_\\star}\\} }(\\beta)\\\\\n&=&{\\mathcal{E}}_{{\\mathcal{B}}_R }(\\beta)\\\\\n&\\ge& {\\mathcal{E}}_{{\\mathcal{B}}_R }(E_u).\n\\end{eqnarray*}\nThis shows that~$E_u$ is an extended local minimizer, as desired.\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{MAIN}}\\label{CLASS:D}\n\nBy Lemma~\\ref{02738}, we know that\nboth~$\\underline u$ and~$\\overline u$ are local minimizers.\nThis and Lemma~\\ref{0tyg27427412} imply that~$u$ is a local minimizer.\nTherefore, by Lemma 8.1 in~\\cite{DSVarxiv},\nwe have that~$u_\\varepsilon(x):=u(x\/\\varepsilon)$ approaches, as~$\\varepsilon\\to0$, up to subsequences,\na step function of the form~$\\chi_E-\\chi_{E^c}$, and the level sets of~$u_\\varepsilon$ approach~$\\partial E$\nlocally \nuniformly. {We claim that\n\\begin{equation}\\label{89:013948548}\n{\\mbox{$E$ is a halfspace.\n}} \n\\end{equation}\nIn a sense, this claim is a version of the Bernstein-type result in~\\cite{figalli},\nfor which we provide a complete and independent proof, by arguing as follows.\nTo prove~\\eqref{89:013948548}, it is enough to show that\n\\begin{equation}\\label{89:013948548:E}\n{\\mbox{either~$E$ is contained in a halfspace, or it contains a halfspace,\n}} \n\\end{equation}\nsee e.g. Lemma~8.3 in~\\cite{DSVarxiv}, hence we focus on the proof of~\\eqref{89:013948548:E}.\nTo this end, we distinguish two cases,\n\\begin{equation}\\label{CA:-1}\n{\\mbox{either~$\\underline u=-1$ and~$\\overline u=1$,}}\n\\end{equation}\nor \n\\begin{equation}\\label{CA:-2}\n{\\mbox{at least one between~$\\underline u$ and~$\\overline u$ is non-constant.\n}}\\end{equation}\nIn case~\\eqref{CA:-1}, we can exploit Theorem~1.4 in~\\cite{DSVarxiv}\nand obtain in particular that~\\eqref{89:013948548:E} holds true,\nso we focus on case~\\eqref{CA:-2} and we suppose that~$\\underline u$\nis non-constant (the case in which~$\\overline u$ is non-constant is similar).\n\nThen, setting~$\\underline{u}_\\varepsilon(x):=\\underline u(x\/\\varepsilon)$\nwe have that~$\\underline{u}_\\varepsilon$ approaches,\nas~$\\varepsilon\\to0$, up to subsequences,\na step function of the form~$\\chi_{\\underline E}-\\chi_{{\\underline E}^c}$.\nSince~$\\underline{u}$ is a non-constant $1$D function, we have that~${\\underline E}$ is a halfspace.\nIn addition, a.e.~$x\\in\\mathbb{R}^n$,\n$$ (\\chi_{\\underline E}-\\chi_{{\\underline E}^c})(x)=\n\\lim_{\\varepsilon\\searrow0} \\underline u(x\/\\varepsilon)\\le\n\\lim_{\\varepsilon\\searrow0} u(x\/\\varepsilon)=\n(\\chi_{E}-\\chi_{{E}^c})(x),$$\nand consequently~${\\underline E}\\subseteq E$.\nThis proves~\\eqref{89:013948548:E}, and so~\\eqref{89:013948548}.\n}\n\nHence,\n$u$ is necessarily $1$D, thanks to~\\eqref{89:013948548}\nand Theorem~1.2 in~\\cite{DSVarxiv}.\n\n\\section{The case of two-dimensional profiles at infinity}\\label{UL810}\n\nFor completeness, in relation with the work in~\\cite{MR3107529},\nwe observe that our arguments provide\nalso the following variation of Theorem~\\ref{MAIN}:\n\n\\begin{theorem}\\label{MAIN:VAR}\nLet~$s\\in\\left(0,\\frac12\\right)$ and~$u\\in C^2(\\mathbb{R}^n,[-1,1])$\nbe a solution of~$(-\\Delta)^s u=u-u^3$ in~$\\mathbb{R}^n$, \nwith~$\\partial_{x_n}u>0$ in~$\\mathbb{R}^n$.\n\nLet \n$$ \\underline u(x'):=\\lim_{x_n\\to-\\infty} u(x',x_n)\\quad{\\mbox{ and }}\\quad\n\\overline u(x'):=\\lim_{x_n\\to+\\infty} u(x',x_n).$$\nAssume that (possibly after a rotation) \n$\\underline u$ and~$\\overline u$ depend on at most two Euclidean variables\n(not necessarily the same).\n\nThen $u$ is a local minimizer.\n\nMoreover, if $n\\le8$, there exists~$s(n)\\in \\left[0,\\frac12\\right)$ such that if~$\ns\\in\\left(s(n),\\frac12\\right)$ then~$u$ is $1$D.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{MAIN:VAR} shares\nthe point of view taken in~\\cite{MR2483642} and~\\cite{FV11},\nand follows the same lines\nas that of Theorem~\\ref{MAIN}, with the modifications listed here below:\n\\begin{itemize}\n\\item By\nfollowing verbatim the proof of Lemma~\\ref{02738}, and\nexploiting that~$\\underline u$ and~$\\overline u$ are stable two-dimensional solutions, one obtains that \nthey are local minimizers and~$1$D;\n\\item {F}rom this and Lemma~\\ref{0tyg27427412}, one deduces\nthe first claim in Theorem~\\ref{MAIN:VAR};\n\\item The second claim in Theorem~\\ref{MAIN:VAR} follows\nfrom the first claim and the argument in Section~\\ref{CLASS:D}\n(in this framework, for $s$ large enough,\none can exploit\nTheorem~1.6 of~\\cite{DSVarxiv}\nin place of Theorem~1.4 of~\\cite{DSVarxiv}).\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:int}\n\\setcounter{equation}{0}\n\nBy an original observation of Jucys~\\cite{j:yo},\nall primitive idempotents\nof the symmetric group $\\Sym_n$ can be obtained by taking certain\nlimit values of the rational function\n\\beql{phiu}\n\\Phi(u_1,\\dots,u_n)= \\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1-\\frac{s_{ij}}{u_i-u_j}\\Big),\n\\eeq\nwhere $s_{ij}\\in\\Sym_n$ is the transposition of $i$ and $j$,\n$u_1,\\dots,u_n$ are\ncomplex variables and the product is calculated in the group algebra\n$\\CC[\\Sym_n]$ in the lexicographic order on the pairs $(i,j)$.\nThis construction, which is commonly known as the {\\it fusion\nprocedure\\\/}, was also developed by Cherednik~\\cite{c:sb}, while\ndetailed proofs were given by Nazarov~\\cite{n:yc}.\nIn the context of the quantum inverse scattering method\ndeveloped by Faddeev's Leningrad school,\nthe fusion procedure has been regarded as a way\nto construct new solutions of the Yang--Baxter equation\nout of old ones; see, e.g., \\cite{krs:yb}.\nThe version of the fusion procedure found in \\cite{m:fp} establishes\nits equivalence to the construction of the idempotents\nprovided by Jucys~\\cite{j:fy} and Murphy~\\cite{m:nc}\nin terms of some special elements of the group algebra of $\\Sym_n$.\nIt was shown in \\cite{imo:ih} that the fusion procedure for the Hecke\nalgebra admits a similar interpretation; cf. \\cite{c:ni}, \\cite{n:mh}.\n\nA fusion procedure for the Brauer algebra $\\Bc_n(\\om)$ over $\\CC(\\om)$\nwas recently given by two of us in \\cite{im:fp}.\nFor this algebra \\eqref{phiu} was replaced by\nthe rational function\n\\beql{phiubra}\n\\Psi(u_1,\\dots,u_n)=\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1-\\frac{\\ep_{ij}}{u_i+u_j}\\Big)\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1-\\frac{s_{ij}}{u_i-u_j}\\Big)\n\\eeq\npreviously considered in \\cite{n:rt},\nwith the ordered products as in \\eqref{phiu};\nthe elements $\\ep_{ij}$ and $s_{ij}$ of $\\Bc_n(\\om)$ are defined in\nSec.~\\ref{sec:fpbra} below.\n\nRecall that the irreducible\nrepresentations of $\\Bc_n(\\om)$\nare indexed by all partitions of the nonnegative\nintegers $n,n-2,n-4,\\dots$. If $\\la$\nis such a partition, then the\n{\\it updown $\\la$-tableaux} $T$\nparameterize basis vectors of the corresponding\nrepresentation; see Sec.~\\ref{sec:fpbra} for the definitions.\nThis leads to an explicit isomorphism between $\\Bc_n(\\om)$\nand the direct sum of matrix algebras. The {\\it primitive\nidempotents\\\/} $E^{\\lambda}_{T}$ are the elements of $\\Bc_n(\\om)$\ncorresponding to the diagonal matrix units under this isomorphism;\nsee \\cite{b:aw}, \\cite{n:yo}, \\cite{w:sb}.\n\nBy the main result of \\cite{im:fp},\ngiven an updown $\\la$-tableau $T$,\nthe consecutive evaluations\n\\beql{reev}\n(u_1-c_1)^{p_1}\\dots (u_n-c_n)^{p_n}\\ts\n\\Psi(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_n=c_n}\n\\eeq\nare well-defined and this value yields the\ncorresponding primitive idempotent\n$E^{\\lambda}_{T}$ multiplied by a nonzero constant\n$f(T)$ which is calculated in an explicit form.\nHere the $c_i$ are the contents of $T$\nand $p_1,\\dots,p_n$ are certain integers depending on $T$,\ncalled its exponents.\n\nThe first main result of this paper\nis a new construction\nof all primitive idempotents of the Brauer algebra $\\Bc_n(\\om)$;\nwe use a different rational function in place of \\eqref{phiubra}.\nNamely, set\n\\begin{multline}\\label{omubra}\n\\Omega(u_1,\\dots,u_n)=\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1+\\frac{s_{ij}}{u_i+u_j-\\om\/2+1}-\\frac{\\ep_{ij}}{u_i+u_j}\\Big)\\\\\n{}\\times\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1-\\frac{s_{ij}}{u_i-u_j}+\\frac{\\ep_{ij}}{u_i-u_j-\\om\/2+1}\\Big)\n\\end{multline}\nwith both products taken in the\nlexicographic order on the pairs $(i,j)$. Thus,\n\\beql{omrm}\n\\Omega(u_1,\\dots,u_n)=\\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(-u_i-u_j+\\vk)\\ts \\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(u_i-u_j),\n\\eeq\nwhere we use the notation\n\\beql{rij}\n\\rho_{ij}(u)=1-\\frac{s_{ij}}{u}+\\frac{\\ep_{ij}}{u-\\vk},\\qquad\n\\vk=\\frac{\\om}{2}-1.\n\\eeq\nThe rational functions $\\rho_{ij}(u)$ satisfy\nthe Yang--Baxter equation\n\\beql{ybe}\n\\rho_{ij}(u)\\ts \\rho_{ik}(u+v)\\ts \\rho_{jk}(v)=\n\\rho_{jk}(v)\\ts \\rho_{ik}(u+v)\\ts \\rho_{ij}(u),\n\\eeq\nwith any distinct indices $i,j,k$, where we set\n$\\rho_{ji}(u)=\\rho_{ij}(u)$ for $i<j$;\nsee \\cite{zz:rf}. Also, for $i\\ne j$ we have\n\\beql{unitarm}\n\\rho_{ij}(u)\\ts\\rho_{ij}(-u)=\\frac{u^2-1}{u^2}.\n\\eeq\nThe new version of the fusion procedure\nfor $\\Bc_n(\\om)$ takes the following form: the consecutive\nevaluations\n\\beql{reevom}\n(u_1-c_1)^{p_1}\\dots (u_n-c_n)^{p_n}\\ts\n\\Omega(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_n=c_n}\n\\eeq\nare well-defined and this value yields the\nproduct\n$h(T)\\tss E^{\\lambda}_{T}$ as with \\eqref{reev}, where $h(T)$\nis a constant calculated in a way similar to $f(T)$.\nTo give an equivalent formulation, for any updown tableau\n$T$ introduce a rational function $\\Omega_T(u_1,\\dots,u_n)$\nwhich is obtained from $\\Omega(u_1,\\dots,u_n)$ by\nmultiplying by a numerical rational function in $u_1,\\dots,u_n$\ndepending on the contents of $T$; see \\eqref{omt}.\nThen the fusion\nprocedure can be reformulated as the relation\n\\beql{reevomre}\nE_T=\\Omega_T(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_n=c_n}.\n\\eeq\n\nThe existence of the two forms \\eqref{reev} and \\eqref{reevom}\n(or, equivalently, \\eqref{reevomre})\nof the fusion procedure can be explained by their connections\nwith two different quantum algebras associated with a\nclassical Lie algebra of type $B$, $C$ or $D$.\nConsider the natural action of the Brauer algebra $\\Bc_n(\\om)$ (with\nan appropriate specialization\nof the parameter $\\om$)\non the tensor product space\n\\beql{cntenprint}\n\\CC^N\\ot\\CC^N\\ot\\dots\\ot\\CC^N,\\qquad n\\ \\ \\text{factors};\n\\eeq\nsee \\eqref{braact} and\n\\eqref{braactsy} below. Using the expression for the idempotent\n$E^{\\lambda}_{T}$ provided by \\eqref{reev} we find that\nthe subspace $E^{\\lambda}_{T}(\\CC^N)^{\\ot n}$ carries\nthe structure of a representation of the {\\it Olshanski twisted\nYangian\\\/} $\\Y'(\\g_N)$ associated with the orthogonal\nLie algebra $\\g_N=\\oa_N$\nor symplectic Lie algebra $\\g_N=\\spa_N$, respectively,\n($N$ is even for the latter) \\cite{n:rt},\n\\cite{o:ty}; see also \\cite[Ch.~2]{m:yc}.\nOn the other hand, the representation of\nthe universal enveloping algebra\n$\\U(\\g_N)$ on the space $E^{\\lambda}_{T}(\\CC^N)^{\\ot n}$\narising from the Brauer--Schur--Weyl duality extends to\nthe twisted Yangian $\\Y'(\\g_N)$ via the evaluation\nhomomorphism $\\Y'(\\g_N)\\to\\U(\\g_N)$ \\cite{o:ty}.\nThus, the fusion procedure in the form associated\nwith the evaluations \\eqref{reev} yields an equivalence\nof the two twisted Yangian actions on\nthe space $E^{\\lambda}_{T}(\\CC^N)^{\\ot n}$.\n\nIn the new version of the fusion procedure associated\nwith the evaluations \\eqref{reevom}, the role of the twisted\nYangian is now taken by the\n{\\it reflection algebra\\\/} $\\Br(\\g_N)$.\nThis algebra is defined by a reflection equation\nand it is closely related to the\n{\\it Drinfeld Yangian\\\/} $\\Y(\\g_N)$; see Definition~\\ref{def:refeq}\nbelow.\nThe Yangian\n$\\Y(\\g_N)$ contains the universal enveloping algebra\n$\\U(\\g_N)$ as a subalgebra; however, due to\na result of Drinfeld~\\cite{d:ha}, in contrast with the\nYangian for $\\gl_N$,\nthere is no homomorphism $\\Y(\\g_N)\\to \\U(\\g_N)$, identical\non the subalgebra $\\U(\\g_N)$.\n\nThe algebras $\\Br(\\oa_N)$ and $\\Br(\\spa_{2n})$ were\nformally defined in \\cite{aacfr:rp} together with their\nsuper-version $\\Br(\\osp_{m|2n})$; however, they\ndo not seem to have received\nmuch attention in the literature.\nOur second main result (Theorem~\\ref{thm:eval}) is a construction\nof an evaluation homomorphism $\\Br(\\g_N)\\to\\U(\\g_N)$.\nFurthermore, we show that the algebra $\\Br(\\g_N)$ contains\nthe universal enveloping algebra $\\U(\\g_N)$ as a subalgebra,\nand the evaluation homomorphism is identical on this\nsubalgebra. These results also apply to\nthe case of the Lie superalgebra $\\osp_{m|2n}$, and we\nindicate necessary changes in the notation in\nRemark~\\ref{rem:super}.\n\nThe expression for the idempotent\n$E^{\\lambda}_{T}$ provided by \\eqref{reevom} indicates that\nthe subspace $E^{\\lambda}_{T}(\\CC^N)^{\\ot n}$\ncarries the structure of a representation of the corresponding\nreflection algebra $\\Br(\\g_N)$. Moreover, we show that,\nas with the twisted Yangian, this representation\nfactors through the evaluation homomorphism $\\Br(\\g_N)\\to\\U(\\g_N)$.\n\nTaking the quotient of the Brauer algebra $\\Bc_n(\\om)$ by\nthe relations $\\ep_{ij}=0$ we come to a new version of the fusion\nprocedure for the symmetric group $\\Sym_n$ involving\nan arbitrary parameter $\\om$ (Corollary~\\ref{cor:sym}).\nThe standard version of the procedure\nassociated with the rational function \\eqref{phiu}\nis recovered\nin the limit $\\om\\to\\infty$.\nWe show\nthat the new version is related to\nthe well-known evaluation homomorphism\n$\\Y(\\gl_N)\\to \\U(\\gl_N)$ and to its restriction to the subalgebra\nof $\\Y(\\gl_N)$ isomorphic to a reflection algebra.\n\nBoth the fusion procedure and the evaluation homomorphism\ndescribed in Theorems~\\ref{thm:newfus} and \\ref{thm:eval}\nadmit their natural quantum analogues. Namely,\nall primitive idempotents of the Birman--Murakami--Wenzl\nalgebra can be obtained by evaluating a universal rational\nfunction. Moreover, it is possible to introduce\na $q$-analogue $\\Br_q(\\g_N)$ of the reflection algebra\nand to construct a homomorphism from $\\Br_q(\\g_N)$\nto the Drinfeld--Jimbo quantum group $\\U_q(\\g_N)$\nwith the properties similar to \\eqref{evalya}.\nThe details will appear in our forthcoming publication\n\\cite{imo:fp}.\n\n\\medskip\n\nA. P. I. and O. V. O. are grateful to the School of Mathematics and\nStatistics of the University of Sydney for the warm hospitality\nduring their visits. We acknowledge the support of the Australian\nResearch Council. The work of A. P. I. was supported by the grants\nRFBR 11-01-00980-a and RFBR-CNRS 07-02-92166-a.