diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjqxx" "b/data_all_eng_slimpj/shuffled/split2/finalzzjqxx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjqxx" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe ATLAS experiment \\cite{ATLAS} at CERN was equipped with the Insertable B-Layer (IBL) \\cite{IBL} between the existing inner pixel layer and a new beam pipe during the long shutdown one to improve the tracking performance. The planar and 3D sensors of the IBL are exposed to a high flux of ionizing radiation because of the close position to the interaction point. The planar sensors are designed to withstand a fluence of $5\\cdot 10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^{2}$. They are produced in n$^+$-in-n sensor technology and have a pixel pitch of $250\\,$\\textmu m $\\times$ $50\\,$\\textmu m \\cite{sensor1, sensor2}. The 80 columns and 336 rows of pixels are read out with the Front-End-I4 readout chip \\cite{FE-I4}.\n\nWhile operating the detector, the radiation damage of the sensors will increase their leakage current and depletion voltage, whereas the charge collection efficiency decreases. In order to continue to obtain a high particle detection efficiency, the bias voltage needs to be increased gradually. Voltages up to $1000\\,$V can be applied to the sensors causing an increase in power consumption. The higher power consumption produces heat which needs to be dissipated by the detectors cooling system to prevent thermal runaway of the sensors' leakage current. Therefore, a higher detection efficiency at lower bias voltage is desirable.\n\nNew pixel implantation shapes were designed in Dortmund to achieve electrical field strength maxima in the pixel and thus increase charge collection and particle detection efficiency at lower voltages after irradiation \\cite{REINER}.\n\nIn the proceeding \\emph{Lab and test beam results of irradiated silicon sensors with modified {ATLAS} pixel implantations} \\cite{meinPaper}, results of these proton and neutron irradiated modules were presented and showed incongruent results. This paper now examines the hypothesis that an annealing process caused the observed differences. \n\n\n\\section{Design of the Pixel Cell}\nThe baseline for the new designs is the IBL pixel design (see figure \\ref{fig:design}, label V0). In the \\mbox{$250\\,$\\textmu m $\\times$ $50$ \\textmu m} pixel cell, the standard n$^+$-implantation is located centrally with rounded corners to create a homogeneous electrical field in the pixel. Moderated p-spray is applied to isolate neighboring pixel cells. In the upper region of the pixel cell the bump bond pad is visible which is the connection to the readout chip. The bias dot with connection to the bias grid is positioned at the other end.\n\nDifferent n$^+$-implantation shapes are realized in the REINER$^2$\\note[2]{\\textbf{RE}designed, \\textbf{IN}novativ, \\textbf{E}xciting and \\textbf{R}ecognizable} pixel designs V1 to V6 (see figure \\ref{fig:design}): For the pixel designs V1 and V4, the n$^+$-implantation is divided in four uniform segments isolated with p-spray. The corners of the n$^+$- implantation of pixel design V1 are rounded, while they are rectangular for pixel design V4. Further division of the n$^+$-implantation are used for pixel designs V2 and V3. No isolation between the segments is used here due to reduced space. A narrowed shape of the metal layer and the n$^+$-implantation are realized in the pixel designs V5 and V6. While in V6 the region for moderated p-spray is the same as for the standard pixel, the section is increased to the inner part of the pixel cell for design V5.\n\nThe connection to the bias dot and bias grid is the same for all pixel designs and causes a loss in efficiency as mentioned in \\cite{effVerlust1, effVerlust2}. This will not be taken into account in this proceeding as it focuses on the influence of the different pixel shapes. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.6\\textwidth]{PixelLayoutQuer.png}\n\\caption{\\label{fig:design} Schematic drawing of the pixel structure and the shape of its components. The $\\text{n}^+$-implantation is shown in blue, the metalization in grey. The areas with high p-spray dose correspond to the nitride-openings as indicated in green. The bump bond pad is depicted in orange \\cite{meinPaper}.}\n\\end{figure}\n\n\\section{Sensors and Modules}\nThe REINER pixel sensor permits measurement of all the described pixel types on one sensor. It is produced in n$^+$-in-n sensor technology with a bulk thickness of $200\\,$\\textmu m. The pixel matrix of 80 columns and 336 rows is read out with the FE-I4 readout chip via bump bonds.\n\nThe sensor is divided in eight structures consisting of ten columns and 336 rows of the same pixel design. The outermost structures on the sensor comprise the standard IBL pixel design (labeled as 05 and V0). In between these standard structures, structures with modified pixel designs (V1 to V6) are placed (see figure \\ref{fig:sensor}, left). Each structure has its own p$^+$-implant and two HV pads. All structures are surrounded by thirteen guard rings beneath the second to last pixel column and the last pixel column (see figure \\ref{fig:sensor}, right). In this way, measurements of the structures independent from each other are possible.\n\nTo compare the performance of the different pixel designs, several sensors are bump bonded to FE-I4 readout chips and tested before and after irradiation in irradiation facilities. The results presented in this work are obtained with modules irradiated with neutrons at the TRIGA reactor in Ljubljana \\cite{Ljubljana} and at the Sandia Annular Core Research Reactor$^4$\\note[4]{https:\/\/www.sandia.gov\/research\/facilities\/annular\\_core\\_research\\_reactor.html}. They are irradiated with neutrons to target fluences of $1\\cdot10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ and $5\\cdot10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{P-Seite2.png}\n\\caption{\\label{fig:sensor} Left: P-side of a REINER pixel sensor with six modified structures and two IBL standard structures (05 and V0). Right: Magnification of the area between two structures. The second to last and last column of each structure are beneath guard rings \\cite{meinPaper}.}\n\\end{figure}\n\n\\section{Test Beam Measurements}\nThe sensors are investigated in test beam measurements at the SPS-Beam Line H6 at CERN with a pion beam energy of $120\\,$GeV and at the beam line 22 at DESY \\cite{DESYPaper} with an electron beam energy of $5\\,\\text{GeV}$. With the EUDET-type beam telescopes ACONITE at CERN and DURANTA at DESY the hit detection efficiency can be measured with a spatial resolution of less than $5\\,$\\textmu m (CERN) and $10\\,$\\textmu m (DESY) \\cite{resolution}. A telescope consists of two arms, each equipped with three Mimosa 26 modules \\cite{mimosa26}. In between the two arms, a cooled box holds the Devices Under Test (DUTs). The box is flushed with nitrogen and cooled using a chiller (CERN) or dry ice (DESY). A non-irradiated IBL-type module is used as a timing reference sensor.\n\nWith this setup the REINER modules are investigated at different positions, bias voltages and tunings of the readout chip. For irradiated REINER modules high efficiencies around $97\\,\\%$ can be reached for all pixel designs with a high bias voltage. Therefore, operation with a lower bias voltage where differences in the performance of the pixel designs occur is desirable.\n\nThe traversing particles produce hits in the telescope modules and the DUTs which are reconstructed to tracks using the software EUTelescope$^5$\\note[5]{http:\/\/eutelescope.web.cern.ch\/}. To get unbiased tracks the hit information of the DUTs is not used for track fitting. About $200\\,\\text{k}$ to $500\\,\\text{k}$ events are collected in one \"run\".\n\nThe hit detection efficiency, as one part of the analysis of the performance of the DUTs, is computed with the software tool TBMon2$^6$\\note[6]{https:\/\/gitlab.cern.ch\/tbmon2}.\n\nAll reconstructed particles measured by the reference sensor and not in a masked, edge or noisy region of the investigated sensor are used for efficiency calculation and called \"tracks\". If a \"track\" is also measured by the investigated sensor, it is defined as a \"hit\". The hit detection efficiency of a sensor $\\epsilon$ is calculated by the quotient of \"hits\" and \"tracks\":\n\n\\begin{equation}\n\\epsilon = \\frac{n_\\text{hits}}{n_\\text{tracks}}, \\quad \\sigma_\\epsilon = \\sqrt{\\frac{\\epsilon \\cdot (1-\\epsilon)}{n_\\text{tracks}}}\n\\end{equation}\n\nTo summarize the efficiencies for runs taken under the same conditions the mean efficiency weighted with the number of tracks is calculated. The fluctuation of the efficiency from run to run is determined with the Clopper-Pearson confidence interval \\cite{clopper-pearson} with a confidence level of $\\gamma=95\\,\\%$. For the lower interval limit the $\\frac{1-\\gamma}{2}$-quantile and for the upper limit the $\\frac{1+\\gamma}{2}$-quantile are calculated \\cite{Andreas}.\n\nFor the following section the efficiency is calculated by using only the four innermost pixel columns of every structure to neglect the influence of guard rings.\n\n\\section{Previous Results}\nThe measurements presented in \\cite{meinPaper} showed that for non-irradiated modules the hit detection efficiency for the different pixel designs is consistent. But for modules irradiated with neutrons or protons the hit detection efficiencies are not consistent. Even the two neutron irradiated sensors R1, irradiated at Sandia, and R3, irradiated in Ljubljana, to the same target fluence of $5\\cdot 10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ and measured at the same voltage and tuning ($3200\\,$e threshold and a ToT response of $6\\,$ at a reference charge of $20\\,$ke) show dissimilar results (see figure \\ref{fig:bild1}, left). For the module R1, irradiated at Sandia, the pixel designs with narrowed n$^+$-implantation V5 and V6 reach the highest efficiencies at $400\\,$V. No efficiency of the pixel design V0 was measured for module R1 as the beam spot was focused on the left side of the module and did not cover pixel design V0.\n\nFor the Ljubljana irradiated module R3, the pixel design with narrowed n$^+$-implantation V6 has the lowest efficiency while all other pixel designs have similar efficiencies of approx. $50\\,\\%$.\n\nThese diverging results might have been caused by the different neutron energy spectra of the two irradiation facilities while another hypothesis is a different temperature of the modules during irradiation leading to annealing of defects. In Ljubljana the maximum temperature of the modules was $45^\\circ$C while they reached about $100^\\circ$C in the irradiation at Sandia (see figure \\ref{fig:bild1}, right).\n\nTo test the annealing hypothesis, the modules R3 and R9 were irradiated with neutrons in Ljubljana to different target fluences and were now annealed in several steps at $80^\\circ$C.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.48\\textwidth]{R1_R3_comparison_3200_iWORID.png}\n\\includegraphics[width=.48\\textwidth]{TempertureProfileIrradiation_IWORID.png}\n\\caption{\\label{fig:bild1} Left: Efficiencies of the different pixel designs for two sensors irradiated with neutrons to a target fluence of $5\\cdot 10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ at $400\\,$V (Tuning: $3200\\,$e threshold, $6\\,$ToT at $20\\,$ke). R1 is irradiated at Sandia while R3 is irradiated in Ljubljana. Right: Temperatur profile during the neutron irradiation of the module R1 at Sandia.}\n\\end{figure}\n\n\\section{First Annealing Results}\nFor the annealing procedure the climate chamber is preheated to $80^\\circ$C before the module is inserted. The temperature of the module is monitored and it is not biased during the annealing. The cooling down afterwards takes place at room temperature and the module is put back in the freezer for storage. After each annealing step IV scans and test beam measurements are performed at DESY.\n \nTo determine if differences in the efficiency occur with annealing at all, the intention was to anneal a module for a long time. Therefore, the first annealing step of module R3 was three hours at $80^\\circ$C. The following annealing steps lasted for two hours each.\nThe results of the hit detection efficiency for the different pixel designs for module R3 after several annealing steps are presented in figure \\ref{fig:bild2}. For these measurements at $300\\,$V the module was tuned to a threshold of $1600\\,$e and a ToT response of $6\\,$ at a reference charge of $20\\,$ke. The standard pixel designs 05 and V0 reach the highest efficiencies for the non-annealed case while the pixel designs with narrowed n$^+$-implantation V5 and V6 are less efficient compared to all other designs. This behavior appears to be independent on the threshold as figure \\ref{fig:bild1} on the left shows where the sensor was measured at a threshold of $3200\\,$e while in figure \\ref{fig:bild2} the module was tuned to a threshold of $1600\\,$e.\n\nWith the first annealing step of three hours the efficiency drops for all pixel designs except for the designs with a narrowed n$^+$-implantation. Compared to the standard pixel design, the hit detection efficiency drops by $8\\,\\%$ while for pixel design V5 the efficiency increases by more than $12\\,\\%$. After this first annealing step, the efficiency of pixel design V5 is even higher than the non-annealed results of the standard designs. With further annealing steps the efficiency of the standard pixel designs is almost constant, but for pixel designs V5 and V6 where the efficiency increases.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.7\\textwidth]{Efficiency_R3_vergleich_0h_3h_5h_7h_80C_1600e_300V_IWORID.png}\n\\caption{\\label{fig:bild2}\nEfficiencies of the different pixel designs for the sensor R3 after various annealing steps at $300\\,$V (Tuning: $1600\\,$e threshold, $6\\,$ToT at $20\\,$ke).}\n\\end{figure}\n\n\\newpage\n\nThese results confirm the hypothesis that annealing caused the narrowed pixel designs to be more efficient compared to the standard pixel design. To determine if this effect is dependent on the particle fluence and to see when the effect becomes relevant, the module R9 was irradiated with neutrons in Ljubljana to a target fluence of $1\\cdot10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ and measured with a shorter annealing step time of five minutes at $80^\\circ$C. The results of the hit detection efficiency for the module R9 at $100\\,$V, tuned to a theshold of $1600\\,$e and a ToT response of $6\\,$ at a reference charge of $20\\,$ke, are presented in \\mbox{figure \\ref{fig:bild3}}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.7\\textwidth]{Efficiency_R9_vergleich_0min_5min_10min_80C_1600e_100V_IWORID.png}\n\\caption{\\label{fig:bild3} Efficiencies of the different pixel designs for the sensor R9 after various annealing steps at $100\\,$V (Tuning: $1600\\,$e threshold, $6\\,$ToT at $20\\,$ke).}\n\\end{figure}\n\nFor the non-annealed measurement the pixel designs with narrowed n$^+$-implantation have the lowest efficiency. Because the beam spot is focused on the left part of the sensor, there is no data point for the pixel design V0. After the first annealing step an increase in efficiency is visible for the pixel designs with a narrowed n$^+$-implantation and the standard pixel design. However, after the second annealing step the efficiency decreases for all designs. Compared to the non-annealed results, the efficiency after the second annealing step is still higher for the standard pixel design and the designs with narrowed n$^+$-implantation.\n\nIt is known, that annealing performed in a short time scale is able to reduce the effective doping concentration in the sensor material and more signal at the same voltage can be measured. This process is called beneficial annealing. In contrast, reverse annealing degrades sensor properties in the long term. For more information on annealing effects see \\cite{Moll}. Beneficial annealing can explain the higher efficiencies after the first annealing step of the module R9 for the shorter annealing time. But with the second step the reverse annealing becomes dominant and the efficiency drops for all designs. In case of the module R3 only the reverse annealing can be observed because of the long term annealing of three hours at $80^\\circ$C.\n\nThe observed changes of the hit detection efficiency with annealing are caused by at least two different effects. With annealing the depletion voltage and the charge collection efficiency change. Thus, if one of these effects depend on the pixel design, different hit detection efficiencies might be observed. Also charge multiplication could explain the higher efficiencies, which was also observed in \\cite{Milovanoivi,Kramberger1}, where neutron irradiated n-in-p micro-strip sensors showed higher electrical fields and therefore charge multiplication close to the strips.\n\n\\section{Summary and Outlook}\nThe results of the Ljubljana neutron irradiated module R3 indicate that long term annealing caused a higher hit detection efficiency for the pixel designs with narrowed n$^+$-implantation compared to all other designs. As a result of this, the higher detection efficiency observed for pixel designs with narrowed n$^+$-implantation on the neutron irradiated sensor R1 at Sandia (see \\cite{meinPaper}) are explained as being due to the higher temperature during the irradiation.\n\nAs this effect was not visible for the module R9, which was irradiated to a lower fluence and annealed in shorter steps, it is not clear if it depends on the neutron fluence. Therefore, the module R9 needs to be further investigated with additional annealing steps.\n\nFor the module R3 further annealing steps will be performed to find the maximum efficiency of pixel design V5 and further research of the performance after reaching the maximum efficiency.\n\nThe higher hit detection efficiencies might be explained with charge multiplication which is observed in n-in-p micro-strip detectors after neutron irradiation and long term annealing \\cite{Milovanoivi,Kramberger1}. To understand if the higher efficiency is caused by charge amplification, which was the intention of the REINER pixel designs, TCT measurements at different voltages and annealing steps are planned for irradiated modules. With this method the pixel designs can be investigated in the range of \\textmu m to see which pixel parts might cause charge multiplication.\n\n\\acknowledgments\nThe authors would like to thank the team at the Sandia Annular Core Research Reactor, especially M. Hoeferkamp and S. Seidel, and the team at the TRIGA reactor in Ljubljana, especially V. Cindro, for their help with irradiation of the sensors.\n\nMany thanks to all participants of the ATLAS ITk pixel test beam campaigns, especially those who develop and maintain the corresponding hardware and software.\n\nThe presented work is carried out within the framework of Forschungsschwerpunkt FSP 103 and supported by the Bundesministerium f\\\"ur Bildung und Forschung BMBF under grant 05H15PECA9.\n\nThis project has received funding from the European Union's Horizon 2020 Research and Innovation programme under Grant Agreement no. 654168.\n\nThe measurements leading to these results have been performed at the Test Beam Facility at DESY Hamburg (Germany), a member of the Helmholtz Association (HGF).\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\hspace{-0.6cm}.\\hspace{0.3cm}#1}} %\n\\newcommand{\\zd}{z^{\\delta}} %\n\\newcommand{\\be}{B_{\\Omega}(z;X)} %\n\\newcommand{\\ca}{C_{\\Omega}(z;X)} %\n\\newcommand{\\ko}{K_{\\Omega}(z;X)} %\n\\newcommand{\\tzd}{\\tilde z^{\\delta}} %\n\\newcommand{\\tjd}{\\tilde \\zeta^{\\delta}} %\n\\newcommand{\\jd}{\\zeta^{\\delta}} %\n\\newcommand{\\ld}{\\lambda_{\\delta}} %\n\\newcommand{\\tj}{\\tilde \\zeta_1} %\n\\newcommand{\\tz}{\\tilde z_1} %\n\\newcommand{\\Cb}{C_b(z_0,z^{\\delta_0})} %\n\\newcommand{\\jdb}{(d\\delta^{\\frac 1\\eta},0,- b\\delta\/2)} %\n\\newcommand{\\dbar}{\\overline{\\partial}} %\n\\newcommand{\\oo}{\\overline \\Omega} %\n\\newcommand{\\ct}{\\Bbb{C}^3} %\n\\newcommand{\\al}{{\\alpha}} %\n\\newcommand{\\ov}{\\overline} %\n\\newcommand{\\dd}{\\partial \\overline{\\partial}} %\n\\newcommand{\\de}{\\delta^{\\frac 1\\eta}} %\n\\newcommand{\\dee}{\\delta^{-\\frac 2\\eta}} %\n\\newcommand{\\bo}{b\\Omega} %\n\\newcommand{\\od}{\\Omega_\\delta} %\n\\newcommand{\\gd}{g_{\\tz,\\delta}} %\n\\newcommand{\\gdd}{g_\\delta} %\n\\newcommand{\\qd}{Q_{\\gamma\\delta}(\\tz)} %\n\\newcommand{\\qad}{Q_{a\\delta}(\\tz)} %\n\\newcommand{\\qdd}{Q_{\\gamma\\delta}(\\tzd)} %\n\\newcommand{\\rd}{R_{\\gamma\\delta}(\\tz)} %\n\\newcommand{\\rbd}{R_{b\\delta}(\\tzd)} %\n\\newcommand{\\qbd}{Q_{b\\delta}(\\tzd)} %\n\\newcommand{\\rdd}{R_{\\gamma\\delta}(\\tjd)} %\n\\newcommand{\\rcd}{R_{c\\delta}(\\jd)} %\n\\newcommand{\\ol}{\\overline L} %\n\\newcommand{\\Pd}{\\Phi_{\\tzd}} %\n\\newcommand{\\td}{\\tau(\\tzd,\\delta)} %\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{defn}[thm]{Definition}\n\n\\numberwithin{equation}{section}\n\n\n\\begin{document}\n\n\\title {Necessary conditions for H\\\"older regularity gain of $\\dbar$ equation in $\\mathbb C^3$}\n\n\\author{Young Hwan You \\thanks{Department of Mathematics, Indiana University East, IN 47374, USA. E-mail:youy@iue.edu\n\\newline 2010 \\textit{Mathematics Subject Classification} Primary 32F45; Secondary 32T25.}}\n\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\n\\noindent Suppose that a smooth holomorphic curve $V$ has order of contact $\\eta$ at a point $w_0$ in the boundary of a pseudoconvex domain $\\Omega$ in $\\mathbb{C}^3.$\nWe show that the maximal gain in H\\\"older regularity for solutions of the $\\bar{\\partial}$-equation is at most $\\frac{1}{\\eta}.$ \n\\end{abstract}\n\n\\section{Introduction}\\label{sec1}\n\n\nLet $\\Omega$ be a given domain in $\\mathbb{C}^n$ and $\\alpha$ be a $\\bar{\\partial}$-closed form of type $(0,1)$ in $\\Omega$. The $\\bar{\\partial}$-problem consists of finding a solution $u$ of $\\bar{\\partial} u = \\alpha$ that satisfies certain boundary regularity estimates as measured by either $L^2$ or $L^p$ norms or in H\\\"older norms.\n\n\nWhen $\\Omega$ is strongly pseudoconvex, in the $L^2$-sense, Kohn \\cite{FK,K1,K2} showed that for any $s\\geq 0$, there is a canonical solution of $\\bar{\\partial}u = \\alpha$ such that \n\\begin{equation}\\label{kohnestimate}\n|||u|||_{s+\\epsilon} \\leq C \\left\\|\\alpha\\right\\|_s \\quad \\mbox{and} \\quad \\ u \\perp A(\\Omega) \\cap L^2(\\Omega),\n\\end{equation}\nwith $\\epsilon = \\frac{1}{2}.$ \n(We say $u$ is the canonical solution if $u \\perp A(\\Omega) \\cap L^2(\\Omega).$)\nHere, $\\left\\|\\cdot\\right\\|^2_s$ is the $L^2$-Sobolev norm of order $s$ and the norm $|||\\cdot |||_{s+\\epsilon}$ measures tangential derivatives near the boundary of order $s+\\epsilon$ in the tangential directions. Kohn showed that if $U$ satisfies\n$\\square U = (\\bar{\\partial}\\bar{\\partial}^* + \\bar{\\partial}^*\\bar{\\partial})U = \\alpha,$ and if $\\bar{\\partial}\\alpha = 0,$ then\n$u = \\bar{\\partial}^*U$ is the canonical solution of $\\bar{\\partial}u = \\alpha.$\nTo prove regularity for this solution, Kohn proved the a priori estimate\n\\begin{equation}\\label{subellipticestimate}\n|||\\phi|||_{\\epsilon}^2 \\leq C(\\left\\|\\bar{\\partial} \\phi \\right\\|^2 + \\left\\|\\bar{\\partial}^* \\phi\\right\\|^2 + \\left\\|\\phi\\right\\|^2)\n\\end{equation} \nwith $\\epsilon = \\frac{1}{2}.$\nHere, $\\phi \\in C_{(0, 1)}^{\\infty}(W) \\cap \\mbox{Dom}(\\bar{\\partial}) \\cap \\mbox{Dom}(\\bar{\\partial}^*) $ is compactly supported in the neighborhood $W$ of the boundary point $w_0.