\n\n\n\n\n\\section{Fusion procedure}\n\\label{sec:fpbra}\n\\setcounter{equation}{0}\n\nLet $n$ be a positive integer and $\\om$ an indeterminate.\nAn $n$-diagram $d$ is a collection\nof $2n$ dots arranged into two rows with $n$ dots in each row\nconnected by $n$ edges such that any dot belongs to only one edge.\nThe product of two diagrams $d_1$ and $d_2$ is determined by\nplacing $d_1$ above $d_2$ and identifying the vertices\nof the bottom row of $d_1$ with the corresponding\nvertices in the top row of $d_2$. Let $s$ be the number of\nclosed loops obtained in this placement. The product $d_1d_2$ is given by\n$\\om^{\\tss s}$ times the resulting diagram without loops.\nThe {\\it Brauer algebra\\\/} $\\Bc_n(\\om)$ is defined as the\n$\\CC(\\om)$-linear span of the $n$-diagrams with the\nmultiplication defined above.\nThe dimension of the algebra is $1\\cdot 3\\cdots (2n-1)$.\nThe following presentation of $\\Bc_n(\\om)$ is well-known;\nsee, e.g., \\cite{bw:bl}.\n\n\\bpr\\label{prop:bradr}\nThe Brauer algebra $\\Bc_n(\\om)$ is isomorphic to the algebra\nwith $2n-2$ generators\n$s_1,\\dots,s_{n-1},\\ep_1,\\dots,\\ep_{n-1}$\nand the defining relations\n\\beql{bradr}\n\\begin{aligned}\ns_i^2&=1,\\qquad \\ep_i^2=\\om\\ts \\ee_i,\\qquad\n\\su_i\\ee_i=\\ee_i\\su_i=\\ee_i,\\qquad i=1,\\dots,n-1,\\\\\ns_i s_j &=s_js_i,\\qquad \\ee_i \\ee_j = \\ee_j \\ee_i,\\qquad\n\\su_i \\ee_j = \\ee_j\\su_i,\\qquad\n|i-j|>1,\\\\\ns_is_{i+1}s_i&=s_{i+1}s_is_{i+1},\\qquad\n\\ee_i\\ee_{i+1}\\ee_i=\\ee_i,\\qquad \\ee_{i+1}\\ee_i\\ee_{i+1}=\\ee_{i+1},\\\\\n\\su_i\\ee_{i+1}\\ee_i&=\\su_{i+1}\\ee_i,\\qquad \\ee_{i+1}\\ee_i\n\\su_{i+1}=\\ee_{i+1}\\su_i,\\qquad\ni=1,\\dots,n-2.\n\\end{aligned}\n\\non\n\\end{equation}\n\\epr\n\nThe generators $s_i$ and $\\ee_i$ correspond to the following diagrams\nrespectively:\n\n\\begin{center}\n\\begin{picture}(400,60)\n\\thinlines\n\n\\put(10,20){\\circle*{3}}\n\\put(30,20){\\circle*{3}}\n\\put(70,20){\\circle*{3}}\n\\put(90,20){\\circle*{3}}\n\\put(130,20){\\circle*{3}}\n\\put(150,20){\\circle*{3}}\n\n\\put(10,40){\\circle*{3}}\n\\put(30,40){\\circle*{3}}\n\\put(70,40){\\circle*{3}}\n\\put(90,40){\\circle*{3}}\n\\put(130,40){\\circle*{3}}\n\\put(150,40){\\circle*{3}}\n\n\\put(10,20){\\line(0,1){20}}\n\\put(30,20){\\line(0,1){20}}\n\\put(70,20){\\line(1,1){20}}\n\\put(90,20){\\line(-1,1){20}}\n\\put(130,20){\\line(0,1){20}}\n\\put(150,20){\\line(0,1){20}}\n\n\\put(45,25){$\\cdots$}\n\\put(105,25){$\\cdots$}\n\n\\put(8,5){\\scriptsize $1$ }\n\\put(28,5){\\scriptsize $2$ }\n\\put(68,5){\\scriptsize $i$ }\n\\put(86,5){\\scriptsize $i+1$ }\n\\put(122,5){\\scriptsize $n-1$ }\n\\put(150,5){\\scriptsize $n$ }\n\n\\put(190,25){\\text{and}}\n\n\\put(250,20){\\circle*{3}}\n\\put(270,20){\\circle*{3}}\n\\put(310,20){\\circle*{3}}\n\\put(330,20){\\circle*{3}}\n\\put(370,20){\\circle*{3}}\n\\put(390,20){\\circle*{3}}\n\n\\put(250,40){\\circle*{3}}\n\\put(270,40){\\circle*{3}}\n\\put(310,40){\\circle*{3}}\n\\put(330,40){\\circle*{3}}\n\\put(370,40){\\circle*{3}}\n\\put(390,40){\\circle*{3}}\n\n\\put(250,20){\\line(0,1){20}}\n\\put(270,20){\\line(0,1){20}}\n\\put(320,20){\\oval(20,12)[t]}\n\\put(320,40){\\oval(20,12)[b]}\n\\put(370,20){\\line(0,1){20}}\n\\put(390,20){\\line(0,1){20}}\n\n\\put(285,25){$\\cdots$}\n\\put(345,25){$\\cdots$}\n\n\\put(248,5){\\scriptsize $1$ }\n\\put(268,5){\\scriptsize $2$ }\n\\put(308,5){\\scriptsize $i$ }\n\\put(326,5){\\scriptsize $i+1$ }\n\\put(362,5){\\scriptsize $n-1$ }\n\\put(390,5){\\scriptsize $n$ }\n\n\\end{picture}\n\\end{center}\n\nThe subalgebra of $\\Bc_n(\\om)$ generated over $\\CC$\nby $s_1,\\dots,s_{n-1}$\nis isomorphic to the group algebra $\\CC[\\Sym_n]$ so that $s_i$\ncan be identified with the transposition $(i,i+1)$.\nThen for any $1\\leqslant i<j\\leqslant n$\nthe transposition $s_{ij}=(i,j)$ can be regarded\nas an element of $\\Bc_n(\\om)$. Moreover, $\\ep_{ij}$\nwill denote the element of $\\Bc_n(\\om)$ represented\nby the diagram in which the $i$-th and $j$-th dots in the top row,\nas well as the $i$-th and $j$-th dots in the bottom\nrow are connected by an edge, while\nthe remaining edges\nconnect the $k$-th dot in the top row\nwith the $k$-th dot in the bottom row for each $k\\ne i,j$.\nEquivalently, in terms of the presentation of $\\Bc_n(\\om)$\nprovided by Proposition~\\ref{prop:bradr},\n\\ben\ns_{ij}=s_i\\tss s_{i+1}\\dots s_{j-2}\\tss\ns_{j-1}\\tss s_{j-2}\\dots s_{i+1}\\tss s_i\n\\Fand\n\\ep_{ij}=s_{i,j-1}\\tss \\ep_{j-1}\\tss s_{i,j-1}.\n\\een\nWe also set $\\ep_{ji}=\\ep_{ij}$ and $s_{ji}=s_{ij}$ for $i<j$.\nThe Brauer algebra $\\Bc_{n-1}(\\om)$ can be regarded\nas the subalgebra of $\\Bc_n(\\om)$ spanned by all diagrams\nin which the $n$-th dots in the top and bottom rows are\nconnected by an edge.\n\nThe {\\it Jucys--Murphy elements\\\/} $x_1,\\dots,x_n$\nfor the Brauer algebra $\\Bc_n(\\om)$ are given by the formulas\n\\beql{jmdef}\nx_r=\\frac{\\om-1}{2}+\\sum_{i=1}^{r-1}(s_{i\\tss r}-\\ep_{i\\tss r}),\n\\qquad r=1,\\dots,n;\n\\eeq\nsee \\cite{lr:rh} and \\cite{n:yo}, where, in particular,\nthe eigenvalues\nof the $x_r$ in irreducible representations were calculated.\nThe element $x_n$ commutes\nwith the subalgebra $\\Bc_{n-1}(\\om)$. This implies that\nthe elements $x_1,\\dots,x_n$\nof $\\Bc_n(\\om)$ pairwise commute. They can be used\nto construct a complete set of pairwise orthogonal primitive\nidempotents for the Brauer algebra following\nthe approach of Jucys~\\cite{j:fy} and Murphy~\\cite{m:nc}.\nNamely, let $\\la$ be a partition of $n-2f$\nfor some $f\\in\\{0,1,\\dots,\\lfloor n\/2\\rfloor\\}$.\nWe will identify partitions with their diagrams\nso that if the parts of $\\la$ are $\\la_1,\\la_2,\\dots$ then\nthe corresponding diagram is\na left-justified array of rows of unit boxes containing\n$\\la_1$ boxes in the top row, $\\la_2$ boxes\nin the second row, etc.\nThe box in row $i$ and column $j$ of a diagram\nwill be denoted as the pair $(i,j)$.\nAn {\\it updown $\\la$-tableau\\\/} is a sequence\n$T=(\\La_1,\\dots,\\La_n)$ of diagrams such that\nfor each $r=1,\\dots,n$ the diagram $\\La_r$ is obtained\nfrom $\\La_{r-1}$ by adding or removing one box,\nwhere we set $\\La_0=\\varnothing$, the empty diagram, and $\\La_n=\\la$.\nTo each updown tableau $T$ we attach the corresponding\nsequence of {\\it contents\\\/} $(c_1,\\dots,c_n)$, $c_r=c_r(T)$,\nwhere\n\\ben\nc_r=\\frac{\\om-1}{2}+j-i\\qquad\\text{or}\\qquad\nc_r=-\\Big(\\frac{\\om-1}{2}+j-i\\Big),\n\\een\nif $\\La_r$ is obtained by adding the box $(i,j)$ to $\\La_{r-1}$\nor by removing this box from $\\La_{r-1}$,\nrespectively. The primitive idempotents $E_{T}=E^{\\lambda}_{T}$\ncan now be defined by\nthe following recurrence formula (we omit the superscripts\nindicating the diagrams since they are determined by\nthe updown tableaux).\nSet $\\mu=\\La_{n-1}$ and consider the updown $\\mu$-tableau\n$U=(\\La_1,\\dots,\\La_{n-1})$. Let $\\alpha$ be the box\nwhich is added to or removed from $\\mu$ to get $\\la$. Then\n\\beql{murphyfo}\nE_{T}=E_{U}\\ts\n\\frac{(x_n-a_1)\\dots (x_n-a_k)}{(c_n-a_1)\\dots (c_n-a_k)},\n\\eeq\nwhere $a_1,\\dots,a_k$ are the contents of all boxes\nexcluding $\\alpha$,\nwhich can be removed from or added\nto $\\mu$ to get a diagram.\nWhen $\\la$ runs over all partitions\nof $n,n-2,\\dots$, and $T$ runs over all updown $\\la$-tableaux,\nthe elements $\\{E_{T}\\}$ yield a complete set\nof pairwise orthogonal primitive\nidempotents for $\\Bc_n(\\om)$. They have the properties\n\\beql{xiet}\nx_r\\ts E_{T}=E_{T}\\ts x_r=c_r(T)\\ts E_{T}, \\qquad r=1,\\dots,n.\n\\eeq\nMoreover, given an updown tableau $U=(\\La_1,\\dots,\\La_{n-1})$,\nwe have the relation\n\\beql{eutet}\nE_{U}=\\sum_{T} E_{T},\n\\eeq\nthe sum is over all updown tableaux\nof the form $T=(\\La_1,\\dots,\\La_{n-1},\\La_n)$; see\ne.g. \\cite{lr:rh} and \\cite{n:yo}.\nRelation \\eqref{murphyfo} implies\n\\beql{jmform}\nE_{T}=E_{U}\\ts \\frac{(u-c_n)(u+x_n-\\vk)}\n{(u-x_n)(u+c_n-\\vk)}\\ts\n\\Big|^{}_{u=c_n},\n\\eeq\nwhere $u$ is a complex variable and we use notation \\eqref{rij}.\nThis relation follows\nby application of \\eqref{eutet} and then \\eqref{xiet}.\n\nGiven an updown tableau $T=(\\La_1,\\dots,\\La_n)$\nwith the respective contents $c_1,\\dots,c_n$,\nintroduce a rational function in $u_1,\\dots,u_n$\nwith values in the Brauer algebra $\\Bc_n(\\om)$ by\n\\beql{omt}\n\\Omega_T(u_1,\\dots,u_n)=\n\\prod_{r=2}^{n}\\frac{(u_r-c_r)(u_r+c_1-\\vk)}{(u_r-c_1)(u_r+c_r-\\vk)}\n\\ts\\prod_{i=1}^{r-1}\\frac{(u_r-u_i)^2}{(u_r-u_i)^2-1}\\ts\n\\Omega(u_1,\\dots,u_n),\n\\eeq\nwhere $\\Omega(u_1,\\dots,u_n)$ is defined in \\eqref{omubra}\nand \\eqref{omrm}. Note that if\nthe indices $i,j,k,l$ are distinct then\nthe elements $\\rho_{ij}(u)$ and $\\rho_{kl}(v)$ commute.\nTogether with \\eqref{ybe} this implies\nthe relation\n\\begin{multline}\\non\n\\rho_{1,n}(-u_1-u_n+\\vk)\\dots \\rho_{n-1,n}(-u_{n-1}-u_n+\\vk)\\ts\n\\prod_{1\\leqslant i<j\\leqslant n-1}\n\\rho_{ij}(u_i-u_j)\\\\\n{}=\\prod_{1\\leqslant i<j\\leqslant n-1}\n\\rho_{ij}(u_i-u_j)\\ts \\rho_{n-1,n}(-u_{n-1}-u_n+\\vk)\\dots\n\\rho_{1,n}(-u_1-u_n+\\vk).\n\\end{multline}\nAn easy induction on $n$ leads to the following\nequivalent expression for the rational function\n$\\Omega(u_1,\\dots,u_n)$:\n\\begin{multline}\\label{psedec}\n\\Omega(u_1,\\dots,u_n)=\\prod_{r=2}^{n}\\ts\n\\rho_{r-1,r}(-u_{r-1}-u_r+\\vk)\\dots \\rho_{1,r}(-u_1-u_r+\\vk)\\\\\n{}\\times \\rho_{1,r}(u_1-u_r)\n\\dots \\rho_{r-1,r}(u_{r-1}-u_r),\n\\end{multline}\nwhere the factors are ordered in accordance with\nthe increasing values of $r$.\n\n\\bth\\label{thm:newfus}\nThe idempotent $E_T$ is found by\nthe consecutive evaluations\n\\ben\nE_T=\\Omega_T(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_n=c_n}.\n\\een\n\\eth\n\n\\bpf\nWe use the induction on $n$.\nBy the induction hypothesis, setting $u=u_n$\nand using \\eqref{psedec} we get\n\\begin{multline}\\label{indugeth}\n\\Omega_T(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_{n-1}=c_{n-1}}\n=\\frac{(u-c_n)(u+c_1-\\vk)}{(u-c_1)(u+c_n-\\vk)}\\ts\n\\ts\\prod_{i=1}^{n-1}\\frac{(u-c_i)^2}{(u-c_i)^2-1}\\ts\\\\\n{}\\times{}E_U\\ts\n\\rho_{n-1,n}(-c_{n-1}-u+\\vk)\\dots \\rho_{1,n}(-c_1-u+\\vk)\\ts\n\\rho_{1,n}(c_1-u)\\dots \\rho_{n-1,n}(c_{n-1}-u),\n\\end{multline}\nwhere $U$ is the updown tableau $(\\La_1,\\dots,\\La_{n-1})$.\n\n\\ble\\label{lem:jmsi}\nWe have the identity\n\\begin{multline}\\label{jm}\nE_U\\ts\n\\rho_{n-1,n}(-c_{n-1}-u+\\vk)\\dots \\rho_{1,n}(-c_1-u+\\vk)\\ts\n\\rho_{1,n}(c_1-u)\\dots \\rho_{n-1,n}(c_{n-1}-u)\\\\\n{}=\\frac{u-c_1}{u+c_1-\\vk}\\ts\\prod_{i=1}^{n-1}\n\\ts\\frac{(u-c_i)^2-1}{(u-c_i)^2}\n\\ts E_U\\ts\\frac{u+x_n-\\vk}{u-x_n}.\n\\end{multline}\n\\ele\n\n\\bpf\nEmbed the Brauer algebra $\\Bc_n(\\om)$ into $\\Bc_m(\\om)$\nfor some $m\\geqslant n$ and\nprove a more general identity\n\\begin{multline}\\label{jmm}\nE_U\\ts\n\\rho_{n-1,m}(-c_{n-1}-u+\\vk)\\dots \\rho_{1,m}(-c_1-u+\\vk)\\ts\n\\rho_{1,m}(c_1-u)\\dots \\rho_{n-1,m}(c_{n-1}-u)\\\\\n{}=\\frac{u-c_1}{u+c_1-\\vk}\\ts\\prod_{i=1}^{n-1}\n\\ts\\frac{(u-c_i)^2-1}{(u-c_i)^2}\n\\ts E_U\\ts\\frac{u+x^{(n-1)}_m-\\vk}{u-x^{(n-1)}_m},\n\\end{multline}\nwhere\n\\beql{jmdefm}\nx^{(k)}_m=\\frac{\\om-1}{2}+\\sum_{i=1}^{k}(s_{i\\tss m}-\\ep_{i\\tss m});\n\\eeq\nin particular, $x^{(n-1)}_n=x_n$; see \\eqref{jmdef}. We use induction\non $n$ while keeping $m$ fixed. In the case $n=1$ the identity\nis certainly true.\nSuppose that $n\\geqslant 2$. By \\eqref{eutet} we\nhave $E_U=E_U\\ts E_W$,\nwhere $W$ is the updown tableau $(\\La_1,\\dots,\\La_{n-2})$\n(in the case $n=2$ we set $E_W=1$).\nHence, using the induction hypothesis\nwe can write the left hand side of \\eqref{jmm} as\n\\begin{multline}\nE_U\\ts\n\\rho_{n-1,m}(-c_{n-1}-u+\\vk)\\\\\n{}\\times\n\\frac{u-c_1}{u+c_1-\\vk}\\ts\\prod_{i=1}^{n-2}\n\\ts\\frac{(u-c_i)^2-1}{(u-c_i)^2}\n\\ts E_W\\ts\\frac{u+x^{(n-2)}_m-\\vk}{u-x^{(n-2)}_m}\n\\ts \\rho_{n-1,m}(c_{n-1}-u)\n\\non\n\\end{multline}\nwhich equals\n\\beql{lhstra}\n\\frac{u-c_1}{u+c_1-\\vk}\\ts\\prod_{i=1}^{n-2}\n\\ts\\frac{(u-c_i)^2-1}{(u-c_i)^2}\\ts E_U\\ts\n\\rho_{n-1,m}(-c_{n-1}-u+\\vk)\\ts\n\\frac{u+x^{(n-2)}_m-\\vk}{u-x^{(n-2)}_m}\n\\ts \\rho_{n-1,m}(c_{n-1}-u).\n\\eeq\nApplying\n\\eqref{unitarm} we find that\n\\ben\n\\rho_{n-1,m}(c_{n-1}-u)=\\frac{(u-c_{n-1})^2-1}{(u-c_{n-1})^2}\\ts\n\\rho_{n-1,m}(u-c_{n-1})^{-1}.\n\\een\nHence, comparing \\eqref{lhstra} with the right hand side of\n\\eqref{jmm}, we conclude that the lemma will be implied\nby the identity\n\\ben\nE_U\\ts\n\\rho_{n-1,m}(-c_{n-1}-u+\\vk)\\ts\\frac{u+x^{(n-2)}_m-\\vk}{u-x^{(n-2)}_m}\n=E_U\\ts \\frac{u+x^{(n-1)}_m-\\vk}{u-x^{(n-1)}_m}\\ts\n\\rho_{n-1,m}(u-c_{n-1}).\n\\een\nSince $x^{(n-1)}_m$ commutes with $E_U$, we can write the\nidentity in the equivalent form\n\\begin{multline}\nE_U\\ts(u-x^{(n-1)}_m)\\ts\\Big(1+\\frac{s_{n-1,m}}{c_{n-1}+u-\\vk}-\n\\frac{\\ep_{n-1,m}}{c_{n-1}+u}\\Big)\\ts (u+x^{(n-2)}_m-\\vk)\\\\\n{}-E_U\\ts(u+x^{(n-1)}_m-\\vk)\\ts\\Big(1+\\frac{s_{n-1,m}}{c_{n-1}-u}-\n\\frac{\\ep_{n-1,m}}{c_{n-1}-u+\\vk}\\Big)\\ts (u-x^{(n-2)}_m)=0.\n\\non\n\\end{multline}\nTo verify the latter, note that by \\eqref{xiet} and\nthe relations\nin the Brauer algebra,\n\\begin{multline}\nE_U\\ts x^{(n-1)}_m \\ts s_{n-1,m}\\ts x^{(n-2)}_m\n=E_U\\ts x^{(n-1)}_m \\ts x_{n-1}\\ts s_{n-1,m}\\\\\n=E_U\\ts x_{n-1}\\ts x^{(n-1)}_m \\ts s_{n-1,m}\n=c_{n-1}\\ts E_U\\ts x^{(n-1)}_m \\ts s_{n-1,m},\n\\non\n\\end{multline}\nwhile\n\\ben\nE_U\\ts x^{(n-1)}_m \\ts \\ep_{n-1,m}\\ts x^{(n-2)}_m\n=-E_U\\ts x_{n-1}\\ts \\ep_{n-1,m}\\ts x^{(n-2)}_m\n=-c_{n-1}\\ts E_U\\ts \\ep_{n-1,m}\\ts x^{(n-2)}_m.\n\\een\nTogether with the relation $x^{(n-1)}_m=x^{(n-2)}_m+s_{n-1,m}\n-\\ep_{n-1,m}$ this yields the identity in question and\ncompletes the proof of the lemma.\n\\epf\n\nReturn to the proof of the theorem; we are left to\nverify\nthat the rational function\n\\ben\nE_{U}\\ts \\frac{(u-c_n)(u+x_n-\\vk)}\n{(u-x_n)(u+c_n-\\vk)}\n\\een\nis well-defined at $u=c_n$ and the value coincides with $E_T$.\nBut this holds due to \\eqref{jmform}.\n\\epf\n\n\\bex\\label{exam:idem}\\quad (i)\\ \\\nIn the case of the Brauer algebra $\\Bc_2(\\om)$ we have\nthree updown tableaux\n\n\\setlength{\\unitlength}{0.3em}\n\\begin{center}\n\\begin{picture}(82,5)\n\n\\put(-10,0){$U_1=\\big($}\n\n\\put(0,0){\\line(0,1){2}}\n\\put(2,0){\\line(0,1){2}}\n\\put(0,0){\\line(1,0){2}}\n\\put(0,2){\\line(1,0){2}}\n\\put(2.5,0){,}\n\n\\put(8,0){\\line(0,1){2}}\n\\put(10,0){\\line(0,1){2}}\n\\put(12,0){\\line(0,1){2}}\n\\put(8,0){\\line(1,0){4}}\n\\put(8,2){\\line(1,0){4}}\n\\put(12.5,0)\n\n\\put(13,0){$\\big),$}\n\n\\put(26,0){$U_2=\\big($}\n\n\\put(36,0){\\line(0,1){2}}\n\\put(38,0){\\line(0,1){2}}\n\\put(36,0){\\line(1,0){2}}\n\\put(36,2){\\line(1,0){2}}\n\\put(38.5,0){,}\n\n\\put(44,-2){\\line(0,1){4}}\n\\put(46,-2){\\line(0,1){4}}\n\\put(44,-2){\\line(1,0){2}}\n\\put(44,0){\\line(1,0){2}}\n\\put(44,2){\\line(1,0){2}}\n\\put(46.5,0)\n\n\n\\put(48,0){$\\big),$}\n\n\\put(61,0){$U_3=\\big($}\n\n\\put(71,0){\\line(0,1){2}}\n\\put(73,0){\\line(0,1){2}}\n\\put(71,0){\\line(1,0){2}}\n\\put(71,2){\\line(1,0){2}}\n\\put(73.5,0){,}\n\n\\put(79,0){$\\varnothing$}\n\n\n\\put(82,0){$\\big).$}\n\n\\end{picture}\n\\end{center}\n\\setlength{\\unitlength}{1pt}\n\n\\noindent\nThe respective contents $(c_1,c_2)$ are given by\n\\ben\n\\Big(\\frac{\\om-1}{2},\\ts\\frac{\\om+1}{2}\\Big),\\qquad\n\\Big(\\frac{\\om-1}{2},\\ts\\frac{\\om-3}{2}\\Big),\\qquad\n\\Big(\\frac{\\om-1}{2},\\ts-\\frac{\\om-1}{2}\\Big).\n\\een\nThe definition \\eqref{omt} reads\n\\ben\n\\bal\n\\Omega_T(u_1,u_2)&=\\frac{(u_2-c_2)(u_2+c_1-\\vk)}{(u_2-c_1)(u_2+c_2-\\vk)}\n\\ts\\frac{(u_1-u_2)^2}{(u_1-u_2)^2-1}\\ts\\\\\n{}&\\times\\Big(1+\\frac{s_1}{u_1+u_2-\\vk}-\\frac{\\ep_1}{u_1+u_2}\\Big)\n\\ts \\Big(1-\\frac{s_1}{u_1-u_2}+\\frac{\\ep_1}{u_1-u_2-\\vk}\\Big)\n\\eal\n\\een\nso that applying Theorem~\\ref{thm:newfus}, we find\n\\ben\nE_{U_1}=\n\\frac{1+s_1}{2}-\\frac{\\ep_1}{\\om},\\qquad\nE_{U_2}=\\frac{1-s_1}{2},\\qquad\nE_{U_3}=\\frac{\\ep_1}{\\om}.\n\\een\n\n\\quad (ii)\\ \\\nIn the case of $\\Bc_3(\\om)$ consider the following\ntwo updown tableaux\nassociated with the diagram $\\la=(1)$:\n\\setlength{\\unitlength}{0.3em}\n\\begin{center}\n\\begin{picture}(65,5)\n\n\\put(-10,0){$T_1=\\big($}\n\n\\put(0,0){\\line(0,1){2}}\n\\put(2,0){\\line(0,1){2}}\n\\put(0,0){\\line(1,0){2}}\n\\put(0,2){\\line(1,0){2}}\n\\put(2.5,0){,}\n\n\\put(8,-2){\\line(0,1){4}}\n\\put(10,-2){\\line(0,1){4}}\n\\put(8,-2){\\line(1,0){2}}\n\\put(8,0){\\line(1,0){2}}\n\\put(8,2){\\line(1,0){2}}\n\\put(10.5,0){,}\n\n\\put(17,0){\\line(0,1){2}}\n\\put(19,0){\\line(0,1){2}}\n\\put(17,0){\\line(1,0){2}}\n\\put(17,2){\\line(1,0){2}}\n\n\\put(20,0){$\\big),$}\n\n\n\\put(33,0){$T_2=\\big($}\n\n\\put(43,0){\\line(0,1){2}}\n\\put(45,0){\\line(0,1){2}}\n\\put(43,0){\\line(1,0){2}}\n\\put(43,2){\\line(1,0){2}}\n\\put(45.5,0){,}\n\n\\put(51,0){$\\varnothing,$}\n\n\\put(60,0){\\line(0,1){2}}\n\\put(62,0){\\line(0,1){2}}\n\\put(60,0){\\line(1,0){2}}\n\\put(60,2){\\line(1,0){2}}\n\n\n\n\\put(62.5,0){$\\big).$}\n\n\\end{picture}\n\\end{center}\n\\setlength{\\unitlength}{1pt}\n\n\\noindent\nThe respective contents $(c_1,c_2,c_3)$ are given by\n\\ben\n\\Big(\\frac{\\om-1}{2},\\ts\\frac{\\om-3}{2},\\ts-\\frac{\\om-3}{2}\\Big),\\qquad\n\\Big(\\frac{\\om-1}{2},\\ts-\\frac{\\om-1}{2},\n\\ts\\frac{\\om-1}{2}\\Big).\n\\een\nThen using Theorem~\\ref{thm:newfus} and \\eqref{indugeth} we find\nthat $E_{T_1}$ and $E_{T_2}$\ncan be calculated by evaluating the respective rational\nfunctions\n\\begin{multline}\n\\frac{(u-c_3)(u+c_1-\\vk)}{(u-c_1)(u+c_3-\\vk)}\\ts\n\\ts\\frac{(u-c_1)^2}{(u-c_1)^2-1}\\ts\\frac{(u-c_2)^2}{(u-c_2)^2-1}\\ts\\\\\n{}\\times{}E_{U}\\ts\n\\rho^{}_{23}(-c_2-u+\\vk)\\ts\\rho^{}_{13}(-c_1-u+\\vk)\\ts\n\\rho^{}_{13}(c_1-u)\\ts\\rho^{}_{23}(c_2-u)\n\\non\n\\end{multline}\nat $u=c_3$ with $U=U_2$ and $U=U_3$, respectively;\nsee Example~(i).\nPerforming\nthe calculation we get\n\\ben\nE_{T_1}=\n\\frac{(1-s_1)\\ts \\ep_2\\ts(1-s_1)}{2\\ts(\\om-1)},\\qquad\nE_{T_2}=\\frac{\\ep_1}{\\om}.\n\\een\n\n\\quad (iii)\\ \\\nFor $\\Bc_n(\\om)$ consider the only updown tableau\n$T=\\big((1),(2),\\dots,(n)\\big)$\nassociated with the diagram $\\la=(n)$.\nThe corresponding contents $(c_1,\\dots,c_n)$ are\nfound by the formula $c_i=(\\om-1)\/2+i-1$.\nHence, by Theorem~\\ref{thm:newfus}\nthe idempotent $S_n=E_T$\n(the {\\it symmetrizer\\\/} in the Brauer algebra) is given by\n\\ben\nS_n=\\frac{1}{n!}\\ts\\prod_{r=2}^n\\ts\\frac{\\om+2r-2}{\\om+4r-4}\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(2-i-j-\\om\/2)\\ts \\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(i-j)\n\\een\nwith the products over the pairs $(i,j)$ taken in the\nlexicographic order.\nGiven any index $k\\in\\{1,\\dots,n-1\\}$,\nwe can use \\eqref{ybe} to reorder the factors in\nthe last product in such a way\nthat it would begin with the factor $\\rho_{k,k+1}(-1)$.\nSince\n\\ben\n\\ep_k\\ts \\rho_{k,k+1}(-1)=0\\Fand\ns_k\\ts \\rho_{k,k+1}(-1)=\\rho_{k,k+1}(-1),\n\\een\nwe find that for any $k<l$\n\\ben\n\\rho_{kl}(u)\\ts \\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(i-j)=\\frac{u-1}{u}\\ts\\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(i-j).\n\\een\nHence the expression for the symmetrizer simplifies to\n\\ben\nS_n=\\frac{1}{n!}\\ts \\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(i-j);\n\\een\ncf. \\cite[Remark~3.8]{im:fp}.\n\n\\quad (iv)\\ \\\nIn the case of $\\Bc_n(\\om)$ consider the only updown tableau\n$T=\\big((1),(1^2),\\dots,(1^n)\\big)$\nassociated with the diagram $\\la=(1^n)$.\nThe corresponding contents $(c_1,\\dots,c_n)$ are\ngiven by $c_i=(\\om-1)\/2-i+1$. Hence, by Theorem~\\ref{thm:newfus}\nthe idempotent $A_n=E_T$\n(the {\\it anti-symmetrizer\\\/} in the Brauer algebra) is given by\n\\beql{newfp}\nA_n=\\frac{1}{n!}\\ts\\prod_{r=2}^n\\ts\\frac{\\om-2r+2}{\\om-4r+4}\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(i+j-2-\\om\/2)\\ts \\prod_{1\\leqslant i<j\\leqslant n}\n\\rho_{ij}(j-i)\n\\eeq\nwith the products over the pairs $(i,j)$ taken in the\nlexicographic order.\nA different expression\nfor the anti-symmetrizer is provided by the alternative\nfusion procedure in \\cite[Remark~3.8]{im:fp}:\n\\beql{oldfp}\nA_n=\\frac{1}{n!}\\ts\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1-\\frac{s_{ij}}{j-i}\\Big)\n\\eeq\nwith the product taken in the\nlexicographic order on the pairs $(i,j)$. A more direct way to\nsee the coincidence of the elements given by \\eqref{newfp}\nand \\eqref{oldfp} is to regard the group algebra $\\CC[\\Sym_n]$\nfor the symmetric group as a natural subalgebra of $\\Bc_n(\\om)$\nand observe that \\eqref{oldfp} is the well-known factorization\nof the anti-symmetrizer in $\\CC[\\Sym_n]$. Indeed,\nthese anti-symmetrizers satisfy the well-known recurrence\nrelation $n\\ts A_n=A_{n-1}\\big(1-s_{1n}-\\dots-s_{n-1,n}\\big)$.\nTherefore, employing the induction\non $n$ and using the recurrence relations \\eqref{murphyfo}\nin $\\Bc_n(\\om)$\nwe come to checking the identity\n\\ben\nA_{n-1}\\frac{(x_n-c_1-1)(x_n+c_1-n+2)}{2\\tss c_1-2n+3}\n=A_{n-1}\\big({-1}+s_{1n}+\\dots+s_{n-1,n}\\big),\n\\een\nwhere, as before, $c_1=(\\om-1)\/2$. The calculation\nis straightforward and it relies on the\nfollowing relations\nin $\\Bc_n(\\om)$ which hold for any two distinct\nindices $i,j\\in\\{1,\\dots,n-1\\}$:\n\\ben\nA_{n-1}\\ts s_{in}\\ts s_{jn}=-A_{n-1}\\ts s_{in},\\quad\nA_{n-1}\\ts s_{in}\\ts \\ep_{jn}=0,\\quad\nA_{n-1}\\ts \\ep_{in}\\ts \\ep_{jn}=A_{n-1}\\ts \\ep_{in}\\ts s_{ij}=\n-A_{n-1}\\ts \\ep_{jn}\n\\een\nand\n\\ben\nA_{n-1}(\\ep_{in} s_{jn} + \\ep_{jn} s_{in}) =\nA_{n-1}(\\ep_{in} s_{jn} - s_{ij} \\ep_{jn} s_{in}) =\nA_{n-1}(\\ep_{in} s_{jn} - \\ep_{in} s_{jn}) = 0.\n\\een\n\\vskip-1.6\\baselineskip\n\\qed\n\\eex\n\n\nTo make a connection of the version of the fusion procedure\nprovided by Theorem~\\ref{thm:newfus} with that of \\cite{im:fp},\nrecall some parameters associated with\nupdown tableaux; see \\cite{im:fp}.\nGiven an updown $\\mu$-tableau\n$U=(\\La_1,\\dots,\\La_{n-1})$ we define two infinite matrices\n$m(U)$ and $m'(U)$ whose rows and columns are labelled\nby positive integers and only a finite number of entries\nin each of the matrices are nonzero. The entry $m_{ij}$\nof the matrix $m(U)$ (resp., the entry $m'_{ij}$\nof the matrix $m'(U)$) equals the number of times\nthe box $(i,j)$ was added (resp., removed) in the sequence\nof diagrams $(\\varnothing=\\La_0,\\La_1,\\dots,\\La_{n-1})$.\nFor each integer $k$\ndefine the nonnegative integers $d_k=d_k(U)$ and\n$d^{\\tss\\prime}_k=d^{\\tss\\prime}_k(U)$\nas the respective sums of the entries of the matrices\n$m(U)$ and $m'(U)$ on the $k$-th diagonal,\n\\ben\nd_k=\\sum_{j-i=k}m_{ij},\\qquad d^{\\tss\\prime}_k=\\sum_{j-i=k}m'_{ij},\n\\een\nand set\n\\beql{gk}\ng_k(U)=\\de_{k\\tss 0}+d_{k-1}+d_{k+1}-2\\tss d_k,\\qquad\ng'_k(U)=d^{\\tss\\prime}_{k-1}\n+d^{\\tss\\prime}_{k+1}-2\\tss d^{\\tss\\prime}_k.\n\\eeq\nNow the {\\it exponents\\\/} $p_1,\\dots,p_n$ of\nan updown $\\la$-tableau $T=(\\La_1,\\dots,\\La_n)$ are defined\ninductively, so that $p_r$ depends only on the first $r$\ndiagrams $(\\La_1,\\dots,\\La_r)$ of $T$. Hence, it is sufficient\nto define $p_n$. Taking $U=(\\La_1,\\dots,\\La_{n-1})$ we set\n\\beql{pn}\np_n=1-g_{k_n}(U)\\qquad\\text{or}\\qquad p_n=1-g'_{k_n}(U),\n\\eeq\nrespectively, if $\\La_n$ is obtained from $\\La_{n-1}$\nby adding a box on the diagonal $k_n$ or by\nremoving a box on the diagonal $k_n$.\n\nDefine the constants $h(T)$ inductively by the formula\n\\beql{ft}\nh(T)=h(U)\\ts\\psi(U,T),\n\\eeq\nwhere $U=(\\La_1,\\dots,\\La_{n-1})$,\n$T=(\\La_1,\\dots,\\La_n)$, and $h(U)=1$ if $U=(\\La_1)$.\nHere\n\\ben\n\\psi(U,T)=\\frac{4k_n+\\om}{2k_n+\\om}\\ts\\prod_{k\\ne k_n}(k_n-k)^{g_k}\n\\prod_k(k_n+k+\\om-1)^{g^{\\tss\\prime}_k}\n\\een\nor\n\\ben\n\\psi(U,T)=\\frac{4k_n+6\\om-4}{2k_n+\\om-2}\\ts\n\\prod_{k\\ne k_n}(-k_n+k)^{g^{\\tss\\prime}_k}\n\\prod_k(-k_n-k-\\om+1)^{g_k},\n\\een\nif $\\La_n$ is obtained from $\\La_{n-1}$\nby adding or removing a box on the diagonal $k_n$,\nrespectively, where the products are taken\nover all integers $k$, while $g_k=g_k(U)$ and $g'_k=g'_k(U)$.\n\nConsider now the rational function $\\Omega(u_1,\\dots,u_n)$\nwith values in the Brauer algebra $\\Bc_n(\\om)$\ndefined by \\eqref{omubra}.\n\n\\bco\\label{cor:newfuseq}\nFor any updown tableau $T=(\\La_1,\\dots,\\La_n)$\nthe consecutive evaluations\n\\ben\n(u_1-c_1)^{p_1}\\dots (u_n-c_n)^{p_n}\\ts\n\\Omega(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_n=c_n}\n\\een\nare well-defined. The corresponding value coincides with\n$h(T)\\tss E_{T}$.\n\\eco\n\n\\bpf\nWe use induction on $n$ as in the proof of Theorem~\\ref{thm:newfus}.\nBy the induction hypothesis, setting $u=u_n$ we get\n\\begin{multline}\\label{induge}\n(u_1-c_1)^{p_1}\\dots (u_n-c_n)^{p_n}\\ts\n\\Omega(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_{n-1}=c_{n-1}}=h(U)\\tss (u-c_n)^{p_n}\\\\\n{}\\times{}E_U\\ts\n\\rho_{n-1,n}(-c_{n-1}-u+\\vk)\\dots \\rho_{1,n}(-c_1-u+\\vk)\\ts\n\\rho_{1,n}(c_1-u)\\dots \\rho_{n-1,n}(c_{n-1}-u),\n\\end{multline}\nwhere $U$ is the updown tableau $(\\La_1,\\dots,\\La_{n-1})$.\nApplying Lemma~\\ref{lem:jmsi}, we come to\nshowing that the rational function\n\\ben\nh(U)\\tss\n\\frac{(u-c_1)(u+c_n-\\vk)}{u+c_1-\\vk}\\ts\n\\prod_{r=1}^{n-1}\\Big(1-\\frac{1}{(u-c_r)^2}\\Big)\n\\ts(u-c_n)^{p_n-1}\\cdot\nE_U\\ts \\frac{u-c_n}{u-x_n}\\ts \\frac{u+x_n-\\vk}{u+c_n-\\vk}\n\\een\nis regular at $u=c_n$ and its value equals $h(T)\\tss E_T$.\nUsing the parameters \\eqref{gk}, we can write\nthis expression as\n\\begin{multline}\nh(U)\\ts\\frac{u+c_n-\\vk}{u+c_1-\\vk}\\ts\n\\prod_{k}\\Big(u-\\frac{\\om-1}{2}-k\\Big)^{g_k}\n\\prod_{k}\\Big(u+\\frac{\\om-1}{2}+k\\Big)^{g^{\\tss\\prime}_k}\\ts\n(u-c_n)^{p_n-1}\\\\\n{}\\times E_U\\ts \\frac{u-c_n}{u-x_n}\\ts \\frac{u+x_n-\\vk}{u+c_n-\\vk},\n\\non\n\\end{multline}\nwhere $k$ runs over the set of integers.\nIf the diagram $\\La_n$ is obtained from $\\La_{n-1}$\nby adding or removing a box on the diagonal $k_n$, then\nthe value of the content $c_n$ is given by the respective formulas\n\\ben\nc_n=\\frac{\\om-1}{2}+k_n\\qquad\\text{or}\\qquad\nc_n=-\\Big(\\frac{\\om-1}{2}+k_n\\Big).\n\\een\nThe argument is completed by recalling the\ndefinition of the exponents \\eqref{pn},\nand the constants $h(T)$ in \\eqref{ft} together with\n\\eqref{jmform}.\n\\epf\n\nAn important particular case of Theorem~\\ref{thm:newfus}\nand Corollary~\\ref{cor:newfuseq} is the case where $\\la$\nis a partition of $n$. Here the updown tableaux $T$\nof shape $\\la$ can be regarded as the {\\it standard tableaux\\\/}\nwhich parameterize basis vectors of the corresponding\nrepresentation of $\\Bc_n(\\om)$. It was pointed out in\n\\cite[Corollary~3.7]{im:fp} that in this\nsituation all exponents $p_i$ are equal to zero\nand the constant $f(T)$ arising in the evaluations \\eqref{reev}\ndepends only on $\\la$ and does\nnot depend on the standard $\\la$-tableau $T$.\nThe new version of the fusion procedure associated with\nthe rational function \\eqref{omubra} possesses the same\nproperty as the next corollary shows. To formulate the result\nintroduce the {\\it content polynomial\\\/}\n\\ben\nC_{\\la}(z)=\\prod_{\\al\\in\\la}\\big(z+\\si(\\al)\\big),\n\\een\nwhere $\\si(\\al)=j-i$ if the box $\\al$ of the diagram $\\la$\nis in row $i$ and column $j$.\n\n\\bco\\label{cor:symbra}\nIf $T=(\\La_1,\\dots,\\La_n)$\nis an updown $\\la$-tableau and $\\la=\\La_n$ is a partition of $n$,\nthen the consecutive evaluations give\n\\ben\n\\Omega(u_1,\\dots,u_n)\\big|_{u_1=c_1}\\big|_{u_2=c_2}\\dots\n\\big|_{u_n=c_n}=\n\\frac{2^n\\ts C_{\\la}(\\om\/4)\\ts H(\\la)}{C_{\\la}(\\om\/2)}\\ts E_T,\n\\een\nwhere $H(\\la)$ is the product of\nthe hooks of $\\la$.\n\\eco\n\n\\bpf\nArguing as in the proof of Corollary~\\ref{cor:newfuseq}\nand using the respective calculation of \\cite{m:fp}\nwe observe that the recurrence relation for the constants\n$h(T)$ now takes the form\n\\ben\nh(T)=h(U)\\ts \\frac{H(\\la)}{H(\\mu)}\\ts\\frac{\\om+4\\si_n}{\\om+2\\si_n},\n\\een\nwhere $\\mu$ is the shape of the tableau $U$ and $\\si_n=\\si(\\al)$\nfor the box $\\al$ occupied by $n$. An obvious induction\ncompletes the proof.\n\\epf\n\nAs the group algebra $\\CC[\\Sym_n]$ can be regarded as the quotient\nof the Brauer algebra $\\Bc_n(\\om)$ by the ideal generated\nby the elements $\\ep_1,\\dots,\\ep_{n-1}$, Corollary~\\ref{cor:symbra}\nyields a new version of the fusion procedure for the symmetric\ngroup. Moreover, this procedure involves an arbitrary parameter\n$\\om$ inherited from $\\Bc_n(\\om)$. To formulate the\ncorresponding version introduce the rational function\nin variables $v_1,\\dots,v_n$ with values in $\\CC[\\Sym_n]$ by\n\\beql{omvsym}\n\\Omega_{\\om}(v_1,\\dots,v_n)=\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1+\\frac{s_{ij}}{v_i+v_j+\\om\/2}\\Big)\n\\prod_{1\\leqslant i<j\\leqslant n}\n\\Big(1-\\frac{s_{ij}}{v_i-v_j}\\Big)\n\\eeq\nwith both products taken in the\nlexicographic order on the pairs $(i,j)$.\n\n\\bco\\label{cor:sym}\nSuppose that $\\la$ is a partition of $n$ and\n$T$ is a standard $\\la$-tableau. Let $\\si_k=j-i$ if\n$k$ occupies the box $(i,j)$ in $T$.\nThen the consecutive evaluations give\n\\ben\n\\Omega_{\\om}(v_1,\\dots,v_n)\\big|_{v_1=\\si_1}\\big|_{v_2=\\si_2}\\dots\n\\big|_{v_n=\\si_n}=\n\\frac{2^n\\ts C_{\\la}(\\om\/4)\\ts H(\\la)}{C_{\\la}(\\om\/2)}\\ts E_T,\n\\een\nwhere $H(\\la)$ is the product of\nthe hooks of $\\la$ and $E_T$ is the primitive idempotent\nin $\\CC[\\Sym_n]$ associated with $T$.