$ Using this estimate and a bootstrap argument, Kohn proved (\\ref{kohnestimate}). Stein and Greiner \\cite{GS} later extended (\\ref{kohnestimate}) to similar estimates in $L^p$ and H\\\"older spaces. For example, if $\\left\\|\\cdot\\right\\|_{\\Lambda^s (\\Omega)}$ is the H\\\"older norm of degree $s$, then Stein and Greiner proved that $u$ satisfies \n\\begin{equation} \\label{holder_estimate}\n\\left\\|u\\right\\|_{\\Lambda^{s+\\epsilon}(\\Omega)} \\leq C \\left\\|\\alpha\\right\\|_{\\Lambda^s (\\Omega)}, \n\\end{equation}\nwith $\\epsilon = \\frac{1}{2}.$\n\n Kohn extended his $L^2$ results to when $\\Omega$ is a regular finite 1-type pseudoconvex domain in $\\mathbb{C}^2.$ To define a regular finite 1-type, we measure the order of contact of a given holomorphic curve at $w_0 \\in b\\Omega.$ Let $V$ be a one-dimensional {\\it smooth} variety parametrized by $\\zeta \\rightarrow \\gamma(\\zeta) = (\\gamma_1(\\zeta), \\cdots, \\gamma_n(\\zeta)),$ where $\\gamma(0) = w_0$ and $\\gamma'(0) \\neq 0.$ We define the order of contact of the curve by $\\nu_o(R\\circ\\gamma),$ where $R$ is a defining function of $\\Omega$ and $\\nu_o(g)$ is just the order of vanishing (an integer at least equal to $2$) of $g$ at $0.$ We then define the type, $T_{\\Omega}^{reg}(w_0) = \\sup\\{\\nu_o (R \\circ \\gamma) ; \\mbox{all} \\ \\gamma \\ \\mbox{with} \\ \\gamma(0) = w_0, \\gamma'(0) \\neq 0\\}.$ Further, we can define the regular type of $\\Omega$ by \n$T^{reg}(\\Omega) = \\sup \\{T_{\\Omega}^{reg}(w_0) ; w_0 \\in b\\Omega\\}.$\n Kohn \\cite{K} proved that if $\\Omega$ is a regular finite 1-type pseudoconvex domain in $\\mathbb{C}^2$, then (\\ref{kohnestimate}) holds for $\\epsilon = \\frac{1}{T^{reg}(\\Omega)}.$ Similarly, Nagel-Rosay-Stein-Wainger \\cite{NRSW} showed that (\\ref{holder_estimate}) also holds for the same $\\epsilon$.\n\n\nIn order to discuss similar estimates in $\\mathbb{C}^n,$ it is important to consider the order of contact of {\\it singular curves}. We define the order of contact of a holomorphic curve parametrized by $\\zeta \\rightarrow \\gamma(\\zeta),$ with $\\gamma(0) = w_0,$ by $C_{\\Omega}(\\gamma, w_0) = \\frac{\\nu_o (R \\circ \\gamma)}{\\nu_o(\\gamma)},$ where $\\nu_o(\\gamma)=\\min\\{\\nu_o (\\gamma_k); k =1, \\cdots, n\\}.$ Define the type of point $w_0$ by $T_{\\Omega}(w_0)= \\sup \\{C_{\\Omega}(\\gamma, w_0); \\mbox{all} \\ \\gamma \\ \\mbox{with} \\ \\gamma(0)= w_0 \\}$ and finally, the type of $\\Omega$ is $T_{\\Omega}= \\sup \\{T_{\\Omega}(w_0); w_0 \\in b\\Omega \\}.$ In the case of the $L^2$-norm, Catlin \\cite{C3} showed that if there is a curve $V$ parametrized by $\\gamma$ through $w_0 \\in b\\Omega$, where $\\Omega \\subset \\mathbb{C}^n$ and (\\ref{subellipticestimate}) holds, then $\\epsilon \\leq \\frac{1}{C_{\\Omega}(\\gamma, w_0)}.$ In H\\\"older norms, McNeal \\cite{Mc} proved that if, with an additional assumption, $\\Omega$ admits a holomorphic support function at $w_0 \\in b\\Omega$ and (\\ref{holder_estimate}) holds, then $\\epsilon \\leq \\frac{1}{C_{\\Omega}(\\gamma, w_0)}$. \n\n\n\n\nThere is the third notion of type, the ``Bloom-Graham\" type, $T_{BG}(w_0).$ It turns out that $T_{BG}(w_0)$ is the maximal order of contact of smooth $(n-1)$-dimensional complex submanifold. Thus, it follows that for any $w_0 \\in b\\Omega,$ $T_{BG}(w_0) \\leq T_{\\Omega}^{reg}(w_0) \\leq T_{\\Omega}(w_0).$ Krantz \\cite{Kr} showed that if $T_{BG}(w_0) = m,$ then $\\epsilon \\leq \\frac{1}{m}.$ \\\\\n\n\nIn this paper we present geometric conditions that must hold if H\\\"older estimate of order $\\epsilon$ is valid in a neighborhood of $w_0 \\in b\\Omega$ in $\\mathbb{C}^3.$ The main result is the following theorem: \n\n\\begin{thm} \\label{main_theorem}\nLet $\\Omega = \\{R(w) < 0\\}$ be a smoothly bounded pseudoconvex domain in $\\mathbb{C}^3.$ Suppose that there is a $1$-dimensional smooth analytic variety $V$ passing through $w_0$ such that for all $w \\in V$, $w$ sufficiently close to $w_0$, $$|R(w)| \\leq C|w-w_0|^\\eta,$$\nwhere $\\eta >0.$\n{\\it If there exists neighborhood $W$ of $w_0$ so that for all $\\alpha \\in L_{\\infty}^{0,1} ({\\Omega})$ with $\\bar{\\partial}\\alpha = 0$, there is a $u \\in \\Lambda^{\\epsilon} (W \\cap \\overline{\\Omega})$ and $C>0$ such that $\\bar{\\partial}u =\\alpha$ and $$ {\\lVert u \\rVert}_{{\\Lambda^\\epsilon}(W \\cap \\overline{\\Omega})} \\leq C{\\lVert \\alpha \\rVert}_{L_{\\infty}(\\Omega)},$$\nthen $\\epsilon \\leq \\frac{1}{\\eta}$}.\n\\end{thm}\n\n\n\\begin{cor} \n$\\epsilon \\leq \\frac{1}{T_{\\Omega}^{reg}(w_0)}.$\n\\end{cor}\n\n\n\\begin{rem} \\\n\\begin{enumerate}[\\normalfont i)]\n \\item If $T_{BG}(w_0) = +\\infty$, Krantz's result \\cite {Kr} holds for any $m > 0$ and we conclude \\\\ $\\epsilon \\leq \\frac{1}{m} \\leq \\frac{1}{\\eta}$ for large $m$. Thus we can assume $T_{BG}(w_0) = m < \\infty.$ Furthermore, since $\\epsilon \\leq \\frac{1}{m}$, we can assume $m < \\eta$ in the rest of this paper.\n \\item Theorem \\ref{main_theorem} improves the results by Krantz \\cite{Kr} and McNeal \\cite{Mc} in the sense that we obtain sharp result since $\\eta > m$ and do not assume the existence of a holomorphic support function. Note that the existence of holomorphic support function is satisfied for restricted domains (see the Kohn-Nirenberg Domain\\cite{KN}).\n \n\\end{enumerate}\n\n\\end{rem}\n\n\nTo prove Theroem \\ref{main_theorem}, the key components are the complete analysis of the local geometry near $w_0 \\in b\\Omega$ (Section \\ref{special coordinates}) and the construction of a bounded holomorphic function with large nontangential derivative near the boundary point (Section \\ref{Sec4}). In Section \\ref{special coordinates}, we construct special holomorphic coordinates about $w_0$ which are adapted to both Bloom-Graham type and the order of contact of $V$. Then, we use the truncation technique developed in \\cite{C} to deal with two dimensional slices of the domain. In Section \\ref{Sec4}, by using the holomorphic function constructed by Catlin \\cite{C2} on two dimensional slice, we construct a bounded holomorphic function $f$ with a large nontangential derivative defined locally up to the boundary in $\\mathbb{C}^3$. Finally, in Section \\ref{Sec5}, we prove Theorem \\ref{main_theorem} by using the constructed holomorphic function.\n\\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Special coordinates}\\label{special coordinates}\n\n\n Let $\\Omega$ be a smoothly bounded pseudoconvex domain in $\\mathbb C^3$ with a smooth defining function $R$ and let $w_0 \\in b\\Omega.$ Since $dR(w_0) \\neq 0$, clearly we can assume that $\\frac{\\partial R}{\\partial w_3}(w) \\neq 0$ for all $w$ in a small neighborhood $W$ about $w_0.$ Furthermore, we may assume that $w_0 = 0.$ In Theorem \\ref{special coordinate}, we construct a special coordinate near $w_0$ which changes the given smooth holomorphic curve into the $z_1$ axis and have a nonzero term along the $z_2$ axis when $z_1 = 0.$\n \n\n\n\n\n\n\\begin{thm} \\label{special coordinate}\n\nLet $\\Omega = \\{ w ; R(w)< 0 \\}$ be a smoothly bounded pseudoconvex domain in $\\mathbb{C}^3$ and let $T_{BG}(0)= m$, where $0 \\in b \\Omega $. Suppose that there is a smooth $1$-dimensional complex analytic variety $V$ passing through $0$ such that for all $w \\in V, w $ sufficiently close to $0$, \n \\begin{equation}\n |R(w)| \\leq C|w|^{\\eta}, \\label{condition of order of contact}\n \\end{equation}\nwhere $\\eta > 0$. Then there is a holomorphic coordinate system $(z_1, z_2, z_3)$ about $0$ with $w = \\Psi(z)$ such that \n\n\\begin{enumerate}[{\\normalfont (i)}]\n \n \\item $r(z)= R \\circ \\Psi(z) = \\mbox{\\normalfont{Re}}{z_3} + \\sum\\limits_ {\\substack{|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta a_{\\alpha, \\beta}{z'}^{\\alpha} {\\bar z}'^{\\beta} + \\mathcal{O}(|z_3||z|+|z'|^{\\eta +1}),$ \\label{special coordinate 1}\n\n \\item $|r(t,0,0)| \\lesssim |t|^{\\eta}$ \\label{changed order of contact}\n \n \n \\item $ {a_{0, \\alpha_2, 0,\\beta_2}} \\neq 0$ with $\\alpha_2+\\beta_2 = m$ for some $\\alpha_2 > 0, \\beta_2 > 0,$ \\label{nonzero term}\n\\end{enumerate}\nwhere $z' = (z_1, z_2),$ and $z = (z_1, z_2, z_3).$\n\n\\end{thm} \n\n\n\n\n\n\nNote that $\\eta$ is a positive integer since $V$ is a smooth 1-dimensional complex analytic variety. \nTo construct the special coordinate in Theorem \\ref{special coordinate}, we start with a similar coordinate about $0$ in $\\mathbb{C}^3$ as in Proposition 1.1 in \\cite{C2}. \n\n\n\n\n\n\n\\begin{prop} \\label{proposition 1}\n\nLet $T_{BG}(0) = m$ and $\\Omega = \\{ w \\in \\mathbb{C}^3 ; R(w)< 0 \\}$.\nThen there is a holomorphic coordinate system $ u = (u_1, u_2, u_3)$ with $ w = \\widetilde{\\Psi}(u)$ such that the function $\\tilde{R}$, given by $\\widetilde{R}(u) = R\\circ \\widetilde{\\Psi}(u)$, satisfies \n\n \\begin{equation}\\label{coordinate 1} \n \\widetilde{R}(u) = \\mbox{{\\normalfont Re}}{u_3} + \\sum_ {\\substack{ |\\alpha|+|\\beta| = m \\\\ | \\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha, \\beta} {u'}^{\\alpha} {\\bar u}'^{\\beta} + \\mathcal{O}(|u_3||u|+|u'|^{\\eta +1}), \n \\end{equation}\nwhere $u'=(u_1,u_2),$ and where $b_{\\alpha, \\beta} \\neq 0$ for some $\\alpha, \\beta$ with $|\\alpha| + |\\beta| = m.$\n\n\\end{prop}\n\n\n\\begin{proof} \nBloom and Graham \\cite{BG} showed that $T_{BG}(w_0) = m$ if and only if there exists coordinate with $w_0$ equal to the origin in $\\mathbb{C}^3$ and $b_{\\alpha, \\beta} \\neq 0$ for some $\\alpha, \\beta$ with $|\\alpha| + |\\beta| = m$ such that \n\n\\begin{equation*}\nR(w) = \\mbox{Re}{w_3} + \\sum\\limits_ {\\substack{|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}} b_{\\alpha, \\beta} {w'}^{\\alpha} {\\bar w}'^{\\beta} + \\mathcal{O}(|w_3||w|+|w'|^{m+1}),\n\\end{equation*}\nwhere $\\alpha = (\\alpha_1, \\alpha_2), \\beta = (\\beta_1, \\beta_2) \\ \\mbox{and} \\ w' = (w_1, w_2).$ \n\nNow assume that we have defined $\\phi^{l} : \\mathbb{C}^3 \\rightarrow \\mathbb{C}^3 $ so that there exist numbers $b_{\\alpha, \\beta}$ for $|\\alpha|, |\\beta|>0$ \nand $|\\alpha|+|\\beta| < l + 1$ with $l > m$ so that $ R_l = R\\circ \\phi^{l}$ satisfies\n\n\\begin{equation} \\label{coordinate 2}\n R_l(v) = \\mbox{{\\normalfont {Re}}}{v_3} + \\sum\\limits_{\\substack{|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}}^{l} b_{\\alpha, \\beta} \n {v'}^{\\alpha}\\bar{v}'^{\\beta} + \\mathcal{O}(|v_3||v|+|v'|^{l+1}), \n\\end{equation}\nwhere $v' = (v_1, v_2)$ and $v = (v_1, v_2, v_3).$\n\nIf we define $$\\phi^{l+1} (u) = \\biggl( u_1, u_2, u_3-\\sum_{|\\alpha|=l+1} {\\frac{2}{\\alpha!}}{\\frac{\\partial^{l+1} R_l}{\\partial {v'}^\\alpha}}(0){u'}^\\alpha \\biggr),$$ then $R_{l+1} = {R_l}\\circ \\phi^{l+1} = R \\circ \\phi^l \\circ \\phi^{l+1}$ satisfies the similar form of (\\ref{coordinate 2}) with $l$ replaced by $l+1$. Therefore, if we take $\\widetilde{\\Psi} = \\phi^l \\circ \\cdots \\circ \\phi^{\\eta}$, then $\\widetilde{R} = R\\circ \\widetilde{\\Psi}$ satisfies \n$$ \\widetilde{R}(u) = \\mbox{Re}{u_3} + \\sum\\limits_{\\substack{ |\\alpha|+|\\beta| =m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha, \\beta} {u'}^{\\alpha} {\\bar u}'^{\\beta} + \\mathcal{O}(|u_3||u|+|u'|^{\\eta +1}).$$ \n\\end{proof}\n\n\n\n\nFrom now on, without loss of generality, we may assume that $\\widetilde{R}$ is $R$ by Proposition \\ref{proposition 1}.\n\n\n\n\n\\begin{lem} \\label{parametrization lemma}\nLet $\\gamma = (\\gamma_1,\\gamma_2,\\gamma_3) :\\mathbb{C} \\to V$ be a local parametrization of a one-dimensional smooth complex analytic variety $V$.\nIf $|R(w)| \\lesssim |w|^{\\eta}$ for $w \\in V$, then we can assume\n $\\gamma =(\\gamma_1,\\gamma_2, 0) $ (i.e., $\\gamma_3$ vanishes to order at least $\\eta$).\n\\end{lem} \n\n\n\\begin{proof}We show $\\gamma_3 $ vanishes to order at least $\\eta$.\nSince $\\gamma(0)= 0$, we know $\\gamma_3$ vanishes to some order $l$.\nIf we suppose $l<\\eta$, then $\\gamma_3(t)= a_l {t}^l+\\mathcal{O}(t^{l+1})$, where $a_l \\neq 0$. Then\n\\begin{align*}\n R(\\gamma(t))&= \\mbox{\\normalfont Re}{\\gamma_3} + \\sum_ {\\substack{|\\alpha|+|\\beta|=m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha,\\beta}\\gamma_1^{\\alpha_1} {{\\bar \\gamma}_1}^{\\beta_1}\\gamma_2^{\\alpha_2} {{\\bar \\gamma}_2}^{\\beta_2}+\\mathcal{O}(|\\gamma_3||\\gamma|+|\\gamma|^{\\eta +1}) \\\\ \n &= \\biggl( {\\frac{a_l}{2}}{t^l}+{\\frac{\\bar a_l}{2}}{\\bar{t}^l} \\biggr) + \\biggl( \\sum_ {\\substack{j + k = m \\\\ j > 0, k > 0}}^\\eta c_{jk} t^j \\bar{t}^k \\biggr) + \\mathcal{O}(|t|^{l+1}).\n\\end{align*}\n\nNote that the first parenthesis consists of order $l$ pure terms and the summation part consists of the mixed terms. The first one is essentially $|t|^l$ with $l<\\eta$, so if we want to improve on the order of contact, then some terms of the summation part must cancel it. However, it is impossible because the summation part has all mixed terms. This contradicts our assumption $|r \\circ \\gamma(t)| \\lesssim |t|^\\eta$. Therefore, $\\gamma_3$ vanishes to order at least $\\eta.$ \n\n\\end{proof}\n\n\nLet $ A(u_1, u_2) = \\sum\\limits_{\\substack {|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}} b_{\\alpha, \\beta} {u}'^{\\alpha} {\\bar u}'^{\\beta}$ be the homogeneous polynomial part of order $m$ in the summation part of (\\ref{coordinate 1}). In the following lemma, we show that there is some nonzero mixed term along some direction in $\\mathbb{C}^2$.\n\n\n\\begin{lem} \\label{z_2 direction} \nConsider $A(hz, z)$ for all $h, z \\in \\mathbb{C}.$ Then there is some $h \\in \\mathbb{C}$ such that\n\\begin{equation*}\n\\frac{\\partial^m A}{{\\partial z^{j}}{\\partial {{\\bar z}^k}}}(0,0) \\neq 0, \\ \\text{ for }\\\nj, k >0.\n\\end{equation*} \n\\end{lem}\n\n\n\n\\begin{proof}\nSuppose that for all $h,$ $A(hz, z) = P(h)z^m + \\overline{P(h)z^m}.$ Since $A(hz, z)$ is a polynomial in $z, {\\bar z}, h$ and ${\\bar h}$ and $\\frac{\\partial^m A}{\\partial z^m} = m! P(h)$, $P(h)$ is a polynomial.\nLet $P(h) = \\sum a_{j,k} h^j {\\bar h}^k.$ Now, we have $A(hz, z)= \\sum a_{j,k} h^j {\\bar h}^k z^m + \\sum {\\bar a}_{j,k} {\\bar h}^j {h}^k {\\bar z}^m.$ Since $u_1 = hz$ and $u_2 = z,$ we have $h = \\frac{u_1}{u_2}$ and $z = u_2$. Therefore, $A(u_1, u_2) = \\sum a_{j,k} (\\frac{u_1}{u_2})^j ({\\frac{{\\bar{u}_1}}{{\\bar{u}_2}}})^k u_2^m + \\sum \\bar{a}_{j,k} ({\\frac{\\bar{u}_1}{{\\bar u}_2}})^j ({\\frac{u_1}{u_2}})^k {\\bar u_2}^m.$ This forces $j$ and $k$ to be $0$ because $A(u_1, u_2)$ is a polynomial. Therefore, we have $A(hz, z)= a_{0,0}z^m + \\bar{a}_{0, 0} \\bar{z}^m.$ This means $A(u_1, u_2)= a_{0,0}{u_2} ^m + \\bar{a}_{0, 0} {\\bar u_2}^m.$\nHowever, this contradicts $b_{\\alpha, \\beta} \\neq 0 $ for some $\\alpha, \\beta$ with $|\\alpha|, |\\beta| > 0 $ and $|\\alpha|+|\\beta|=m $ in (\\ref{coordinate 1}). \n\\end{proof}\n\n\nNow, we prove Theorem \\ref{special coordinate}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{special coordinate}]\nWe may assume ${\\gamma_1}' (0) \\neq 0$, and hence, after reparametrization, we can write $\\gamma(t) = (t, \\gamma_2(t), 0)$. Now, define \n\\begin{equation*}\n u = \\Psi_1 (v) = (v_1, v_2 + \\gamma_2(v_1), v_3).\n\\end{equation*}\nSince $\\gamma_2(t) = \\mathcal{O}(|t|)$ is holomorphic, (\\ref{coordinate 1}) means \n\\begin{align*}\n r_1 (v) &= R \\circ \\Psi_1 (v) \n = \\mbox{Re}{v_3}+\\sum_ {\\substack{|\\alpha|+|\\beta|=m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha, \\beta} v_1^{\\alpha_1}{\\bar v}_1^{\\beta_1}(v_2+\\gamma_2(v_1))^{\\alpha_2} {\\overline {(v_2+\\gamma_2(v_1)) }}^{\\beta_2} \n + E_1 (v) \\\\\n &=\\mbox{Re}{v_3}+\\sum_ {\\substack{|\\alpha|+|\\beta|=m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta c_{\\alpha, \\beta} v_1^{\\alpha_1}{\\bar v_1}^{\\beta_1}v_2^{\\alpha_2} {\\bar v_2}^{\\beta_2}+ E_1(v), \\ \\text{ where }\\ E_1(v) = \\mathcal{O}(|v_3||v|+|v'|^{\\eta +1}).\n\\end{align*}\nNote that $T_{BG} = m$ means $c_{\\alpha, \\beta} \\neq 0$ for some $\\alpha, \\beta > 0$ with $|\\alpha|+|\\beta| = m.$ Now, we fix $h$ in lemma \\ref{z_2 direction} and define \n\n\\begin{equation*}\n v = \\Psi_2 (z) = (z_1 + h z_2, z_2, z_3).\n\\end{equation*}\nThen, we have \n \\begin{align} \n r(z) &= r_1 \\circ \\Psi_2(z) = R \\circ \\Psi_1 \\circ \\Psi_2 (z) \\nonumber \\\\\n &= \\mbox{Re}z_3 +\\sum_ {\\substack{|\\alpha|+|\\beta|=m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta c_{\\alpha, \\beta} (z_1+hz_2)^{\\alpha_1}{\\overline {(z_1+hz_2)}}^{\\beta_1}z_2^{\\alpha_2} {\\bar z_2}^{\\beta_2}+ E_1(z), \\label{eqn2} \\\\\n &=\\mbox{Re}z_3 + \\sum_ {\\substack{|\\alpha|+|\\beta|=m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta a_{\\alpha, \\beta} z_1^{\\alpha_1}{\\bar z_1}^{\\beta_1}z_2^{\\alpha_2} {\\bar z_2}^{\\beta_2}+ E_1(z), \\label{eqn3}\n\\end{align}\nwhere $a_{\\alpha, \\beta}$ is a polynomial of $h$ and ${\\bar h},$ and where $E_1(z) = \\mathcal{O}(|z_3||z|+|z'|^{\\eta +1}).$\nLet $\\Psi = \\Psi_1 \\circ \\Psi_2.$ Then we have $r(z) = R \\circ \\Psi$ and (\\ref{eqn3}) shows (\\ref{special coordinate 1}) of Theorem \\ref{special coordinate}. Furthermore, since $|r(t, 0, 0)| = |R \\circ \\Psi(t, 0, 0)|= |R(\\gamma(t))| \\lesssim |t|^\\eta,$ this proves part (\\ref{changed order of contact}). For (\\ref{nonzero term}), if we consider $r(0, z_2, 0)$ and (\\ref{eqn2}), we have\n\\begin{equation*} \n r(0, z_2, 0) = A(hz_2, z_2) + \\sum\\limits_ {\\substack{|\\alpha|+|\\beta| = m +1 \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta c_{\\alpha,\\beta}{(hz_2)}^{\\alpha_1}{\\overline{(hz_2)}}^{\\beta_1}z_2^{\\alpha_2} {\\bar z}_2^{\\beta_2} +\\mathcal{O}(|z_2|^{\\eta +1}).\n\\end{equation*}\nThen Lemma \\ref{z_2 direction} means\n\\begin{equation*}\n \\frac{\\partial^m r }{{\\partial {z_2}^{\\alpha_2}}{\\partial {\\bar z}_2}^{\\beta_2}}(0) = \\frac{\\partial^m A }{{\\partial {z_2}^{\\alpha_2}}{\\partial {\\bar z}_2}^{\\beta_2}}(0,0) \\neq 0\n\\end{equation*} \nfor some $\\alpha_2, \\beta_2 > 0$ with $\\alpha_2 + \\beta_2 = m .$\nSince $\\frac{\\partial^m r }{{\\partial {z_2}^{\\alpha_2}}{\\partial {\\bar z}_2}^{\\beta_2}}(0) = \\alpha_2 ! \\beta_2 ! a_{0, \\alpha_2, 0, \\beta_2}$ in (\\ref{eqn3}), this completes the proof. \n\\end{proof}\n\n\n\n\nCatlin \\cite{C2} constructed a bounded holomorphic funtion with a large derivative near a finite type point in the boundary of pseudoconvex domain in $\\mathbb{C}^2$. To construct a similar function in $\\mathbb{C}^3$, we will use the function constructed by Catlin. In order to achieve this goal, as a first step, we need to consider two dimensional slice with respect to the $z_2$ and $z_3$ variables when $z_1$ is fixed at some point. For this, we consider the representative terms in the summation part of (\\ref{special coordinate 1}) of Theorem \\ref{special coordinate}.\n\nLet \n\\begin{align*}\n \\Gamma &= \\{(\\alpha, \\beta); a_{\\alpha, \\beta} \\neq 0, m \\leq |\\alpha|+|\\beta| \\leq \\eta \\ \\mbox{and} \\ |\\alpha|, |\\beta| > 0 \\} \\\\\n S &= \\{(p, q); \\alpha_1 + \\beta_1 = p, \\alpha_2 + \\beta_2 = q \\ \\mbox{for some} \\ (\\alpha, \\beta) \\in \\Gamma \\} \\cup \\{(\\eta, 0)\\}.\n\\end{align*}\nThen there is an positive integer $N$ such that $(p_\\nu, q_\\nu) \\in S$ for $\\nu = 0,\\cdots, N $ and $\\eta_\\nu, \\lambda_\\nu > 0$ for $\\nu = 1, \\cdots, N$ satisfying\n \n\\begin{enumerate}[(1)]\n\t\\item $(p_0, q_0)=(\\eta, 0), (p_{N}, q_{N})= (0, m) , \\lambda_N = m, \\eta_1 = \\eta,$ \\label{condition1}\n\t\\item $p_0 > p_1 > \\cdots > p_{N}$ and $ q_0 < q_1 < \\cdots < q_{N},$ \\label{condition2}\n\t\\item $\\lambda_1 < \\lambda_2 <\\cdots < \\lambda_N$ and $\\eta_1 > \\eta_2 > \\cdots > \\eta_N,$ \\label{condition3}\n\t\\item $\\frac{p_{\\nu-1}}{\\eta_\\nu} + \\frac{q_{{\\nu-1}}}{\\lambda_\\nu} = 1$ and $\\frac{p_{\\nu}}{\\eta_\\nu} + \\frac{q_{\\nu}}{\\lambda_\\nu}=1$ \\label{condition4} and\n\t\\item $a_{\\alpha, \\beta}=0$ if $\\frac{\\alpha_1 + \\beta_1}{\\eta_\\nu}+ \\frac{\\alpha_2 + \\beta_2}{\\lambda_\\nu} < 1$ for each $\\nu = 1,\\cdots,N.$\\label{condition5}\n\\end{enumerate}\n\nNote that if $1 \\leq l \\leq m,$ then $q_{\\nu - 1} < l \\leq q_\\nu$ for some $\\nu = 1, \\cdots, N.$\nLet $L_\\nu$ be the line segment from $(p_{\\nu - 1}, q_{\\nu - 1})$ to $(p_{\\nu}, q_{\\nu})$ for each $\\nu = 1, \\cdots, N$ and set $L =\\ L_1 \\cup L_2 \\cup \\cdots \\cup L_{N}.$ Define\n\n\\begin{itemize}\n\t\\item $\\Gamma_L = \\{(\\alpha, \\beta) \\in \\Gamma; \\alpha+\\beta \\in L \\}.$\n\t\\item $t_l = \\begin{cases}\n\t \\eta & \\text{if $l = 0$} \\\\\n\t \\eta_\\nu \\biggl(1 - \\frac{l}{\\lambda_\\nu} \\biggr) & \\text{if $q_{\\nu-1} < l \\leq q_\\nu$ for some $\\nu$.} \n\t \\end{cases}$\n\\end{itemize}\nNote that $(p_{\\nu-1}, q_{\\nu-1}), (t_l, l)$ and $(p_\\nu, q_\\nu)$ are collinear points in the first quadrant of the plane and $\\eta_\\nu$ and $\\lambda_\\nu$ are the $x$, $y$-intercepts of the line. \\\\\n\nNow, we want to show that for each element $(p_\\nu, q_\\nu)$ with $\\nu = 1, \\cdots, N$, there is some $(\\alpha, \\beta)$ allowing a mixed term in the $z_2$ variable. To show this, we need to use a variant of the notations and the results from Lemma 4.1 and Proposition 4.4 in \\cite{C}.\nFor $t$ with $0 0, \\beta_2^\\nu > 0$\nand $\\alpha^\\nu + \\beta^\\nu = (p_\\nu, q_\\nu).$\n\n\\end{lem}\n\n\n\\begin{proof}\nConsider ${\\widetilde r}^\\nu$, which is plurisubharmonic. Now, consider $${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1} = \\lim\\limits_{t \\to 0} t^{-1}({H_t^{\\nu+1}}^* {\\widetilde r^\\nu }).$$ This is also plurisubharmonic. Since $(p_\\nu, q_\\nu)$ is the unique point with $L_\\nu \\cap L_{\\nu + 1}$ (i.e., $\\frac{p_{\\nu}}{\\eta_\\nu} + \\frac{q_{{\\nu}}}{\\lambda_\\nu} = 1$ and $\\frac{p_{\\nu}}{\\eta_{\\nu+1}} + \\frac{q_{\\nu}}{\\lambda_{\\nu+1}}=1$), we have \n \n\\begin{equation}\\label{formoftrucated}\n {\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1} = \\mbox{Re}{z_3} + \\sum_ {\\substack{\\alpha + \\beta = (p_\\nu, q_\\nu) \\\\ (\\alpha, \\beta) \\in \\Gamma_L}} a_{\\alpha_1,\\alpha_2 ,\\beta_1,\\beta_2} z_1^{\\alpha_1} {\\bar z}_1^{\\beta_1}z_2^{\\alpha_2} {\\bar z}_2^{\\beta_2}. \n \n\\end{equation}\nIn particular, $(\\alpha, \\beta) \\in \\Gamma_L$ means $|\\alpha|, |\\beta| > 0.$\nSuppose that ${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}$ has no terms with both $\\alpha_2 > 0$ and $\\beta_2>0$ in (\\ref{formoftrucated}) (i.e., no mixed terms in $z_2$ variable). Thus\n$${\\widetilde{(\\widetilde{r^\\nu})}}^{\\nu+1} = \\mbox{Re}{z_3} + P_{q_\\nu}(z_1){z_2}^{q_\\nu} +\\overline{P_{q_\\nu}(z_1){z_2}^{q_\\nu}}$$ \nwhere $P_{q_\\nu}(z_1) = \\sum\\limits_{\\alpha_1 +\\beta_1 = p_\\nu} c_{\\alpha_i,\\beta_i} {z_1}^{\\alpha_1} {\\bar z_1}^{\\beta_1}$ with $\\beta_1 > 0$. \nBy the plurisubharmonicity of ${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}$, \n $${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{11} {\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{22}- {\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{12}{\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{21} = - \\lvert {q_\\nu} \\frac{\\partial {P_{q_\\nu}}}{\\partial{\\bar z_1}}(z_1) {z_2}^{q_\\nu -1} \\rvert^2 \\geq 0,$$\nwhere ${{\\widetilde{(\\widetilde{r^\\nu})}}^{\\nu+1}}_{ij} = \\frac{\\partial^{2}{\\widetilde{({\\widetilde r}^\\nu)}}^{\\nu+1}}{{\\partial z_i}{\\partial \\bar{z_j}}} $ for $i, j = 1, 2.$ \nTherefore, we have $\\frac{\\partial {P_{q_\\nu}}}{\\partial{\\bar{z_1}}}(z_1) = 0.$ This means $P_{q_\\nu} (z_1)$ is holomorphic. \nThis contradicts the fact that $P_{q_\\nu}(z_1) = \\sum\\limits_{\\alpha_1 +\\beta_1 = p_\\nu} c_{\\alpha_i,\\beta_i} {z_1}^{\\alpha_1} {{\\bar {z_1}}^{\\beta_1}}$ with $\\beta_1 > 0$. \n\\end{proof} \n\n\n\nNow, we define these special terms with respect to the $z_2$ variable. Let\n\\begin{equation*}\n\\Lambda = \\{(\\alpha, \\beta) \\in \\Gamma_L ; \\alpha+\\beta = (p_\\nu, q_\\nu), \\alpha_2 > 0, \\beta_2 > 0, \\nu = 1, \\cdots, N \\}.