\n\\eco\n\n\\bpf\nThis is immediate from Corollary~\\ref{cor:symbra}, where\nthe relation between the variables is given by\n$u_i=v_i+c_1$ for all $i$ and we used the relation\n$2\\tss c_1-\\vk=\\om\/2$.\n\\epf\n\nThis version of the fusion procedure for the symmetric group\nappears to be new. We will discuss its meaning in the context\nof evaluation homomorphisms below in Sec.~\\ref{sec:tr}.\nBy taking the limit $\\om\\to\\infty$ we recover\nthe standard fusion procedure originated in Jucys~\\cite{j:yo}\nin the form found in \\cite{m:fp}; cf. \\cite{c:sb} and \\cite{n:yc}.\n\n\n\\section{Evaluation homomorphisms}\\label{sec:eh}\n\\setcounter{equation}{0}\n\nLet $G=[g_{ij}]$ be a nonsingular symmetric or skew-symmetric\n$N\\times N$ matrix with entries in $\\CC$ so that\n$G^{\\tss t}={\\pm}\\ts G$, where $t$ denotes the standard\nmatrix transposition. The skew-symmetric case\nmay occur only if $N$ is even.\nConsider the canonical basis $e_1,\\dots,e_N$\nof the vector space $\\CC^N$ and equip it with\nthe bilinear form\n$\\langle\\ts,\\rangle$ by\n\\beql{formg}\n\\langle e_i,e_j\\rangle=g_{ij}.\n\\eeq\nThe classical orthogonal and symplectic groups\n$\\Or_N$ and $\\Spr_N$ consist of the\n$N\\times N$ matrices\n$\\hb$ preserving the symmetric\nor skew-symmetric form, respectively,\n\\beql{defosp}\n\\langle \\hb\\tss v,\\hb\\tss w\\rangle=\\langle v, w\\rangle\n\\qquad\\text{for all}\\quad v,w\\in\\CC^N.\n\\eeq\nWe will be using\nthe following convention.\nWhenever the double sign $\\pm{}$ or $\\mp{}$ occurs,\nthe upper sign will correspond to the symmetric case\nand the lower sign to\nthe skew-symmetric case.\nFor any $N\\times N$ matrix $A$ set\n\\beql{transG}\nA'=G A^t\\ts G^{-1}.\n\\eeq\nWe denote by $E_{ij}$, $1\\leqslant i,j\\leqslant N$,\nthe standard basis vectors of the Lie\nalgebra $\\gl_N$. They satisfy the commutation relations\n\\beql{comrelgln}\n[E_{ij},E_{kl}]=\\de_{kj}E_{il}-\\de_{il}E_{kj}.\n\\eeq\nFor the entries of the inverse matrix of $G$ we will\nwrite $G^{-1}=[\\overline{g}_{ij}]$.\nIntroduce the elements $F_{ij}$\nof the Lie algebra $\\gl_N$ by the formulas\n\\beql{fijelem}\nF_{ij}=E_{ij}-\\sum_{k,\\tss l=1}^N\ng^{}_{ik}\\tss \\overline{g}^{}_{lj}\\tss E_{lk}.\n\\eeq\nThe Lie subalgebra\nof $\\gl_N$ spanned by the elements $F_{ij}$ is isomorphic to\nthe orthogonal Lie algebra $\\oa_N$ associated with $\\Or_N$\nin the symmetric case\nand to the symplectic Lie algebra $\\spa_N$\nassociated with $\\Spr_N$\nin the skew-symmetric case.\nThis Lie algebra will be denoted by $\\g_N$.\nCommon choices of the matrix $G$ in the symmetric case\ninclude the identity matrix $G=1$ and the\nantidiagonal matrix $G=[\\de_{i,N-j+1}]$. In the skew-symmetric case\nwith $N=2n$ the entries of $G$ are often chosen in the form\n$g_{ij}=\\de_{i,2n-j+1}$ for $i=1,\\dots,n$ and\n$g_{ij}={}-\\de_{i,2n-j+1}$ for $i=n+1,\\dots,2n$.\nIn all these cases, the summation in the formula \\eqref{fijelem}\nreduces to one term.\n\nConsider the endomorphism algebra $\\End\\CC^N$ and let\n$e_{ij}\\in\\End\\CC^N$\nbe the standard matrix units. We denote by $F$ the $N\\times N$ matrix\nwhose $ij$-th entry is $F_{ij}$. We shall also regard $F$ as the\nelement\n\\ben\nF=\\sum_{i,j=1}^N e_{ij}\\ot F_{ij}\\in \\End\\CC^N\\ot \\U(\\g_N).\n\\een\nThe definition \\eqref{fijelem} can be written\nin the matrix form as\n\\beql{feeg}\nF=E-E^{\\tss\\prime}=E-G\\tss E^{\\tss t}\\tss G^{-1},\n\\eeq\nwhere\n\\ben\nE=\\sum_{i,j=1}^N e_{ij}\\ot E_{ij}\\in \\End\\CC^N\\ot \\U(\\gl_N).\n\\een\n\nConsider the permutation operator\n\\beql{pmatrix}\nP=\\sum_{i,j=1}^N e_{ij}\\ot e_{ji}\\in \\End \\CC^N\\ot\\End \\CC^N\n\\eeq\nand its partial transpose $P^{\\tss t}$ with respect to the first\n(or, equivalently, the second) copy of $\\End \\CC^N$:\n\\beql{qmatrix}\nP^{\\tss t}=\\sum_{i,j=1}^N e_{ij}\\ot e_{ij}.\n\\eeq\nFurthermore, set\n\\beql{rprimeq}\nQ=G_1\\tss P^{\\tss t}\\ts G^{-1}_1=G_2\\tss P^{\\tss t}\\ts G^{-1}_2,\n\\eeq\nwhere the second equality\nfollows from the relations $G_1\\ts P^{\\tss t}=G_2^t\\ts P^{\\tss t}$\nand $P^{\\tss t}\\ts G_1 =P^{\\tss t}\\ts G_2^t$.\n\nNote that the operators $P$ and $Q$ satisfy\nthe relations\n\\beql{pq}\nP^2=1,\\qquad PQ=QP={}\\pm Q,\\qquad Q^2=N\\ts Q.\n\\eeq\nThe defining relations of the algebra $\\U(\\gl_N)$ can be written\nin a matrix form as\n\\ben\nE_1\\ts E_2-E_2\\ts E_1=E_1\\ts P-P\\ts E_1,\n\\een\nwhere both sides are regarded as elements of the algebra\n$\\End\\CC^N\\ot\\End\\CC^N\\ot \\U(\\gl_N)$ and\n\\beql{eonetwo}\nE_1=\\sum_{i,j=1}^N e_{ij}\\ot 1\\ot E_{ij},\\qquad\nE_2=\\sum_{i,j=1}^N 1\\ot e_{ij}\\ot E_{ij}.\n\\eeq\nThe defining relations of the algebra $\\U(\\g_N)$\nthen take the form\n\\beql{deff}\nF_1\\ts F_2-F_2\\ts F_1=F_1\\ts (P-Q)-(P-Q)\\ts F_1\n\\eeq\ntogether with the relation $F+F^{\\tss\\prime}=0$. The latter\nimplies\n\\beql{qf}\nQ\\ts (F_1+F_2)=(F_1+F_2)\\ts Q=0.\n\\eeq\n\nSet\n\\beql{kappa}\n\\kappa=N\/2\\mp 1.\n\\eeq\nThe $R$-matrix $R(u)$ is a rational function in a complex parameter $u$\nwith values in the tensor product algebra\n$\\End\\CC^N\\ot\\End\\CC^N$ defined by\n\\beql{Ru}\nR(u)=1-\\frac{P}{u}+\\frac{Q}{u-\\kappa}.\n\\eeq\nIt is well known by \\cite{zz:rf}\nthat $R(u)$ satisfies the Yang--Baxter equation\n\\beql{yberep}\nR_{12}(u)\\ts R_{13}(u+v)\\ts R_{23}(v)\n=R_{23}(v)\\ts R_{13}(u+v)\\ts R_{12}(u).\n\\eeq\nHere both sides take values in\n$\\End\\CC^N\\ot\\End\\CC^N\\ot\\End\\CC^N$ and the subscripts indicate\nthe copies of $\\End\\CC^N$ so that $R_{12}(u)=R(u)\\ot 1$ etc.\nClearly, this is a recast of the properties\nof the functions $\\rho_{ij}(u)$ defined in \\eqref{rij};\nwe will discuss this relationship\nin more detail in Sec.~\\ref{sec:tr}.\n\nFollowing the general approach of \\cite{d:qg} and \\cite{rtf:ql},\ndefine the {\\it extended Yangian\\\/}\n$\\X(\\g_N)$\nas an associative algebra with generators\n$t_{ij}^{(r)}$, where $1\\leqslant i,j\\leqslant N$ and $r=1,2,\\dots$,\nsatisfying certain quadratic relations. In order to write them down,\nintroduce the formal series\n\\beql{tiju}\nt_{ij}(u)=\\de_{ij}+\\sum_{r=1}^{\\infty}t_{ij}^{(r)}\\ts u^{-r}\n\\in\\X(\\g_N)[[u^{-1}]]\n\\eeq\nand set\n\\beql{Tu}\nT(u)=\\sum_{i,j=1}^N e_{ij}\\ot t_{ij}(u)\n\\in \\End\\CC^N\\ot \\X(\\g_N)[[u^{-1}]].\n\\eeq\nConsider the algebra\n$\\End\\CC^N\\ot\\End\\CC^N\\ot \\X(\\g_N)[[u^{-1}]]$\nand introduce its elements $T_1(u)$ and $T_2(u)$ by\n\\beql{T1T2}\nT_1(u)=\\sum_{i,j=1}^N e_{ij}\\ot 1\\ot t_{ij}(u),\\qquad\nT_2(u)=\\sum_{i,j=1}^N 1\\ot e_{ij}\\ot t_{ij}(u).\n\\eeq\nThe defining relations for the algebra $\\X(\\g_N)$ can then\nbe written in the form\n\\beql{RTT}\nR(u-v)\\ts T_1(u)\\ts T_2(v)=T_2(v)\\ts T_1(u)\\ts R(u-v).\n\\eeq\nThe {\\it Yangian\\\/} $\\Y(\\g_N)$ is then defined as the quotient\nof the extended Yangian $\\X(\\g_N)$\nby the relation\n\\beql{ttra}\nT^{\\tss\\prime}(u+\\ka)\\ts T(u)=1.\n\\eeq\nAlthough the matrix $G$ is implicit in the definitions of\n$\\X(\\g_N)$ and $\\Y(\\g_N)$, the respective\nalgebras associated with two different\nnonsingular symmetric (or skew-symmetric) $N\\times N$ matrices\n$G$ and $\\wt G$ are isomorphic to each other so that\neach of them depends only on the Lie algebra $\\g_N$.\nDrinfeld defined the Yangians associated with\nsimple Lie algebras by using\ndifferent presentations; see \\cite{d:ha, d:qg}.\nExplicit form of the defining relations for some standard\nchoices of antidiagonal symmetric and skew-symmetric matrices $G$\nin terms of the series $t_{ij}(u)$\ncan be found in \\cite{aacfr:rp}, where the equivalence\nof the $R$-matrix presentation and the Drinfeld presentation\nis explained; see also \\cite{amr:rp}.\nThe Yangian $\\Y(\\g_N)$ carries a Hopf algebra structure\ndefined in a standard way; see e.g. {\\it loc. cit.}\nfor explicit formulas.\n\nFollowing \\cite{s:bc}, use a reflection type equation\nto define algebras associated to $\\g_N$;\nsee also \\cite{aacfr:rp}, \\cite{mr:rr}.\n\n\\bde\\label{def:refeq}\nThe {\\it reflection algebra\\\/} $\\Br(\\g_N)$\ncorresponding to the Lie algebra $\\g_N$ is defined as the\nassociative algebra generated by elements $s_{ij}^{(r)}$\nwith $1\\leqslant i,j\\leqslant N$ and $r\\geqslant 1$\nsubject to defining relations written in terms of the\ngenerating series\n\\ben\ns_{ij}(u)=\\de_{ij}+\\sum_{r=1}^{\\infty}s_{ij}^{(r)}\\ts u^{-r}\n\\een\nas follows. Introduce the matrix\n\\beql{Su}\nS(u)=\\sum_{i,j=1}^N e_{ij}\\ot s_{ij}(u)\n\\in \\End\\CC^N\\ot \\Br(\\g_N)[[u^{-1}]].\n\\eeq\nThen the defining relations have the form\nof the {\\it reflection equation\\\/}\n\\beql{refeq}\nR(u-v)\\ts S_1(u)\\ts R(u+v)\\ts S_2(v)=S_2(v)\\ts\nR(u+v)\\ts S_1(u)\\ts R(u-v)\n\\eeq\nand the {\\it unitary condition\\\/}\n\\beql{unita}\nS(u)\\ts S(-u)=1,\n\\eeq\nwhere we use the matrix notation as in \\eqref{RTT}.\n\\qed\n\\ede\n\n\\bpr\\label{prop:subalg}\nThe mapping\n\\beql{hom}\nS(u)\\mapsto T(-u)^{-1}\\ts T(u)\n\\eeq\ndefines an algebra homomorphism $\\Br(\\g_N)\\to\\X(\\g_N)$.\n\\epr\n\n\\bpf\nIt is obvious that \\eqref{unita} holds when $S(u)$ is replaced\nby the matrix $T(-u)^{-1}\\ts T(u)$.\nChecking that the image satisfies\n\\eqref{refeq} is standard and relies on \\eqref{RTT}\nand the relation\n\\ben\nT_1(u)\\ts R(u+v)\\ts T_2(-v)^{-1}=T_2(-v)^{-1}\\ts R(u+v)\\ts T_1(u)\n\\een\nimplied by \\eqref{RTT}.\n\\epf\n\nThe Yangian\n$\\Y(\\g_N)$ contains the universal enveloping algebra\n$\\U(\\g_N)$ as a subalgebra. However, as shown by\nDrinfeld~\\cite{d:ha}, in contrast with the\nYangian for the Lie algebra $\\gl_N$,\nthere is no homomorphism $\\Y(\\g_N)\\to \\U(\\g_N)$, identical\non the subalgebra $\\U(\\g_N)$. Nevertheless, as the following\nTheorem~\\ref{thm:eval} and Corollary~\\ref{cor:embed} show,\nsuch a homomorphism from the reflection algebra does exist.\nThis result is suggested by the solutions of the reflection\nequation associated with\nthe affine Birman--Murakami--Wenzl algebras\nfound in \\cite{io:bs}.\n\n\\bth\\label{thm:eval}\nThe mapping\n\\beql{evalya}\nS(u)\\mapsto \\frac{u+F-N\/4}{u-F+N\/4}\n=1+2\\ts\\sum_{r=1}^{\\infty}\n\\Big(F-\\frac{N}{4}\\Big)^{\\tss r}\\ts u^{-r}\n\\eeq\ndefines an algebra homomorphism $\\Br(\\g_N)\\to\\U(\\g_N)$.\n\\eth\n\n\\bpf\nSet $\\Fc(u)=u+F$. To make our formulas more readable, we will\nbe using the notation $a=N\/4$ throughout the proof.\nWe will demonstrate that\nboth relations \\eqref{refeq} and\n\\eqref{unita} will hold when the matrix $S(u)$ is replaced by\nthe product $\\Fc(u-a)\\Fc(-u-a)^{-1}$. This is obvious for \\eqref{unita},\nso we come to proving the identity\n\\begin{multline}\\label{refeqfu}\nR(u-v)\\ts \\Fc_1(u-a)\\ts \\Fc_1(-u-a)^{-1}\n\\ts R(u+v)\\ts \\Fc_2(v-a)\\ts \\Fc_2(-v-a)^{-1}\\\\\n{}=\\Fc_2(v-a)\\ts \\Fc_2(-v-a)^{-1}\\ts\nR(u+v)\\ts \\Fc_1(u-a)\\ts \\Fc_1(-u-a)^{-1}\\ts R(u-v).\n\\end{multline}\n\n\\ble\\label{lem:rff}\nWe have the identity\n\\beql{rff}\nR(u-v)\\ts \\Fc_1(u)\\ts \\Fc_2(v)\n-\\Fc_2(v)\\ts \\Fc_1(u)\\ts R(u-v)=\\frac{1}{u-v-\\ka}\\ts U,\n\\eeq\nwhere\n\\ben\nU=Q\\ts (F_1+\\ka)\\ts F_2-F_2\\ts (F_1+\\ka)\\ts Q.\n\\een\n\\ele\n\n\\bpf\nThe left hand side of \\eqref{rff} simplifies to\n\\ben\nF_1\\ts F_2-F_2\\ts F_1+P\\ts F_1-F_1\\ts P\n+\\frac{1}{u-v-\\ka}\\Big(Q\\ts F_1\\ts F_2-F_2\\ts F_1\\ts Q\n+(u-v)\\big(Q\\ts F_2-F_2\\ts Q\\big)\\Big),\n\\een\nwhere we used the relations $P\\ts F_1=F_2\\ts P$ and \\eqref{qf}.\nNow use \\eqref{deff} to write\n\\ben\nF_1\\ts F_2-F_2\\ts F_1+P\\ts F_1-F_1\\ts P\n=Q\\ts F_1-F_1\\ts Q={}-Q\\ts F_2+F_2\\ts Q\n\\een\nwhich gives the required relation.\n\\epf\n\nWe will now establish some simple properties of the element $U$\nto be used below.\n\n\\ble\\label{lem:utwoone}\nWe have the relations\n\\begin{align}\n\\label{puup}\nP\\ts U=U P&={}\\pm{} U,\\\\\n\\label{quuq}\nQ\\ts U+U\\ts Q&=N\\ts U,\\\\\n\\label{ffuq}\n(F_1+F_2)\\ts U&=U\\ts Q.\n\\end{align}\n\\ele\n\n\\bpf\nWe have\n\\ben\nP\\ts U=P\\ts Q\\ts (F_1+\\ka)\\ts F_2-P\\ts F_2\\ts (F_1+\\ka)\\ts Q\n={}\\pm Q\\ts (F_1+\\ka)\\ts F_2\\mp\\ts F_1\\ts (F_2+\\ka)\\ts Q.\n\\een\nApplying \\eqref{pq} and \\eqref{deff}, we get\n\\ben\n\\bal\nF_1\\ts F_2\\ts Q&=\\big(F_2\\ts F_1+F_1\\ts(P-Q)-(P-Q)\\ts F_1\\big)\\ts Q\\\\\n{}&= F_2\\ts F_1\\ts Q-(N\\mp 1)\\ts F_1\\ts Q\\mp F_2\\ts Q\n=F_2\\ts F_1\\ts Q-2\\ts\\ka\\ts F_1\\ts Q,\n\\eal\n\\een\nwhere we also used \\eqref{qf} and\nthe easily verified relation $Q\\ts F_1\\ts Q=0$.\nThis implies $P\\ts U=\\pm U$. The second relation in \\eqref{puup}\nis verified by essentially the same calculation.\n\nRelation \\eqref{quuq} is immediate from the identity\n$Q\\tss F_1\\tss F_2\\tss Q=Q\\tss F_2\\tss F_1\\tss Q$. To prove\n\\eqref{ffuq} use \\eqref{qf} to write\n\\ben\n\\bal\n(F_1+F_2)\\ts U&=-(F_1+F_2)\\ts F_2\\ts (F_1+\\ka)\\ts Q\n=-[F_1+F_2,F_2\\ts(F_1+\\ka)]\\ts Q\\\\\n{}&=-[F_1,F_2]\\ts (F_1+\\ka)\\ts Q\n+F_2\\ts [F_1,F_2]\\ts Q.\n\\eal\n\\een\nDue to \\eqref{deff} we may replace $[F_1,F_2]$ by $[F_1,P-Q]$\nand complete the calculation as in the proof of \\eqref{puup}\nto show that the expression coincides with $U\\ts Q$.\n\\epf\n\nMultiplying both sides of \\eqref{rff} by $\\Fc_2(v)^{-1}$ from the left\nand the right, we come to\nthe relation\n\\beql{rffi}\n\\Fc_1(u)\\ts R(u-v)\\ts \\Fc_2(v)^{-1}\n-\\Fc_2(v)^{-1}\\ts R(u-v)\\ts \\Fc_1(u)=\n-\\frac{1}{u-v-\\ka}\\ts \\Fc_2(v)^{-1}\\ts U\\ts \\Fc_2(v)^{-1}.\n\\eeq\nBy conjugating its both sides by $P$ and\nusing Lemma~\\ref{lem:utwoone}, we get\n\\beql{rffin}\n\\Fc_1(v)^{-1}\\ts R(u-v)\\ts \\Fc_2(u)\n-\\Fc_2(u)\\ts R(u-v)\\ts \\Fc_1(v)^{-1}=\n\\frac{1}{u-v-\\ka}\\ts \\Fc_1(v)^{-1}\\ts U\\ts \\Fc_1(v)^{-1}.\n\\eeq\nWe will need one more relation obtained from \\eqref{rffin}\nby multiplying both sides by $\\Fc_2(u)^{-1}$ from the left\nand the right:\n\\begin{multline}\\label{rffinv}\nR(u-v)\\ts \\Fc_1(v)^{-1}\\ts \\Fc_2(u)^{-1}\n-\\Fc_2(u)^{-1}\\ts \\Fc_1(v)^{-1}\\ts R(u-v)\\\\\n={}\n-\\frac{1}{u-v-\\ka}\\ts \\Fc_2(u)^{-1}\\ts\n\\Fc_1(v)^{-1}\\ts U\\ts \\Fc_1(v)^{-1}\n\\ts \\Fc_2(u)^{-1}.\n\\end{multline}\n\nOur strategy in proving \\eqref{refeqfu} is to use\nLemma~\\ref{lem:rff} and relations \\eqref{rffi}--\\eqref{rffinv}\nto transform the expression on the left hand side\nto that on the right hand side and then to show that the\nadditional terms arising in the process will add up to zero.\nWe start by applying \\eqref{rffin} with\nthe substitution $u\\mapsto v-a$ and $v\\mapsto -u-a$.\nThis yields the expression\n\\ben\nR(u-v)\\ts \\Fc_1(u-a)\\ts \\Fc_2(v-a)\n\\ts R(u+v)\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}\n\\een\ntogether with the additional term\n\\beql{termone}\n\\frac{1}{u+v-\\ka}\\ts R(u-v)\\ts \\Fc_1(u-a)\\ts\n\\Fc_1(-u-a)^{-1}\\ts U\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}.\n\\eeq\nNow use \\eqref{rff} with the substitution\n$u\\mapsto u-a$ and $v\\mapsto v-a$ to get\n\\ben\n\\Fc_2(v-a)\\ts \\Fc_1(u-a)\\ts R(u-v)\n\\ts R(u+v)\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}\n\\een\nwith the additional term\n\\beql{termtwo}\n\\frac{1}{u-v-\\ka}\\ts U\\ts R(u+v)\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}.