\n\\end{equation*} \nThen we represent the expression of $r$ in terms of these terms. \\\\\n\n\n\n\\begin{prop}\\label{final expression of r}\n The defining function $r$ can be expressed as \n\\begin{equation} \\label{repre of r}\n r(z) = \\mbox{\\normalfont Re}{z_3} + \\sum_ {\\Gamma_L - \\Lambda} a_{\\alpha, \\beta} {z'}^{\\alpha} {\\bar z}'^{\\beta} + {\\sum_{\\nu = 1}^N}\\sum_{\\substack{\\alpha_2 + \\beta_2= q_\\nu \\\\ \\alpha_2 > 0, \\beta_2 > 0}} M_{\\alpha_2, \\beta_2}(z_1) {z_2}^{\\alpha_2} {\\bar z}_2^{\\beta_2}+E_2(z),\n\\end{equation} \nwhere $M_{\\alpha_2, \\beta_2}(z_1) = \\sum\\limits_{\\alpha_1+\\beta_1 = p_\\nu} a_{\\alpha, \\beta} z_1^{\\alpha_1} \\bar{z}_1^{\\beta_1}$ and $E_2(z) = \\mathcal{O}(|z_3||z|+\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |z_1|^{[t_l]+1}|z_2|^l+|z_2|^{m+1}).$\n\n\\end{prop}\n\n\n\n\\begin{proof}\nBy theorem \\ref{special coordinate}, we have \n\\begin{equation} \\label{rform}\n r(z) = {\\mbox{Re}}{z_3}+ \\sum_{\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar z}'^{\\beta}+ \\sum_{\\Gamma-\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar z}'^{\\beta}+\\mathcal{O}(|z_3||z|+|z'|^{\\eta +1}). \n\\end{equation}\nSuppose that $(k, l) = (\\alpha_1 + \\beta_1, \\alpha_2 + \\beta_2)$ for some $(\\alpha, \\beta)\\in \\Gamma-\\Gamma_L.$ Then, we consider two cases; $1\\leq l \\leq m$ and $m < l < \\eta.$ If $1\\leq l \\leq m$, there is a unique $\\nu =1, \\cdots, N$ so that $q_{\\nu-1} < l \\leq q_\\nu$ and $t_l = \\eta_\\nu \\biggl(1 - \\frac{l}{\\lambda_\\nu} \\biggr).$ Since $(k, l) = (\\alpha_1 + \\beta_1, \\alpha_2 + \\beta_2)$ for some $(\\alpha, \\beta) \\in \\Gamma -\\Gamma_L,$ $ \\frac{k}{\\eta_\\nu}+\\frac{l}{\\lambda_\\nu} > 1.$ This gives $t_l = \\eta_\\nu \\biggl(1 - \\frac{l}{\\lambda_\\nu}\\biggr) < k.$ Since $k$ is an integer, $[t_l]+1 \\leq k.$ Thus, we have $|z_1|^k|z_2|^l \\leq |z_1|^{[t_l]+1}|z_2|^l$ for each $l = 1, \\cdots, m.$ \nOn the other hand, if $(k, l)=(\\alpha_1 + \\beta_1, \\alpha_2 + \\beta_2)$ for some $(\\alpha, \\beta)\\in \\Gamma-\\Gamma_L$ and $m < l < \\eta,$ then $|z_1|^k|z_2|^l \\leq |z_1|^k|z_2|^{m+1} \\leq |z_2|^{m+1}$ for small $z_1$ and $z_2.$\nSince $|z'|^{\\eta +1} \\approx |z_1|^{\\eta +1} + |z_2|^{\\eta +1},$ it follows that\n$\\sum_{\\Gamma -\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar{z'}}^{\\beta}+\\mathcal{O}(|z_3||z|+|z'|^{\\eta +1})= \\mathcal{O}(|z_3||z|+\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |z_1|^{[t_l]+1}|z_2|^l+|z_2|^{m+1}).$ Therefore, $r(z)$ in (\\ref{rform}) is represented as \n\\begin{equation} \\label{r form 2}\n{\\mbox{Re}}{z_3}+ \\sum_{\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar z}'^{\\beta} + \\mathcal{O}(|z_3||z|+\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |z_1|^{[t_l]+1}|z_2|^l+|z_2|^{m+1}).\n\\end{equation}\nNow, apply $\\Gamma_L = (\\Gamma_L - \\Lambda) \\cup \\Lambda$ for the second part of summation in (\\ref{rform}). \n\\end{proof}\n\n\n\\begin{rem}\\label{sizeofM} \\\n\\begin{enumerate}[\\normalfont i)] \n\t\\item $M_{\\alpha_2, \\beta_2}(z_1)$ is not identically zero for $\\alpha_2 + \\beta_2 = q_\\nu$ and the homogeneous polynomial is of order $p_\\nu$ for each $\\nu = 1, \\cdots, N-1.$\n\t\\item If $\\nu = N,$ then $|M_{\\alpha_2, \\beta_2}(z_1)|$ is a nonzero constant for all $\\alpha_2, \\beta_2 > 0$ with $\\alpha_2+\\beta_2 = m = q_N$ since $ p_N = 0.$\n\t\\item Since $M_{\\alpha_2, \\beta_2}(z_1)$ is a homogeneous polynomial of order $p_\\nu, \\nu =1, \\cdots, N,$ in $z_1$-variable, there are $\\theta_0 \\in [0, 2\\pi]$ and a small constant $c > 0$ such that $|M_{\\alpha_2, \\beta_2}(\\tau e^{i\\theta})| \\neq 0$ for all $|\\theta - \\theta_0| < c$ and $0 < \\tau \\leq 1.$ \n\t In particular, if we take $d = e^{i\\theta_0}$ and $\\tau = \\delta^{\\frac{1}{\\eta}}$ we have $|M_{\\alpha_2, \\beta_2}(d \\delta^{\\frac{1}{\\eta}})| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}$ for all $\\alpha_2+ \\beta_2 = q_\\nu$ with all $\\nu = 1, \\cdots, N.$\n\\end{enumerate}\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The construction of bounded holomorphic function with large derivative near the boundary}\\label{Sec4}\n\nLet $z_1 = d\\delta^{\\frac{1}{\\eta}}.$ Then, we get a complex two dimensional slice. After the holomorphic coordinate change as Proposition 1.1 in \\cite{C2}, we can define a bounded holomorphic function with a large nontangential derivative as in \\cite{C2} on the slice. In this section, first, we construct a holomorphic coordinate system in $\\mathbb{C}^3$ to exactly fit the holomorphic coordinate system as in proposition 1.1 of \\cite{C2} when $z_1$ is fixed as $d\\delta^{\\frac{1}{\\eta}}$. Second, we show that the holomorphic function defined on the slice is also well-defined on a family of slices along the small neighborhood of $z_1 = d\\delta^{\\frac{1}{\\eta}}.$ To show the well-definedness of the holomorphic function up to boundary in $\\mathbb{C}^3,$ we need the estimates of derivatives. Let's denote $ U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}} = U \\cap \\{(d\\delta^{\\frac{1}{\\eta}}, z_2, z_3)\\}$ and let $ \\widetilde{e}_\\delta = (d\\delta^{\\frac{1}{\\eta}}, 0, e_\\delta)$ satisfy $r(\\widetilde{e}_\\delta) = 0.$ Since $\\frac{\\partial r}{\\partial z_3}(0) \\neq 0,$ clearly $\\frac{\\partial r}{\\partial z_3}(\\widetilde{e}_\\delta) \\neq 0.$ We start with the similar argument as Proposition 1.1 in \\cite{C2}.\\\\\n\n\n\\begin{prop} \\label{coordinatechangeinc2}\nFor $ \\widetilde{e}_\\delta \\in U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}},$ there exists a holomorphic coordinate system $ (z_2, z_3)= \\Phi_{\\widetilde{e}_\\delta}(\\zeta '') = (\\zeta_2 , \\Phi_3 (\\zeta'')))$ such that in the new coordinate $\\zeta'' = (\\zeta_2, \\zeta_3)$ defined by\n\\begin{equation}\n\\Phi_{\\widetilde{e}_\\delta}(\\zeta '')= \\biggl(\\zeta_2, \\ e_\\delta + \\biggl(\\frac{\\partial r}{\\partial z_3}(\\widetilde{e_\\delta}) \\biggr)^{-1} \\biggl(\\frac{\\zeta_3}{2}- \\sum_{l=2}^m c_l({\\widetilde e}_\\delta) \\zeta_2^l - \\frac{\\partial r}{\\partial z_2} ({\\widetilde e}_\\delta){\\zeta_2} \\biggr) \\biggr), \\label{catlin version coordinate chanage}\n\\end{equation}\nthe function $\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = r(d\\delta^{\\frac{1}{\\eta}}, z'') \\circ \\Phi_{\\widetilde{e}_\\delta}(\\zeta'')$ satisfies \n\\begin{equation} \\label{catlin_defining_expression}\n\\rho(d\\delta^{\\frac{1}{\\eta}},\\zeta'')= \\mbox{\\normalfont{Re}}\\zeta_3 + \\sum\\limits_{\\substack{j+k=2\\\\j, k > 0}}^{m} a_{j,k}(\\widetilde{e_\\delta}) \\zeta_2^j {\\bar \\zeta_2}^k + \\mathcal{O}(|\\zeta_3||\\zeta''|+|\\zeta_2|^{m+1} ),\n\\end{equation}\nwhere $z'' = (z_2, z_3)$.\n\\end{prop} \n\n\n\\begin{proof} For $ {\\widetilde e}_\\delta \\in U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}},$ define\n\\begin{equation}\\label{rho2changeofcoordinate}\n \\Phi_{\\widetilde{e}_\\delta}^1(w'') =\\biggl( w_2, \\ e_\\delta + \\biggl(\\frac{\\partial r}{\\partial z_3}({\\widetilde e}_\\delta) \\biggr)^{-1}\\biggl(\\frac{w_3}{2} - \\frac{\\partial r}{\\partial z_2} ({\\widetilde e}_\\delta)w_2 \\biggr)\\biggr). \n\\end{equation}\nThen we have \n\\begin{equation}\\label{rho2expression}\n\\rho_2(d\\delta^{\\frac{1}{\\eta}}, w'') = r(d\\delta^{\\frac{1}{\\eta}}, z'') \\circ \\Phi_{\\widetilde{e}_\\delta}^1(w'') = \\mbox{Re} w_3 + \\mathcal{O}(|w''|^2),\n\\end{equation} \nwhere $w'' = (w_2, w_3).$\nNow assume that we have defined $\\Phi_{\\widetilde{e}_\\delta}^{l-1} : \\mathbb{C}^2 \\rightarrow \\mathbb{C}^2 $ so that there exist numbers $a_{j, k}$ for $j, k > 0$ and $j+k < l$ so that $\\rho_l(d\\delta^{\\frac{1}{\\eta}}, w'') = r(d\\delta^{\\frac{1}{\\eta}}, z'')\\circ \\Phi_{\\widetilde{e}_\\delta}^{l-1}(w'')$ satisfies\n$$\\rho_l(d\\delta^{\\frac{1}{\\eta}}, w'') = \\mbox{Re}w_3 + \\sum_{\\substack{j+k=2\\\\j, k > 0}}^{l-1} a_{j,k}({\\widetilde e}_\\delta) w_2^j {\\bar w_2}^k + \\mathcal{O}(|w_3||w''|+|w_2|^l), $$\nwhere $w'' = (w_2, w_3).$\nIf we define $\\Phi_{\\widetilde{e}_\\delta}^l = \\Phi_{\\widetilde{e}_\\delta}^{l-1}\\circ \\phi^l,$ where \n\\begin{equation}\\label{lthchangeofvariable}\n\\phi^l(\\zeta'') = \\biggl(\\zeta_2, \\ \\zeta_3 - \\frac{2}{l!} \\frac{\\partial^l \\rho_l}{\\partial w_2^l}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\zeta_2^l \\biggr).\n\\end{equation} \nthen \n\\begin{equation}\\label{rholexpression}\n\\rho_{l+1}(d\\delta^{\\frac{1}{\\eta}}, \\zeta'')= \\rho_l \\circ \\phi^l (\\zeta'') = r(d\\delta^{\\frac{1}{\\eta}}, z'')\\circ \\Phi_{\\widetilde{e}_\\delta}^l(\\zeta'')\n\\end{equation} satisfies\n$$\\rho_{l+1}(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = \\mbox{Re}\\zeta_3 + \\sum\\limits_{\\substack{j+k=2\\\\j, k > 0}}^{l} a_{j,k}({\\widetilde e}_\\delta) \\zeta_2^j {\\bar \\zeta_2}^k + \\mathcal{O}(|\\zeta_3||\\zeta''|+|\\zeta_2|^{l+1}),$$ where $\\zeta'' = (\\zeta_2, \\zeta_3).$\nTherefore, if we choose $\\Phi_{\\widetilde{e}_\\delta} = \\Phi_{\\widetilde{e}_\\delta}^m = \\Phi_{\\widetilde{e}_\\delta}^{m-1} \\circ \\phi^m = \\cdots = \\Phi_{\\widetilde{e}_\\delta}^1 \\circ \\phi^2 \\circ \\cdots \\circ \\phi^m,$ then $\\rho = \\rho_{m+1} = \\rho_m \\circ \\phi^m = r \\circ \\Phi_{\\widetilde{e}_\\delta}.$ \nThis shows (\\ref{catlin version coordinate chanage}) and (\\ref{catlin_defining_expression}), where $ c_l({\\widetilde e}_\\delta)$ is defined by\n\\begin{equation}\\label{clexpression}\n c_l({\\widetilde e}_\\delta) = \\frac{1}{l!}\\frac{\\partial^l \\rho_l}{\\partial w_2^l}(d\\delta^{\\frac{1}{\\eta}}, 0, 0). \n\\end{equation}\n\\end{proof}\n\nAs in \\cite{C2}, we set\n\\begin{equation}\\label{def of Al}\nA_l ({\\widetilde e}_\\delta) = \\mbox{max} \\{|a_{j,k}({\\widetilde e}_\\delta)|; j+k = l \\}, \\hspace{0.3in} l=2, \\cdots, m \n\\end{equation}\nand \n\\begin{equation}\\label{taudef}\n\\tau({\\widetilde e}_\\delta, \\delta) = \\min \\biggl\\{\\biggl(\\frac{\\delta}{A_l({\\widetilde e}_\\delta)} \\biggr)^{1\/l}; 2 \\leq l \\leq m \\biggr\\}\n\\end{equation}\nAs we will see later (Remark \\ref{Amnonzero}), we have $A_m ({\\widetilde e}_\\delta) \\neq 0$ since $ |A_m ({\\widetilde e}_\\delta)| \\geq c_m > 0,$ where $\\delta > 0$ is sufficiently small. This means $$\\tau({\\widetilde e}_\\delta, \\delta) \\lesssim \\delta^{\\frac{1}{m}}.$$\nDefine\n\\begin{equation}\\label{rbox}\n R_\\delta ({\\widetilde e}_\\delta)= \\{\\zeta'' \\in \\mathbb{C}^2; |\\zeta_2| < \\tau({\\widetilde e}_\\delta, \\delta), |\\zeta_3| < \\delta \\}.\n\\end{equation} \\\\\n\n\nBefore estimating the derivative of $r$, we estimate the size of $e_\\delta$.\nSince $r({\\widetilde e}_\\delta)= 0,$ Taylor's theorem in $z_3$ about $e_\\delta$ gives\n$$r(d\\delta^{\\frac{1}{\\eta}}, 0, z_3)= 2\\mbox{Re}\\biggl( \\frac{\\partial r}{\\partial z_3}(d\\delta^{\\frac{1}{\\eta}}, 0, e_\\delta)(z_3 - e_\\delta) \\biggr)+ \\mathcal{O}(|z_3 - e_\\delta|^2).$$\nIf we take $z_3 = 0,$ then $|r(d\\delta^{\\frac{1}{\\eta}}, 0, 0)|= \\left|2\\mbox{Re}\\biggl( \\frac{\\partial r}{\\partial z_3}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)(- e_\\delta) \\biggr)+ \\mathcal{O}(|e_\\delta|^2)\\right| \\approx |e_\\delta|$ since $|e_\\delta| \\ll 1$ and $|\\frac{\\partial r}{\\partial z_3}| \\approx 1$ near $0.$ Therefore \\ref{changed order of contact}) of Theorem \\ref{special coordinate} means $|e_\\delta| \\lesssim \\delta.$ \\\\\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{rderivative}\nLet $ l = 1, 2, \\cdots, m$ and let $\\alpha_2^\\nu$ and $\\beta_2^\\nu$ be positive numbers as given in Lemma \\ref{existence of mixed term in z_2} for $\\nu = 1, \\cdots, N.$ Then the function $r$ satisfies\n\\begin{enumerate}[\\normalfont (i)]\n\t\\item $\\biggl|\\frac{\\partial^l r}{{\\partial z_2^{\\alpha_2}}{\\partial {\\bar{z}_2}}^{\\beta_2}}({\\widetilde e}_\\delta)\\biggr| \\lesssim \\delta^{\\frac{t_l}{\\eta}},$\n\t \\quad \\text{where $\\alpha_2, \\beta_2 \\geq 0.$} \\label{rderivative1}\n\t\\item $\\biggl| \\frac{\\partial^{q_\\nu} r }{{\\partial {z_2}^{\\alpha_2^\\nu}}{\\partial {\\bar z_2}^{\\beta_2^\\nu}} }({\\widetilde e}_\\delta)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}},$ \\quad \\text{where $\\alpha_2^\\nu > 0$ and $\\beta_2^\\nu > 0.$}\\label{rderivative2}\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\nBy (\\ref{r form 2}) and $t_l < [t_l]+1$, we have \n\\begin{equation}\\label{rdifferentiation}\n \\biggl|\\frac{\\partial^l r}{{\\partial z_2^{\\alpha_2}}{\\partial {\\bar{z}_2}}^{\\beta_2}}({\\widetilde e}_\\delta)\\biggr| \\lesssim \\delta^{\\frac{t_l}{\\eta}} + |e_\\delta| + \\delta^{\\frac{[t_l]+1}{\\eta}} \\lesssim \\delta^{\\frac{t_l}{\\eta}}.\n\\end{equation} \nFor (\\ref{rderivative2}), note that if $l = q_\\nu,$ then $t_l = p_\\nu$. Therefore,\n(\\ref{repre of r}) gives\n\\begin{equation*}\n |M_{\\alpha_2^\\nu, \\beta_2^\\nu}(d\\delta^{\\frac{1}{\\eta}})| - C_1(|e_\\delta| + \\delta^{\\frac{p_\\nu + 1}{\\eta}}) \\leq \\biggl|\\frac{1}{{\\alpha_2^\\nu}!{\\beta_2^\\nu}!}\\frac{\\partial^{q_\\nu} r }{{\\partial {z_2}^{\\alpha_2^\\nu}}{\\partial {\\bar z_2}^{\\beta_2^\\nu}} }(\\widetilde{e_\\delta})\\biggr| \\leq |M_{\\alpha_2^\\nu, \\beta_2^\\nu}(d\\delta^{\\frac{1}{\\eta}})| + C_1(|e_\\delta| + \\delta^{\\frac{{p_\\nu} + 1}{\\eta}})\n\\end{equation*}\nfor some constant $C_1.$\nSince Remark \\ref{sizeofM} means $|M_{\\alpha_2^\\nu, \\beta_2^\\nu}(d\\delta^{\\frac{1}{\\eta}})| \\approx \\delta^{\\frac{p_\\nu}{\\eta}},$ we have $$\\biggl|\\frac{\\partial^{q_\\nu} r }{{\\partial {z_2}^{\\alpha_2^\\nu}}{\\partial {\\bar z_2}^{\\beta_2^\\nu}} }({\\widetilde e}_\\delta)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.$$ \n\\end{proof}\n\n\n\n\n\n\\begin{lem} \\label{rhoderivative}\nLet $\\rho_l, \\phi^l$ and $\\Phi^l$ be given as in (\\ref{rho2changeofcoordinate})-(\\ref{rholexpression}) for $l=2, \\cdots, m+1$ and $\\alpha_2^\\nu$ and $\\beta_2^\\nu$ be positive numbers as given in Lemma \\ref{existence of mixed term in z_2} for $\\nu = 1, \\cdots, N.$ Then\n\\begin{enumerate}[\\normalfont (i)] \n \\item $\\biggl|\\frac{\\partial^k \\rho_l}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial {{\\bar \\zeta}_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{t_k}{\\eta}} \\quad \\text{for each } \\ k = 1, \\cdots, m.$\n \\item $\\biggl|\\frac{\\partial^{q_\\nu} \\rho_l }{{\\partial {\\zeta_2}^{\\alpha_2^\\nu}}{\\partial {\\bar \\zeta_2}^{\\beta_2^\\nu}} }(d\\delta^{\\frac{1}{\\eta}}, 0, 0) \\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}} \\quad \\text{for each } \\ \\nu = 1, \\cdots, N.$\n\\end{enumerate} \n \nIn particular, $|c_l({\\widetilde e}_\\delta)| \\lesssim \\delta^{\\frac{t_l}{\\eta}},$ where $c_l({\\widetilde e}_\\delta)$ is given in (\\ref{clexpression}).\n\\end{lem} \n\n\n\\begin{proof}\nBy induction, we prove both (i) and (ii). For part (i), let $l = 2.$ Since $\\rho_2(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = r(d\\delta^{\\frac{1}{\\eta}}, z'') \\circ \\Phi_{\\widetilde{e}_\\delta}^1 (\\zeta''),$ by chain rule and Lemma \\ref{rderivative}, we have\n\\begin{equation*}\n\\biggl|\\frac{\\partial^k \\rho_2}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial \\bar{\\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \n \\lesssim \\biggl|\\frac{\\partial^k r}{{\\partial z_2^{\\alpha_2}}{\\partial {\\bar z_2}^{\\beta_2}}}({\\widetilde e}_\\delta)\\biggr| \n + \\biggl|\\frac{\\partial r}{\\partial z_2}({\\widetilde e}_\\delta) \\biggr| \n \\lesssim \\delta^{\\frac{t_k}{\\eta}}+\\delta^{\\frac{t_1}{\\eta}} \\lesssim \\delta^{\\frac{t_k}{\\eta}}.\n\\end{equation*}\nfor all $ k = 1, \\cdots, m.$\nThis proves for the case $l = 2.$ \nNow, by induction, we assume\n$$\\biggl|\\frac{\\partial^k \\rho_l}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial \\bar{\\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{t_k}{\\eta}}$$ for all $k = 1, \\cdots, m$ and $l = 2, \\cdots, j.$ \\\nNote that\n\\begin{equation}\\label{rholrealexpression}\n\\rho_{j+1}(d\\delta^{\\frac{1}{\\eta}}, \\zeta_2, \\zeta_3) = \\rho_j(d\\delta^{\\frac{1}{\\eta}},\\ \\zeta_2,\\ \\zeta_3 - 2 c_j({\\widetilde e}_\\delta)\\zeta_2^j).\n\\end{equation}\nIf $k < j,$ the inductive assumption gives\n$$\\biggl|\\frac{\\partial^k \\rho_{j+1}}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial {\\bar \\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| = \\biggl| \\frac{\\partial^k \\rho_{j}}{{\\partial w_2^{\\alpha_2}}{\\partial {\\bar w_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{t_k}{\\eta}}.$$\nNow, let $k = j.$ If $\\alpha_2 > 0$ and $\\beta_2 > 0,$ we have the same result as the previous one. Otherwise, $\\frac{\\partial^j \\rho_{j+1}}{\\partial \\zeta_2^j}(d\\delta^{\\frac{1}{\\eta}}, 0, 0) = \\frac{\\partial^j \\rho_{j}}{\\partial w_2^{j}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0) - 2j!c_j({\\widetilde e}_\\delta)\\frac{\\partial \\rho_j}{\\partial w_3}(d\\delta^{\\frac{1}{\\eta}}, 0, 0) = 0.$ \nIf $k > j,$ the inductive assumption gives\n\\begin{equation*}\n\\biggl|\\frac{\\partial^k \\rho_{j+1}}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial \\bar{\\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\biggl| \\frac{\\partial^k \\rho_{j}}{{\\partial w_2^{\\alpha_2}}{\\partial \\bar{w_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| + |c_j({\\widetilde e}_\\delta)| \\lesssim \\delta^{\\frac{t_k}{\\eta}}+\\delta^{\\frac{t_j}{\\eta}} \\lesssim \\delta^{\\frac{t_k}{\\eta}}.\n\\end{equation*} \n\n\nFor part (ii), let $l = 2$ and apply the chain rule again to $\\rho_2,$ we have \n\\begin{equation*}\n\\biggl|\\frac{\\partial^{q_\\nu} r}{{\\partial z_2^{\\alpha_2^\\nu}}{\\partial \\bar{z_2}^{\\beta_2^\\nu}}}(\\widetilde{e_\\delta})\\biggl| - C\\biggl|\\frac{\\partial r}{\\partial z_2}(\\widetilde{e_\\delta}) \\biggr| \\leq \\biggl|\\frac{\\partial^{q_\\nu} \\rho_2}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\leq \\biggl|\\frac{\\partial^{q_\\nu} r}{{\\partial z_2^{\\alpha_2^\\nu}}{\\partial \\bar{z_2}^{\\beta_2^\\nu}}}(\\widetilde{e_\\delta})\\biggl| + C\\biggl|\\frac{\\partial r}{\\partial z_2}(\\widetilde{e_\\delta}) \\biggr| \n\\end{equation*}\nfor some constant $C.$ Then, Lemma \\ref{rderivative} means \n\\begin{equation}\\label{lowerbound2}\n\\delta^{\\frac{p_\\nu}{\\eta}} - \\delta^{\\frac{t_1}{\\eta}} \\lesssim \\biggl|\\frac{\\partial^{q_\\nu} \\rho_2}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{p_\\nu}{\\eta}} + \\delta^{\\frac{t_1}{\\eta}}.\n\\end{equation}\nSince $1 < q_\\nu$ for each $\\nu = 1, \\cdots, N,$ it gives $p_\\nu = t_{q_\\nu} < t_1.$ Therefore, we have \n\\begin{equation*}\n\\biggl|\\frac{\\partial^{q_\\nu} \\rho_2}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.\n\\end{equation*}\nThis proves the statement for the case $l = 2.$ By induction, assume\n$\\biggl|\\frac{\\partial^{q_\\nu} \\rho_l}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.$\nFirst, consider the case when $q_\\nu \\leq l.$ Since $\\alpha_2^\\nu > 0$ and $\\beta_2^\\nu > 0,$ by the similar argument as in the proof of (i) and the by inductive assumption, we have $$\\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| = \\biggl| \\frac{\\partial^{q_\\nu} \\rho_{l}}{{\\partial w_2^{\\alpha_2^\\nu}}{\\partial {{\\bar w}_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{t_{q_\\nu}}{\\eta}} = \\delta^{\\frac{{p_\\nu}}{\\eta}} .$$ \nNow, consider the case when $q_\\nu > l,$ If we take the derivative of $\\rho_{l+1}$ in (\\ref{rholrealexpression}) about $\\zeta_2,$ the derivative related to the third component involves $c_l(\\widetilde{e_\\delta}).$ Therefore, we have\n\\begin{align*}\n\\biggl| \\frac{\\partial^{q_\\nu} \\rho_{l}}{{\\partial w_2^{\\alpha_2^\\nu}}{\\partial \\bar{w_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| - C'|c_l(\\widetilde{e_\\delta})| \\leq \\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| & \\leq \\biggl| \\frac{\\partial^{q_\\nu} \\rho_{l}}{{\\partial w_2^{\\alpha_2^\\nu}}{\\partial \\bar{w_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\\\ \n& + C'|c_l(\\widetilde{e_\\delta})|\n\\end{align*}\nfor some constant $C'$. \nTherefore, the inductive assumption and part (i) means \n\\begin{equation}\\label{lowerboundl}\n\\delta^{\\frac{p_\\nu}{\\eta}} - \\delta^{\\frac{t_l}{\\eta}} \\lesssim \\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{p_\\nu}{\\eta}} - \\delta^{\\frac{t_l}{\\eta}}\n\\end{equation}\nSince $q_\\nu > l$, it means $p_\\nu = t_{q_\\nu} < t_l.$ Thus, we have \n$\\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.$\n\n\\end{proof}\n\nFinally, we show that the derivatives of $\\rho$ can be bounded from below.\n\n\n\n\\begin{rem}\\label{Amnonzero}\nTake $\\nu = N.$ Since $\\biggl|\\frac{\\partial^{q_\\nu} \\rho }{{\\partial {\\zeta_2}^{\\alpha_2^\\nu}}{\\partial {\\bar \\zeta_2}^{\\beta_2^\\nu}} }(d\\delta^{\\frac{1}{\\eta}}, 0, 0) \\biggr| \\approx |A_m ({\\widetilde e}_\\delta)| ,$ Lemma \\ref{rhoderivative} means $|A_m (\\widetilde{e_\\delta})| \\approx 1.$\n\\end{rem} \n\n\nNow, we recall some facts in \\cite{C2} before showing the holomorphic function defined in the complex two dimensional slice(i.e $z_1$ is fixed) is well-defined when we move $z_1$ in a small neighborhood of $z_1 = d\\delta^{\\frac{1}{\\eta}}.$\n\n\\begin{thm}[\\bf Catlin]\\label{existenceofholomorphic}\nSuppose the defining function $\\rho$ for a pseudoconvex domain in $b\\Omega \\subset \\mathbb{C}^2$ has the following form: $$\\rho(\\zeta) = \\mbox{\\normalfont{Re}}\\zeta_2 + \\sum_{\\substack{j+k=2 \\\\ j,k > 0 }}^m a_{j,k}{\\zeta_1}^j{\\bar{\\zeta_1}}^k + \\mathcal{O}(|\\zeta_2||\\zeta|+|\\zeta_1|^{m+1}).$$\nSet $$A_l = \\max \\{|a_{j, k}| ; j+k =l \\}, \\qquad l = 2, \\cdots, m.$$ and $$J_\\delta(\\zeta) = (\\delta^2 + |\\zeta_2|^2 + \\sum\\limits_{k=2}^m (A_k)^2 |\\zeta_1|^{2k} )^{\\frac{1}{2}}.$$\nDefine\n$$\\Omega_{a, \\delta}^{\\epsilon_0} = \\{\\zeta ; |\\zeta_1| < a , |\\zeta_2| 0.$}$$\nIf we have $|A_m| \\geq c_m > 0$ for some positive constant $c_m,$\nthen there exist small constants $a, \\epsilon_0 > 0$ so that for any sufficiently small $\\delta>0,$ there is a $L^2$ holomorphic function $f \\in A(\\Omega_{a, \\delta}^{\\epsilon_0})$ satisfying $\\biggl|\\frac{\\partial f}{\\partial\\zeta_2}(0, -\\frac{b\\delta}{2}) \\biggr| \\geq \\frac{1}{2\\delta}$ for some small constant $b$. Moreover, the values $a$ and $\\epsilon_0$ depend only on the constant $c_m$ and $C_{m+1} = \\left\\| \\rho \\right\\|_{C^{m+1}(U)},$ where $U$ is a small neighborhood of $0.$\n\\end{thm}\n \nThe result stated in \\cite{C2} applies to a more restricted situation, but a careful examination of the proof actually implies the above result. To apply theorem \\ref{existenceofholomorphic} to the complex two dimensional slice, we consider the pushed out domain about ${\\widetilde e}_\\delta.$ Let $\\Phi_{\\widetilde{e}_\\delta}$ be the map associated with ${\\widetilde e}_\\delta$ as in (\\ref{coordinatechangeinc2}). Set ${U'' \\big|}_{z_1 = d\\delta^{\\frac{1}{\\eta}}} = \\{\\zeta''=(\\zeta_2, \\zeta_3) ; \\Phi_{\\widetilde{e}_\\delta} (\\zeta'') \\in U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}}\\}.$ For all small $\\delta,$ define \n\\begin{equation} \\label{def of Jdelta}\n J_\\delta (\\zeta'') = \\biggl(\\delta^2 + |\\zeta_3|^2 + \\sum_{k = 2}^{m} (A_k ({\\widetilde e}_\\delta))^2 |\\zeta_2|^{2k} \\biggr)^{\\frac{1}{2}}\n\\end{equation} \nand the pushed-out domain with respect to the slice \n\\begin{equation} \\label{working domain} \n \\Omega_{a, \\delta}^{\\epsilon_0} = \\{(\\zeta_2, \\zeta_3) ; |\\zeta_2| < a, |\\zeta_3| < a \\ \\mbox{and} \\ \\rho (d\\delta^{\\frac{1}{\\eta}}, \\zeta'') < {\\epsilon_0} J_\\delta(\\zeta'') \\}. \n\\end{equation} \nBy Theorem \\ref{existenceofholomorphic}, we have a $L^2$ holomorphic function $f$ in $ \\Omega_{a, \\delta}^{\\epsilon_0}$ satisfying\n\\begin{equation}\\label{large derivative}\n \\biggl| \\frac{\\partial f}{\\partial \\zeta_3} ( 0, -\\frac{b\\delta}{2})\\biggr| \\geq \\frac{1}{2\\delta}.\n\\end{equation}\nIn order to show the well-definedness of the holomorphic function $f$ when $z_1$ moves in a small neighborhood of $z_1 = d\\delta^{\\frac{1}{\\eta}}$, we use $\\Phi_{\\widetilde{e}_\\delta} $ given as in (\\ref{coordinatechangeinc2}) and define\n\\begin{equation*}\n\\Phi(\\zeta_1, \\zeta_2, \\zeta_3) = (\\zeta_1, \\zeta_2, \\Phi_3 (\\zeta)), \n\\end{equation*}\nwhere $\\Phi_3 (\\zeta)$ is defined by\n\n\\begin{equation}\\label{phitobeused}\n \\Phi_3 (\\zeta) = e_\\delta + \\biggl(\\frac{\\partial r}{\\partial z_3}({\\widetilde e}_\\delta) \\biggr)^{-1}\\biggl(\\frac{\\zeta_3}{2}- \\sum_{l=2}^m c_l({\\widetilde e}_\\delta)\\zeta_2^l - \\frac{\\partial r}{\\partial z_2} ({\\widetilde e}_\\delta){\\zeta_2} \\biggr) \n\\end{equation}\nand define\n\n\\begin{equation}\\label{rhotobeused} \n \\rho(\\zeta_1, \\zeta_2, \\zeta_3) = r(z_1, z_2, z_3)\\circ \\Phi(\\zeta_1, \\zeta_2, \\zeta_3).\n\\end{equation}\nIn particular, when we fix $z_1 = d\\delta^{\\frac{1}{\\eta}},$ we have the holomophic function $f$ defined in the slice $\\Omega_{a, \\delta}^{\\epsilon_0}$ satisfying (\\ref{large derivative}). Now, we consider the domain given by the family of the pushed out domains of the slice along with $\\zeta_1$ axis and the domain in the new coordinate of $\\Omega$ by $\\Phi$. \nDefine $$\\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0} = \\{\\zeta \\in \\mathbb{C}^3; |\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a, |\\zeta_3| < a \\ \\mbox{and} \\ \\rho (d \\delta^{\\frac{1}{\\eta}}, \\zeta'') < {\\epsilon_0} J_\\delta(\\zeta'') \\}$$ and \n$${\\Omega}_{a, \\delta, \\zeta_1} = \\{\\zeta \\in \\mathbb{C}^3; |\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a, |\\zeta_3| < a \\ \\mbox{and} \\ \\rho (\\zeta_1, \\zeta'') < 0 \\} $$ for some small $c > 0$ only depending on $\\epsilon_0.$ \nSince the holomorphic function $f(\\zeta_2, \\zeta_3)$ defined in $\\Omega_{a, \\delta}^{\\epsilon_0}$ is independent of $\\zeta_1,$ $f$ is the well-defined holomophic function in $\\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0}.$ \nWe want to show $f$ is well-defined holomorphic function in ${\\Omega}_{a, \\delta, \\zeta_1}.$ Therefore, it is enough to show ${\\Omega}_{a, \\delta, \\zeta_1} \\subset \\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0}$ for the well-definedness of $f$ in ${\\Omega}_{a, \\delta, \\zeta_1}$. More specifically, \n\\begin{align*}\n{\\Omega}_{a, \\delta, \\zeta_1} \\subset \\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0}\n \n &\\ \\Leftrightarrow \\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') - \\rho (\\zeta_1, \\zeta'') < {\\epsilon_0}J_\\delta(\\zeta''),\n\\end{align*}\nwhere $\\zeta'' = (\\zeta_2, \\zeta_3)$ and $|\\zeta_1-d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a \\ \\mbox{and} \\ |\\zeta_3| < a$. \n\n\n\n\\begin{prop} \\label{well-defined property}\nGiven any small $\\epsilon \\leq {\\epsilon_0},$ there is a small $c > 0$ such that if $|\\zeta_1-d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a \\ \\mbox{and} \\ |\\zeta_3| < a,$ then\n $$|\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') - \\rho (\\zeta_1, \\zeta'')| \\lesssim {\\epsilon}J_\\delta (\\zeta'').$$ \n\\end{prop}\n\n\nBefore proving Proposition \\ref{well-defined property}, we note that from the standard interpolation method, we have the following fact: Let $(p_1, q_1), (p, q)$ and $(p_2, q_2)$ be collinear points in the first quadrant of the plane, and $ p_1 \\leq p \\leq p_2, q_2 \\leq q \\leq q_1.$ Then, we have\n$$|\\zeta_1|^{p}|\\zeta_2|^q \\leq |\\zeta_1|^{p_1}|\\zeta_2|^{q_1} + |\\zeta_1|^{p_2}|\\zeta_2|^{q_2}$$ for sufficiently small $\\zeta_1 , \\zeta_2 \\in \\mathbb{C}$. In particular, this means that if $(\\alpha, \\beta) \\in \\Gamma_L,$ then \n\\begin{equation} \\label{interpolation}\n|\\zeta_1|^{\\alpha_1 + \\beta_1} |\\zeta_2|^{\\alpha_2 + \\beta_2} \\lesssim |\\zeta_1|^{p_{\\nu-1}} |\\zeta_2|^{q_{\\nu-1}} + |\\zeta_1|^{p_{\\nu}} |\\zeta_2|^{q_{\\nu}} \n\\end{equation} for some $\\nu = 1, \\cdots, N.$ \n \n\\begin{proof}[Proof of Proposition \\ref{well-defined property}]\nDefine \n $${J_\\delta}^\\nu(\\zeta'') = \\delta + |\\zeta_3| + \\sum_{\\nu = 1}^N {\\delta^{\\frac{p_\\nu}{\\eta}}}|\\zeta_2|^{q_\\nu}.$$\n\nIn order to show the proposition, it is enough to show ${J_\\delta}^\\nu(\\zeta'') \\lesssim J_\\delta (\\zeta'')$ and $|\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta_2, \\zeta_3) - \\rho (\\zeta_1, \\zeta_2, \\zeta_3)| \\lesssim {\\epsilon}{J_\\delta}^\\nu (\\zeta''),$ where $|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a \\ \\mbox{and} \\ |\\zeta_3| < a.$ \nBy (\\ref{def of Al}) and $a_{j,k}(\\widetilde{e_\\delta}) = j!k! \\frac{\\partial^{j+k} \\rho}{{\\partial {\\zeta_2}^j}{\\partial {\\bar{\\zeta_2}}^k}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0),$ we have $$|\\frac{\\partial^{j+k} \\rho}{{\\partial {\\zeta_2}^j}{\\partial {\\bar{\\zeta_2}}^k}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)| \\lesssim |A_l(\\widetilde{e_\\delta})| $$ for $ j+k = l$ with $l =2,\\cdots,m.$\nTherefore, Lemma \\ref{rhoderivative} means that \n$$ \\delta^{\\frac{p_\\nu}{\\eta}} \\approx \\biggl|\\frac{\\partial^{q_\\nu} \\rho }{{\\partial {\\zeta_2}^{\\alpha_2^\\nu}}{\\partial {\\bar \\zeta_2}^{\\beta_2^\\nu}} }(d\\delta^{\\frac{1}{\\eta}}, 0, 0) \\biggr| \\lesssim |A_{q_\\nu}(\\widetilde{e_\\delta})|,$$\nwhere $\\alpha_2^\\nu + \\beta_2^\\nu = q_\\nu, \\alpha_2^\\nu \\ \\mbox{and} \\ \\beta_2^\\nu > 0.$ This shows ${J_\\delta}^\\nu(\\zeta'') \\lesssim J_\\delta (\\zeta'').$ \\\\\n\nLet's estimate $|\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') - \\rho (\\zeta_1, \\zeta'')|$. Let $D_1$ denote the differential operator either $\\frac{\\partial}{\\partial \\zeta_1}$ or $\\frac{\\partial}{\\partial {\\overline \\zeta}_1}.$ Then,\n\\begin{equation} \\label{originalestimate}\n|\\rho (\\zeta_1, \\zeta'')- \\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'')| \n \\leq c \\delta^{\\frac{1}{\\eta}} \\max\\limits_{|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}} |D_1 \\rho (\\zeta_1, \\zeta'')|.\n\\end{equation}\nLet's estimate $ D_1 \\rho (\\zeta_1, \\zeta'').$ By (\\ref{r form 2}), (\\ref{phitobeused}) and (\\ref{rhotobeused}), we know\n\\begin{align*}\n\\rho(\\zeta_1, \\zeta'') &= \\mbox{Re}(\\Phi_3 (\\zeta) ) + \\sum_{\\Gamma_L} a_{\\alpha, \\beta}{\\zeta_1}^{\\alpha_1}{\\bar{\\zeta}_1}^{\\beta_1}{\\zeta_2}^{\\alpha_2}{\\bar{\\zeta}_2}^{\\beta_2} + \\mathcal{O}(|\\Phi_3(\\zeta)||(\\zeta_1, \\zeta_2, \\Phi_3(\\zeta))| \\\\\n &\\ \\qquad +\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |\\zeta_1|^{[t_l]+1}|\\zeta_2|^l + |\\zeta_2|^{m+1}).\n\\end{align*}\nSince $|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}$ and $\\Phi_3$ is independent of $\\zeta_1$, we have\n\\begin{equation} \\label{rhozeta1derivative}\n |{D_1 \\rho}(\\zeta_1, \\zeta'')|\n \\lesssim \\sum_{\\Gamma_L}{\\delta}^{\\frac{\\alpha_1 + \\beta_1 -1}{\\eta}} |\\zeta_2|^{\\alpha_2 + \\beta_2} + |\\Phi_3 (\\zeta)| + \\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} \\delta^{\\frac{[t_l]}{\\eta}}|\\zeta_2|^l . \n\\end{equation}\nCombining (\\ref{originalestimate}) with (\\ref{rhozeta1derivative}), we obtain\n\\begin{equation}\\label{finalestimatetobeused} \n|\\rho (\\zeta_1, \\zeta'')- \\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'')| \\lesssim c\\biggl( \\sum_{\\Gamma_L}{\\delta}^{\\frac{\\alpha_1 + \\beta_1}{\\eta}}|\\zeta_2|^{\\alpha_2 + \\beta_2} + |\\Phi_3 (\\zeta)| \\nonumber + \\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} \\delta^{\\frac{[t_l]+1}{\\eta}}|\\zeta_2|^l \\biggr) \n\\end{equation}\n\n\n\\noindent With $\\zeta_1 = d\\delta^{\\frac{1}{\\eta}}$, (\\ref{interpolation}) means $\\sum\\limits_{\\Gamma_L}{\\delta}^{\\frac{\\alpha_1 + \\beta_1}{\\eta}}|\\zeta_2|^{\\alpha_2 + \\beta_2} \\lesssim {J_\\delta}^\\nu (\\zeta'').$ \nAlso, (\\ref{clexpression}) and Lemma \\ref{rhoderivative} gives $|\\Phi_3(\\zeta)| \\lesssim |e_\\delta| + |\\zeta_3| + \\sum_{l = 1}^m |c_l(\\widetilde{e_\\delta})||\\zeta_2|^l \\lesssim \\delta + |\\zeta_3| + \\sum_{l = 1}^m \\delta^{\\frac{t_l}{\\eta}}|\\zeta_2|^l.$ Since $(t_l, l ) \\in L_\\nu$ for some $\\nu = 1, \\cdots, N,$ again, (\\ref{interpolation}) gives $|\\Phi_3(\\zeta)| \\lesssim {J_\\delta}^\\nu (\\zeta'').$\nFurthermore, since $\\delta^{\\frac{[t_l]+1}{\\eta}}|\\zeta_2|^l \\lesssim \\delta^{\\frac{t_l}{\\eta}}|\\zeta_2|^l$, the same argument as before gives $\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} \\delta^{\\frac{[t_l]+1}{\\eta}}|\\zeta_2|^l \\lesssim {J_\\delta}^\\nu (\\zeta'').$ \n\\end{proof} \n\n\\vspace{0.5cm}\n\nNow, we know that there is a holomorphic function $f(\\zeta_1, \\zeta_2, \\zeta_3)= f(\\zeta_2, \\zeta_3)$ defined on ${\\Omega}_{a, \\delta, \\zeta_1}^{\\epsilon_0}$ such that\n\\begin{enumerate}[i)]\n\t\\item ${\\Omega}_{a, \\delta, \\zeta_1} \\subset {\\Omega}_{a, \\delta, \\zeta_1}^{\\epsilon_0}$\n\n\t\\item $\\biggl| \\frac{\\partial f}{\\partial \\zeta_3} (0, -\\frac{b\\delta)}{2}\\biggr| \\geq \\frac{1}{2\\delta}$ for a small constant $b > 0.$ \n\\end{enumerate}\n\nWithout loss of generality, we can assume ${\\Omega}_{a, \\delta, \\zeta_1} \\subset {\\Omega}_{a, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}} \\subset {\\Omega}_{a, \\delta, \\zeta}^{\\epsilon_0}.$ For the boundedness of $f$ in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}},$ we follow the same argument as Chapter 7 (p 462) in \\cite{C2}. Before showing the boundedness, we define a polydisc $P_{a_1} ({\\zeta''_0})$ by\n$$P_{a_1} (\\zeta''_0) = \\{\\zeta'' = (\\zeta_2, \\zeta_3); |\\zeta_2 - {\\zeta_2^0}| < \\tau (\\widetilde{e_\\delta}, a_1 J_\\delta (\\zeta''_0)) \\ \\mbox{and} \\\n |\\zeta_3 - {\\zeta_3^0}| < a_1 J_\\delta (\\zeta''_0)\\}, $$ \nwhere $\\zeta''_0 = (\\zeta_2^0, \\zeta_3^0)$ and $a_1 > 0.$\n\n\n\n\n\n\n\\begin{thm}\\label{boundedholomorphicfunction}\n$f$ is bounded holomorphic function in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}}$ such that \n\\begin{equation}\\label{largederivative}\n\\biggl| \\frac{\\partial f}{\\partial \\zeta_3} \\biggl(0, -\\frac{b\\delta}{2}\\biggr)\\biggr| \\geq \\frac{1}{2\\delta} \\ \\text{for a small constant $b > 0$}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nSince $f$ is a $L^2$ holomorphic function in ${\\Omega}_{a, \\delta, \\zeta_1}^{\\epsilon_0}$ with (\\ref{largederivative}), it is enough to show $f$ is bounded in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}}$. Let $(\\zeta_2^0, \\zeta_3^0) \\in \\{\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = \\frac{\\epsilon_0}{2}J_\\delta(\\zeta''), |\\zeta_2| < \\frac{3a}{4}, |\\zeta_3|<\\frac{3a}{4}\\} \\subset {\\Omega}_{a, \\delta, \\zeta}^{\\epsilon_0}.$ By the similar property as (iii) of Proposition 4.3 in \\cite{C2}, if $\\zeta''_0 = (\\zeta_2^0, \\zeta_3^0) \\in \\{\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = \\frac{\\epsilon_0}{2}J_\\delta(\\zeta''), |\\zeta_2| < \\frac{3a}{4}, |\\zeta_3|<\\frac{3a}{4}\\}$, then\n$$ P_{a_1} (\\zeta''_0) \\subset {\\Omega}_{a, \\delta, \\zeta}^{\\epsilon_0},$$\nfor some small constant $a_1 > 0.$ We can apply the same argument as Chapter 7 (p 462) in \\cite{C2} to obtain $|f(\\zeta_2^0, \\zeta_3^0)| \\lesssim 1$. \nFor all others points on the boundary and interior of ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}}$, we can choose the polydics with fixed radius which is contained in ${\\Omega}_{{a}, \\delta, \\zeta_1}^{{\\epsilon_0}}$ and apply the same argument as Chapter 7 in \\cite{C2}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem 1.1} \\label{Sec5}\n\nIn this section, we prove our main theorem. Before proving the Theorem, let's recall the notations for H\\\"older norm and H\\\"older space. For $U \\in \\mathbb{C}^n$, we denote by ${\\lVert u \\rVert}_{L_{\\infty}(U)}$ the essential supremum of $u \\in L_{\\infty}(U)$ in $U$. For a real $0 < \\epsilon < 1$, set\n \n $${\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U)} = {\\lVert u \\rVert}_{L_{\\infty}(U)} + \\mbox{sup}_{z,w \\in U} \\frac{|u(w)-u(z)|}{|w-z|^\\epsilon}, $$ \n $$ \\Lambda^{\\epsilon} (U) = \\{ u : {\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U)} < \\infty \\} $$\nIn here, ${\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U)}$ denote the H\\\"older norm of order $\\epsilon$. \n\nBy theorem \\ref{special coordinate}, we can assume $\\Omega = \\{z \\in \\mathbb{C}^3; r(z)< 0\\}$ and restate Theorem \\ref{main_theorem}: \n\n\n\\begin{thm}\nLet $\\Omega = \\{ r(z) < 0\\}$ be a smoothly bounded pseudoconvex domain in $\\mathbb{C}^3,$ where $r$ given by theorem \\ref{special coordinate}. Furthermore,\nif there exists a neighborhood $U$ of $0$ so that for all $\\alpha \\in L_{\\infty}^{0,1} ({\\Omega})$ with $\\bar{\\partial}\\alpha = 0$, there is a $u \\in \\Lambda_{\\epsilon} (U \\cap \\overline{\\Omega})$ and $C>0$ such that $\\bar{\\partial}u =\\alpha$ and \n\n\\begin{equation} \\label{holder estimate in z}\n{\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U \\cap \\overline{\\Omega})} \\leq C{\\lVert \\alpha \\rVert}_{L_{\\infty}(\\Omega),}\n\\end{equation}\nthen $\\epsilon \\leq \\frac{1}{\\eta}$.\n\\end{thm}\n\n\n\n\n\n\\begin{proof}Let us consider $U' = \\{(\\zeta_1, \\zeta_2, \\zeta_3) ; \\Phi(\\zeta_1, \\zeta_2, \\zeta_3) \\in U \\}$ and $\\rho = r \\circ \\Phi$ as (\\ref{phitobeused}) and (\\ref{rhotobeused}). Let's choose $\\beta =\\bar{\\partial}(\\phi(\\frac{|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}|}{c\\delta^{\\frac{1}{\\eta}}})\\phi(\\frac{|\\zeta_2|}{a\/2})\\phi(\\frac{|\\zeta_3|}{a\/2})f(\\zeta_2, \\zeta_3))$, where \n\n\\begin{displaymath}\n\\phi (t) = \\left \\{\n \\begin{array}{lr}\n 1 & , |t| \\leq \\frac{1}{2}\\\\\n 0 & , |t| \\geq \\frac{3}{4}\n \\end{array}\n \\right.\n\\end{displaymath} \nNote that $f$ is the well-defined bounded holomorphic function in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon}{2}}$ by Theorem \\ref{boundedholomorphicfunction}.\nIf we define $\\alpha = (\\Phi^{-1})^* \\beta,$ then $\\bar{\\partial}(\\Phi^* u) = \\Phi^* \\bar{\\partial} u = \\Phi^* \\alpha = \\beta$. Therefore, if we set $U_1 = \\Phi^* u =u\\circ \\Phi$, (\\ref{holder estimate in z}) means \n\\begin{equation} \\label{holder estimate in zeta}\n{\\lVert U_1 \\rVert}_{\\Lambda^{\\epsilon}(U' \\cap \\overline{\\Omega})} \\leq C{\\lVert \\beta \\rVert}_{L_{\\infty}} \n\\end{equation}\nIn here, we note that the definition of $\\beta$ means\n\\begin{equation}\\label{supnorminofbeta}\n{\\lVert \\beta \\rVert}_{L^\\infty} \\lesssim \\delta^{-\\frac{1}{\\eta}}\n\\end{equation}\nNow, let $h(\\zeta_1, \\zeta_2, \\zeta_3) = U_1(\\zeta_1, \\zeta_2, \\zeta_3) - \\phi(\\frac{|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}|}{c\\delta^{\\frac{1}{\\eta}}})\\phi(\\frac{|\\zeta_2|}{a\/2})\\phi(\\frac{|\\zeta_3|}{a\/2})f(\\zeta_2, \\zeta_3).$ Then $\\bar{\\partial} U_1 = \\beta$ means $h$ is holomorphic.\nSet $ q_1^\\delta(\\theta)= (d\\delta^{\\frac{1}{\\eta}}+\\frac{4}{5}c\\delta^{\\frac{1}{\\eta}} e^{i\\theta}, 0, -\\frac{b\\delta}{2}) \\ \\mbox{and} \\ q_2^\\delta(\\theta) = ( d\\delta^{\\frac{1}{\\eta}}+\\frac{4}{5}c\\delta^{\\frac{1}{\\eta}}e^{i\\theta}, 0, -b\\delta)$, where $\\theta \\in \\mathbb{R}$.\nFrom now on, we estimate the lower bound and upper bound of the integral \n\n\\begin{equation*}\n H_{\\delta} = \\biggl| \\frac{1}{2\\pi} \\int_0^{2\\pi} [h(q_1^\\delta(\\theta))-h(q_2^\\delta(\\theta))] d\\theta \\biggr|. \n\\end{equation*}\nFrom the definition of $\\phi,$ (\\ref{holder estimate in zeta}), and (\\ref{supnorminofbeta}) we have \n\\begin{equation} \\label{upperbound} \n H_{\\delta} = \\biggl|\\frac{1}{2\\pi} \\int_0^{2\\pi} [U_1(q_1^\\delta (\\theta))-U_1(q_2^\\delta (\\theta))] d\\theta \\biggr| \\lesssim \\delta^{\\epsilon} {\\lVert \\beta \\rVert}_{L^\\infty} \\lesssim \\delta^{\\epsilon-\\frac{1}{\\eta}} \n\\end{equation} \n\n\n\n\nOn the other hand, for the lower bound estimate, we start with an estimate of the holomorphic function $f$ with a large nontangential derivative we constructed in theorem \\ref{boundedholomorphicfunction}. The Taylor's theorem of $f$ in $\\zeta_3$ and Cauchy's estimate means \n\n $$f(0, \\zeta_3) = f(0, -\\frac{b\\delta}{2}) + \\frac{{\\partial{f}}}{{\\partial{\\zeta_3}}}(0, -\\frac{b\\delta}{2})(\\zeta_3 + \\frac{b\\delta}{2})\n + \\mathcal{O}(|\\zeta_3 + \\frac{b\\delta}{2}|^2). $$\nNow, if we take $\\zeta_3 = -b\\delta$, we have\n $$ f(0, -b\\delta) - f(0, -\\frac{b\\delta}{2})= \\frac{{\\partial{f}}}{{\\partial{\\zeta_3}}}(0, -\\frac{b\\delta}{2})(-\\frac{b\\delta}{2})\n + \\mathcal{O}(\\delta^2).$$ \nSince $|\\frac{\\partial{f}}{\\partial{z_3}} (0, -\\frac{b\\delta}{2} )| \\geq \\frac{1}{2\\delta},$ we know\n\\begin{equation} \\label{contradictionequation}\n |f(0, -b\\delta) - f(0, -\\frac{b\\delta}{2})| = \\biggl|\\frac{{\\partial{f}}}{{\\partial{\\zeta_3}}}(0, -\\frac{b\\delta}{2})(-\\frac{b\\delta}{2})\n + \\mathcal{O}(\\delta^2)\\biggr| \\gtrsim 1 \n\\end{equation} \nfor all sufficiently small $\\delta > 0$.\nReturning to the lower bound estimate of $H_{\\delta},$ the Mean Value Property, (\\ref{holder estimate in zeta}), (\\ref{supnorminofbeta}), and (\\ref{contradictionequation}) give\n\\begin{align}\n H_{\\delta} &= \\biggl| \\frac{1}{2\\pi} \\int_0^{2\\pi} [h(q_1^\\delta (\\theta))) \n -h(q_2^\\delta (\\theta)) ]d\\theta \\biggr| = \\left|h(d\\delta^{\\frac{1}{\\eta}}, 0, -\\frac{b\\delta}{2} )- h(d\\delta^{\\frac{1}{\\eta}}, 0, -b\\delta) )\\right| \\nonumber \\\\\n &= \\left|U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -\\frac{b\\delta}{2}) - f(0, -\\frac{b\\delta}{2})- U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -b\\delta) + f(0, -b\\delta)\\right| \\nonumber \\\\\n &\\geq \\left|f(0, -b\\delta)-f(0, -\\frac{b\\delta}{2})| -|U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -\\frac{b\\delta}{2})-U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -b\\delta)\\right| \\nonumber \\\\\n &\\gtrsim 1 - \\delta^{\\epsilon-\\frac{1}{\\eta}} \\label{lowerbound} \n \\end{align}\nIf we combine (\\ref{upperbound}) with (\\ref{lowerbound}), we have \n \n\\begin{equation} \\label{last_estimate}\n 1 \\lesssim \\delta^{\\epsilon-\\frac{1}{\\eta}}. \n\\end{equation}\nIf we assume $\\epsilon > \\frac{1}{\\eta}$ and $\\delta \\rightarrow 0$, (\\ref{last_estimate}) will be a contradiction. Therefore, $\\epsilon \\leq \\frac{1}{\\eta}.$ \n\\\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nModern advancement of machine learning is heavily driven by the advancement of computational power and techniques. Nowadays, it is not unusual to train a single model using hundreds of computational devices such as GPUs. As a result, scaling up training algorithms in the distributed setting has attracted intensive interests over the years. One important direction is communication efficient distributed training, which enhances the scalability of the training system by reducing the communication cost. Example techniques include quantization~\\citep{pmlr-v70-zhang17e,Wangni2018-ux}, decentralization~\\citep{Lian2017-ni, Koloskova*2020Decentralized, NIPS2018_8028}, and asynchronous communication~\\citep{DBLP:journals\/corr\/ZhengMWCYML16, NIPS2015_6031}.\n\nOne widely used strategy for alleviating the communication overhead is gradient compression, Before communication, the original gradient $\\bm{g}$ will be compressed into $\\mathcal{C}_{\\omega}[\\bm{g}]$, where $\\mathcal{C}_{\\omega}[\\cdot]$ {\\footnote{$\\mathcal{C}_{\\omega}[\\cdot]$ could also include randomness.}} is the compress operator. As a result the communication volume could be greatly reduced. However, this gradient compression could slow down the convergence speed because important information might get lost during the compression. To recover this information lost, error-compensated compression strategy was proposed: Instead of compressing the gradient at $t$-th iteration directly, we would first add back the compression error from the last step and then do the compression. Recent studies \\citep{martinmemory}\nobserved that by using error-compensated compression, the asymptotic convergence speed remains unchanged for \\textbf{SGD} even using 1-bit compression.\n\n\nOn the other hand, many state-of-the-art models have to be trained using a more complicated variant, \\textbf{Adam} \\citep{adam}. For example, to train models such as BERT, one has to resort to the \\textbf{Adam} optimizer, since training it with vanilla\/momentum \\textbf{SGD} has been shown to be less effective. Unfortunately, we find that error-compensated compression does not work for \\textbf{Adam}, because \\textbf{Adam} is non-linearly dependent on the gradient which affects the error compensation mechanism (see Section~\\ref{sec:moti-convergence} and~\\ref{intuition:why_adam_fails} for more details).\n\nIn this paper, we first analyze the limitation of directly applying existing compression technique to \\textbf{Adam}. One of our key findings is that Adam's variance (the non-linear term) becomes stable at early stage of training (Section~\\ref{sec:moti-variance}). This motivates us to design a new 2-stage algorithm, {{{\\textbf{1-bit Adam}}}}, which uses \\textbf{Adam} (warmup stage) to ``pre-condition'' a communication compressed momentum \\textbf{SGD} algoirthm (compression stage). We provide theoretical analysis on communication compressed momentum \\textbf{SGD}, which is the core component of {{{\\textbf{1-bit Adam}}}}. We design a custom collective primitive using MPI to transfer the $5\\times$ communication volume reduction (achieved by our algorithm) into actual runtime speedup, which is hard to accomplish using existing DL framework libraries. Experiments with BERT-Base, BERT-Large, SQuAD 1.1 and ResNet-18 training tasks on up to 256 GPUs show that {{{{\\textbf{1-bit Adam}}}}} converges as fast as uncompressed \\textbf{Adam}, and runs up to $3.3\\times$ faster than uncompressed algorithms.\n\n{\\bf (Contributions)}\nWe make the following contributions:\n\\vspace{-0.3cm}\n\\begin{itemize}\n\\item We propose a new algorithm, {{{\\textbf{1-bit Adam}}}}, a communication efficient momentum \\textbf{SGD} algorithm pre-conditioned with \\textbf{Adam} optimizer, which to the best of our knowledge is the first work that apply a pre-conditioned strategy for compressed momentum \\textbf{SGD}. We present theoretical analysis on the convergence of {{{\\textbf{1-bit Adam}}}}, and show that it admits the same asymptotic convergence rate as the uncompressed one.\n\n\\item We conduct experiments on large scale ML tasks that are currently challenging for \\textbf{SGD} to train. We show that on both BERT pre-training, SQuAD fine-tuning and ResNet-18, {{{\\textbf{1-bit Adam}}}} is able to achieve the same convergence behaviour and final accuracy as \\textbf{Adam}, together with up to $5\\times$ less communication volume and $3.3\\times$ faster end-to-end throughput (including the full-precision warmup stage). To our best knowledge, this is the first distributed learning algorithm with communication compression that can train a model as demanding as BERT.\n\n\\item We implement a custom collective communication primitive using Message Passing Interface (MPI) to provide a scalable and efficient communication system for 1-bit Adam. The communication as well as the 1-bit Adam optimizer has been open sourced in a deep learning optimization library called DeepSpeed\\footnote{https:\/\/github.com\/microsoft\/DeepSpeed, https:\/\/www.deepspeed.ai\/}.\n\\end{itemize}\n\n\\section{Related Work}\n\\paragraph{Communication-efficient distributed learning:}\nTo further reduce the communication overhead, one promising direction is to compress the variables that are sent between different workers ~\\citep{NIPS2019_8694,NIPS2019_9473}. Previous work has applied a\nrange of techniques such as quantizaiton,\nsparsification, and sketching\n~\\citep{Alistarh2017-yh,Agarwal2018-hg,Spring2019-ep,Ye2018-mf}.