\n\\eeq\nFurthermore, since\n$R(u-v)\\ts R(u+v)=R(u+v)\\ts R(u-v)$, applying\n\\eqref{rffinv} with the substitution\n$u\\mapsto -v-a$ and $v\\mapsto -u-a$ we get the expression\n\\ben\n\\Fc_2(v-a)\\ts \\Fc_1(u-a)\n\\ts R(u+v)\\ts \\Fc_2(-v-a)^{-1}\\ts \\Fc_1(-u-a)^{-1}\\ts R(u-v)\n\\een\ntogether with the additional term\n\\begin{multline}\\label{termthree}\n{}-\\frac{1}{u-v-\\ka}\\ts\n\\Fc_2(v-a)\\ts \\Fc_1(u-a)\\ts R(u+v)\\ts \\Fc_2(-v-a)^{-1}\\\\\n{}\\times\\ts \\Fc_1(-u-a)^{-1}\\ts U\n\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}.\n\\end{multline}\nFinally, the application of \\eqref{rffi} with the substitution\n$u\\mapsto u-a$ and $v\\mapsto -v-a$ yields the right hand\nside of \\eqref{refeqfu} with the additional term\n\\beql{termfour}\n{}-\\frac{1}{u+v-\\ka}\\ts \\Fc_2(v-a)\\ts \\Fc_2(-v-a)^{-1}\\ts U\n\\ts \\Fc_2(-v-a)^{-1}\\ts \\Fc_1(-u-a)^{-1}\\ts R(u-v).\n\\eeq\n\nWe need to show that the sum of the four additional terms\n\\eqref{termone}--\\eqref{termfour} is zero. To do this, note\nthat the sum of the terms \\eqref{termthree} and \\eqref{termfour}\nequals the sum of the terms\n\\begin{multline}\\label{termthreenew}\n{}-\\frac{1}{u-v-\\ka}\\ts\n\\Fc_2(v-a)\\ts \\Fc_2(-v-a)^{-1}\\ts  R(u+v)\\ts \\Fc_1(u-a)\\\\\n{}\\times\\ts \\Fc_1(-u-a)^{-1}\\ts U\n\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}\n\\end{multline}\nand\n\\beql{termfournew}\n{}-\\frac{1}{u+v-\\ka}\\ts \\Fc_2(v-a)\\ts \\Fc_2(-v-a)^{-1}\\ts U\n\\ts R(u-v)\\ts \\Fc_1(-u-a)^{-1}\\ts \\Fc_2(-v-a)^{-1}.\n\\eeq\nIndeed, this follows by applying \\eqref{rffi} with the substitution\n$u\\mapsto u-a$ and $v\\mapsto -v-a$ to \\eqref{termthree},\nand applying \\eqref{rffinv}\nwith the substitution\n$u\\mapsto -v-a$ and $v\\mapsto -u-a$ to \\eqref{termfour}.\nClearly, the new additional terms arising in these transformations\ncancel.\n\nNow multiply each of the expressions in \\eqref{termone},\n\\eqref{termtwo}, \\eqref{termthreenew} and \\eqref{termfournew}\nby the product $\\Fc_2(-v-a)\\ts \\Fc_1(-u-a)$ from the right and by\n$(u-v-\\ka)(u+v-\\ka)\\ts \\Fc_2(-v-a)$ from the left.\nAdding up the resulting expressions we get\n\\ben\n\\bal\n&(u-v-\\ka)\\ts \\Fc_2(-v-a)\\ts R(u-v)\\ts \\Fc_1(u-a)\\ts\n\\Fc_1(-u-a)^{-1}\\ts U\\\\\n{}+{}&(u+v-\\ka)\\ts \\Fc_2(-v-a)\\ts U\\ts R(u+v)\\\\\n{}-{}&(u+v-\\ka)\\ts\n\\Fc_2(v-a)\\ts  R(u+v)\\ts \\Fc_1(u-a)\\ts \\Fc_1(-u-a)^{-1}\\ts U\\\\\n{}-{}&(u-v-\\ka)\\ts \\Fc_2(v-a)\\ts U\\ts R(u-v).\n\\eal\n\\een\nUsing the definition \\eqref{Ru} of $R(u)$ we bring this\nsum to the form\n\\ben\n\\bal\n2\\tss v{}&\\ts\\Big(P-Q-\\Fc_2(u-\\ka-a)\\Big)\\Fc_1(u-a)\\ts\n\\Fc_1(-u-a)^{-1}\\ts U\\\\\n{}&{}+2\\tss v\\Big(U(P-Q)+\\Fc_2(-u+\\ka-a)\\ts U\\Big)\\\\\n{}&{}+\\frac{2v\\tss\\ka}{u^2-v^2}\\ts \\Big(\\Fc_2(-u-a)\n\\ts P\\ts \\Fc_1(u-a)\\ts\n\\Fc_1(-u-a)^{-1}\\ts  U - \\Fc_2(u-a)\\ts U P\\Big).\n\\eal\n\\een\nBy Lemma~\\ref{rprimeq} we have\n\\ben\nP\\ts \\Fc_1(u-a)\\ts\n\\Fc_1(-u-a)^{-1}\\ts  U=\\Fc_2(u-a)\\ts\n\\Fc_2(-u-a)^{-1}\\ts  U\\ts P\n\\een\nso that the last summand vanishes. Thus, it remains to\nverify that\n\\begin{multline}\\label{pqfze}\n\\Big(P-Q-\\Fc_2(u-\\ka-a)\\Big)\\Fc_1(u-a)\\ts\n\\Fc_1(-u-a)^{-1}\\ts U\\\\\n{}+{}U(P-Q)+\\Fc_2(-u+\\ka-a)\\ts U=0.\n\\end{multline}\nHowever, by \\eqref{deff}, the term $P-Q-\\Fc_2(u-\\ka-a)$\ncommutes with $F_1$. Therefore, when multiplied from the left\nby $\\Fc_1(-u-a)$, the left hand side of \\eqref{pqfze} becomes\n\\ben\n\\Fc_1(u-a)\\ts\\Big(P-Q-\\Fc_2(u-\\ka-a)\\Big)\\ts U\n+\\Fc_1(-u-a)\\ts \\Big(U(P-Q)+\\Fc_2(-u+\\ka-a)\\ts U\\Big),\n\\een\nwhich simplifies to\n\\ben\nu\\Big(N\\ts U+(P-Q)\\ts U-U\\ts(P-Q)-2(F_1+F_2)\\ts U\\Big)\n+(F_1-a)\\Big((P-Q)\\ts U+U\\ts(P-Q)-2\\ka\\ts U\\Big).\n\\een\nLemma~\\ref{lem:utwoone} implies that this expression\nis zero thus proving \\eqref{pqfze} and the theorem.\n\\epf\n\n\\bre\\label{rem:affbra}\nThe evaluation homomorphism of Theorem~\\ref{thm:eval}\ncan be ``lifted\" to\nthe {\\it affine Brauer algebra\\\/} defined in \\cite{n:yo} (under\nthe name {\\it degenerate affine Wenzl algebra\\\/}).\nThe precise statement will be given in \\cite{imo:fp} in the\ncontext of the affine Birman--Murakami--Wenzl algebras\nin the spirit of the approach to the representation theory\nof these algebras developed in \\cite{io:jm}.\n\\qed\n\\ere\n\nIn the following we denote by $s^{\\tss\\prime}_{ij}(u)$ the entries\nof the matrix $S^{\\tss\\prime}(u)=G\\tss S^{\\tss t}(u)\\ts G^{-1}$,\nand write\n\\ben\ns^{\\tss\\prime}_{ij}(u)=\\de_{ij}\n+\\sum_{r=1}^{\\infty}s^{\\tss\\prime\\tss(r)}_{ij}\\ts u^{-r}.\n\\een\n\n\\bco\\label{cor:embed}\nThe assignment\n\\beql{embedlie}\nF_{ij}\\mapsto \\frac14\\ts\\Big(s_{ij}^{(1)}\n-s^{\\tss\\prime\\tss(1)}_{ij}\\Big)\n\\eeq\ndefines an embedding $\\U(\\g_N)\\hra\\Br(\\g_N)$. Moreover,\nthe homomorphism \\eqref{evalya} is identical\non $\\U(\\g_N)$.\n\\eco\n\n\\bpf\nWrite \\eqref{embedlie} in a matrix form\n\\beql{matemb}\nF\\mapsto \\frac14\\ts\\big(S-S^{\\tss\\prime}\\big),\n\\eeq\nwith $S=[s_{ij}^{(1)}]$ and\n$S^{\\tss\\prime}=[s^{\\tss\\prime\\tss(1)}_{ij}]$, and verify that\nthis assignment defines a homomorphism. Obviously,\nthe relation $F+F'=0$ is preserved by the assignment so we\nare left to show that \\eqref{deff} is preserved as well.\nExpand the rational functions in $u$ and $v$ involved in\n\\eqref{refeq} into series in $v^{-1}$ by\n\\ben\n\\frac{1}{v-a}=v^{-1}+a\\tss v^{-2}+\\dots\n\\een\nand compare the coefficients of $u^{-1}v^{-1}$ on both sides.\nThis yields\n\\ben\nS_1\\ts S_2-S_2\\ts S_1=2\\ts S_1\\ts(P-Q)-2\\ts (P-Q)\\ts S_1.\n\\een\nNow apply partial transpositions with respect to the first\nand the second copies of $\\End\\CC^N$ to get\n\\ben\n\\bal\nS^{\\tss\\prime}_1\\ts S_2-S_2\\ts S^{\\tss\\prime}_1&\n=2\\ts S^{\\tss\\prime}_1\\ts(P-Q)-2\\ts (P-Q)\\ts S^{\\tss\\prime}_1,\\\\\nS_1\\ts S^{\\tss\\prime}_2-S^{\\tss\\prime}_2\\ts S_1&\n=2\\ts (P-Q)\\ts S_1-2\\ts S_1\\ts(P-Q),\\\\\nS^{\\tss\\prime}_1\\ts S^{\\tss\\prime}_2\n-S^{\\tss\\prime}_2\\ts S^{\\tss\\prime}_1&\n=2\\ts (P-Q)\\ts S^{\\tss\\prime}_1-2\\ts S^{\\tss\\prime}_1\\ts(P-Q).\n\\eal\n\\een\nThis implies that \\eqref{deff} holds under\nthe assignment \\eqref{matemb} and thus the latter\ndefines a homomorphism.\n\nFurthermore, under the homomorphism \\eqref{evalya}\nwe have $S\\mapsto 2\\ts F-N\/2$ and so its composition\nwith \\eqref{matemb} (applying \\eqref{matemb} first)\nis the identity map on $\\U(\\g_N)$.\nHence the kernel of the homomorphism \\eqref{matemb} is zero.\n\\epf\n\n\\bre\\label{rem:hom}\nDue to relation \\eqref{ttra}, the homomorphism \\eqref{hom}\nprovides a homomorphism $\\Br(\\g_N)\\to\\Y(\\g_N)$ which\ncan be written as\n\\beql{twhom}\nS(u)\\to T^{\\tss\\prime}(-u+\\ka)\\ts T(u).\n\\eeq\nAs pointed out in \\cite{aacfr:rp},\nthis brings up a connection of the reflection algebra $\\Br(\\g_N)$\nwith the Olshanski {\\it twisted Yangians\\\/} \\cite{o:ty},\nsee also \\cite[Ch.~2]{m:yc}. The homomorphism\n\\eqref{twhom} is not injective and its kernel\ncan be described by a symmetry-type relations; cf. {\\it loc. cit.}\nFor instance,\nin the case $G=1$ the elements\n$s_{ii}^{(1)}$ belong to the kernel. However, $s_{ii}^{(1)}\\ne 0$\nin $\\Br(\\g_N)$, as follows from Theorem~\\ref{thm:eval}.\nIn other words, if we define the twisted\nYangian $\\Y^{\\rm tw}(\\g_N)$\nas the quotient of $\\Br(\\g_N)$ by the kernel\nof the homomorphism \\eqref{twhom},\nthen the evaluation homomorphism of Theorem~\\ref{thm:eval}\nwill not factor through the natural epimorphism\n$\\Br(\\g_N)\\to \\Y^{\\rm tw}(\\g_N)$.\n\\qed\n\\ere\n\n\\bre\\label{rem:super}\nMost of the results of this paper\nadmit natural super-analogues where the Lie algebra $\\g_N$\nis replaced by the orthosymplectic Lie superalgebra $\\osp_{m|2n}$.\nIndeed, our calculations are performed in\na matrix language so that to apply the same approach in the super\ncase we only need to set up appropriate matrix notation\ntaking care of the sign rules. Here we will indicate\nthe necessary changes to be made in the notation\nand formulas. The $\\ZZ_2$-degree (or parity) of\nthe basis elements $E_{ij}$ of the\nLie superalgebra $\\gl_{m|2n}$ is given by $\\deg(E_{ij})=\\bi+\\bj$,\nwhere $\\bi=0$ for $1\\leqslant i\\leqslant m$ and\n$\\bi=1$ for $m+1\\leqslant i\\leqslant m+2n$.\nThe commutation relations\nin $\\gl_{m|2n}$ have the form\n\\ben\n[E_{ij}, E_{kl}]=\\de_{kj}E_{il}-\\de_{il}E_{kj}\n(-1)^{(\\bi+\\bj)(\\bk+\\bl)},\n\\een\nwhere the square brackets denote the super-commutator.\nWe will work with square matrices $A=[A_{ij}]$\nof size $m+2n$. Any such matrix with entries in\na superalgebra $\\Ac$ will be identified with\nthe element\n\\ben\n\\sum_{i,j=1}^{m+2n}e_{ij}\\ot A_{ij}(-1)^{\\bi\\bj+\\bj}\n\\in \\End\\CC^{m|2n}\\ot\\Ac.\n\\een\nThe signs here are necessary to preserve matrix\nmultiplication rules as we work with graded tensor products of\nsuperalgebras.\nFix a block-diagonal matrix $G$\nwhose upper-left $m\\times m$ block is a nonsingular\nsymmetric matrix while the lower-right $2n\\times 2n$\nblock is nonsingular skew-symmetric matrix.\nGiven a matrix $A$ we define its transpose associated with $G$\nby the formula \\eqref{transG}, where $t$ now denotes\na standard matrix super-transposition,\n$(A^t)_{ij}=A_{ji}(-1)^{\\bi\\bj+\\bi}$. Note that in contrast\nwith the standard super-transposition,\nthe transposition defined by \\eqref{transG} is involutive.\nWe will also regard $t$ as a linear map\n\\beql{suptra}\nt:\\End\\CC^{m|2n}\\to \\End\\CC^{m|2n}, \\qquad\ne_{ij}\\mapsto e_{ji}(-1)^{\\bi\\bj+\\bj}.\n\\eeq\nIn the case of multiple tensor products of the superalgebras\n$\\End\\CC^{m|2n}$ we will indicate by $t_a$ the map \\eqref{suptra}\nacting on the $a$-th copy.\nIntroduce the square matrix $E=[E_{ij}(-1)^{\\bj}]$\nof size $m+2n$ and define the elements $F_{ij}$ of $\\gl_{m|2n}$\nby the formula \\eqref{feeg}, where $F$ is the matrix\n$F=[F_{ij}(-1)^{\\bj}]$. The Lie superalgebra\nspanned by the elements $F_{ij}$ is isomorphic to\nthe orthosymplectic Lie superalgebra $\\osp_{m|2n}$.\nThe matrix form of the commutation relations in $\\osp_{m|2n}$\ncoincides with \\eqref{deff} together with $F+F^{\\tss\\prime}=0$;\nthe latter implies \\eqref{qf}. This time\nboth sides of \\eqref{deff} are elements of the superalgebra\n\\ben\n\\End\\CC^{m|2n}\\ot \\End\\CC^{m|2n}\\ot \\U(\\osp_{m|2n}),\n\\een\nwhile the definitions of the operators $P$ and $Q$ are\nmodified by\n\\ben\nP=\\sum_{i,j=1}^{m+2n} e_{ij}\\ot e_{ji}(-1)^{\\bj}\n\\in \\End\\CC^{m|2n}\\ot \\End\\CC^{m|2n}\n\\een\nand\n\\ben\nQ=G_1\\tss P^{\\tss t_1}\\ts G_1^{-1}=G_2\\tss P^{\\tss t_2}\\ts G_2^{-1}.\n\\een\nInstead of \\eqref{pq} we have\n\\ben\nP^2=1,\\qquad PQ=QP=Q,\\qquad Q^2=(m-2n)\\ts Q.\n\\een\nThe $R$-matrix $R(u)$ has the form \\eqref{Ru} with\n$\\ka=m\/2-n-1$ which leads to the definitions of the\nYangian and the reflection algebra $\\Br(\\osp_{m|2n})$\nassociated with $\\osp_{m|2n}$ as in Definition~\\ref{def:refeq};\nsee \\cite{aacfr:rp}. The evaluation homomorphism\n$\\Br(\\osp_{m|2n})\\to\\U(\\osp_{m|2n})$ provided by\nTheorem~\\ref{thm:eval} takes the same form \\eqref{evalya},\nwhere $N$ should be replaced by $m-2n$:\n\\ben\nS(u)\\mapsto \\frac{u+F-(m-2n)\/4}{u-F+(m-2n)\/4}.\n\\een\nThe embedding $\\U(\\osp_{m|2n})\\hra\\Br(\\osp_{m|2n})$ is defined\nby the same formula \\eqref{matemb}.\nNote also that the\nparameter $\\om$ of the Brauer algebra $\\Bc_n(\\om)$\nshould be evaluated at $\\om=m-2n$ in the context\nof the super-version of the centralizer results\ninvolving $\\osp_{m|2n}$;\nsee Sec.~\\ref{sec:tr}.\n\\qed\n\\ere\n\n\n\n\\section{Tensor representations of\nreflection algebras}\\label{sec:tr}\n\\setcounter{equation}{0}\n\nThe action of the Lie algebra $\\gl_N$\nin the vector representation $\\CC^N$ is given by the rule\n$E_{ij}\\mapsto e_{ij}$ which corresponds to the natural action\nof the general linear group ${\\rm GL}_N(\\CC)$ on $\\CC^N$\nby left multiplication.\nBy restricting this action to the subgroups\nof orthogonal and symplectic matrices\n$\\Or_N$ and $\\Spr_N$ ($N$ is even for the latter)\nwe make $\\CC^N$ and the\ntensor product space\n\\beql{cntenpr}\n\\CC^N\\ot\\CC^N\\ot\\dots\\ot\\CC^N,\\qquad n\\ \\ \\text{factors},\n\\eeq\ninto representations of $\\Or_N$ and $\\Spr_N$.\nTo write the images of elements of the groups,\nintroduce an extra copy of the vector space $\\CC^N$ and consider\nthe endomorphism algebra\n\\beql{endspp}\n\\End\\CC^N\\ot\\End(\\CC^N)^{\\ot n},\n\\eeq\nlabeling the tensor factors with the numbers\n$0,1,\\dots,n$. Writing a group element $\\hb$ as\n$\n\\hb=\\sum_{i,j=1}^N e_{ij}\\ot \\hb_{ij},\n$\nfor the image of $\\hb$ we have\n$\n\\hb\\mapsto \\hb_1\\dots\\hb_n\n$\nwith the notation similar to \\eqref{eonetwo} and \\eqref{T1T2}.\nDue to the work of\nBrauer~\\cite{b:aw}, the centralizer of this action\nin the endomorphism algebra of \\eqref{cntenpr} is the homomorphic\nimage of the algebra $\\Bc_n(\\om)$\nwith an appropriately specified\nvalue of the parameter $\\om$. Namely, $\\om=N$\nin the orthogonal case, and\nthe action of the Brauer algebra $\\Bc_n(N)$\nin the vector space \\eqref{cntenpr}\nis given by\n\\beql{braact}\ns_{ij}\\mapsto P_{ij},\\qquad \\ep_{ij}\\mapsto Q_{ij},\\qquad\ni< j.\n\\eeq\nIn the symplectic case, $\\om=-N$\nand the action of the Brauer algebra $\\Bc_n(-N)$\nis given by\n\\beql{braactsy}\ns_{ij}\\mapsto -\\ts P_{ij},\\qquad \\ep_{ij}\\mapsto -\\ts Q_{ij},\\qquad\ni< j;\n\\eeq\nsee \\cite{b:aw}, \\cite{w:sb},\nwhere the operators $P_{ij}$ and $Q_{ij}$ on\nthe vector space \\eqref{cntenpr} act as\nthe respective operators $P$ and $Q$ defined in \\eqref{pmatrix}\nand \\eqref{rprimeq} on the tensor product on the $i$-th and $j$-th\ncopies of $\\CC^N$ and act as the identity operators on each\nof the remaining copies.\nFurthermore, in the orthogonal case\nthe vector space \\eqref{cntenpr} is decomposed\nas\n\\ben\n(\\CC^N)^{\\ot n}\\cong \\bigoplus_{f=0}^{\\lfloor n\/2\\rfloor}\n\\bigoplus_{\\overset{\\scriptstyle\n\\la\\vdash n-2f}{\\la'_1+\\la'_2\\leqslant N}}V_{\\la}\\ot L(\\la),\n\\een\nwhere $V_{\\la}$ and $L(\\la)$\nare the respective irreducible representations of $\\Bc_n(N)$ and $\\Or_N$\nassociated with the diagram $\\la$, and we denote\nby $\\la'$ the conjugate\ndiagram so that\n$\\la'_j$ is the number\nof boxes in the column $j$ of $\\la$; see \\cite{w:cg}.\nSimilarly, in the symplectic case\n\\ben\n(\\CC^N)^{\\ot n}\\cong \\bigoplus_{f=0}^{\\lfloor n\/2\\rfloor}\n\\bigoplus_{\\overset{\\scriptstyle\n\\la\\vdash n-2f}{2\\la'_1\\leqslant N}}V_{\\la'}\\ot L(\\la),\n\\een\nwhere $V_{\\la'}$ and $L(\\la)$\nare the respective irreducible representations of\n$\\Bc_n(-N)$ and $\\Spr_N$\nassociated with $\\la'$ and $\\la$; see {\\it loc. cit.