\nThe compression is mostly assumed to be unbiased ~\\citep{Wangni2018-ux,pmlr-v80-shen18a,pmlr-v70-zhang17e,NIPS2017_6749,NIPS2018_7519}.\nA general theoretical analysis of centralized compressed parallel \\textbf{SGD} can be found in ~\\citet{Alistarh2017-yh}. Beyond this, some biased compressing methods are also proposed and proven to be quite efficient in reducing the communication cost. One example is the \\textbf{1-bit SGD} ~\\citep{1-bitexp}, which compresses the entries in gradient vector into $\\pm 1$ depends on its sign. \n\n\\paragraph{Error-compensated compression:}\nThe idea of using error compensation for compression is proposed in ~\\citet{1-bitexp}, where they find that by using error compensation the training could still achieves a very good speed even using $1$-bit compression. Recent study indicates that this strategy admits the same asymptotic convergence rate as the uncompressed one~\\citep{martinmemory}, which means that the influence of compression is trivial. More importantly, by using error compensation, it has been proved that we can use almost any compression methods~\\citep{martinmemory}, whereas naive compression could only converge when the compression is unbiased (the expectation of the compressed tensor is the same as the original).This method can be combined with decentralized training \\citep{ec_decentralize}, local SGD \\citep{ec_local}, accelerated algorithms \\citep{ec_linearly}. Due to the promising efficiency of this method, error compensation has been applied into many related area ~\\citep{NIPS2019_9321,9051706,NIPS2019_8694,8884924,NIPS2019_9473,NIPS2019_8598,NIPS2019_9610,NIPS2019_9571} in order to reduce the communication cost. \n\n\\paragraph{\\textbf{Adam}:} \\textbf{Adam}~\\citep{Kingma2015AdamAM} has shown\npromising speed for many deep learning tasks, and also admits a very good robustness to the choice of the hyper-parameters, such as learning rate. \nIt can be viewed as an adaptive method that scales the learning rate with the magnitude of the gradients on each coordinate when running \\textbf{SGD}. Beyond \\textbf{Adam}, many other strategies that that shares the same idea of changing learning rate dynamically was studied. For example, \\citet{JMLR:v12:duchi11a} (\\textbf{Adagrad}) and \\citep{rmsprop} (\\textbf{RESprop}), use the gradient, instead of momentum, for updating the parameters; \\textbf{Adadelta}~\\citep{DBLP:journals\/corr\/abs-1212-5701} changes the variance term of \\textbf{Adam} into a non-decreasing updating rule; \\citet{luo2018adaptive} proposed \\textbf{AdaBound} that gives both upper bound and lower bound for the variance term. In \\citet{adam_theoretical,adam_liu2020adam} authors develop a novel analysis for the convergence rate of \\textbf{Adam}. \n\n\\section{Motivation and Insights}\n\\subsection{Communication overhead affects the efficiency of distributed training}\n\\label{sec:moti-profile}\nTo demonstrate the opportunity for communication compression, we conduct performance profiling experiments that measures the impact of communication time with respect to the total training time per step. Here we use BERT-Large pre-training task as an example (sequence length 128, detailed training parameters can be found at Section~\\ref{sec:bert-eval}), since BERT and transformer models in general are the state-of-the-art approaches in natural language processing and many other areas. We evaluate two different kinds of clusters: the first cluster has 4 NVIDIA Tesla V100 GPUs per node, and different nodes are connected by 40 Gigabit Ethernet (effective bandwidth is 4.1 Gbps based on iperf benchmark); the second cluster has 8 V100 GPUs per node, and different nodes are connected by 100 Gigabit InfiniBand EDR (effective bandwidth is close to theoretical peak based on microbenchmark). We perform BERT-Large pre-training using the two clusters with different number of nodes and GPUs, batch sizes, and gradient accumulation steps. And we measure the average latency of forward, backward (allreduce and everything else), and step function calls. Table~\\ref{table_comm_overhead} presents the profiling results.\n\nResults show that allreduce communication contributes to a great portion of the training time per step, up to 94\\% and 75\\% for our experiments on two different kinds of inter-node networks. As expected, communication overhead is proportionally larger when the number of nodes is larger, when the batch size\/gradient accumulation step is smaller, and when the network bandwidth is lower. These are the situations where communication compression could provide the most benefit.\n\n\\begin{table*}\n \\footnotesize\n \\caption{BERT-Large pre-training sequence 128 profiling results.}\\label{table_comm_overhead}\n \\centering\n \\begin{tabular}{rrrrrrrrrrr}\n \\hline\n Cluster& Num.& Num.& Batch& Batch& Grad& Forward& Backward& Backward& Step& allreduce\\% \\\\\n Network& node& GPU& size per& size& accum.& (ms)& allreduce& everything& (ms)& \\\\\n Type& & & GPU& & step& & (ms)& else (ms)& & \\\\\n \\hline\n Ethernet& 16& 64& 1& 64& 1& 36.65& 2205.86& 33.63& 74.96& \\textbf{94\\%} \\\\\n Ethernet& 16& 64& 16& 1024& 1& 35.71& 2275.43& 60.81& 75.59& 93\\% \\\\\n Ethernet& 16& 64& 16& 4096& 4& 137.80& 2259.36& 243.72& 74.92& 83\\% \\\\\n Ethernet& 8& 32& 16& 512& 1& 37.91& 2173.35& 60.71& 75.63& 93\\% \\\\\n Ethernet& 4& 16& 16& 256& 1& 36.94& 2133.24& 62.82& 76.85& 92\\% \\\\\n Ethernet& 2& 8& 16& 128& 1& 34.95& 1897.21& 61.23& 75.26& 92\\% \\\\\n Ethernet& 1& 4& 16& 64& 1& 35.99& 239.76& 59.95& 74.21& 58\\% \\\\\n \\hline\n InfiniBand& 8& 64& 1& 64& 1& 25.36& 316.18& 23.25& 58.49& \\textbf{75\\%} \\\\\n InfiniBand& 8& 64& 16& 1024& 1& 32.81& 336.40& 59.99& 57.79& 69\\% \\\\\n InfiniBand& 8& 64& 16& 4096& 4& 131.04& 339.52& 237.92& 56.91& 44\\% \\\\\n InfiniBand& 4& 32& 16& 512& 1& 33.45& 297.28& 56.81& 57.98& 67\\% \\\\\n InfiniBand& 2& 16& 16& 256& 1& 32.86& 183.74& 56.49& 58.60& 55\\% \\\\\n InfiniBand& 1& 8& 16& 128& 1& 32.74& 28.18& 59.73& 57.29& 16\\% \\\\\n \\hline\n \\end{tabular}\\vspace{-0.1cm}\n\\end{table*}\n\n\\subsection{Basic compression affects \\textbf{Adam}'s convergence}\n\\label{sec:moti-convergence}\nGiven the great opportunity for communication compression, we investigate whether existing Error-Compensated gradient compression strategy can be applied to \\textbf{Adam}, an important optimization algorithm for large model distributed training. We implement a basic compression strategy for \\textbf{Adam} based on the compression-based \\textbf{SGD} approach~\\citep{martinmemory}, where we perform error-compensated 1-bit compression over the gradient, and update both the momentum and variance based on the compressed gradient. We compare the BERT-Large pre-training (sequence 128) training loss when using vanilla \\textbf{Adam} and \\textbf{Adam} with our basic compression strategy in Figure~\\ref{fig:moti_loss}.\n\nResults show that basic compression based on existing work greatly affects the convergence speed for Adam. The main reason is that \\textbf{Adam} is non-linearly dependent to the gradients (see Section~\\ref{intuition:why_adam_fails} for more details). This motivates us to look for new compression strategy that overcomes the non-linear gradient dependency challenge, and at the same time achieves the same convergence speed as \\textbf{Adam}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{moti_loss.pdf}\n\\caption{Training loss for BERT-Large pre-training using vanilla Adam and Adam with error compensated gradient compression.}\\label{fig:moti_loss}\\vspace{-0.1cm}\n\\end{figure}\n\n\\subsection{\\textbf{Adam}'s variance becomes stable during training}\n\\label{sec:moti-variance}\nUnlike \\textbf{SGD}, which directly uses the gradient $\\bm{g}$ to update the model $\\bm{x}$, \\textbf{Adam} uses two auxiliary variables $\\bm{m}$ and $\\bm{v}$ for the update. The mathematical updating rule of original \\textbf{Adam} can be summarized as:\n\\begin{align*}\n\\bm{m}_{t+1} =& \\beta_1\\bm{m}_t + (1-\\beta_1)\\bm{g}_t\\\\\n\\bm{v}_{t+1} =& \\beta_2\\bm{v}_t + (1-\\beta_2)(\\bm{g}_t)^2,\\numberthis\\label{alg:v}\\\\\n\\bm{\\bm{x}}_{t+1} =& \\bm{x}_t - \\gamma\\frac{\\bm{m}_{t+1}}{\\sqrt{\\bm{v}_{t+1} + \\eta}}\n\\end{align*}\nHere $\\bm{x}_t$ is the model at $t$-iteration, $\\bm{g}_t = \\nabla F(\\bm{x}_t;\\bm{\\zeta}_t)$ is the stochastic gradient, $\\gamma$ is the learning rate, $\\eta$ usually is a very small constant, $\\beta_1$ and $\\beta_2$ are decaying factor that controls the speed of forgetting history information. Notice here we disable the bias correction term in the original \\textbf{Adam}, which is consistent with exact optimizer for training BERT \\citep{bert}.\n\nHere we refer $\\bm{m}_t$ as the momentum term and $\\bm{v}_t$ as the variance term. Notice that when $\\bm{v}_t$ is changed into a constant $\\bm{v}$, then \\textbf{Adam} becomes equivalent to \\textbf{Momentum SGD} under a coordinate-dependent learning rate $\\frac{\\gamma}{\\sqrt{\\bm{v}} + \\eta}$.\n\n\nTo investigate the non-linear gradient dependency challenge, we analyze \\textbf{Adam}'s variance during BERT-Large pre-training (sequence 128). At each step, we fuse the variance of all parameters, and calculate the norm of the fused variance. Figure~\\ref{fig:moti_var_norm_log} presents this fused variance norm at each step. Results show that the variance norm becomes stable after around $23K$ steps. This motivates our approach {{{\\textbf{1-bit Adam}}}} to ``freeze'' the Adam variance after it becomes stable, and then use it as a precondition during 1-bit compression stage.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{moti_var_norm_log.pdf}\n\\caption{Norm of fused variance for BERT-Large pre-training using vanilla Adam. The y-axis is in log scale.}\\label{fig:moti_var_norm_log}\\vspace{-0.1cm}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{{{\\textbf{1-bit Adam}}}} Algorithm}\nIn this section, we start with some background introduction for error compensated compression and why it is incompatible with \\textbf{Adam}. Then we give full description of {{{\\textbf{1-bit Adam}}}}.\n\n\\paragraph*{Problem setting} In this paper, we focus on the following optimization task and rely on the following notions and definitions:\n\\vspace{-0.3cm}\n\\begin{equation}\n\\min_{\\bm{x}\\in\\mathcal{R}^d}\\quad f(\\bm{x}) = \\frac{1}{n} \\sum_{i=1}^n \\underbrace{\\mathbb{E}_{\\bm{\\zeta}^{(i)}\\sim\\mathcal{D}_i}F(\\bm{x}; \\bm{\\bm{\\zeta}}^{(i)})}_{:=f_i(\\bm{x})},\\label{eq:main}\n\\end{equation}\nwhere $d$ is the dimension of the input model $\\bm{x}$, $\\mathcal{D}_i$ is the data distribution of individual data sample $\\bm{\\zeta}^{(i)}$ on the $i$-th worker, $F(\\bm{x};\\bm{\\zeta})$ is the loss function.\n\n\\paragraph{Notations and definitions}\nThroughout this paper, we use the following notations:\n\\begin{itemize}\n\\item $\\nabla f(\\cdot)$ denotes the gradient of a function $f$.\n\\item $f^{*}$ denotes the optimal value of the minimization problem \\eqref{eq:main}.\n\\item $f_i(\\bm{x}) := \\mathbb{E}_{\\bm{\\zeta}^{(i)}\\sim\\mathcal{D}_i}F(\\bm{x}; \\bm{\\zeta}^{(i)})$.\n\\item $\\|\\cdot\\|$ denotes the $\\ell_2$ norm for vectors and the spectral norm for matrices.\n\\item $\\|X\\|_A:=\\text{Tr}(X^{\\top}AX)$.\n\\item $\\bm{C}_{\\omega}(\\cdot)$ denotes the randomized compressing operator, where $\\omega$ denotes the random variable. One example is the randomized quantization operator, for example, $\\bm{C}_{\\omega}(0.7) = 1$ with probability $0.7$ and $\\bm{C}_{\\omega}(0.7) = 0$ with probability $0.3$. \n\\item { $\\sqrt{\\cdot}$ denotes the square root of the argument. In this paper if the argument is a vector, then it returns a vector taking the element-wise square root.}\n\\item $(\\bm{x})^2$ denotes the element-wise square operation if $\\bm{x}$ is a vector.\n\\item $\\frac{\\bm{a}}{\\bm{b}}$ or $\\bm{a}\/\\bm{b}$ denotes the element-wise division operation if both $\\bm{a}$ and $\\bm{b}$ are vectors and their dimension matches.\n\\end{itemize}\n\n\n\\subsection{Why error compensation works for \\textbf{SGD}}\nFor \\textbf{SGD} , since the update is linearly dependent to the gradient, using error compensation could potentially remove the side-effect of the history compression error. The updating rule of \\textbf{vanilla SGD} follows\n\\begin{align*}\n\\bm{x}_{t+1} =& \\bm{x}_t - \\gamma \\bm{g}_t = \\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s.\\numberthis\\label{intuition:sgd_eq1}\n\\end{align*}\nWhen directly compressing the gradient without error compensation, the updating rule becomes\n\\begin{align*}\n\\bm{x}_{t+1} =& \\bm{x}_t - \\gamma C_\\omega[\\bm{g}_t] = \\bm{x}_t - \\gamma (\\bm{g}_t-\\bm{\\delta}_t)\\\\\n= &\\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s + \\underbrace{\\gamma\\sum_{s=0}^t \\bm{\\delta}_s}_{\\text{history compression error}}.\\numberthis\\label{intuition:sgd_eq2}\n\\end{align*}\nAs we can see in \\eqref{intuition:sgd_eq2}, the history compression error would get accumulated and therefore slow down the convergence rate. Moreover, previous work \\citep{Alistarh2017-yh} indicates that when using biased compression operator, the training convergence cannot be guaranteed. \n\nNow if we apply error compensation at each compression step, the updating rule becomes\n\\begin{align*}\n\\bm{x}_{t+1} =& \\bm{x}_t - \\gamma C_\\omega[\\bm{g}_t + \\bm{\\delta}_{t-1}] = \\bm{x}_t - \\gamma (\\bm{g}_t-\\underbrace{\\bm{\\delta}_t +\\bm{\\delta}_{t-1}}_{\\text{error cancellation}})\\\\\n=& \\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s + \\gamma\\sum_{s=0}^t(\\bm{\\delta}_s - \\bm{\\delta}_{s-1})\\\\\n=& \\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s + \\gamma\\bm{\\delta}_t.\\numberthis\\label{intuition:sgd_eq3}\n\\end{align*}\n\n\nThis demonstrates that by using error compensation, each step's compression error would get cancelled in the next step instead of getting accumulated over steps. To make the error compensation work correctly, it is necessary that we ensure an error cancellation term $\\bm{\\delta}_t +\\bm{\\delta}_{t-1}$ in the updating rule. Below we are going to see that this cannot be achieved for \\textbf{Adam}.\n\n\n\\subsection{Why \\textbf{Adam} cannot be combined with error compensation}\\label{intuition:why_adam_fails}\nAs we can see, \\textbf{Adam} is non-linearly dependent to the gradient, and this non-linearity is widely believed to be essential for the superiority of \\textbf{Adam}. Below we are going to first intuitively explain why error compensation works well for \\textbf{SGD}, and then discuss two major reasons why this non-linearity makes \\textbf{Adam} incompatible with error compensation.\n\n\n\n\n\\paragraph{Difficulty for estimating the variance term $\\bm{v}$.} Notice that for \\textbf{Adam}, it is necessary to communicate the gradient $\\bm{g}_t$ or momentum $\\bm{m}_t$, and the variance term can be updated using $\\bm{g}_t$. However, when using error-compensated gradient to update $\\bm{v}_t$, the updating rule follows:\n\\begin{align*}\n\\bm{v}_{t+1} = &\\beta_2 \\bm{v}_t + (1-\\beta_2)\\left(C_\\omega[\\bm{g}_t + \\bm{\\delta}_{t-1}] \\right)^2\\\\\n=& \\beta_2 \\bm{v}_t + (1-\\beta_2)\\left(\\bm{g}_t + \\bm{\\delta}_{t-1} - \\bm{\\delta}_t \\right)^2\\\\\n= & \\beta_2 \\bm{v}_t + (1-\\beta_2)\\left(\\bm{g}_t \\right)^2 + \\underbrace{\\left( \\bm{\\delta}_{t-1} - \\bm{\\delta}_t \\right)^2}_{\\text{non-linear error correction}} + 2 \\langle \\bm{g}_t,\\bm{\\delta}_{t-1} - \\bm{\\delta}_t\\rangle.\n\\end{align*}\nHere the quadratic term $\\left( \\bm{\\delta}_{t-1} - \\bm{\\delta}_t \\right)^2$ cannot be cancelled by itself, therefore it will be hard to get an accurate estimation of $\\bm{v}_t$ with history error being cancelled.\n\\paragraph{Difficulty for setting the correction factor.} Another problem is that for \\textbf{SGD} , when applying error compensation under a time varying learning rate $\\gamma_t$, we need to compensate the history error using\n\\begin{align*}\nC\\left[ \\bm{g}_t + \\frac{\\gamma_t}{\\gamma_{t-1}}\\bm{\\delta}_{t-1}\\right],\n\\end{align*} instead of adding back $\\bm{\\delta}_{t-1}$ directly. In this case,\nif we view $\\frac{\\gamma}{\\sqrt{\\bm{v}_t} + \\eta}$ as a coordinate-dependent learning rate, which makes \\textbf{Adam} equivalent to \\textbf{Momentum SGD} with time-varying learning rate, we need to apply the scale factor according to \n\\begin{align*}\n\\bm{m}_{t+1} = C_\\omega\\left[\\beta_1\\bm{m}_t + (1-\\beta_1)\\bm{g}_t + \\frac{\\sqrt{\\bm{v}_{t-1}} + \\eta}{\\sqrt{\\bm{v}_{t}} + \\eta}\\bm{\\delta}_{t-1}\\right].\n\\end{align*}\nThe problem is that we cannot get the value of $\\bm{v}_{t}$ after the compression, which makes it impossible to set the scale factor for error compensation.\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[\\scriptsize \\textbf{Gather step}: Each worker sends its $i$-th chunk to worker $i$.]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{gather.pdf}\n\\end{minipage}\n}\\quad\n\\subfigure[\\scriptsize \\textbf{Average step}: Each worker averages all chunks it receives.]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{average.pdf}\n\\end{minipage}%\n}\\quad\n\\subfigure[\\scriptsize \\textbf{Scatter step}: Each worker receives the $i$-th chunk from worker $i$.]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{scatter.pdf}\n\\end{minipage}%\n}%\n\\centering\n\\caption{Efficient system design for communication (compressed\\_allreduce)}\\label{allreduce}\\vspace{-0.1cm}\n\\label{fig:allreduce}\n\\end{figure*}\n\n\\subsection{{{{\\textbf{1-bit Adam}}}}}\\label{alg:description}\nBased on our findings (Section~\\ref{sec:moti-variance}) that \\textbf{Adam}'s variance term becomes stable at an early stage, we propose {{{\\textbf{1-bit Adam}}}} summarized in Algorithm \\ref{alg:de_ec}. First we use vanilla \\textbf{Adam} for a few epochs as a warm-up. After the warm-up stage, the compression stage starts and we stop updating the variance term $\\bm{v}$ and use it as a fixed precondition. At the compression stage, we communicate based on the momentum applied with error-compensated 1-bit compression. The momentums are quantized into 1-bit representation (the sign of each element). Accompanying the vector, a scaling factor is computed as $\\frac{\\text{magnitude of compensated gradient}}{\\text{magnitude of quantized gradient}}.$ This scaling factor ensures that the compressed momentum has the same magnitude as the uncompressed momentum. This 1-bit compression could reduce the $97\\%$ communication cost of the original for float32 type training and $94\\%$ for float16 type training. \n\n\n\n\n\n\n\n\\begin{algorithm}[t!]\\caption{{{{\\textbf{1-bit Adam}}}}}\n\\begin{algorithmic}[1]\n\\footnotesize\n\\STATE {\\bfseries Initialize}: $\\bm{x}_0$, learning rate $\\gamma$, initial error $\\bm{\\delta} = \\boldsymbol{0}$, $\\bm{m}_0 = \\boldsymbol{0}$, $\\bm{v}_0 = \\boldsymbol{0}$, number of total iterations $T$, warm-up steps $T_{w}$, two decaying factor $\\beta_1$ and $\\beta_2$ for \\textbf{Adam}.\n\n\\STATE Running the original \\textbf{Adam} for $T_{w}$ steps, then store the variance term (defined as $\\bm{v}_t$ in \\eqref{alg:v}) $\\bm{v}_{_{ T_w}}$.\n\\FOR {$t=T_w,\\ldots,T$}\n\n\\STATE \\textbf{(On $i$-th node)}\n\\STATE Randomly sample $\\bm{\\xi}_t^{(i)}$ and compute local stochastic gradient $\\bm{g}_t^{(i)} := \\nabla F_i(\\bm{x}_t^{(i)}, \\bm{\\xi}_t^{(i)})$.\n\n\\STATE Update the local momentum variable $\\bm{m}_{t-1}$ according to\n$\n\\bm{m}_t^{(i)} = \\beta_1\\bm{m}_{t-1} + (1 - \\beta_1)\\bm{g}_t^{(i)}.\n$\n\\STATE Compress $\\bm{m}_t^{(i)}$ into $\\hat{\\bm{m}}_t^{(i)} = \\bm{C}_\\omega\\left[\\bm{m}_t^{(i)} + \\bm{\\delta}_{t-1}^{(i)}\\right]$, and update the compression error by $\\bm{\\delta}_t^{(i)} = \\bm{m}_t^{(i)} + \\bm{\\delta}_{t-1}^{(i)} - \\hat{\\bm{m}}_t^{(i)}$.\n\\STATE Send the $\\hat{\\bm{m}}_t^{(i)}$ to the server.\n\\STATE \\textbf{(On server)}\n\\STATE Take the average over all $\\hat{\\bm{m}}_t^{(i)}$ it receives and compress it into\n$\n\\overline{\\bm{m}}_t =\\bm{C}_\\omega\\left[ \\frac{1}{n}\\sum_{i=1}^n\\hat{\\bm{m}}_t^{(i)} + \\overline{\\bm{\\delta}}_{t-1}\\right],\n$\n and update the compression error accordingly by $\\overline{\\bm{\\delta}}_t = \\frac{1}{n}\\sum_{j=1}^n \\bm{C}_\\omega\\left[\\bm{m}_t^{(i)}\\right] + \\overline{\\bm{\\delta}}_{t-1} - \\overline{\\bm{m}}_t$.\n \\STATE Send $\\overline{\\bm{m}}_t$ to all the workers.\n \\STATE \\textbf{(On $i$-th node)}\n \\STATE Set $\\bm{m}_t = \\overline{\\bm{m}}_t$ , and update local model $\\bm{x}_{t+1} = \\bm{x}_t - \\gamma \\bm{m}_t\/\\sqrt{\\bm{v}_{_{\\tiny T_w}}}$.\n\\ENDFOR\n\\STATE {\\bfseries Output}: $\\bm{x}$.\n\\end{algorithmic}\\label{alg:de_ec}\n\\end{algorithm}\n\n\n\\section{Theoretical Analysis}\nNotice that for {{{\\textbf{1-bit Adam}}}}, we only use original \\textbf{Adam} at warm-up, and then we essentially run error-compensated momentum \\textbf{SGD} with coordinate-dependent learning rate $\\frac{\\gamma}{\\sqrt{\\bm{v}_{_{T_w}}}}$. Therefore here we consider the \\textbf{Adam}-based warm-up phase as a way to find a good precondition variance term $\\bm{v}_{_{T_w}}$ to be used in the compression phase. Below we are going to introduce the convergence rate for the compression phase after warm-up. We first introduce some necessary assumptions, then we present the theoretical guarantee of the convergence rate for {{{\\textbf{1-bit Adam}}}}.\n\n\n\n\n\n\n\n\n\n\\begin{assumption}\\label{ass:global}\nWe make the following assumptions:\n\\begin{enumerate}\n\\item \\textbf{Lipschitzian gradient:} $f(\\cdot)$ is assumed to be with $L$-Lipschitzian gradients, which means\n \\begin{align*}\n \\|\\nabla f(\\bm{x}) - \\nabla f(\\bm{y}) \\| \\leq L \\|\\bm{x} - \\bm{y} \\|,\\quad \\forall \\bm{x},\\forall \\bm{y},\n \\end{align*}\n \\item\\label{ass:var} \\textbf{Bounded variance:}\nThe variance of the stochastic gradient is bounded\n\\begin{align*}\n\\mathbb E_{\\bm{\\zeta}^{(i)}\\sim\\mathcal{D}_i}\\|\\nabla F(\\bm{x};\\bm{\\zeta}^{(i)}) - \\nabla f(\\bm{x})\\|^2 \\leq \\sigma^2,\\quad\\forall \\bm{x},\\forall i.\n\\end{align*}\n\\item \\textbf{Bounded magnitude of error for $\\mathcal{C}_{\\omega}[\\cdot]$:}\nThe magnitude of worker's local errors $\\bm{\\delta}_t^{(i)}$ and the server's global error $\\overline{\\bm{\\delta}}_t$, are assumed to be bounded by a constant $\\epsilon$\n\\begin{align*}\n\\sum_{k=1}^n\\mathbb E_{\\omega} \\left\\|\\bm{\\delta}_t^{(i)}\\right\\|\\leq \\frac{\\epsilon}{2},\\quad\n\\sum_{i=1}^n\\mathbb E_{\\omega}\\left\\|\\overline{\\bm{\\delta}}_t\\right\\|\\leq \\frac{\\epsilon}{2},\\quad\\forall t,\\forall i.\n\\end{align*}\n\\end{enumerate}\n\\end{assumption}\n\n\nNext we present the main theorem for {{{\\textbf{1-bit Adam}}}}.\n\\begin{theorem}\\label{theo:global}\n Under Assumption~\\ref{ass:global}, for {{{\\textbf{1-bit Adam}}}}, we have the following convergence rate\n \\begin{align*}\n &\\left(1-\\frac{\\gamma L}{v_{\\min}} - \\frac{2\\gamma^2 L^2}{(1-\\beta)^2v_{\\min}^2} \\right)\\sum_{t=0}^T \\mathbb E\\|\\nabla f(\\bm{x}_t)\\|^2_{V}\\\\\n \\leq & \\frac{2\\mathbb E f(\\bm{x}_{0}) - 2\\mathbb Ef(\\bm{x}^*)}{\\gamma} + \\frac{6\\gamma^2L^2\\epsilon^2 T}{(1-\\beta)^2v_{\\min}^3} + \\frac{L\\gamma \\sigma^2T}{nv_{\\min}} + \\frac{2\\gamma^2L^2\\sigma^2 T}{n(1-\\beta)^2v_{\\min}^2},\\numberthis\\label{main:theo:eq}\n\\end{align*}\nwhere $V= \\text{diag}\\left(1\/\\bm{v}_{T_w}^{(1)},1\/\\bm{v}_{T_w}^{(2)},\\cdots,1\/\\bm{v}_{T_w}^{(d)}\\right)$ is a diagonal matrix spanned by $\\bm{v}_{_{T_w}}$ and $v_{\\min} = \\min\\{\\bm{v}_{T_w}^{(1)},\\bm{v}_{T_w}^{(2)},\\cdots,\\bm{v}_{T_w}^{(d)}\\}$ is the mimimum value in $\\bm{v}_{T_w}$\n\\end{theorem}\n\nGiven the generic result in Theorem~\\ref{theo:global}, we obtain the convergence rate for {{{\\textbf{1-bit Adam}}}} with appropriately chosen learning rate $\\gamma$.