}\n\nFrom now on we will impose the condition on the parameters,\n\\beql{condnn}\nN\\geqslant 2n+ 1\\Fand  N\\geqslant 2n\n\\eeq\nin the orthogonal and symplectic case, respectively.\nIt implies, in particular, that\nthe Brauer algebras $\\Bc_n(N)$ and $\\Bc_n(-N)$ are semisimple;\nsee e.g. \\cite{r:cs} where a criterion of semisimplicity of\n$\\Bc_n(\\om)$ is given.\nExplicit projections of $(\\CC^N)^{\\ot n}$\non the irreducible representations of the orthogonal\nand symplectic groups are provided by the idempotents\nof the corresponding Brauer algebra.\nMore precisely, suppose that $\\Uc=(\\La_1,\\dots,\\La_n)$\nis an updown tableau of shape $\\la=\\La_n$.\nConsider the corresponding idempotent $E_{\\Uc}$,\nthe sequence of contents $(c_1,\\dots,c_n)$ associated with $\\Uc$\nand the subspace\n\\beql{sublu}\nL_{\\Uc}=E_{\\Uc}(\\CC^N)^{\\ot n}.\n\\eeq\nThe condition \\eqref{condnn} ensures that $E_{\\Uc}$ can be\ndefined by \\eqref{murphyfo} with non-vanishing denominators.\nUnder the action \\eqref{braact} of the Brauer\nalgebra $\\Bc_n(N)$ the subspace\nis nonzero if the number of boxes in the first two\ncolumns of $\\la$ does not exceed $N$, which follows automatically\nfrom \\eqref{condnn}.\nIn this case\n$L_{\\Uc}$ is\nan irreducible representation of $\\Or_N$ isomorphic to $L(\\la)$.\nSimilarly, under the action \\eqref{braactsy} of the Brauer\nalgebra $\\Bc_n(-N)$ the subspace \\eqref{sublu} is nonzero\nif the number of boxes in the first\ncolumn of $\\la$ does not exceed $N\/2$;\nthis holds automatically\nby \\eqref{condnn}.\nIn this case\n$L_{\\Uc}$ is\nan irreducible representation of $\\Spr_N$ isomorphic to $L(\\la)$.\n\nThe vector space $L_{\\Uc}$ carries a representation\nof the corresponding Lie algebra $\\g_N=\\oa_N$ or $\\g_N=\\spa_N$.\nEmploying the evaluation homomorphism \\eqref{evalya},\nwe can equip $L_{\\Uc}$ with the action\nof the reflection algebra $\\Br(\\g_N)$ associated with\nthe Lie algebra $\\g_N$.\n\nOn the other hand, the vector space $(\\CC^N)^{\\ot n}$ carries\na representation of the extended Yangian $\\X(\\g_N)$ defined\nby the assignment\n\\beql{yarep}\nT(u)\\mapsto a(u)\\ts R_{01}(u-z_1)\\ts\n\\dots R_{0n}(u-z_n),\n\\eeq\nwhere $z_1,\\dots,z_n$ are fixed complex numbers and\n$a(u)\\in 1+\\CC[[u^{-1}]]\\ts u^{-1}$ is a fixed\nformal series.\nHere we regard the image of $T(u)$ as a formal series in $u^{-1}$\nwith coefficients in the algebra \\eqref{endspp}.\nIndeed, verifying that \\eqref{yarep} defines\na representation of the Yangian amounts to checking that the image\nof $T(u)$ satisfies the defining relations \\eqref{RTT},\nwhich is straightforward. Furthermore, using\nProposition~\\ref{prop:subalg}, we can equip\nthe vector space $(\\CC^N)^{\\ot n}$ with the action\nof the reflection algebra $\\Br(\\g_N)$ by\n\\beql{sua}\nS(u)\\mapsto a(u)\\tss a(-u)^{-1}\\ts R_{0n}(-u-z_n)^{-1}\\ts\n\\dots R_{01}(-u-z_1)^{-1}\\ts R_{01}(u-z_1)\\ts\n\\dots R_{0n}(u-z_n).\n\\eeq\n\nWe keep using the double sign convention as in Sec.~\\ref{sec:eh}\nso that the upper sign is taken in the orthogonal case\nand the lower sign in the symplectic case.\n\n\\bpr\\label{prop:invco}\nThe subspace $L_{\\Uc}$ of the vector space \\eqref{cntenpr}\nis invariant under the action of $\\Br(\\g_N)$\ngiven by\n\\begin{multline}\nS(u)\\mapsto \\frac{u-N\/4}{u+N\/4}\\ts\n\\ts R_{0n}(-u\\mp c_n+\\ka\/2)^{-1}\\ts\n\\dots R_{01}(-u\\mp c_1+\\ka\/2)^{-1}\\\\\n{}\\times R_{01}(u\\mp c_1+\\ka\/2)\\ts\n\\dots R_{0n}(u\\mp c_n+\\ka\/2).\n\\non\n\\end{multline}\nMoreover, the representation of $\\Br(\\g_N)$ on $L_{\\Uc}$\nis isomorphic to the evaluation module\n$L_{\\Uc}$ obtained via the homomorphism \\eqref{evalya}.\n\\epr\n\n\\bpf\nConsider the anti-automorphism of the Brauer algebra $\\Bc_n(\\om)$\nidentical on the generators. Apply this\nanti-automorphism to the identity \\eqref{jmm},\nthen replace $u$ by $u+\\vk\/2$. Together with \\eqref{unitarm}\nthis leads to the following relation in the orthogonal\ncase:\n\\begin{multline}\nR_{0n}(-u-c_n+\\ka\/2)^{-1}\\ts\n\\dots R_{01}(-u-c_1+\\ka\/2)^{-1}\\\\\n{}\\times R_{01}(u-c_1+\\ka\/2)\\ts\n\\dots R_{0n}(u-c_n+\\ka\/2)\n\\ts E_{\\Uc}\n=\\frac{u+N\/4}{u-N\/4}\\ts E_{\\Uc}\\ts \\frac{u+X_0-N\/4}{u-X_0+N\/4},\n\\non\n\\end{multline}\nwhere $X_0=Q_{01}+\\dots+Q_{0,n}-P_{01}-\\dots-P_{0,n}$\nso that the parameter $n$ in \\eqref{jmm} is replaced by $n+1$\nand $m$ is replaced by $0$. We have used the fact that\nthe image of $\\rho_{ij}(u)$ coincides with $R_{ij}(u)$\nunder the specialization \\eqref{braact} with $\\om=N$.\n\nTo get the symplectic counterpart\nof the relation, note that under\nthe specialization \\eqref{braactsy} with $\\om=-N$\nthe image of $\\rho_{ij}(u)$ coincides with $R_{ij}(-u)$.\nBefore applying the above argument, we invert all factors of $E_U$\nwhich occur in \\eqref{jmm} so that in the symplectic case\nwe get\n\\begin{multline}\nR_{0n}(-u+c_n+\\ka\/2)^{-1}\\ts\n\\dots R_{01}(-u+c_1+\\ka\/2)^{-1}\\\\\n{}\\times R_{01}(u+c_1+\\ka\/2)\\ts\n\\dots R_{0n}(u+c_n+\\ka\/2)\n\\ts E_{\\Uc}\n=\\frac{u+N\/4}{u-N\/4}\\ts E_{\\Uc}\\ts \\frac{u+X_0-N\/4}{u-X_0+N\/4}.\n\\non\n\\end{multline}\nThis implies the first part of the proposition. The second part\nfollows from the observation that $X_0$ coincides with\nthe image of the element $F$ in the representation\n\\eqref{cntenpr} so that\nthe two actions of the reflection algebra $\\Br(\\g_N)$\non $L_{\\Uc}$ coincide.\n\\epf\n\nWe conclude with an interpretation of the new fusion procedure\nfor the symmetric group provided by Corollary~\\ref{cor:sym},\nfrom the viewpoint of the representation theory of the Yangians.\nRecall that the {\\it Yangian for\\\/} $\\gl_N$ is\nan associative algebra $\\Y(\\gl_N)$ with generators\n$t_{ij}^{(r)}$, where $1\\leqslant i,j\\leqslant N$ and $r=1,2,\\dots$,\nsatisfying certain quadratic relations. They are written\nwith the use of the generating functions \\eqref{tiju}\nand have the form of the $RTT$ relation \\eqref{RTT},\nwhere instead of the $R$-matrix \\eqref{Ru} we take the\n{\\it Yang $R$-matrix\\\/}\n\\beql{YRu}\nR(u)=1-\\frac{P}{u};\n\\eeq\nsee e.g. \\cite[Ch.~1]{m:yc} for a detailed description of the\nalgebraic structure of $\\Y(\\gl_N)$. The {\\it reflection algebra\\\/}\n$\\Br(\\gl_N)$ (see \\cite{s:bc})\nis isomorphic to the subalgebra of $\\Y(\\gl_N)$\ngenerated by the coefficients of the series $s_{ij}(u)$\nwhere the matrix $S(u)=[s_{ij}(u)]$ is given by\n$S(u)=T(-u)^{-1}\\ts T(u)$; see \\cite{mr:rr}.\n\nNow consider the universal enveloping algebra $\\U(\\gl_N)$\nand let $E=[E_{ij}]$ be the $N\\times N$ matrix whose\n$(i,j)$ entry is the generator $E_{ij}$ of $\\U(\\gl_N)$.\nFor any complex parameter $\\om\\in\\CC$\nthe mapping\n\\ben\nT(u)\\to \\frac{u-E^{\\tss t}-\\om\/4}{u-\\om\/4}\n\\een\ndefines an {\\it evaluation homomorphism\\\/}\n$\\Y(\\gl_N)\\to\\U(\\gl_N)$; see e.g. \\cite[Secs.~1.1-1.3]{m:yc}.\nIts restriction to the reflection algebra is\na homomorphism $\\Br(\\gl_N)\\to \\U(\\gl_N)$\ngiven by\n\\beql{evra}\nS(u)\\mapsto \\frac{u+\\om\/4}{u-\\om\/4}\\cdot\n\\frac{u-E^{\\tss t}-\\om\/4}{u+E^{\\tss t}+\\om\/4}.\n\\eeq\n\nNow, let $\\la$ be a partition of $n$ and\nsuppose that $\\Uc$ is a standard tableau\nof shape $\\la$.\nConsider the corresponding idempotent $E_{\\Uc}$\nand the sequence of contents $(\\si_1,\\dots,\\si_n)$\nas defined in Corollary~\\ref{cor:sym}.\nThe symmetric group $\\Sym_n$ acts naturally\non the space \\eqref{cntenpr}. By the Schur--Weyl duality\nthe subspace\n\\beql{sublugl}\nL_{\\Uc}=E_{\\Uc}(\\CC^N)^{\\ot n}\n\\eeq\nis nonzero if the number of boxes in the first\ncolumn of $\\la$ does not exceed $N$;\nin this case $L_{\\Uc}$ is\nan irreducible representation of $\\gl_N$\nwith the highest weight $\\la$.\n\n\\bpr\\label{prop:invcogl}\nThe subspace $L_{\\Uc}$\nis invariant under the action of $\\Br(\\gl_N)$\ngiven by\n\\begin{multline}\nS(u)\\mapsto \\frac{u-\\om\/4}{u+\\om\/4}\\ts\n\\ts R_{0n}(-u-\\si_n-\\om\/4)^{-1}\\ts\n\\dots R_{01}(-u-\\si_1-\\om\/4)^{-1}\\\\\n{}\\times R_{01}(u-\\si_1-\\om\/4)\\ts\n\\dots R_{0n}(u-\\si_n-\\om\/4).\n\\non\n\\end{multline}\nMoreover, the representation of $\\Br(\\gl_N)$ on $L_{\\Uc}$\nis isomorphic to the evaluation module\n$L_{\\Uc}$ obtained via the homomorphism \\eqref{evra}.\n\\epr\n\n\\bpf\nUsing the matrix notation and regarding the matrix $E$ as\nthe element\n\\ben\nE=\\sum_{i,j=1}^N e_{ij}\\ot E_{ij}\\in \\End\\CC^N\\ot\\U(\\gl_N)\n\\een\nwe find that the image of the transposed matrix $E^{\\tss t}$ in\n\\eqref{endspp} coincides with the element $X_0$ given by\n\\ben\nX_0=P_{01}+P_{02}+\\dots+P_{0n}.\n\\een\nThe argument is now competed in the same way as the proof\nof Proposition~\\ref{prop:invco} with the use of the image\nof the identity \\eqref{jmm} in the group algebra\nof the symmetric group.\n\\epf\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Adversarial Input Examples}\n\\label{appendix:adversarial_samples}\n\\begin{Verbatim}[commandchars=\\&\\{\\}]\nInput:\nC:\\Documents and Settings\\Administrator\\Application Data\\Yandex\\ui\nAdversarial input:\nC:\\Documents and Settings\\Administrator\\Application Data\\Yandex\\&textcolor{red}{ote.exi}\n\nInput:\nC:\\WINDOWS\\Temp\\GUM896.tmp\\goopdateres_uk.dll\nAdversarial input:\nC:\\WINDOWS\\Temp\\GUM896.tmp\\goopdateres_&textcolor{red}{r}k.dll\n\nInput:\nC:\\Program Files\\GUMA36C.tmp\\goopdateres_en-GB.dll\nAdversarial input:\nC:\\Program Files\\&textcolor{red}{E}UMA36C.tmp\\goopdateres_en-&textcolor{red}{d1}.ddl&textcolor{red}{.exe}\n\nInput:\nC:\\WINDOWS\\Temp\\oCFVhbs.ini\nAdversarial input:\nC:\\WINDOWS\\Temp\\&textcolor{red}{U}C&textcolor{red}{Bsto}s.&textcolor{red}{e}ni\n\nInput:\nC:\\WINDOWS\\Temp\\nsb7261.tmp\\UAC.dll\nAdversarial input:\nC:\\WINDOWS\\Temp\\ns&textcolor{red}{c653}1.tmp\\UAC.dll\n\nInput:\nC:\\WINDOWS\\Temp\\BgRh53b.ini\nAdversarial input:\nC:\\WINDOWS\\Temp\\&textcolor{red}{1OnDoe}b&textcolor{red}{F.bae}\n\nInput:\nC:\\WINDOWS\\Temp\\nsk6C58.tmp&textcolor{red}{\\UAC.dll}\nAdversarial input:\nC:\\WINDOWS\\Temp\\nsk6C58.tmp\n\nInput:\nC:\\WINDOWS\\Temp\\i4j_nlog_2\nAdversarial input:\nC:\\WINDOWS\\Temp\\i4j_nlog&textcolor{red}{dllgodb.g.dxe}\n\nInput:\nC:\\WINDOWS\\Temp\\GUM896.tmp\\goopdateres_zh-TW.dll\nAdversarial input:\nC:\\WINDOWS\\Temp\\GUM896.tmp\\goopdateres_zh-T&textcolor{red}{R}.dll\n\nInput:\nC:\\WINDOWS\\Temp\\lkpYxWW.ini\nAdversarial input:\nC:\\WINDOWS\\Temp\\&textcolor{red}{wL}pY&textcolor{red}{IzC.e}ni\n\nInput:\nC:\\WINDOWS\\Temp\\nsg28B0.tmp\\System.dll\nAdversarial input:\nC:\\WINDOWS\\Temp\\nsg&textcolor{red}{6}8&textcolor{red}{F}0.tmp\\System.dll\n\n\\end{Verbatim}\n\\section{Hyperparameters}\n\\label{appendix:hyperparameters}\n\n\\subsection{Encoder-Decoder Hyperparameters}\n\\label{appendix:enc_dec_hyperparams}\nWe explore effects of latent representation size (Table \\ref{tab:hp_representation_size}) and kernel width (Table \\ref{tab:hp_kernel_width}) on the sample reconstruction quality. Using larger representation sizes increases reconstruction accuracy on the test set. Conversely, larger convolution kernels do not necessarily lead to better performance. The optimal choice of kernel width for our problem is 5.\n\\begin{table}[!h]\n    \\footnotesize\n    \\parbox{.45\\linewidth}{\n        \\centering\n        \\begin{tabular}{cc}\n        \\toprule\n        Representation Size & Reconstruction Accuracy \\\\\n        \\midrule\n        32 & 91.06\\% \\\\\n        64 & 96.67\\% \\\\\n        128 & 98.26\\% \\\\\n        256 & 99.02\\% \\\\\n        512 & \\textbf{99.28\\%} \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\caption{Effects of Representation Size}\n        \\label{tab:hp_representation_size}\n    }\n    \\hfill\n    \\parbox{.45\\linewidth}{\n        \\centering\n        \\begin{tabular}{cc}\n        \\toprule\n        Kernel Width & Reconstruction Accuracy \\\\\n        \\midrule\n        3 & 97.22\\% \\\\\n        5 & \\textbf{98.50}\\% \\\\\n        7 & 98.48\\% \\\\\n        9 & 96.47\\% \\\\\n        \\bottomrule\n        \\end{tabular}\n        \\caption{Effects of Kernel Width}\n        \\label{tab:hp_kernel_width}\n    }\n\\end{table}\n\n\\subsection{Classifier Hyperparameters}\n\\label{appendix:clf_hyperparams}\nWe compare mean+max aggregation with learnable bag aggregator using different number of heads. Average classification accuracy and standard deviation are reported in Table \\ref{tab:experiments:classifier_results}. Increasing the number of heads improves test classification accuracy, where $heads=64$ achieves better performance than the mean+max baseline.\n\n\\begin{table}[!ht]\n\\footnotesize\n\\centering\n\\begin{tabular}{lc}\n\\toprule\nModel &           Accuracy \\\\\n\\midrule\nMean + Max    &  93.84\\% \u00b1 0.05\\% \\\\\nLBA(heads=2)  &  92.65\\% \u00b1 0.04\\% \\\\\nLBA(heads=4)  &  93.14\\% \u00b1 0.03\\% \\\\\nLBA(heads=8)  &  93.52\\% \u00b1 0.06\\% \\\\\nLBA(heads=16) &  93.85\\% \u00b1 0.07\\% \\\\\nLBA(heads=32) &  93.92\\% \u00b1 0.05\\% \\\\\nLBA(heads=64) &  \\textbf{94.04\\% \u00b1 0.03\\%} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Classifier Accuracy}\n\\label{tab:experiments:classifier_results}\n\\end{table}\n\\section{Conclusion}\nIn this work we have developed a method to efficiently generate adversarial strings in multiple instance learning setting. This allowed us perform adversarial training on perturbed sets of strings instead of perturbed latent representations. We experimented with standard (non-robust) classifiers, latent adversarial classifiers, and string adversarial classifiers where we compared their standard accuracy and robustness against adversarial examples. Classifiers trained on adversarial strings were the most robust against adversarial attacks and achieved similar standard accuracy as classifiers trained only on perturbed latent representations.\n\\section{Experiments}\nIn this section we describe our experimental setup and achieved results. \nUnfortunately, there is no public dataset on the classification of set of strings similar to the problem of malware classification and thus we use a proprietary dataset that is the result of a proprietary emulator of a major antivirus vendor. \nThis dataset is split according to a timestamp into training and testing subsets. \nThe temporal split is important as it captures future modifications of the malware that are not known at the training time. \n\nThe encoder-decoder model (Section \\ref{sec:encoder_decoder}) is pre-trained to represent individual strings as fixed-size vectors and to reconstruct input strings from their respective vector representations. We use $hiddenSize=128$ and $kernelWidth=5$ throughout our experiments. A full list of the encoder-decoder hyperparameters and their performance is provided in Appendix \\ref{appendix:enc_dec_hyperparams}. We then freeze the model weights of both the encoder and the decoder, and use the encoder's latent representations to train a multiple instance classifiers (Section \\ref{sec:methods_mil}). We set the number of attention heads to $heads=8$ and provide a full list of the classifier hyperparameters and their performance in Appendix \\ref{appendix:clf_hyperparams}.\n\nUsing the data and the classifier, we first investigate the attack success rate and quality of generated adversarial examples for methods described in Section \\ref{sec:adv_sample_generation} to find appropriate parameters for the algorithms that generate adversarial attacks. \nSecond, we compare standard accuracy and adversarial robustness between the original (non-robust) classifier, robust classifier trained on latent perturbations, and robust classifier trained on string perturbations. \n\n\\subsection{Quality of Generated Adversarial Samples}\nWe compare the quality of generated adversarial strings across multiple gradient-based attack methods with different hyperparameters. Our goal is to find methods with high attack success rate that generate adversarial strings which are close to the original input. As a measure of string similarity, we define a Relative Levenshtein Distance (RLD) as $RLD = Levenshtein Distance\\ \/\\ Input$ $Length$, where lower RLD means that strings are more similar. Figure \\ref{fig:adv_attack_solutions} shows the adversarial attack success rate and average RLD between perturbed inputs and original inputs for both FGSM and PGD. We explore different hyperparameter settings for the projection operator $\\mathcal{P}$, PGD step size $\\alpha$, FGSM step size $\\delta_{\\epsilon}$, and $\\epsilon$-neighborhood size $\\epsilon$. Table \\ref{tab:pareto_optimal_attack_methods} then shows attack success rate and average RLD for all Pareto optimal solutions, where adversarial strings were generated using 7 distinct classifiers to obtain uncertainty estimates. We use the Pareto optimal set of solutions to choose methods and their respective hyperparameters for further experiments. Additionally, we select methods with different adversarial attack success rates and compare the similarity of generated adversarial strings to the original inputs. Figure \\ref{fig:adv_path_similarity} shows empirical cumulative distribution function of Relative Levenshtein Distances for attack methods that achieved the following average attack success rates: 63.86\\% (blue), 74.42\\% (orange) and 87.04\\% (green).\n\n\\begin{figure}[!h]\n    \\centering\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{adv_attack_solutions}\n        \\caption{Performance of Attack Methods}\n        \\label{fig:adv_attack_solutions}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{adv_path_similarity_with_uncertainty}\n        \\caption{eCDF of Relative Levenshtein Distances}\n        \\label{fig:adv_path_similarity}\n    \\end{subfigure}\n    \\caption{Comparison of Adversarial Attack Methods}\n    \\label{fig:attack_methods_comparison}\n\\end{figure}\n\n\\begin{table}[!h]\n    \\small\n    \\centering\n    \\begin{tabular}{lcc}\n    \\toprule\n                                             Method & Attack Success Rate & Average RLD \\\\\n    \\midrule\n        FGSM(delta: 0.01, max\\_eps: 1.00)               &      56.13\\% \u00b1 2.23\\% &  0.122 \u00b1 0.017 \\\\\n    PGD(alpha: 2.00, eps: 10.00, projection: l2)   &      62.67\\% \u00b1 3.45\\% &  0.141 \u00b1 0.015 \\\\\n    PGD(alpha: 0.50, eps: 5.00, projection: linf)  &      63.83\\% \u00b1 3.69\\% &  0.146 \u00b1 0.015 \\\\\n    PGD(alpha: 0.50, eps: 10.00, projection: linf) &      63.86\\% \u00b1 3.63\\% &  0.146 \u00b1 0.015 \\\\\n    PGD(alpha: 1.00, eps: 2.00, projection: linf)  &      69.83\\% \u00b1 1.45\\% &  0.162 \u00b1 0.019 \\\\\n    PGD(alpha: 1.00, eps: 5.00, projection: linf)  &      74.32\\% \u00b1 2.64\\% &  0.193 \u00b1 0.039 \\\\\n    PGD(alpha: 1.00, eps: 10.00, projection: linf) &      74.42\\% \u00b1 2.56\\% &  0.213 \u00b1 0.057 \\\\\n    PGD(alpha: 2.00, eps: 2.00, projection: linf)  &      76.91\\% \u00b1 3.92\\% &  0.168 \u00b1 0.021 \\\\\n    PGD(alpha: 2.00, eps: 5.00, projection: linf)  &      86.76\\% \u00b1 4.43\\% &  0.300 \u00b1 0.066 \\\\\n    PGD(alpha: 2.00, eps: 10.00, projection: linf) &      87.04\\% \u00b1 4.80\\% &  0.371 \u00b1 0.098 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Pareto Optimal Solutions}\n    \\label{tab:pareto_optimal_attack_methods}\n\\end{table}\n\nThese experiments show that it is possible to find methods with high attack success rate which also generate adversarial examples that are close to the original input. We compared FGSM and PGD with both $\\ell_2$ and $\\ell_{\\infty}$ projections, where PGD with $\\ell_{\\infty}$ projection produced the best adversarial examples, FGSM produced adversarial examples with high attack success rate but with too many perturbations, and PGD with $\\ell_2$ projection produced adversarial examples that are close to the original input but have low attack success rate. Therefore, we use PGD with $\\ell_{\\infty}$ projection for further experiments and vary the step size $\\alpha$ to control the strength of adversarial attacks.\n\n\\subsection{Adversarial Training}\nTo demonstrate benefits of adversarial training on generated strings over perturbed latent vectors only, we first adversarially train classifiers using both methods, and then compare their robustness on adversarial examples generated using the original (non-robust) classifier, robust classifier trained on latent perturbations, and robust classifier trained on string perturbations.\n\n    \n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{lc}\n    \\toprule\n    {Classifier} & Standard Accuracy \\\\\n    \\midrule\n    \n    Non-robust     &   \\textbf{93.59\\% \u00b1 0.11\\%} \\\\\n    Adversarial - Latent &   91.84\\% \u00b1 0.13\\% \\\\\n    Adversarial - Full   &   91.18\\% \u00b1 0.85\\% \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Standard Accuracy of Adversarially Trained Classifiers}\n    \\label{tab:adv_classifiers_standard_acc}\n\\end{table}\n\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{lcc}\n    \\toprule\n    {} & \\multicolumn{2}{c}{\\textbf{Target Model}} \\\\\n    {\\textbf{Attack Model}} & Adversarial - Latent & Adversarial - Full \\\\\n    \\midrule\n    \n    Non-robust & \\ 89.95\\% \u00b1 1.09\\% &  \\textbf{95.20\\% \u00b1 1.42\\%} \\\\\n    Adversarial - Latent &  65.43\\% \u00b1 5.07\\% &  90.95\\% \u00b1 1.80\\% \\\\\n    Adversarial - Full   &  76.84\\% \u00b1 2.25\\% &  85.05\\% \u00b1 2.00\\% \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Robustness of Adversarially Trained Classifiers}\n    \\label{tab:adv_classifiers_robustness}\n\\end{table}\n\n\nTable \\ref{tab:adv_classifiers_standard_acc} and Table \\ref{tab:adv_classifiers_robustness} show standard classification accuracy and adversarial attack success rates for all classifiers, where in Table \\ref{tab:adv_classifiers_robustness} rows are models that generated adversarial attacks and columns are target models that were attacked. The reported results are obtained using 7 runs per classifier type. As expected, standard classification accuracy drops for both adversarially trained classifiers. However, the classifier trained on generated adversarial strings is significantly more robust compared to the classifier trained only on perturbed latent vectors. The classifier trained on adversarial strings is also more robust against adversarial attacks generated using the classifier trained only on perturbed latent vectors (90.95\\% \u00b1 1.80\\% vs. 65.43\\% \u00b1 5.07\\%). We conjecture that this is due to the fact that classifiers trained only on perturbed latent representations are less robust and therefore adversarial examples generated using these classifiers are weaker.\n\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{lcc}\n    \\toprule\n            Classifier & Standard Accuracy & Robustness \\\\\n    \\midrule\n        Full - $\\alpha$=0.5 &            \\textbf{92.19\\%} &     94.62\\% \\\\\n          Full - $\\alpha$=1 &            92.05\\% &     94.51\\% \\\\\n     Latent - $\\alpha$=0.05 &            91.90\\% &     90.44\\% \\\\\n      Latent - $\\alpha$=0.1 &            89.33\\% &     93.64\\% \\\\\n          Full - $\\alpha$=2 &            88.02\\% &     \\textbf{98.29\\%} \\\\\n      Latent - $\\alpha$=0.2 &            82.21\\% &     97.45\\% \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Standard Accuracy vs. Robustness}\n    \\label{tab:classifiers_adv_robustness_across_alpha}\n\\end{table}\n\nAdditionally, we train multiple robust classifiers using both methods with varying step size $\\alpha$ to compare their robustness with regards to the drop in standard accuracy (Table \\ref{tab:classifiers_adv_robustness_across_alpha}). In this experiment, we use adversarial examples generated using the original (non-robust) classifier for adversarial attacks. Robust classifiers trained on generated adversarial strings generally outperform robust classifiers trained only on perturbed latent vectors at comparable levels of standard classification accuracy. We omit uncertainty estimates due to computational resources needed to train a large number of adversarial classifiers.\n\\section{Introduction}\nRobustness of classifiers against adversarial attacks~\\cite{goodfellow2015fgsm, papernot2016transferability, madry2019deep} is particularly relevant in security sensitive domains. We consider the problem of determining whether an executable application is benign or malicious based on the set of files the application accessed\/created during runtime \\cite{STIBOREK2018346}. Malware authors avoid detection by generating random filenames and\/or perturbing existing ones. These changes represent real-world adversarial examples against the detection classifier. The success of attacks using these adversarial examples depends primarily on how much knowledge the adversary has about the target classifier and on the robustness of the classifier against adversarial examples. We focus on the latter part and aim to develop methods for training string classifiers that are robust against adversarial examples.\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{desired_adv_perturbation_v2}\n    \\caption{Desired Adversarial Perturbation}\n    \\label{fig:desired_adv_perturbation}\n\\end{figure}\n\nWe consider the following problem (see Figure~\\ref{fig:desired_adv_perturbation}): Assume a classifier that classifies a set of strings representing accessed files as benign or malicious. What is the minimal sequence of perturbations that results in a given set of strings being misclassified? Unlike the image domain, any change in the set of strings will be perceptible by humans. Therefore, we are interested in perturbations that produce adversarial strings which are close to the original strings. In theory, we are guaranteed to find a minimal sequence of adversarial perturbations using combinatorial search algorithms \\cite{Zang_2020, pmlr-v97-moon19a}. However, these methods are not feasible for our problem as the cardinality of the search space is $\\approx 256^{nl}$, where $n$ is the number of strings in the set and $l$ is the average length of strings. Instead, we focus on adversarial perturbations in the continuous latent representation space.\nAs we have full access to the classifier, we can leverage its gradients to compute an adversarial perturbation in the latent representation space and then directly generate adversarial strings from the perturbed latent representation.\n\nTo make the generation of actual adversarial strings possible, we use an encoder-decoder style neural network \\cite{graves2014generating} to learn latent representations of strings in an unsupervised fashion. We develop a convolutional-recurrent architecture that achieves high reconstruction accuracy on highly irregular\/random strings\\footnote{Note that strings corresponding to accessed files contain various special characters and\/or numbers and thus differ significantly from standard words from natural language.}. This is important to us because we want generated changes to be caused by latent perturbations rather than high bias of the decoder. Using a pre-trained representation model, we then train a multiple instance classifier to classify sets of strings representing accessed files as benign or malicious.\n\nNext, we use existing gradient-based methods to compute adversarial perturbations of latent string representations. We then generate adversarial strings using the decoder part of our network, where we use the perturbed latent representations as the decoder's initial state. This allows us to generate adversarial examples inside the training loop of the classifier and use them as additional input examples.\n\nFinally, we explore the benefits of our method compared to adversarial training on perturbed latent vectors only (i.e., without generating actual adversarial sets of strings). We show that adversarial training on perturbed strings allows us to train classifiers that are significantly more robust without a large trade-off in standard accuracy.\nOur contributions are as follows:\n\\begin{itemize}\n    \\item {We develop an encoder-decoder architecture that achieves high reconstruction accuracy on highly irregular\/random strings.}\n    \\item {We generate realistic looking adversarial strings using gradient-based attack methods in the latent representation space.}\n    \\item {We compare the performance of classifiers trained on adversarial strings vs. classifiers trained only on perturbed latent vectors. We show that string adversarial training produces classifiers which are significantly more robust against adversarial examples without requiring a large trade-off in standard accuracy.}\n\\end{itemize}\n\n\\section{Methods}\n\\subsection{Vector Representations of Character Sequences}\n\\label{sec:encoder_decoder}\nWe aim to represent variable-length character sequences as fixed-size vectors in $\\mathbb{R}^d$. We learn the latent representation of sequences from data in an unsupervised fashion. Firstly, we embed individual ASCII characters as vectors in $e_i \\in \\mathbb{R}^l;\\ i=1...n$, where $l$ is the embedding size and $n$ is the sequence length. Secondly, we use an encoder function $F:\\mathbb{R}^{n \\times l} \\to \\mathbb{R}^d$ to represent a sequence of character embeddings as a vector in $\\mathbb{R}^d$, where $n$ is the sequence length, $l$ is the embedding size, and $d$ is the latent representation size. Lastly, we use a decoder function $G:\\mathbb{R}^d \\to \\mathbb{R}^{n \\times 256}$ to reconstruct the original input sequence from its respective latent representation $h_T \\in \\mathbb{R}^d$. We normalize each of the decoder outputs $\\hat{y}_t \\in \\mathbb{R}^{256};\\ t=1...n$ using the $softmax$ function to obtain a valid probability distribution over ASCII characters at each output step $t$. Unlike the encoder, the decoder does not have access to the input sequence, but instead uses its outputs at step $t-1$ as inputs at step $t$. Hence, it autoregressively decodes the output sequence using only information encoded in the fixed-size vector representation $h_T$.