\n\n\n\\begin{corollary}\\label{coro:global}\nUnder Assumption~\\ref{ass:global}, for {{{\\textbf{1-bit Adam}}}}, choosing\n$\n\\gamma = \\frac{1}{4L(v_{\\min})^{-1} + \\sigma\\sqrt{\\frac{ T}{n}} + \\epsilon^{\\frac{2}{3}} T^{\\frac{1}{3}}(v_{\\min})^{-1} },\n$\nwe have the following convergence rate\n\\begin{align*}\n\\frac{1}{Tv_{\\min}}\\sum_{t=0}^{T-1}\\mathbb{E}\\|\\nabla f(\\bm{x}_t)\\|^2_V \\lesssim \\frac{\\sigma}{\\sqrt{nT}} + \\frac{\\epsilon^{\\frac{2}{3}}}{T^{\\frac{2}{3}}} + \\frac{1}{ T},\n\\end{align*}\nwhere we treat $f(\\bm{x}_1) - f^*$, $\\beta$ and $L$ as constants.\n\\end{corollary}\n\n\nThis result suggests that: {{{\\textbf{1-bit Adam}}}} essentially admits the same convergence rate as distributed \\textbf{SGD} in the sense that both of them admit the asymptotical convergence rate $O(1\/\\sqrt{nT})$, which means we can still achieve linear speedup w.r.t. the number of workers $n$.\n\n\n\\section{Efficient system design for compressed communication}\nNVIDIA NCCL is an efficient and widely used communication library that has been tightly integrated in DL frameworks like PyTorch and TensorFlow. However, NCCL library cannot be used directly for performing communication based on 1-bit compression. This is because the collective communication primitives like Allreduce and Allgather are at a higher level of abstraction and can only perform data movement and\/or simple operations like sum, min, max etc. In addition, NCCL library (before v2.7) did not expose either an Alltoall primitive or any point-to-point (send\/recv) communication primitives that can be used to implement an Alltoall. Thus for {{{\\textbf{1-bit Adam}}}}, we designed a custom collective primitive using Message Passing Interface (MPI). We call it ``compressed allreduce'' and it has three phases as shown in Figure~\\ref{fig:allreduce}: 1) The gather step, which we have implemented using the MPI\\_Alltoall (personalized exchange) primitive, 2) The average step, where {{{\\textbf{1-bit Adam}}}} computes the average of compressed local momentums, and 3) The scatter step, which we implement using MPI\\_Allgather. We develop two versions of compressed allreduce: 1) CUDA-Aware version that exploits GPUDirect features and requires CUDA-Aware libraries like MVAPICH2-GDR and 2) Basic version that can be used with any MPI library but copies data between GPU and CPU buffers. The CUDA-Aware version works only on systems with InfiniBand whereas the basic version can run on any system with Ethernet interconnect. \n\n\\section{Experiments}\n\nWe evaluate {{{{\\textbf{1-bit Adam}}}}} and existing approaches using BERT-Base, BERT-Large, SQuAD 1.1 and ResNet-18 training tasks on up to 256 GPUs. We show that {{{{\\textbf{1-bit Adam}}}}} converges as fast as uncompressed \\textbf{Adam}, and runs up to 3.3 times faster than uncompressed algorithms under limited bandwidth.\n\n \n \n\n\\subsection{BERT pre-training and fine-tuning}\n\\label{sec:bert-eval}\n\\paragraph{Dataset and models} We evaluate the convergence and performance of {{{\\textbf{1-bit Adam}}}} and uncompressed \\textbf{Adam} for BERT-Base ($L=12$, $H=768$, $A=12$, $110M$ params) and BERT-Large ($L=24$, $H=1024$, $A=16$, $340M$ params) pre-training tasks. We use the same dataset as \\citet{bert}, which is a concatenation of Wikipedia and BooksCorpus with $2.5B$ and $800M$ words respectively. We use the GLUE fine-tuning benchmark\\citep{glue} to evaluate the convergence of the BERT models trained by \\textbf{Adam} and {{{\\textbf{1-bit Adam}}}}.\n\nIn addition, we also evaluate the convergence and performance of {{{\\textbf{1-bit Adam}}}} for SQuAD 1.1 fine-tuning task\\footnote{https:\/\/rajpurkar.github.io\/SQuAD-explorer\/} using a pre-trained BERT model checkpoint from HuggingFace\\footnote{https:\/\/github.com\/huggingface\/transformers}.\n\n\\paragraph{Hardware} We use the two clusters described in Section~\\ref{sec:moti-profile}. We use up to 256 GPUs for pre-training tasks and up to 32 GPUs for fine-tuning tasks.\n\n\\paragraph{Training parameters} For BERT pre-training, the learning rate linearly increases to $4\\times 10^{-4}$ as a warmup in the first $12.5K$ steps, then decays into $0.99$ of the original after every $520$ steps. We set the two parameters in Algorithm~\\ref{alg:de_ec} as $\\beta_1 = 0.9$ and $\\beta_2 = 0.999$ for {{{\\textbf{1-bit Adam}}}} and \\textbf{Adam}. For convergence test, we set total batch size as $4K$ for BERT-Base and BERT-Large. For performance test, we test different batch sizes. Table~\\ref{table_bert_steps} summarizes the total number of steps for BERT sequence length 128 and 512 phases, together with the number of warmup steps for {{{\\textbf{1-bit Adam}}}}.\n\nFor GLUE benchmarks we use original \\textbf{Adam} optimizer and perform single-task training on the dev set. We search over the hyperparameter space with batch sizes $\\in\\{8,16\\}$ and learning rates $\\in\\{1\\times 10^{-5},3\\times 10^{-5},5\\times 10^{-5},8\\times 10^{-5}\\}$. Other setting are the same as pre-training task.\n\nFor SQuAD fine-tuning we use the same parameters as published by HuggingFace (batch size = $24$, learning rate=$3e-5$, dropout=$0.1$, 2 epochs), except that we increase the batch size to $96$ (using $32$ GPUs). The first $400$ steps out of total $1848$ steps are used as the warmup stage for {{{\\textbf{1-bit Adam}}}}.\n\n\\begin{table}[t]\n \\footnotesize\n \\caption{Number of steps for BERT pre-training tasks.}\\label{table_bert_steps}\n \\centering\n \\begin{tabular}{lll}\n \\hline\n & Seqlen 128& Seqlen 512\\\\\n & (warmup)& (warmup)\\\\\n \\hline\n BERT-Base \\textbf{Adam}& $118K$ (N\/A)& $22K$ (N\/A) \\\\\n BERT-Base {{{\\textbf{1-bit Adam}}}}& $118K$ ($16K$)& $22K$ ($1.5K$) \\\\\n BERT-Large \\textbf{Adam}& $152K$ (N\/A)& $10K$ (N\/A) \\\\\n BERT-Large {{{\\textbf{1-bit Adam}}}}& $152K$ ($23K$)& $10K$ ($1.5K$) \\\\\n \\hline\n \\end{tabular}\\vspace{-0.1cm}\n\\end{table}\n\n\\paragraph{Convergence results}\nFigure~\\ref{fig:bert} presents the sample-wise convergence results. We use the BertAdam \\citep{bert} optimizer as the uncompressed baseline. For both BERT-Base and BERT-Large and for both sequence length phases, we find that {{{\\textbf{1-bit Adam}}}} provides the same convergence speed as baseline, while the communication volume is reduced into $6\\%$ of the original during the compression stage.\n\nTable~\\ref{table1} presents the GLUE results using the checkpoints from our pre-training experiments. {{{\\textbf{1-bit Adam}}}} achieves similar accuracy compared to the uncompressed baseline and the numbers reported in previous work.\n\nFor SQuAD 1.1 fine-tuning task using checkpoint from HuggingFace, {{{\\textbf{1-bit Adam}}}} achieves similar F1 score (93.32) compared to the score reported by HuggingFace (93.33) using same number of samples and trainig parameters. \n \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{large_loss.pdf}\n\\caption{Epoch-wise convergence speed for BERT-Large pre-training sequence length 128. {{{\\textbf{1-bit Adam}}}} and \\textbf{Adam} also achieve the same convergence speed for BERT-Base pre-training.}\\label{fig:bert}\\vspace{-0.1cm}\n\\end{figure}\n\n\\begin{table*}[t]\n\\footnotesize\n \\caption{GLUE development set results. BERT-Base\/Large(original) results are from \\citet{bert}. BERT-Base\/Large (uncompressed) results use the full-precision \\textbf{BertAdam} with the same training parameters as the {{{\\textbf{1-bit Adam}}}} case. BERT-Base\/Large (compressed) are the results using {{{\\textbf{1-bit Adam}}}}. The scores are the median scores over 10 runs.}\\label{table1}\n \\centering\n \\begin{tabular}{lccccccc}\n \\hline \n \\textbf{Model}& RTE& MRPC& CoLA & SST-2& QNLI& QQP& MNLI-(m\/mm) \\\\\n \\hline \n BERT-Base (original) & 66.4 & 84.8 & 52.1 & 93.5 & 90.5& 89.2& 84.6\/83.4\\\\\n BERT-Base (uncompressed) & 68.2 & 84.8 & 56.8 & 91.8 & 90.9& 90.9& 83.6\/83.5\\\\\n BERT-Base (compressed) & 69.0& 84.8 & 55.6 & 91.6 & 90.8& 90.9& 83.6\/83.9\\\\\n \\hline\n BERT-Large (original) & 70.1& 85.4 & 60.5 & 94.9 & 92.7& 89.3& 86.7\/85.9\\\\\n BERT-Large (uncompressed) & 70.3& 86.0 & 60.3 & 93.1 & 92.2& 91.4& 86.1\/86.2\\\\\n BERT-Large (compressed) & 70.4& 86.1 & 62.0 & 93.8 & 91.9& 91.5& 85.7\/85.4\\\\\n \\hline\n \\end{tabular}\\vspace{-0.1cm}\n\\end{table*}\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[Bert-Large pre-training, batch size = number of GPUs $\\times$ 16]{\n\\begin{minipage}[t]{0.33\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{eval_bert_accum1.pdf}\\label{fig:e2e-1}\n\\end{minipage}\n}\n\\subfigure[Bert-Large pre-training, batch size = 4K]{\n\\begin{minipage}[t]{0.33\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{eval_bert_bsz4k.pdf}\\label{fig:e2e-2}\n\\end{minipage}\n}\n\\subfigure[SQuAD fine-tuning, batch size = number of GPUs $\\times$ 3]{\n\\begin{minipage}[t]{0.28\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{eval_squad_accum1.pdf}\\label{fig:e2e-3}\n\\end{minipage}%\n}\n\\centering\n\\caption{Scalability of {{{\\textbf{1-bit Adam}}}} for BERT-Large pre-training sequence length 128 and SQuAD 1.1 fine-tuning on V100 GPUs. \\textbf{Adam} lines represent the throughput at {{{\\textbf{1-bit Adam}}}}'s warmup stage (i.e., baseline \\textbf{Adam}'s throughput). {{{\\textbf{1-bit Adam}}}} lines represent the throughput at compression stage. Annotations represent the highest speedup achieved in each figure. Note that this is the speedup between warmup and compression stage. The end-to-end speedup also depends on the percentage of warmup.}\\label{fig:e2e}\\vspace{-0.1cm}\n\\end{figure*}\n\n\n\\paragraph{Performance results}\nComputed as 1\/(warmup ratio + (1 - warmup ratio)\/16) for FP16 training, {{{\\textbf{1-bit Adam}}}} offers up to 5x less end-to-end communication volume for BERT-Base and BERT-Large. This leads to to 3.3x higher throughput for BERT-Large sequence length 128 pre-training and up to 2.9x higher throughput for SQuAD fine-tuning. This end-to-end throughput improvement is enabled by the 5.48x (Figure~\\ref{fig:e2e-1}) and 6.17x (Figure~\\ref{fig:e2e-3}) speedup observed during the compression stage. Figure~\\ref{fig:e2e-2} shows that {{{\\textbf{1-bit Adam}}}} also provides better scalability: \\textbf{Adam}'s throughput reaches peak at 32 GPUs on Ehternet, while {{{\\textbf{1-bit Adam}}}}'s throughput keeps increasing until 128 GPUs. It is also worth mentioning that {{{\\textbf{1-bit Adam}}}} on Ethernet (4.1 Gbps effective bandwidth, 4 GPUs per node) is able to achieve comparable throughput as \\textbf{Adam} on InfiniBand (near 100 Gbps effective bandwidth, 8 GPUs per node), which demonstrates {{{\\textbf{1-bit Adam}}}}'s efficiency considering the hardware differences.\n\n\n\n\\subsection{ResNet on CIFAR10}\\label{resnet}\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Training loss]{\n\\begin{minipage}[t]{0.35\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{resnet_loss.pdf}\n\\end{minipage}\n}\n\\subfigure[Testing accuracy]{\n\\begin{minipage}[t]{0.25\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{resnet_acc.pdf}\n\\end{minipage}\n}\n\\centering\n\\caption{Epoch-wise convergence speed for ResNet-18.}\\label{fig:resnet}\\vspace{-0.1cm}\n\\end{figure}\n\nTo further evaluate the convergence speed of {{{\\textbf{1-bit Adam}}}} and related works, we train CIFAR10 using ResNet-18\\citep{7780459}. The dataset has a training set of 50000 images and a test set of 10000 images, where each image is given one of the 10 labels. We run the experiments on $8$ 1080Ti GPUs where each GPU is used as one worker. The batch size on each worker is $128$ and the total batch size is $1024$.\n\nWe evaluate five implementations for comparison: 1) Original \\textbf{SGD}. 2) Original \\textbf{Adam} \\citep{adam}. 3) {{{\\textbf{1-bit Adam}}}} where we use $13$ out of $200$ epochs as warmup. 4) {{{\\textbf{1-bit Adam}}}}\\textbf{(32-bits)} where we do not compress the momentum while still freezing the varaince. 5) \\textbf{Adam(1-bit Naive)} where we compress the gradient instead of momentum, and don't freeze the variance. We set the learning rate as $1\\times 10^{-1} $ for \\textbf{SGD} and $1\\times 10^{-4}$ for the other 4 cases. For all five cases, the learning rate is decayed into $10\\%$ of the original after every $100$ epochs.\n\n\n\n\n\n\n\nAs illustrated in Figure~\\ref{fig:resnet}, {{{\\textbf{1-bit Adam}}}} achieves similar convergence speed as \\textbf{Adam} and {{{\\textbf{1-bit Adam}}}}\\textbf{(32-bits)}. \\textbf{SGD} has a slightly slower convergence speed while \\textbf{Adam(1-bit Naive)} is much worse. This and Section~\\ref{sec:moti-convergence} demonstrate that existing compression method doesn't work for \\textbf{Adam}. In the supplementary materials we further compare {{{\\textbf{1-bit Adam}}}} with other related works using ResNet-18.\n\n\\section{Conclusions}\nIn this paper, we propose an error-compensated \\textbf{Adam} preconditioned momentum SGD algorithm, {{{\\textbf{1-bit Adam}}}}, which provides both communication efficiency and \\textbf{Adam}'s convergence speed. Our theoretical analysis demonstrates that ${{\\textbf{1-bit Adam}}}$ admits a linear speed w.r.t the number of workers in the network, and is robust to any compression method. We validate the performance of {{{\\textbf{1-bit Adam}}}} empirically on BERT, SQuAD and ResNet training tasks on up to 256 GPUs. Results show that {{{\\textbf{1-bit Adam}}}} converges as fast as uncompressed \\textbf{Adam}, reduces communication volume by up to 5x, and runs up to 3.3 times faster than uncompressed algorithms.\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n An American Mathematical Monthly problem posed within relatively recent memory [{\\bf{1}}] sought the\nevaluation\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi\/2}\\left\\{\\rule{0mm}{4mm}\\log(\\,2\\sin(x)\\,)\\right\\}^{2}dx\\,=\\,\\frac{\\,\\,\\,\\pi^{\\,3}\\,}{\\,24\\,}\\;.\n\\end{equation}\nOne mode of solution depended upon integration of an analytic function around the periphery $\\,\\Omega\\,$ of a semi-infinite\nvertical strip with no singularities enclosed, the quadrature having thus a null outcome\\footnote{Both\ncontour $\\,\\Omega\\,,$ a vertical rectangle of unlimited height, and the notion of integrating an analytic function thereon so\nas to obtain a null result, imitate a similar ploy utilized in [{\\bf{2}}] on behalf of (2) and still further attributed there\nto Ernst Lindel{\\\"{o}}f.} known in advance on the strength of Cauchy's theorem.\nEvaluation (1) emerged automatically by setting to zero the real part of that integral,\\footnote{Two solutions for (1) were submitted by\nthe undersigned, one involving contour integration in the manner suggested, and the other based upon a Fourier series.\nThe Bernoulli recurrence (5) emerged as a spontaneous by-product of an ancillary, null-quadrature calculation upon\nthat same contour $\\,\\Omega\\,,$ initially aimed only at evaluating the log-sine integrals (4). This note embodies the content\nof that collateral calculation, slightly rephrased so as to highlight the newly recovered Bernoulli number sum identity.}\nwhereas the complementary requirement that the imaginary part likewise vanish brought into play, and successfully so, the known quadratures\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\log(\\,\\sin(x)\\,)\\,dx\\,=\\,-\\,\\pi\\log(2)\n\\end{equation}\nand\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}x\\log(\\,\\sin(x)\\,)\\,dx\\,=\\,-\\frac{\\,\\,\\,\\pi^{\\,2}\\,}{\\,2\\,}\\log(2)\\;.\n\\end{equation}\nWith (2) and (3) in plain view, a temptation arose to provide for them, too, an {\\em{ab initio}} verification, and,\nmore even than that, to evaluate the entire hierarchy of log-sine integrals\\footnote{If that were the only goal then we should\nassuredly stop dead in our tracks, simply because, on the one hand, {\\bf{MATHEMATICA}} provides all such evaluations\non demand, with great aplomb, and this even in its symbolic mode, while, on the other, a relatively painless\nderivation can be based upon a Fourier series, one which emerges in its turn from the power series for $\\,\\log(1-z)\\,$\nwhen argument $\\,z\\,$ is forced to lie upon the unit circle. This Fourier series underlies in addition an essentially zinger\nverification of (1). All such manifold benefits of the Fourier option are sketched in an Appendix. Moreover, it goes without\nsaying that both contour-based (14) and Fourier-based (27) evaluations of (4), even though they may be of secondary interest\nin the present context, do stand in complete agreement.} \n\\begin{equation}\nI_{n}\\,=\\,\\int_{\\,0}^{\\,\\pi}x^{n}\\log(\\,\\sin(x)\\,)\\,dx\n\\end{equation}\nas the power of $\\,x\\,$ roams over all non-negative integers $\\,n\\,=\\,0,\\,1,\\,2,\\,3,\\,\\ldots\\,{\\rm{ad\\;inf}}\\,.$\nNot only was this fresh ambition, digressive and self-indulgent though it may have been, easy to satisfy via quadrature\non the same contour as before, but it also exposed to view once more the fundamental Bernoulli number recurrence\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{n-1} \\left(\\!\\!\\begin{array}{c}\n n \\\\\n k\n \\end{array}\\!\\!\\right)\nB_{k}\\,=\\,0\n\\end{equation}\nwhich is valid for $\\,n\\geq 2\\,$ and, together with the initial condition $\\,B_{0}\\,=\\,1\\,$ and the self-consistent choice\n$\\,B_{1}\\,=\\,-1\/2\\,,$ is adequate to populate the entire Bernoulli ladder, complete with null entries at all odd\nindices beyond $\\,k\\,=\\,1\\,,$ {\\em{viz.,}} $\\,B_{2l+1}\\,=\\,0\\,$ whenever $\\,l\\geq1\\,.$ Source material on the\nBernoulli numbers and the related Bernoulli polynomials is ubiquitous, and can be sampled, for example, in [\\textbf{3-5}].\nReferences [\\textbf{6,7}] provide a valuable overview all at once of their\nmathematical properties and historical genesis in com-\n\\newpage\n\\mbox{ }\n\\newline\n\\newline\n\\newline\nputing sums of finite progressions of successive integers raised to\nfixed positive powers. Equally valuable is online Reference [{\\bf{8}}], which cites a rich literature and\ncovers besides a vast panorama of diverse mathematical knowledge.\n\n\nBernoulli identity (5), which is the principal object of our present concern, emerges thus by setting to\nzero the imaginary part of the analytic quadrature (6), below, around contour $\\,\\Omega\\,,$\nwith the corresponding null value requirement on its real part providing an evaluation of the general term from sequence (4),\nlisted in (14). No claim whatsoever is made here as to any ultimate novelty in outcome (14), which is available\nin symbolic form at any desired index $\\,n\\,$ through routine demand from \\textbf{MATHEMATICA}. Outcome (14),\nexpressed here as a finite sum of Riemann zeta functions at odd integer arguments, continues to attract the attention\nof contemporary research focused upon polylogarithms [\\textbf{9-12}]. But the formulae thus\nmade available are subordinated in [\\textbf{11,12}] and elsewhere to the task of evaluating a variety of dissimilar\nquantities, and appear to be tangled in thickets of notation. From this\nstandpoint, formula (14) (and its identical twin (27) derived in an even more elementary fashion)\nmay perhaps still provide the modest service of a stand-alone, encapsulated result, easily derived and\neasily surveyed. In particular, the canonical method of derivation evolved in [\\textbf{9}] and repeatedly\nalluded to in [\\textbf{11,12}] requires rather strenuous differentiations of Gamma function ratios,\nand results finally in a recurrence on the individual $\\,I_{n}\\,$ (or else an equivalent generating function).\nTo be sure, while the work in [\\textbf{9}] is immensely elegant, it is at the same time immensely more intricate\nthan either of our independent derivations culminating in (14) and (27).\n\n\n On the other hand, it does appear to have escaped previous notice that the Bernoulli recurrence (5), which\nis ancient and foundational in its own right, should likewise re\\\"{e}merge (via (16)) from the same\nquadrature around contour $\\,\\Omega\\,$ when one insists that the\ncorresponding imaginary part also vanish. And, just as is the case with (14), formula (16), too, emerges\nfrom a contact with Riemann zeta functions, but evaluated this time at even integer arguments, which latter\ncircumstance, by virtue of the celebrated Euler connection, opens the portal to entry by the similarly\nindexed Bernoulli numbers. It is of course none of our purpose here to compete with, let alone to supplant\nin any way the standard derivations of (5). Rather, we seek merely to highlight its re\\\"{e}mergence in what surely\nmust be conceded to be an unexpected setting. \n\n We round out this note with an appendix wherein contour integration cedes place to the more elementary\nsetting of a Fourier series on whose basis (14) is recovered yet again (as (27)) through repeated integration by\nparts. That same Fourier series provides moreover an exceedingly short and simple confirmation of (1),\ncomplementary to the contour integral method, an option to which allusion has already been made in Footnote 3.\nOf course, at this point, no further light can, nor need be shed upon (5) \\textit{per se}.\n\\vspace{-4mm}\n\n\n \n\\section{Null Quadratures on Contour $\\,\\Omega$}\n\\vspace{-2mm}\n\n Guided by the cited example in [{\\bf{2}}], we consider for $\\,n\\,\\geq\\,0\\,$ the sequence of numbers\n\\begin{equation}\nK_{n}\\,=\\,\\int_{\\,\\Omega}z^{n}\\log\\left(1-e^{\\,2\\,i\\,z}\\right)\\,dz\\,=\\,0\\;\\,,\n\\end{equation}\nall of them annulled by virtue of closed contour $\\,\\Omega\\,$ being required to\nlie within a domain of analyticity for $\\,\\log\\left(1-e^{\\,2\\,i\\,z}\\right)\\,$\nin the plane of complex $z=x+iy.$ Save for quarter-circle indentations of vanishing radius\n$\\,\\delta\\,$ around $\\,z\\,=\\,0\\,$ and $\\,z\\,=\\,\\pi\\,,$ contour $\\,\\Omega\\,$\nbounds a semi-infinite vertical strip, with a left leg having $\\,x\\,=\\,0\\,$\n\\newpage\n\\mbox{ }\n\\newline\n\\newline\n\\newline\nfixed and descending from $\\,y\\,=\\,\\infty\\,$ to $\\,y\\,=\\,\\delta\\,$ (quadrature\ncontribution $\\,L_{n}\\,$), and a right leg at a fixed $\\,x\\,=\\,\\pi\\,$ ascending from\n$\\,y\\,=\\,\\delta\\,$ to $\\,y\\,=\\,\\infty\\,$ (quadrature contribution $\\,R_{\\,n}\\,$),\nlinked at their bottom by a horizontal segment with $\\,y\\,=\\,0\\,$ and\n$\\,\\delta\\,\\leq\\,x\\,\\leq\\,\\pi-\\,\\delta\\,$ (quadrature contribution $\\,H_{n}\\,$).\nIn what follows it will be readily apparent that the limit $\\,\\delta\\!\\downarrow\\!0+\\,$\nmay be enforced with full impunity, a gesture whose {\\em{fait accompli}} status will\nbe taken for granted. Likewise passed over without additional\ncomment will be the fact that no contribution is to be sought from contour completion\nby a retrograde horizontal segment $\\,\\pi\\,\\geq\\,x\\,\\geq\\,0\\,$ at\ninfinite remove, $\\,y\\,\\rightarrow\\,\\infty\\,.$\n\n\n We now find\n\\begin{equation}\nL_{n}\\,=\\,-\\,i^{\\,n+1}\\int_{\\,0}^{\\,\\infty}y^{\\,n}\\log\\left(1-e^{-2\\,y}\\right)\\,dy\\;\\,,\n\\end{equation}\n\\begin{equation}\nR_{\\,n}\\,=\\,+\\,i\\int_{\\,0}^{\\,\\infty}\\left(\\,\\pi\\,+\\,i\\,y\\,\\right)^{n}\\log\\left(1-e^{-2\\,y}\\right)\\,dy\\;\\,,\n\\end{equation}\nand\n\\begin{eqnarray}\nH_{n} & = & \\int_{\\,0}^{\\,\\pi}x^{n}\\left[\\rule{0mm}{4mm}\\log(2)\\,-\\,\n \\frac{\\,i\\,\\pi\\,}{\\,2\\,}\\,+\\,i\\,x\\,+\\,\\log(\\,\\sin(x)\\,)\\,\\right]\\,dx \\nonumber \\\\\n & = & \\frac{\\,\\pi^{\\,n+1}\\,}{\\,n\\,+\\,1\\,}\\log(2)\\,-\\,i\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,2\\,(\\,n\\,+\\,1\\,)\\,}\\,+\\,\ni\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,\\,n\\,+\\,2\\,}\\,+\\,\\int_{\\,0}^{\\,\\pi}x^{n}\\log(\\,\\sin(x)\\,)\\,dx\\;\\,.