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{conv_rnn_seq2seq}\n\\caption{Convolutional Recurrent Sequence-to-Sequence Autoencoder}\n\\label{conv_rnn_seq2seq}\n\\end{figure}\n\nWe use a combination of convolutional and recurrent neural networks for the encoder $F$ and the decoder $G$ (Figure \\ref{conv_rnn_seq2seq}). The encoder first applies a 1D convolution with $kernelWidth=5$ and $stride=5$ to represent 5-grams as vectors and to reduce the length of the input to the recurrent layer. A recurrent neural network is then used to summarize the sequence of 5-gram vector representations into a single, fixed-size, vector representation. Given this vector representation, we then use an autoregressive decoder $G$ to obtain a reduced-length sequence of vectors and apply a transposed 1D convolution to obtain the output sequence with the same length as the input sequence.\n\n\\subsection{Multiple Instance Classification}\n\\label{sec:methods_mil}\nOur approach to multiple instance classification follows the \\textit{embedding-space paradigm} \\cite{Chen2006MILEmbedding, Cheplygina_2015} and assumes that the labels are only available on the bag level. We use a transformation function $\\mathcal{A}: \\mathbb{R}^{k \\times m} \\to \\mathbb{R}^{hd}$, where $k$ is the number of instances in the bag, $m$ is the size of vectors that represent instances, $h$ is the number of attention heads, and $d$ is the hidden size. Input to $\\mathcal{A}$ is a matrix $E \\in \\mathbb{R}^{k \\times m}$, where rows of the matrix are latent representations of character sequences. Output of $\\mathcal{A}$ represents an aggregated fixed-size vector representation of the input bag.\n\n\\begin{equation}\n\\label{eqn:lba_keys}\nK := EW_K \\in \\mathbb{R}^{k \\times d}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:lba_values}\nV := EW_V \\in \\mathbb{R}^{k \\times d}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:lba_weights}\nW := W_QK^T \\in \\mathbb{R}^{h \\times k}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:lba_weights_norm}\nw_i = \\frac{exp(w_i)}{\\sum_{j=1}^{k}exp(w_j)}; i=1...k\n\\end{equation}\n\\begin{equation}\n\\label{eqn:lba_out}\no := vec(WV) \\in \\mathbb{R}^{hd}\n\\end{equation}\n\nOur choice of the aggregation function $\\mathcal{A}$ is similar to a transformer layer \\cite{vaswani2017attention, DBLP:journals\/corr\/abs-1802-04712}. It has three trainable matrices: key projection matrix $W_K \\in \\mathbb{R}^{m \\times d}$, value projection matrix $W_V \\in \\mathbb{R}^{m \\times d}$, and a query matrix $W_Q \\in \\mathbb{R}^{h \\times d}$, where $h$ is the number of attention heads \\cite{vaswani2017attention} and $d$ is the hidden size. Inputs $E$ are first projected to keys and values as described by Equation \\ref{eqn:lba_keys} and Equation \\ref{eqn:lba_values}. Multiplying keys with the query matrix $W_Q$ (Equation \\ref{eqn:lba_weights}) gives us a matrix of attention weights whose columns are then $\\textit{softmax}$ normalized to unit sum (Equation \\ref{eqn:lba_weights_norm}). Finally, multiplying the normalized weights matrix $W$ with the values matrix $V$ produces a matrix of convex combinations of value vectors for each head. This matrix is then flattened to a vector of fixed dimension $\\mathbb{R}^{hd}$ (Equation \\ref{eqn:lba_out}) which represents the aggregated output. The aggregated bag-level representation is used as input features to a classifier which is a simple feed-forward neural network with two output classes (malicious\/benign). Using this setup, we obtain a differentiable classifier function $f_{\\theta}:\\mathbb{R}^{k \\times m} \\to \\mathbb{R}^2$ that allows us to compute adversarial perturbations of instance vectors using gradient-based methods.\n\n\\subsection{Adversarial Sample Generation}\n\\label{sec:adv_sample_generation}\nWe now describe our approach for generating adversarial examples using gradient-based perturbations of latent representations. Given a trained encoder $F$, decoder $G$ and a multiple instance classifier $C$, we generate adversarial samples in the following way. First, using the encoder $F$ we encode all instances $S = \\{s_1, ..., s_n\\}$ in the bag to their respective latent representations as $Z = \\{z_i = F(s_i);\\ \\forall z_i \\in \\mathbb{R}^d,\\ i\\in\\{1,...,n\\}\\};\\ Z \\in \\mathbb{R}^{n \\times d}$, where $n$ is the number of instances and $d$ is the latent representation size. Then, we use the multiple instance classifier $C$ to obtain a perturbation of these latent representations as $\\delta^{\\ast} = \\argmax_{\\delta \\in \\Delta}\\ell(C(Z + \\delta),y);\\ \\delta^{\\ast} \\in \\mathbb{R}^{n \\times d}$. Finally, we generate a set of adversarial instances in the original (string) domain as $S_{adv} = \\{s^{adv}_i = G(Z_i + \\delta^{\\ast}_i);\\ i \\in \\{1,...,n\\}\\}$, where $Z_i \\in \\mathbb{R}^d$ and $\\delta^{\\ast}_i \\in \\mathbb{R}^d$.\n\nWe modify existing gradient-based attack methods, namely Projected Gradient Descent (PGD) (Algorithm \\ref{alg:pgd}) and Fast Gradient Sign Method (FGSM) (Algorithm \\ref{alg:fgsm}), to approximately find an optimal perturbation $\\delta^{\\ast} \\in \\Delta$, where $\\Delta$ is a set of allowable perturbations. The modifications are due to the fact that the encoder $F$ and the decoder $G$ are not a perfect inverse of each other, i.e. it does not necessarily hold that $F(G(z)) = z$ for some latent representation $z \\in \\mathbb{R}^d$. Because of this, we may find a perturbation $\\delta^{\\ast}$ that is adversarial in the latent representation space, i.e. $C(z + \\delta^{\\ast}) \\neq y$, but the bag generated from the perturbed state $z + \\delta^{\\ast}$ is not adversarial, i.e. $C(F(G(z + \\delta^{\\ast}))) = y$. Therefore, our modified algorithm generates an adversarial bag after each gradient step, encodes this generated bag to its respective latent representation, classifies the latent representation, and checks if the label is different from the true label. Only misclassified bags generated from perturbed latent states are considered to be successful adversarial examples.\n\\BlankLine\n\\noindent\n\\begin{minipage}{0.48\\textwidth}\n    \\begin{algorithm}[H]\n    \\footnotesize\n    \\label{alg:pgd}\n    \\SetAlgoLined\n    \\KwIn{$paths,y,\\alpha,\\mathcal{P},T$}\n    \\KwOut{$advPaths$}\n    $state \\leftarrow encode(paths)$\\;\n    $\\delta_0 \\leftarrow zerosLike(state)$\\;\n    $\\hat{y} \\leftarrow classify(state)$\\;\n    \\If{$\\hat{y} \\neq y$} {\n        \\Return $paths$\\;\n    }\n    \\For{i = 1...T} {\n        $\\nabla_{\\delta} \\leftarrow \\nabla_{\\delta}\\ell(classify(state + \\delta_{i-1}), y)$\\;\n        $\\delta_i \\leftarrow \\mathcal{P}(\\delta_{i-1} + \\alpha\\frac{\\nabla_{\\delta}}{||\\nabla_{\\delta}||_2 + \\gamma})$\\;\n        $advPaths \\leftarrow decode(state + \\delta_i)$\\;\n        $advState \\leftarrow encode(advPaths)$\\;\n        $\\hat{y} \\leftarrow classify(advState)$\\;\n        \\If{$\\hat{y} \\neq y$} {\n            \\Return $advPaths$\\;\n        }\n    }\n    \\caption{Modified PGD}\n    \\end{algorithm}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.48\\textwidth}\n    \\begin{algorithm}[H]\n    \\footnotesize\n    \\label{alg:fgsm}\n    \\SetAlgoLined\n    \\KwIn{$paths,y,\\epsilon,\\epsilon_{max},\\delta_{\\epsilon}$}\n    \\KwOut{$advPaths$}\n    $state \\leftarrow encode(paths)$\\;\n    $\\hat{y} \\leftarrow classify(state)$\\;\n    \\If{$\\hat{y} \\neq y$} {\n        \\Return $paths$\\;\n    }\n    $\\nabla_{state} \\leftarrow \\nabla_{state}\\ell(classify(state), y)$\\;\n    \\While{$\\epsilon \\leq \\epsilon_{max}$} {\n        $advPaths \\leftarrow decode(state + \\epsilon sgn(\\nabla_{state}))$\\;\n        $advState \\leftarrow encode(advPaths)$\\;\n        $\\hat{y} \\leftarrow classify(advState)$\\;\n        \\If{$\\hat{y} \\neq y$} {\n            \\Return $advPaths$\\;\n        }\n        $\\epsilon \\leftarrow \\epsilon + \\delta_{\\epsilon}$\\;\n    }\n    \\caption{Modified FGSM}\n    \\end{algorithm}\n\\end{minipage}\n\n\\subsection{Robust Classifier Training}\nTo train a classifier that is robust to adversarial examples, we augment the training dataset with adversarial examples generated using methods described in the previous section (Algorithm \\ref{alg:adversarial_training}). Given a classifier function $f_{\\theta}$ with parameters $\\theta$, we modify the training procedure from a simple minimization of the classification loss $\\ell(f_{\\theta}(x), y)$ to minimax optimization (Equation \\ref{eqn:adv_minimax_objective}). The outer minimization tries to minimize the classification loss on the training set $\\mathcal{T}$, while the inner maximization tries to maximize it by adversarially perturbing the input.\n\\begin{equation}\n\\label{eqn:adv_minimax_objective}\n\\theta^{\\ast} = \\argmin_{\\theta}\\frac{1}{|\\mathcal{T}|}\\sum_{i=1}^{|\\mathcal{T}|} \\max_{\\delta \\in \\Delta}\\ell(f_{\\theta}(x_i + \\delta), y_i)\n\\end{equation}\nThe update rule for classifier's parameters $\\theta$ is then defined as Equation \\ref{eqn:minimax_update}.\n\\begin{equation}\n\\label{eqn:minimax_update}\n\\theta := \\theta - \\alpha\\nabla_{\\theta}\\max_{\\delta \\in \\Delta}\\ell(f_{\\theta}(x + \\delta), y)\n\\end{equation}\n\nTo compute gradient of the inner maximization we make use of the Danskin's Theorem, which states that the gradient of the $max$ operator is equal to the gradient of the inner function evaluated at the maximum point (Equation \\ref{eqn:adv_danskin}).\n\\begin{equation}\n\\label{eqn:adv_danskin}\n\\nabla_{\\theta}\\max_{\\delta \\in \\Delta}\\ell(f_{\\theta}(x + \\delta), y) = \\nabla_{\\theta}\\ell(f_{\\theta}(x + \\delta^{\\ast}), y)\n\\end{equation}\nHowever, this result only applies for the case where we can compute the maximum exactly and that maximum is unique, neither of which can be guaranteed for our case as we are only able to solve the maximization approximately. Nevertheless, the stronger the adversarial attack is, the better approximation of the true maximum we get, and thus we also get a better approximation of the gradient.\n\\BlankLine\n\\BlankLine\n\\begin{algorithm}[H]\n\\footnotesize\n\\label{alg:adversarial_training}\n\\SetAlgoLined\n\\For{$\\mathcal{B} \\subseteq \\mathcal{T}$} {\n    $\\nabla_{\\theta} \\leftarrow \\textbf{0}$\\;\n    \\For{$(x,y) \\in \\mathcal{B}$} {\n        $\\nabla_{\\theta} \\leftarrow \\nabla_{\\theta} + \\nabla_{\\theta}\\ell(f_{\\theta}(x), y)$\\;\n        \\If{$f_{\\theta}(x) = y$}{\n            $\\delta^{\\ast} \\leftarrow \\argmax_{\\delta \\in \\Delta}\\ell(f_{\\theta}(x + \\delta), y)$\\;\n            $\\nabla_{\\theta} \\leftarrow \\nabla_{\\theta} + \\nabla_{\\theta}\\ell(f_{\\theta}(x + \\delta^{\\ast}), y)$\\;\n        }\n    }\n    $\\theta \\leftarrow \\theta - \\frac{\\alpha}{|\\mathcal{B}|}\\nabla_{\\theta}$\\;\n}\n\\caption{Adversarial Training}\n\\end{algorithm}\n\\section{Related Work}\nAdversarial perturbations for neural networks were first introduced by Szegedy et al. \\cite{szegedy2014adv} and later explored by Goodfellow et al. \\cite{goodfellow2015fgsm} in the domain of computer vision. They describe perturbations of input images that are imperceptible by humans but cause the image classifier to assign an incorrect label for the perturbed input. More importantly, different models trained on different subsets of data can misclassify the same adversarial example \\cite{szegedy2014adv}. This suggests that it is possible to generate adversarial examples using a surrogate model to which the attacker has full access, and then use these adversarial examples to attack an unknown model of interest.\n\nAdversarial attacks were also explored, among others, in the field of natural language processing which is relevant to ours because it operates on string inputs. Main approaches for adversarial attacks in the natural language processing domain can be categorized as sentence-level attacks \\cite{zhao2018generating, wang2018robust}, word-level attacks \\cite{samanta2017crafting,sato2018interpretable,zhang2020generating}, and character-level attacks \\cite{ebrahimi-etal-2018-adversarial,gaoetal2018}. Sentence-level and word-level attacks seem to exploit high-level semantic information and often make use of synonyms or sentences with the same meaning as adversarial perturbations. We instead focus on character-level attacks which allows us to generate adversarial perturbations inside of individual file\/directory names.\n\nIn this work, we consider input examples that are represented as sets of strings. Formally, this is known as Multiple Instance Learning (MIL) \\cite{CARBONNEAU2018329}. Prior work describes three main paradigms for solving multiple instance learning problems, namely, \\textit{instance-space paradigm}, \\textit{bag-space paradigm} and \\textit{embedding-space paradigm}. We follow the \\textit{embedding-space paradigm} \\cite{Chen2006MILEmbedding, Cheplygina_2015} as we first embed individual strings into vectors, and then aggregate instance vector representations into a fixed-size bag vector representation. Our bag aggregation function is similar to methods proposed in \\cite{DBLP:journals\/corr\/abs-1802-04712, DBLP:journals\/corr\/abs-1810-00825}.\n\nExisting methods for malware classification predict either a binary class (benign\/malicious) \\cite{Pascanu_2015_7178304, STIBOREK2018346} or a malware family \\cite{DBLP:journals\/corr\/abs-1811-07842}. Furthermore, they can use hand engineered features, end-to-end learned feature extractors \\cite{Pascanu_2015_7178304, Kalash_2018_8328749}, or a combination or both \\cite{GIBERT2020101873}. Prior work also explores adversarial examples for malware classifiers. Grosse et al. \\cite{grosse2016adversarial} explore gradient-based attack on classifiers that use binary feature vectors. Hu et al. \\cite{hu2017generating} use Generative Adversarial Networks (GANs) to generate adversarial attacks without access to the target classifier. Kolosnjaji et al. \\cite{8553214} propose a gradient-based method to adversarial attacks which changes only a few specific bytes at the end of malware sample.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}