\n\\end{eqnarray}\nSeries expansion of the logarithm further gives\n\\begin{equation}\nL_{n}\\, = \\, +\\,i^{\\,n+1}\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l\\,}\\int_{\\,0}^{\\,\\infty}y^{\\,n}e^{-2\\,l\\,y}\\,dy \\,=\\,+\\,\ni^{\\,n+1}\\,\\frac{\\,n\\,!\\,}{\\,2^{\\,n+1}\\,}\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,n+2}\\,} \\,,\n\\end{equation}\nthe interchange in summation and integration being legitimated by Beppo Levi's monotone convergence theorem, and similarly\n\\begin{equation}\nR_{\\,n}\\, = \\, -\\,i\\sum_{k\\,=\\,0}^{n}\\left(\\!\\!\\begin{array}{c}\n n \\\\\n k\n \\end{array}\\!\\!\\right)\\pi^{\\,n-k}\\,i^{\\,k}\n\\frac{\\,k\\,!\\,}{\\,2^{\\,k+1}\\,}\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,k+2}\\,}\\;\\,,\n\\end{equation}\nin both of which there insinuates itself the Riemann zeta function \n\\begin{equation}\n\\zeta\\,(s)\\,=\\,\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,s}\\,}\n\\end{equation}\nat a variety of its argument values $\\,s.$\\footnote{This canonical\ndefinition implies a guarantee of series convergence, assured by the requirement that $\\,\\Re\\,s\\,>\\,1\\,.$\nA robust arsenal of knowledge exists for continuing $\\,\\zeta(s)\\,$ across the entire plane of\ncomplex variable $\\,s\\,=\\,\\sigma\\,+\\,i\\,t\\,,$ with a simple pole emerging at $\\,s\\,=\\,1\\,.$}\nSo armed, we proceed next to set\n\\begin{equation}\nK_{n}\\,=\\,L_{n}\\,+\\,H_{n}\\,+\\,R_{\\,n}\\,=\\,0\n\\end{equation}\nand remark that, regardless of the parity of index $\\,n\\,,$ $\\,L_{n}\\,$ {\\em{per se}}\nis always absorbed by the contribution from the highest power $\\,y^{\\,n}\\,$ within the\nintegrand for $\\,R_{\\,n}\\,.$ This circumstance accounts for the imminent appearance of\nthe floor function affecting the highest value\nof summation index $\\,k\\,$ in Eqs. (14)-(16) and (19) below.\n\\newpage\n\\mbox{ }\n\\newline\n\n\n A requirement that the real part of (13) vanish provides now the following string of valuable\nlog-sine quadrature formulae\n\\begin{eqnarray}\n\\int_{\\,0}^{\\,\\pi}x^{n}\\log(\\,\\sin(x)\\,)\\,dx & = & -\\,\\frac{\\,\\pi^{\\,n+1}\\,}{\\,n\\,+\\,1\\,}\\log(2)\\,+ \\nonumber \\\\\n & & \\rule{-2.3cm}{0mm} +\\,\\frac{\\,n\\,!\\,}{\\,2^{\\,n+1}}\\!\\sum_{\\,k\\,=\\,1}^{\\lfloor n\/2 \\rfloor}\\,(-1)^{\\,k}\\, \n \\frac{\\,(2\\pi)^{n\\,-\\,2k\\,+\\,1}\\,} {\\,(n\\,-\\,2k\\,+\\,1\\,)\\,!\\,}\\,\\zeta\\,(\\,2k+1\\,) \\;\\,,\n\\end{eqnarray}\nof which the first two, at $\\,n\\,=\\,0\\,$ and $\\,n\\,=\\,1\\,,$ with the sum on the\nright missing, validate (2) and (3), and are in any event widely tabulated. And again, as was\nfirst stated in Footnote 3, Eq. (14) is consistently reaffirmed by {\\bf{MATHEMATICA}},\neven when harnessed in its symbolic mode. We note in passing the self-evident fact that,\nunlike the corresponding prescriptions found in [\\textbf{9,10}],\nformula (14) is fully explicit, needing to rely neither upon a generating function nor\na recurrence, even though, naturally, such recurrence arrives at a final rendezvous with identically\nthe same result.\n\n\n A close prelude to identity (5) follows next from the co\\\"{e}xisting requirement that\nthe imaginary part of (13) vanish. This requirement takes the initial form\n\\begin{eqnarray}\n-\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,2\\,(\\,n\\,+\\,1\\,)\\,}\\,+\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,\\,n\\,+\\,2\\,}\\,- \\rule{5.2cm}{0mm} & & \\nonumber \\\\\n-\\,\\sum_{k\\,=\\,0}^{\\lfloor \\frac{n-1}{2} \\rfloor }\\left(\\!\\!\\begin{array}{c}\n n \\\\\n 2k\n \\end{array}\\!\\!\\right)\\pi^{\\,n-2k}(-1)^{\\,k}\n\\frac{\\,(2k)\\,!\\,}{\\,2^{\\,2k\\,+\\,1}\\,}\\,\\zeta\\,(\\,2k+2\\,) \\rule{0.0cm}{0mm} & = & 0 \\rule{8mm}{0mm} \n\\end{eqnarray}\nand is subsequently moulded into the shape\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{\\lfloor \\frac{n-1}{2} \\rfloor }\\left(\\!\\!\\begin{array}{c}\n n \\\\\n 2k\n \\end{array}\\!\\!\\right)\n\\frac{\\,B_{2k\\,+\\,2}\\,}{\\,(\\,k\\,+\\,1\\,)(\\,2k\\,+\\,1\\,)\\,}\\,=\\,\\frac{\\,n\\,}{\\,(\\,n\\,+\\,1\\,)(\\,n\\,+\\,2\\,)\\,} \n\\end{equation}\non taking note of Euler's\ncelebrated connection [{\\bf{3-8}}]\n\\begin{equation}\n\\zeta\\,(2k)\\,=\\,(-1)^{\\,k\\,+\\,1}(2\\,\\pi)^{2k}\\frac{\\,B_{2k}\\,}{\\,2\\,(2k)\\,!\\,} \\;\\;\\;(\\,k\\,=\\,1\\,,\\,2\\,,\\,3\\,,\\,\\ldots\\,)\n\\end{equation}\nallowing us to displace attention from the even-argument values of Riemann's zeta\nto the correspondingly indexed Bernoulli numbers $\\,B_{2k}\\,.$\n\\parindent=0.25in\n\n\n\\section{Recurrence Reduction}\n\n Recurrence (16) is not quite yet in the desired form (5), but it is easily steered\ntoward this goal. That process begins by noting that\n\\begin{equation}\n\\left(\\!\\!\\begin{array}{c}\n n \\\\\n 2k\n \\end{array}\\!\\!\\right)\n\\frac{\\,1\\,}{\\,(\\,k\\,+\\,1\\,)(\\,2k\\,+\\,1\\,)\\,}\\,=\\,\n\\left(\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n 2k+2\n \\end{array}\\!\\!\\right)\\frac{\\,2\\,}{\\,(\\,n\\,+\\,1\\,)(\\,n\\,+\\,2\\,)\\,}\\;,\n\\end{equation}\n\\newpage\n\\mbox{ }\n\\newline\n\\newline\n\\newline\nwhereupon (16) becomes\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{\\lfloor \\frac{n-1}{2} \\rfloor}\\left(\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n 2k+2\n \\end{array}\\!\\!\\right)B_{2k\\,+\\,2}\\,=\\,\\frac{\\,n\\,}{\\,2\\,}\\;.\n\\end{equation}\nNow the advance of index $\\,2k\\,$ in steps of two means that it reaches a maximum value $\\,M=n-1\\,$ when $\\,n\\,$ is odd,\nand one offset instead by two below $n,$ $M=n-2,$ when $\\,n\\,$ is even. At the same time the accepted {\\mbox{null value of odd-index}}\nBernoulli numbers starting with $\\,B_{3}=0\\,$ means that we are free, and self-consistently so, to intercalate all missing indices in steps of one\nand to entertain a common maximum $\\,M=n-1\\,,$ regardless of the parity of $n.$ Altogether then, (19) re{\\\"{e}}merges as\n\\begin{equation}\n\\sum_{k\\,=\\,2}^{n+1}\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n k\n \\end{array}\\!\\!\\right)B_{k}\\,=\\,\\frac{\\,n\\,}{\\,2\\,}\\;,\n\\end{equation}\nor else\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{n+1}\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n k\n \\end{array}\\!\\!\\right)B_{k}\\,=\\,\\frac{\\,n\\,}{\\,2\\,}\\,+\\,\\left\\{\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n 0\n \\end{array}\\!\\!\\right)B_{0}\\,+\\,\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n 1\n \\end{array}\\!\\!\\right)B_{1}\\right\\}\\;.\n\\end{equation}\nBut now we find that\n\\begin{equation}\n\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n 0\n \\end{array}\\!\\!\\right)B_{0}\\,+\\,\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n 1\n \\end{array}\\!\\!\\right)B_{1}\\,=\\,1\\,-\\,\\frac{\\,n+2\\,}{\\,2\\,}\\,=\\,-\\frac{\\,n\\,}{\\,2\\,}\\,,\n\\end{equation}\nwith the effect of reducing (21) to just\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{n+1}\\left(\\!\\!\\!\\begin{array}{c}\n \\,n+2 \\\\\n k\n \\end{array}\\!\\!\\right)B_{k}\\,=\\,0\\;,\n\\end{equation}\nwhich is nothing other than (5).\n\n\n\\section{Appendix: A Fourier Series Grace Note}\n\n\n A somewhat more pedestrian derivation of (14) rests upon consideration of the power series\n\\begin{equation}\n\\log\\,(\\,1-z\\,)\\,=\\,-\\,\\sum_{l\\,=\\,1}^{\\infty}\\,\\frac{\\,z^{\\,l}\\,}{\\,l\\,}\n\\end{equation}\nalong the unit circle $\\,z\\,=\\,e^{\\,i\\,\\vartheta}.$\nSeparation into real and imaginary parts emerges as a pair of Fourier series\n\\begin{equation}\n\\log\\left(\\rule{0mm}{5mm}2\\left|\\,\\sin\\,\\left\\{\\frac{\\,\\vartheta\\,}{\\,2\\,}\\right\\}\\right|\\right)\\,=\\,\n-\\sum_{l\\,=\\,1}^{\\infty}\\,\\frac{\\,\\cos\\,(l\\,\\vartheta)\\,}{\\,l\\,}\n\\end{equation}\nand\n\\begin{equation}\n\\left\\{\\rule{0mm}{5mm}\\frac{\\,\\vartheta-\\pi\\,}{\\,2\\,}\\,,\\,{\\rm{mod}}\\,2\\,\\pi\\right\\}\\,\n=\\,-\\sum_{l\\,=\\,1}^{\\infty}\\,\\frac{\\,\\sin\\,(l\\,\\vartheta)\\,}{\\,l\\,}\\;\\;,\n\\end{equation}\nof which the second is of no interest {\\em{vis-\\`{a}-vis}} our immediate objective. We repress all scruples\nhenceforth as to the divergence of series (25) whenever $\\,\\vartheta\\,=\\,0\\;\\,{\\rm{mod}}\\;2\\,\\pi\\,.$\n\n\n\\newpage\n\\mbox{ }\n\\newline\n\n\n Repeated integration by parts {\\em{vis-\\`{a}-vis}} the first of these Fourier series,\nwhen multiplied by the argument power $\\,\\vartheta^{\\,n}\\,,$\nadvances by $\\,\\cos\\rightarrow\\sin\\rightarrow\\cos\\,$ couplets, with end-point contributions\narising only on the second beat, and the argument powers falling\nin steps of two.\\footnote{In particular, this quadrature cadence provides a\nmotivation, alternative to that previously given,\nas to why it is that the floor function affects the upper index cutoff\n$\\,\\lfloor n\/2 \\rfloor\\,$ in both (14) and (27), allowing for unit growth in that\ncutoff only when $\\,n\\,$ {\\em{per se}} advances by two.} One assembles in this manner the general formula\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\,\\vartheta^{\\,n}\\log(\\,\\sin(\\vartheta)\\,)\\,d\\,\\vartheta \\, = \\, \n-\\,\\frac{\\,\\pi^{\\,n\\,+\\,1}\\,}{\\,n\\,+\\,1\\,}\\,\\log(2)\\,+\\,\\frac{\\,n\\,!\\,}{\\,2^{\\,n\\,+\\,1}\\,}\n\\,\\sum_{\\,k\\,=\\,1}^{\\lfloor n\/2 \\rfloor}\\,(-\\,1)^{\\,k}\\, \n\\frac{\\,(\\,2\\,\\pi\\,)^{\\,n\\,-\\,2k\\,+\\,1}\\,}\n{\\,(\\,n\\,-\\,2k\\,+\\,1\\,)\\,!\\,}\\,\\zeta\\,(\\,2k\\,+\\,1\\,\n\\end{equation}\nholding good unrestrictedly for $\\,n\\,$ even or odd, and agreeing in every respect with (14). The only\nwrinkle to notice, perhaps, is that the sequence of integrations by parts which underlies (27) terminates,\nat each summation index $\\,l\\,$ in (25), with a term proportional to either\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\cos(2l\\vartheta)\\,d\\,\\vartheta\\,=\\,0\n\\end{equation}\nin the event that $\\,n\\,$ is even, or\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\vartheta \\cos(2l\\vartheta)\\,d\\,\\vartheta\\,=\\,0\n\\end{equation} \notherwise. Equation (28) is of course obvious whereas (29), while equally true and\nwelcome as such, is, at first blush, mildly surprising. All in all the derivation which underlies (14) is\nfar smoother and less apt to inflict bookkeeping stress, even if it is (27) which seems to rest\non a more elementary underpinning.\n\n It would be truly disappointing were we not able to utilize (25) so as to give an\nessentially one-line, zinger-style proof of (1). This anticipation is readily met simply\nby squaring both sides of (25), with summation indices $\\,l\\,$ and $\\,l\\,'\\,$ figuring\nnow on its right, and noting that when, as here, both $\\,l\\,\\geq\\,1\\,$ and\n$\\,l\\,'\\,\\geq\\,1\\,,$\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\cos(\\,2\\,l\\,\\vartheta\\,)\\cos(\\,2\\,l\\,'\\vartheta\\,)\\,d\\,\\vartheta\\,=\\,\\frac{\\,\\pi\\,}{\\,2\\,}\\,\n\\delta^{\\,l}_{\\,l\\,'}\\;\\,,\n\\end{equation}\nwith $\\,\\delta^{\\,l}_{\\,l\\,'}\\,$ being the Kronecker delta, unity when its indices match, and zero otherwise.\nIt follows immediately that\n\\begin{equation}\nI\\,=\\,\\frac{\\,1\\,}{\\,2\\,}\n\\int_{\\,0}^{\\,\\pi}\\left\\{\\rule{0mm}{4mm}\\,\\log(\\,2\\,\\sin(\\vartheta)\\,)\\,\\right\\}^{\\,2}\\!d\\vartheta\\,=\\,\n\\frac{\\,\\pi\\,}{\\,4\\,}\\sum_{l=1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,2}\\,}\\,=\\,\\frac{\\;\\;\\pi^{\\,3}\\,}{\\,24\\,}\\;\\,,\n\\end{equation}\nand we are done.\n\n\n\\parindent=0.0in\n\n\n\\section{References}\n\n1.\tOmran Kouba, Problem No. 11639, {\\bf{The American Mathematical Monthly}}, Vol. 119, No. 4, April 2012, p. 345.\n\n2.\tLars V. Ahlfors, {\\bf{Complex Analysis}}, McGraw-Hill Book Company, Inc., New York, 1953, pp. 130-131.\n\\newpage\n\\mbox{ }\n\\newline\n\\newline\n\\newline\n3.\tTom M. Apostol, {\\bf{Introduction to Analytic Number Theory}}, Springer-Verlag, New York, 1976, p. 266.\n\n4.\tHans Rademacher, {\\bf{Topics in Analytic Number Theory}}, Springer-Verlag, New York, 1973, p. 16.\n\n5.\tHerbert S. Wilf, {\\bf{Mathematics for the Physical Sciences}}, John Wiley \\& Sons, Inc., New York, 1962, pp. 114-116.\n\n6.\tTom M. Apostol, {\\bf{A Primer on Bernoulli Numbers and Polynomials}}, Mathematics Magazine, Vol. 81, No. 3, June 2008, pp. 178-190.\n\n7.\tOmran Kouba, {\\bf{Lecture Notes: Bernoulli Polynomials and Applications}}, arXiv:1309.7560v1 [math.CA] 29 Sep 2013.\n\n8.\tEric W. Weisstein, {\\bf{Bernoulli Number}}, from {\\em{MathWorld}}--A Wolfram Web Resource available\n@ {\\bf{http:\/\/mathworld.wolfram.com\/BernoulliNumber.html}}\n\n9.\tL. Lewin, {\\bf{On the evaluation of log-sine integrals}}, {\\em{The Mathematical Gazette}}, Vol. 42, 1958, pp. 125-128.\n\n10.\tL. Lewin, {\\bf{Polylogarithms and associated functions}}, North Holland, 1981.\n\n11.\tJonathan M. Borwein and Armin Straub, {\\bf{Special values of generalized log-sine integrals}},\n{\\em{Proceedings of ISSAC 2011 (36th International Symposium on Symbolic and Algebraic Computation)}}, 2011, pp. 43-50.\n\n\n12.\tJonathan M. Borwein and Armin Straub, {\\bf{Mahler measures, short walks and log-sine integrals}}, {\\em{Theoretical Computer Science\n(Special issue on Symbolic and Numeric Computation)}}, Vol. 479, No. 1, 2013, pp. 4-21.\n\n\n\n\n\\end{document}\n \t\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{}\n\n\n\\tableofcontents\n\n\\section{Introduction}\n\nA large class of interesting and useful asymptotically locally anti-de Sitter (AlAdS) spacetimes have been constructed by starting with AdS in Poincar\\'e coordinates, in which the spacetime is foliated by slices on which the metric is conformal to the Minkowski metric $\\eta_{ab}$, and replacing $\\eta_{ab}$ with any Ricci-flat metric $\\gamma_{ab}$. Thus~$(D-1)$-dimensional vacuum solutions to Einstein's equations straightforwardly give rise to new $D$-dimensional solutions to Einstein's equation with a negative cosmological constant. For example, taking $\\gamma_{ab}$ to be the Schwarzchild black hole yields a black cigar \\cite{Chamblin:1999by}, and $\\gamma_{ab}$ was taken to be a vacuum $pp$-wave in \\cite{Chamblin:1999cj} to construct a wave in the far field of an AdS-brane spacetime. Furthermore, the AlAdS solutions so generated are of particular interest in light of the AdS\/CFT correspondence~\\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}, since they are dual to a large-$N$, strongly coupled conformal field theory (CFT) that lives on the spacetime~$\\gamma_{ab}$ (or a conformally rescaled version thereof). For instance, \\cite{Engelhardt:2013jda,Engelhardt:2014mea} took~$\\gamma_{ab}$ to be a vacuum Kasner metric in order to study a cosmological singularity by computing the entanglement entropy and Wightman functions of the CFT\\footnote{In fact, in~\\cite{Engelhardt:2014mea} the CFT lived on a singularity-free conformally rescaled version of Kasner.}.\n\nIn this paper we generalize this construction and show how non-vacuum $(D-1)$-dimensional spacetimes can be used to give AlAdS spacetimes with nonzero stress-energy. Specifically, if $\\gamma_{ab}$ is a solution to the Einstein equations in $(D-1)$-dimensions with stress-energy tensor $\\widehat{T}_{ab}$, we show that replacing $\\eta_{ab}$ on the Poincar\\'e slices with $\\gamma_{ab}$ gives a $D$-dimensional AdS solution with stress-energy $T_{ab}$ that satisfies\n\\be\nT_{\\mu\\nu} = \\widehat{T}_{\\mu\\nu},\n\\ee\nwhere~$\\mu,\\nu = 0,\\ldots,D-1$. By appropriate choice of~$\\widehat{T}_{ab}$, we use this construction to find new AlAdS spacetimes that have a physically sensible stress-energy. We also show that these spacetimes can be ``solitonized'' \\cite{Haehl:2012tw} by adding a compact dimension that shrinks smoothly to zero in the AdS bulk. A variation of this non-vacuum construction was performed in \\cite{Cvetic:2000gj,Lu:2000xc,Park:2001jh}, which studied getting supergravity gauge fields ``on the brane'' by doing a Kaluza-Klein reduction of a supergravity theory in the higher dimensional AdS spacetime. A related\nconstruction starting with supergravity fields in ten dimensions was used to explore properties of time dependent boundaries in references\n \\cite{Das:2006dz,Das:2006pw,Awad:2007fj,Awad:2008jf}.\n The relation between their higher and lower dimensional matter theories differs from what we find here, as will be clarified in Section~\\ref{derive}. \n\nAs a special example, we will take~$\\gamma_{ab}$ to be a Friedman-Robertson-Walker (FRW) cosmology\\footnote{From the CFT side, such solutions can be thought of as a generalization of those in~\\cite{Koyama:2001rf}, which took the boundary metric to be a conformally flat FRW geometry.}. The cosmological stress-energy is an isotropic perfect fluid with energy density $\\tilde{\\rho} (t)$ and pressure $ \\tilde{p} (t) $, which are related by an equation of state $\\tilde{p} (t) = w \\tilde{\\rho} (t) $. We will show that in the AdS spacetime, the fluid is still isotropic on the cosmological slices with the same equation of state $p= w\\rho $ and that the pressure in the AdS radial direction is given by $p_y= (3w - 1)\\rho \/2 $. The pressures and density decay towards the AdS boundary as well as in time as the universe expands. A case of special interest is a free, massless scalar field which in a $(D-1)$-dimensional FRW spacetime has the equation of state $\\tilde{p} = \\tilde{\\rho} $, that is, $w=1$. Hence the scalar field generates a $D$-dimensional AdS cosmology which is isotropic in all spatial directions and has corresponding equation of state $p=\\rho$. According to the AdS\/CFT prescription, such a scalar field in the bulk AdS is dual to to a scalar operator in the CFT with vanishing expectation value but nonzero source.\n \nWe will focus on FRW metrics with negatively curved spatial slices, in which case~$\\gamma_{ab}$ approaches the future Milne wedge of Minkowski space at late time, so long as $w>-1\/3$. The resulting AlAdS solution therefore approaches either the Poincar\\'e patch of AdS or the AdS soliton at late times, which we interpret as an approach to equilibrium. We use these solutions to perturbatively study the approach to equilibrium of the boundary stress tensor and the ADM charges. \nInterestingly, we find that the latter \\textit{decrease} to their equilibrium values at late times, with the time dependent correction proportional to the dimensionless density parameter of the universe $\\Omega$. \nFor example, the mass of the solitonized cosmology decays as\n\\be\n {\\cal M} = \\left( 1-{1\\over 2} \\Omega \\right) {\\cal M}^{(0)} + \\cdots,\n \\ee\n where ${\\cal M}^{(0)} $ is the mass of the static soliton and~$\\cdots$ stands for subleading terms at late times.\nIn the context of spacetimes approaching the AdS soliton, this result is consistent with the energy conjecture of \\cite{Horowitz:1998ha} that the AdS soliton is the lowest energy spacetime with the prescribed asymptotic structure.\n\nA second application of our cosmological AdS solutions will be to compute the behavior of the entanglement entropy $S$ of a spherical region in the CFT as the spacetime evolves to equilibrium. We use the covariant prescription of~\\cite{Hubeny:2007xt}, which is a generalization of the static prescription~\\cite{Ryu:2006ef}. This states that the entanglement entropy~$S_\\mathcal{R}$ of a region~$\\mathcal{R}$ of a holographic CFT is related to the area of a special bulk surface~$\\Sigma$. In general,~$S_\\mathcal{R}$ is UV-divergent, but it can be regulated and the behavior of this regulated entropy~$S_\\mathrm{ren}$ is studied. We find that at late times,~$S_\\mathrm{ren}$ decays as a power law in the proper time of an asymptotically static observer.\n\nOur results add to and complement the substantial body of work in the literature on vacuum AlAdS spacetimes in which the metric on the AdS boundary is time dependent. For instance,~\\cite{Fischler:2013fba} constructed an elegant solution in which the metric on each Poincar\\'e slice is a de Sitter cosmology. Several studies in the general category of holographic cosmology apply coordinate transformations to AdS black holes to produce cosmological boundaries \\cite{Lidsey:2009xz,Erdmenger:2012yh,Ghoroku:2012vi,Banerjee:2012dw,\nCompere:2008us,Binetruy:1999hy,Kajantie:2008hh,Apostolopoulos:2008ru}, and resulting metrics have been analyzed as describing an expanding boost-invariant plasma \\cite{Janik:2005zt,Janik:2006ft,Kajantie:2008jz,Culetu:2009xm,Pedraza:2014moa}.\nSignificant analytical work has also been done on out of equilibrium thermal properties of field theories using various AdS black hole spacetimes, including \\cite{Janik:2010we,Lamprou:2011sa,Figueras:2009iu,\nTetradis:2009bk,Heller:2011ju,Fischetti:2012ps,Fischetti:2012vt,Beuf:2009cx}. Discussion and further references can be found in \\cite{Marolf:2013ioa}. Our work adds a set of new \nnon-vacuum AlAdS spacetimes which allow a wide range of boundary metrics.\n \nThis paper is organized as follows. Section \\ref{derive} contains the derivation of the new AlAdS solutions, as well as an analysis of the scalar field and perfect fluid cases. In section \\ref{sec:bst}, the leading time dependent corrections to the boundary stress tensor and the ADM charges for a solitonic cosmology in an open universe are found. In section \\ref{entropy} the perturbation to the entanglement entropy is calculated, and section \\ref{conclusion} contains discussion and concluding remarks. Unless otherwise specified, we take Newton's constant~$G_N = 1$. \n\n\n\n\\section{AdS and AdS soliton cosmologies}\\label{derive}\n\nWe start by considering AlAdS spacetimes of the general form\n\\be\n\\label{metric}\nds_D^2 = dy^2 + e^{2y\/l}\\gamma_{\\mu\\nu } (x^\\alpha ) dx^\\mu dx^\\nu,\n\\ee\nwhere~$l$ is the AdS length and as before~$\\mu,\\nu = 0,1,\\dots, D-1$. For $\\gamma_{\\mu\\nu} = \\eta_{\\mu\\nu}$ this is AdS in Poincar\\'e coordinates with cosmological constant given by \n\\be\n\\Lambda = -{(D-1)(D-2)\\over 2l^2},\n\\ee\nso we will refer to hypersurfaces~$y = \\mathrm{const.}$ as ``Poincar\\'e slices''. As mentioned above, it is well known that the Einstein equations with cosmological constant $\\Lambda$ are still satisfied for any Ricci-flat $\\gamma_{\\mu\\nu}(x^\\rho)$. In the particular case where~$\\gamma_{\\mu\\nu}$ is a cosmological metric, we refer to \\eqref{metric} as an AdS cosmology. \n\nIn general, spacetimes of the form~\\eqref{metric} with~$\\gamma_{\\mu\\nu} \\neq \\eta_{\\mu\\nu}$ suffer from a singularity at the Poincar\\'e horizon~$y \\to -\\infty$. This singularity can be resolved by introducing an additional compact direction~$v$:\n\\be\n\\label{solmetric}\nds_D^2 = {dy^2 \\over F(y) } + e^{2y\/l}\\left( F(y) dv^2 + \\gamma_{\\mu\\nu } (x^\\rho ) dx^\\mu dx^\\nu \\right),\n\\ee\nwhere $F(y) = 1- e^{ -(D-1)(y-y_+)\/l }$, and now~$\\mu,\\nu= 0,1,\\dots,D-2$. The metric~\\eqref{solmetric} is capped off at~$y = y_+$, so that the Poincar\\'e horizon (and its possibly singular behavior) is removed. Regularity at this cap fixes the period of~$v$ to be\n\\be\\label{period}\nv \\sim v + \\frac{4\\pi l e^{-y_+\/l}}{D-1}.\n\\ee\nNow, if $\\gamma_{\\mu\\nu } = \\eta_{\\mu\\nu}$, then~\\eqref{solmetric} is the usual AdS soliton metric~\\cite{Horowitz:1998ha}\\footnote{Although this is no longer Poincar\\'e AdS, we will continue to call surfaces of~$y = \\mathrm{const.}$,~$v = \\mathrm{const.}$ Poincar\\'e slices.}. However, it was noted in \\cite{Haehl:2012tw} that the Einstein equations with negative cosmological constant $\\Lambda$ will still be satisfied for any Ricci-flat~$\\gamma_{\\mu\\nu}$. In analogy with~\\eqref{metric}, if~$\\gamma_{\\mu\\nu}$ is a cosmological metric, we will refer to~\\eqref{solmetric} as an AdS soliton cosmology.\n\nThe solutions~\\eqref{metric} and~\\eqref{solmetric} provide a simple construction of AlAdS spacetimes with any desired Ricci-flat boundary metric~$\\gamma_{\\mu\\nu}$\\footnote{Technically, the boundary of~\\eqref{solmetric} is~$\\gamma_{\\mu\\nu}$ cross the circle direction~$v$.}. Our goal is to generalize the above results to isotropic FRW cosmological metrics $\\gamma_{\\mu\\nu}$; as such cosmologies are not (in general) Ricci-flat, we will require the introduction of matter fields.\n\n\n\n\\subsection{Massless Scalar Field}\n\nWe obtain a direct analogue of the results for vacuum metrics by considering a free massless scalar field $\\phi$ in the AdS and AdS soliton spacetimes above. The full $D$-dimensional Einstein-massless scalar equations are\n\\be\n\\label{einstein}\nG_{ab}=-\\Lambda g_{ab}+8\\pi T_{ab},\\qquad \\nabla^2\\phi=0,\n\\ee\nwhere~$G_{ab}$ is the Einstein tensor and \n\\be\nT_{ab}={1\\over 8\\pi}[(\\nabla_a\\phi)\\left(\\nabla_b\\phi\\right)-{1\\over 2}g_{ab}g^{cd}(\\nabla_c\\phi)\\left(\\nabla_d\\phi\\right)]\n\\ee\nis the stress-energy of a free massless scalar field in any dimension. \nConsider a lower-dimensional metric $\\gamma_{\\mu\\nu}(x^\\rho)$ and scalar field configuration $\\phi(x^\\mu)$ that solve the Einstein-scalar equations\n\\be\n\\label{slice}\n{\\widehat G}_{\\mu\\nu} = 8\\pi{\\widehat T}_{\\mu\\nu}, \\qquad \\widehat\\nabla^2\\phi=0,\n\\ee\nwhere hatted objects are computed with respect to the metric~$\\gamma_{\\mu\\nu}$. Furthermore, let\n\\be\\label{smetricdef}\ns_{\\mu\\nu}= e^{2y\/l} \\gamma_{\\mu\\nu}\n\\ee\nbe the induced metric on a Poincar\\'e slice. Finally, we pause to note that the scalar field stress energy satisfies the important property that from the full~$D$-dimensional point of view, the induced stress tensor on each Poincar\\'e slice is equal to the lower-dimensional stress tensor of the scalar field on~$\\gamma_{\\mu\\nu}$:\n\\be\n\\label{Tcondition}\nT_{\\mu\\nu} = \\widehat{T}_{\\mu\\nu}.\n\\ee\n\nNow, consider first the metric (\\ref{metric}). The $D$-dimensional Ricci tensor for $g_{ab}$ is related to the Ricci tensor of $s_{\\mu\\nu}$ by\n %\n\\be\n\\label{ricci}\nR_{\\mu\\nu} [g] = R_{\\mu\\nu} [s] - \\frac{D-1}{l^2} \\, s_{\\mu\\nu}, \\quad R_{yy} = \\frac{D-1}{l^2}.\n\\ee\n When these components are assembled into the $D$-dimensional Einstein tensor and substituted into the left hand side of the Einstein field equation \\eqref{einstein}, one sees that the terms which do not involve the curvature of $s_{\\mu\\nu}$ are equal to the cosmological constant term on the right hand side. If $\\gamma_{\\mu\\nu}$ is Ricci flat, then the metric (\\ref{metric}) is a solution with $T_{ab}=0$. If instead $\\gamma_{\\mu\\nu}$ is a solution to (\\ref{slice})\n with nonzero ${\\widehat T}_{\\mu\\nu}$,\n then it is then straightforward to show that the metric~\\eqref{metric} constructed from~$\\gamma_{\\mu\\nu}$ will satisfy the full equations of motion \\eqref{einstein}, with the full bulk scalar field taken to be~$\\phi(x^\\mu)$ (which, in particular, is independent of~$y$). The additional nonzero component of the\n stress-energy tensor is $8\\pi T_{yy}=-{1\\over 2} s^{\\mu\\nu} \\nabla_\\mu \\phi \\nabla_\\nu \\phi$. The construction with the solitonized metric~\\eqref{solmetric} proceeds in a similar way, and one finds that $T_{yy}$ is the same and $T_{vv} = g_{vv} T_{yy}$.\n\nOne may naturally ask if such a straightforward foliation can be extended to other types of matter as well. For instance, one might hope to replace the scalar field with a Maxwell field and obtain multi-black hole solutions analogous to those of~\\cite{Kastor:1992nn}. This is not the case: a key ingredient in the proof was the property \\eqref{Tcondition} of the scalar field stress-energy. \nThis property holds for the massless scalar field stress tensor but not {\\it e.g.} for Maxwell fields, or even for a scalar field with nonzero potential $V(\\phi)$.\n\nA massless scalar field that depends only on time can serve as the source for an FRW cosmology on the Poincar\\'e slices of either \\eqref{metric} or the soliton metric \\eqref{solmetric}. For instance, setting $d\\hat s^2=\\gamma_{\\mu\\nu}dx^\\mu dx^\\nu$ and specializing to $4$-dimensional cosmologies with flat spatial sections we have\n\\be\nd\\hat s^2= -dt^2 + \\left({t\\over t_0}\\right)^{2\/3}(dx^2+dy^2+dz^2),\\quad \\phi=-\\sqrt{{2\\over 3}} \\, \\ln\\left({t\\over t_0}\\right),\n\\ee\nwith corresponding stress tensor equal to that of a perfect fluid obeying the stiff matter equation of state $\\tilde{p}=\\tilde{\\rho}$. From the holographic perspective, the AdS\/CFT dictionary tells us that the bulk scalar field is dual to a scalar operator in the CFT. To be specific, the near-boundary behavior of a massless scalar field in AdS takes the form\n\\be\n\\phi(y) = \\left(\\phi_0 + \\cdots\\right) + e^{-(D-1)y\/l} \\left(\\phi_{(D-1)} + \\cdots \\right),\n\\ee\nwhere~$\\phi_0$ and~$\\phi_{(D-1)}$ are independent parameters that are fixed by the boundary conditions, and~$\\cdots$ represent subleading terms in~$e^{-y\/l}$. The coefficient~$\\phi_0$ should be interpreted as the source of a scalar operator~$\\mathcal{O}$ of dimension~$D-1$, whose expectation value is~$\\left\\langle \\mathcal{O} \\right\\rangle = \\phi_{(D-1)}$. Our solutions correspond to the special case~$\\phi_{(D-1)} = 0$. Note that this is unconventional: the operator~$\\mathcal{O}$ is being sourced, but nevertheless has a zero expectation value.\n\n\n\\subsection{Perfect Fluid Matter}\n\nNoting that property~\\eqref{Tcondition} of the scalar field stress tensor was the key element in the above construction, we may extend the range of AdS and AdS soliton cosmologies by considering more general types of stress-energy that satisfy this condition. We will shortly focus on perfect fluids, but we begin by assuming just that the metric $\\gamma_{\\mu\\nu}$ satisfies Einstein's equation on a Poincar\\'e slice with some stress-energy ${\\widehat T}_{\\mu\\nu}$. We can then analyze the content of the full $D$ dimensional Einstein equations,\n beginning with the AdS type metrics (\\ref{metric}), in the following way. \n\nUsing the relations between the components of the Ricci tensor (\\ref{ricci}) as in the previous subsection, we find that\n the AdS-type metric \\eqref{metric} solves the Einstein equation~\\eqref{einstein} with stress-energy given by\n\\be\n\\label{fullads}\nT_{\\mu\\nu} = {\\widehat T}_{\\mu\\nu},\\qquad T_{yy}={1\\over D-3}\\, T ,\n\\ee\nwhere $T=s^{\\mu\\nu}\\,T_{\\mu\\nu}$. \n A similar analysis for the AdS soliton-type metric \\eqref{solmetric} shows that the Einstein equation \\eqref{einstein} is solved with stress-energy given by\n\\be\n\\label{fullsoliton}\nT_{\\mu\\nu} = {\\widehat T}_{\\mu\\nu},\\qquad T_{yy}={1\\over D-4} \\, g_{yy}T, \\qquad T_{vv}={1\\over D-4} \\, g_{vv}T.\n\\ee\nFor example, one could embed a textbook example of a four-dimensional spherical static star into AdS. According to\n(\\ref{fullsoliton}) the pressures in the radial AdS and compact soliton directions of this cigar-star will be equal to each other,\nbut different from the radial pressure in the Poincar\\'e plane.\n\nWe now specialize to the case of AdS and AdS soliton cosmologies, taking\nthe metric $\\gamma_{\\mu\\nu}$ to have the FRW form\n\\be\n\\label{cosmometric}\nd\\hat s^2 = -dt^2 + a^2 (t)\\, d\\Sigma _k ^2,\n\\ee\nwhere $d\\Sigma _k ^2 $ is a metric on a space with constant curvature $k=0,\\pm 1$. We also restrict our attention to $4$-dimensional Poincar\\'e slices, so that the AdS cosmologies \\eqref{metric} have overall dimension $D=5$ and the AdS soliton cosmologies \\eqref{solmetric} have dimension $D=6$. \n Finally, we assume that the stress-energy ${\\widehat T}_{\\mu\\nu}$ on the slice has the perfect fluid form\n\\be\n{\\widehat T}_{\\mu\\nu} = (\\hat\\rho+\\hat p)\\hat{u}_\\mu \\hat{u}_\\nu +\\hat p \\gamma_{\\mu\\nu},\n\\ee\nwith $\\gamma^{\\mu\\nu}\\hat{u}_\\mu \\hat{u}_\\nu=-1$ and equation of state $\\hat p=w\\hat\\rho$. Note that the strong energy condition requires~$w \\geq -1\/3$. Important special cases are $w=0$ for dust, $w=1\/3$ for radiation, and $w=1$ for the massless free scalar field; indeed, note that such a stress tensor obeys the condition~\\eqref{Tcondition}\\footnote{Values of $w$ different from~1 could be obtained from an interacting scalar field, but as such interactions would require the introduction of a scalar potential, they are not compatible with our ansatz. We will keep $w$ general, with the understanding that this is a bulk, hydrodynamic description.}.\n\nThe cosmological scale factor on the Poincar\\'e slices evolves according to the Friedmann equations\n\\be\n\\label{frweqs}\nd(\\hat \\rho\\, a^3 ) = - \\hat p\\, d(a^3 ) , \\qquad\n\\left( {\\dot a\\over a}\\right)^2 ={8\\pi \\hat\\rho\\over 3}-{k\\over a^2}.\n\\ee\nThe full stress energy tensor $T_{ab}$ for AdS cosmologies, given by \\eqref{fullads}, now has the form of an anisotropic fluid, with a distinct equation of state parameter for the pressure in the $y$-direction. Moreover, the energy density and pressures depend on the radial coordinate~$y$, as well as on time. One finds that the energy density is given by $\\rho= e^{-2y\/l}\\hat\\rho$, while the pressures tangent to the Poincar\\'e slices satisfy an equation of state in the $D$-dimensions of the same form as in $(D-1)$, namely $p=w\\rho$. In the $y$-direction one finds that $p_y = w_y\\rho$ with\n\\be\nw_y ={ (3w -1)\\over 2}.\n\\ee\nWith a soliton, equation \\eqref{fullsoliton} implies that the pressure in the compact $v$-direction is equal to $p_y$, so also\n$p_v= w_y\\rho$. To summarize, the stress-energy for the AdS soliton cosmology is\n\\be\n\\rho= e^{-2y\/l}\\hat\\rho (t), \\quad p=w\\rho, \\quad p_y =p_v = { (3w -1)\\over 2} \\, \\rho.\n\\ee\nSome observations are as follows. For $w=1$, which corresponds to the massless scalar field discussed above, $w_y=1$ as well so\nthe pressure in the full spacetime is isotropic. For radiation ($w=1\/3$), the stress-energy on the Poincar\\'e slices is traceless and the pressure orthogonal to the slices vanishes, so the stress tensor remains traceless. For $w<1\/3$, the orthogonal pressure is negative.\n\n\n\\subsection{Open AdS and AdS Soliton Cosmologies}\n\nWe will be particularly interested in AdS and AdS soliton cosmologies with open ($k=-1$) FRW universes on the Poincar\\'e slices. In this case, provided that the equation of state parameter is in the range $w>-1\/3$ (that is, that the strong energy condition holds), the energy density $\\hat\\rho$ will fall off faster than $1\/a^2$ and at late times the scale factor will grow linearly in time. At sufficiently late times, the metric $\\gamma_{\\mu\\nu}$ on the Poincare slices then approaches\n\\be\nd\\hat s^2_\\mathrm{late} = -dt^2 + t^2\\, d\\Sigma _{-1} ^2,\n\\ee\nwhich is flat spacetime in Milne coordinates.\nThe full AdS and AdS soliton cosmological metrics \\eqref{metric} and \\eqref{solmetric} then respectively approach the AdS or AdS soliton metrics at late times. The late-time behavior of these cosmologies can therefore be thought of as an approach to equilibrium; in particular, the CFT dual can be thought of as an expanding isotropic plasma equilibrating at late time. The solutions~\\eqref{metric} and~\\eqref{solmetric} correspond to the plasma being in a deconfined or confined phase, respectively.\n\n\n\n\\section{ADM Mass and Boundary Stress Tensor for AdS Soliton Cosmologies}\n\\label{sec:bst} \n\nFrom the field theoretic side, the late-time behavior of the AdS cosmology with open spatial slices described above is interpreted as a relaxation of the CFT to the vacuum state. This relaxation can be studied by computing the late-time behavior of CFT observables. \nAs a first examination of the properties of these AdS cosmologies, we look at how the cosmological expansion impacts the boundary stress tensor and the\nADM mass and tensions of the AdS soliton (which corresponds to the confined phase of the dual field theory; see e.g.~\\cite{Mateos:2007ay}). In general we lack a definition of the ADM charges that will apply at the boundary $y=\\infty$ with a time dependent boundary metric. However, as we will see the special case of an open cosmology with matter obeying the strong energy condition~$w>-1\/3$ allows for a perturbative computation of how the ADM charges of the soliton approach their static values at late times.\n\nThe static AdS soliton has negative ADM mass, reflecting the negative Casimir energy of the boundary field theory with a compact direction, and is conjectured to be the lowest energy solution among spacetimes with these asymptotics \\cite{Horowitz:1998ha}.\nIn addition to its mass, the AdS soliton has nonzero ADM tensions \\cite{El-Menoufi:2013pza,El-Menoufi:2013tca}. The tension along the compact $v$-direction in (\\ref{solmetric}) is found to be large and positive, while the other three spatial tensions have negative values, such that the trace of the ADM charges (sum of the mass and the tensions) vanishes. One can think of the static soliton solution as an equilibrium configuration. In this section we will compute the approach to equilibrium of the boundary stress tensor, as well as the mass and tensions for an open AdS soliton cosmology. We will see that the mass decreases to the static soliton value, a result that is consistent with the minimum mass conjecture with matter obeying the strong energy condition.\n\nAs noted above, the FRW boundary metric does not have a time-translation symmetry and therefore the ADM mass is not defined in the usual sense. However, at late times the FRW cosmologies with negatively curved spatial slices approach Minkowski spacetime. We can then define a time dependent ADM mass in this late time limit by writing the metric as static AdS plus time dependent perturbations that decay to zero. These perturbations to the metric determine the late time corrections to the asymptotic constant value of the mass of the soliton. \n\n\nConsider an AdS soliton cosmology \\eqref{solmetric} with an open FRW metric \n\\be\n\\label{openfrw}\nd\\hat s^2 = - dt^2 + a^2 (t) \\left( d\\chi ^2 + \\sinh ^2 \\chi \\, d\\Omega_{(2)} ^2 \\right)\n\\ee\non the Poincar\\'e slices. We assume that $w>-1\/3$, so that in the late time limit $a(t)\\simeq t$. Define new coordinates on the slices according to\n\\be\n\\label{coordtrans}\nT =a(t ) \\cosh \\chi, \\quad R= a(t) \\sinh \\chi.\n\\ee\nNote that since $R\/T =\\tanh \\chi$ it follows that $R\/T \\leq 1$ with equality when $\\chi \\rightarrow \\infty$.\nIn terms of these new coordinates the AdS soliton cosmology has the form\n\\be\n\\label{latemetric}\nds^2 = {dy^2 \\over F(y) } + e^{2y\/l} \\left[\\, F(y)\\, dv^2 -dT^2 (1-\\delta \\tilde{g}_{TT} ) +dR^2 (1+ \\delta \\tilde{g}_{RR} ) + 2 \\, \\delta \\tilde{g}_{TR}\\, dR\\, dT +\nR^2 d\\Omega_{(2)} ^2 \\right]\n\\ee\nwhere \n$F(y)$ is given in (\\ref{solmetric}) and the functions $\\delta \\tilde{g}_{TT}$, $\\delta \\tilde{g}_{RR}$ and $\\delta \\tilde{g}_{TR}$, which give the deviance of the metric on the Poincar\\'e slices from flat, may be written as\n\\be\n\\label{fndef}\n\\delta \\tilde{g}_{TT} = \\Omega \\, \\frac{1}{1-(R\/T)^2}, \\quad\n \\delta \\tilde{g}_{RR} = \\Omega \\, \\frac{(R\/T)^2}{1-(R\/T)^2}, \\quad \\delta \\tilde{g}_{TR} = \\Omega \\, \\frac{R\/T}{1-(R\/T)^2}.\n\\ee\nHere $\\Omega$ is the dimensionless density parameter of the open FRW metric,\n\\be\\label{omegadef}\n\\Omega = {8\\pi \\hat \\rho \\over 3 H^2} = \\left(1- {1\\over \\dot{a} ^2 } \\right),\n\\ee\nand $H=\\dot a\/a$ is the Hubble parameter. For an open universe $\\Omega<1$ and approaches zero in the far future. Hence, the metric \\eqref{latemetric} approaches the AdS soliton at late times. We emphasize that the expressions in \\eqref{fndef} are exact up to this point.\n\n\nTo proceed further, the density parameter $\\Omega$ must be expressed in terms of the asymptotically Minkowski coordinates $(T,R)$, which requires the expression for the scale factor $a(t)$ at late times. To obtain this expression, first we substitute the equation of state~$p=w \\rho$ into the Friedman equations \\eqref{frweqs}, which allows the energy density to be solved for in terms of the scale factor, giving\n\\be\\label{laterho}\n\\hat\\rho (t)= {3\\bar{\\Omega}_* H_*^2\\over 8\\pi (H_*a(t))^{3(1+w)}} \\ , \\mbox{ where } \\bar{\\Omega}_* \\equiv {\\Omega_* \\over (1- \\Omega_* )^{3(w+1)\/2 }}\n\\ee\nand $H_*$ and $\\Omega_*$ are the Hubble and density parameters evaluated at a fiducial time $t=t_*$.\nThe density and scale factor evaluated at $t_*$ are given by $\\hat{ \\rho}_* = 3\\Omega_* H_*^2\/8\\pi$ and $a_* = 1\/ ( H_* \\sqrt{1-\\Omega_* }) $ respectively. The equation for the scale factor then reduces to\n\\be\n\\dot a^2 = 1 +{ \\bar\\Omega_*H_*^2 a_*^{3(1+w)}\\over a^{1+3w}}.\n\\ee\nFor $w>-1\/3$, this reduces in the limit of large scale factor to $\\dot a^2 \\simeq1$, giving $a(t)\\simeq t$ in the late time limit. Including a subleading correction of the form~$a(t) \\simeq t + \\alpha t^\\beta$ yields\n\\begin{subequations}\n\\label{latea}\n\\bea\na(t) &\\simeq t - {\\bar\\Omega_* \\over 6wH_* ( H_* t ) ^{3w } }, \\quad w\\neq 0, \\\\\na(t) &\\simeq t + {\\bar\\Omega_* \\over 2 H_* } \\ln \\left( {t \\over H_* } \\right) ,\\quad w = 0,\n\\eea\n\\end{subequations}\nwhich by~\\eqref{omegadef} yield\n\\be\\label{omegat}\n\\Omega (t) \\simeq {\\bar\\Omega_*\\over (H_* t) ^{(1+3w )}}.\n\\ee\n\nThe expressions~\\eqref{latea} can be inverted and combined with the transformation to~$(R,T)$ coordinates to yield the coordinate transformation from~$t$ to~$(R,T)$, valid at late times, including terms up to order $R^2 \/ T^2$,\n\\begin{subequations}\n\\bea\nt&\\simeq T+{\\bar\\Omega_*\\over 6wH_*(H_*T)^{3w}}-{R^2\\over 2T},\\qquad w\\neq 0, \\\\\nt&\\simeq T -{\\bar\\Omega_*\\over 2H_*} \\ln\\left( {T \\over H_* } \\right) -{R^2\\over 2T},\\qquad w= 0,\n\\eea\n\\end{subequations}\nwhich then give $\\Omega$ as a function of $T$ and $R$:\n\\begin{subequations}\n\\label{lateomega}\n\\bea\n\\Omega &\\simeq {\\bar\\Omega_*\\over ( H_* T) ^{3w+1 } } \n\\left(1- {(1+3w)\\bar\\Omega_*\\over 6w (H_* T ) ^{3w +1 } } + {(1+3w ) R^2 \\over 2T^2 } \\right), \\quad w\\neq 0, \\\\\n\\Omega &\\simeq {\\bar\\Omega_*\\over H_* T} \\left( 1 +\n {\\hat\\Omega_* \\over 2 H_* T } \\ln\\left( {T \\over H_* } \\right) +{ R^2 \\over 2 T^2} \\right), \\quad w = 0.\n\\eea\n\\end{subequations}\nThis is our desired result.\n\n\nWe are now prepared to compute the leading late time corrections to the boundary stress tensor density, which we will denote by\n $\\tau _{\\mu\\nu}$. Let $K _{\\mu\\nu}$ be the extrinsic curvature of the AdS boundary. In the boundary stress tensor formalism a boundary\n action is defined that includes an integral over $K$ plus\n geometrical counterterms that are constructed from the metric on the boundary $s_{\\mu\\nu}$, defined in equation \\eqref{smetricdef}.\n These terms include a cosmological constant,\n the scalar curvature of $s_{\\mu\\nu}$, and potentially higher derivative counter terms as needed. \n The stress tensor density results from \nvarying the boundary action with respect to $s^{\\mu\\nu}$. The coefficients of the counter terms are\n chosen to cancel divergences that occur in $\\tau_{\\mu\\nu}$ and are dimension dependent. One finds the result \\cite{Balasubramanian:1999re,Myers:1999psa,de Haro:2000xn}\n\\be\n\\label{bst}\n8\\pi \\tau_{\\mu\\nu} = \\sqrt{-s} \\left( K _{\\mu\\nu} - Ks_{\\mu\\nu} +{ D-2 \\over l} s_{\\mu\\nu} +{1\\over D-3} G _{\\mu\\nu} [s] + \\cdots \\right),\n\\ee\nwhere the $\\cdots$ indicate higher derivative terms in the Riemann tensor of $s_{\\mu\\nu}$, which we will show are subdominant at late times.\nWe work with the boundary stress tensor density because the volume element of the late time metric changes at leading order, and \nalso because this is the appropriate quantity to integrate to get the ADM charges.\n\n In the case of the static AdS soliton, the metric on the boundary is flat and the terms in (\\ref{bst})\n depending on the curvature of $s_{\\mu\\nu}$ all vanish. This is no longer true for\n the cosmological AdS spacetimes. \n The Einstein tensor term in (\\ref{bst}) \n contributes a time-dependent piece to $\\tau_{\\mu\\nu}$ which goes to zero at late times like the energy density $\\hat\\rho$ in (\\ref{laterho}). Additional time dependence\n in $\\tau_{\\mu\\nu}$ comes from the volume element in (\\ref{bst}) which goes like\n %\n \\be\\label{volchange}\n \\sqrt{-s} = \\left(1-{1\\over 2} \\, \\Omega \\right) \\sqrt{-s_{(0)} },\n\\ee\nwhere $\\sqrt{-s_{(0)} }$ denotes the volume element in the static AdS soliton, and the late time behavior of the density parameter\n$\\Omega$ is given in (\\ref{omegat}).\nComparing the decay rates of $\\hat\\rho$ and $\\Omega$, one finds that the \ncontribution of $G_{\\mu\\nu}$ to the boundary stress tensor density is subdominant at late times compared to that of the volume element. The contributions of higher derivative terms in (\\ref{bst}) will decay\n even more rapidly. The leading contributions to the boundary stress tensor density are then readily found by combining\n the results for the static AdS soliton in \\cite{El-Menoufi:2013pza} with equation \\eqref{volchange} giving\n \n %\n \\be\\label{solitonbst}\n \\tau_{\\mu\\nu } \n = {e^{5y_+\/l} \\over 16\\pi l} \\left( 1-{1\\over 2} \\, \\Omega \\right) \\mathrm{diag} (-1, 1 ,1, 1, -4),\n \\ee\n %\n where the coordinates are ordered according to $(t, x_1, x_2 , x_3 , v)$. \nHence the decaying time dependent corrections to the static values of $\\tau_{\\mu\\nu } $ are simply proportional to $\\Omega$, the density parameter of the cosmology.\n \n \nWe now use the results above to determine the ADM mass and tensions for the AdS soliton cosmologies. Comparison with \\cite{El-Menoufi:2013pza} shows that the integrands of the ADM charges in AdS coincide with the first three terms in the boundary stress tensor in equation (\\ref{bst}), and the components of $\\tau_{\\mu\\nu} $ above then just need to be integrated to obtain the ADM charges. In the static coordinates, the density parameter $\\Omega$ depends on $R$ as well as $T$, so the integrand is not a constant. This does not mean that $R=0$ is a special point, since any location in the homogeneous open cosmology could equally well be chosen as the origin. For the static AdS soliton, the ADM charges are made finite by taking the planar geometry to be periodically identified, with {\\ $-L_j \/2 \\leq x^j \\leq L_j \/2$. For notational brevity let the asymptotic volume be $V= L_1 L_2 L_3 L_v$ where $L_v$ is the range of compact coordinate $v$ given in (\\ref{period}). In the limit that the plane is infinite, the relevant energy is the mass per unit volume obtained by dividing the total mass by $V$, and similarly for the spatial tensions.\n \nFinally, it is important to note that the static radial coordinate $R$ has the range $0\\leq R\\leq T$, with the upper limit corresponding to $\\chi \\rightarrow \\infty$ in the coordinate transformation (\\ref{coordtrans}).\nThe integrals for the ADM charges are then over a box of length $L< T$, and at the end we divide out the volume of the box.\nDefine the spatial average of the density parameter $\\Omega$ at time $T$ by\n\\be\n\\label{avomega}\n\\ev{\\Omega} \n= {\\bar\\Omega_*\\over V ( H_* T) ^{3w+1 } } \\int dx_1 dx_2 dx_3 dv \\, \\Omega (R , T).\n\\ee\nIn the late time limit, we substitute the approximate expression for~$\\Omega$ given in (\\ref{lateomega}).\nFor the general case $w\\neq 0$, this yields\n\\be\\label{intomega}\n\\ev{\\Omega} = {\\bar\\Omega_*\\over V ( H_* T) ^{3w+1 } } \\left(1- {c_1\\over T ^{3w +1 } } + { c_2L^2 \\over T^2 }\\right),\n\\ee \nwhere the coefficients $c_1 , c_2$ can be read off of the expansion of $\\Omega$ in equation (\\ref{lateomega}). One sees that\nthe terms proportional to $c_1$ and $c_2$ make increasingly small contributions and so will be dropped in subsequent formulae. This also allows us to treat the cases~$w = 0$,~$w \\neq 0$ simultaneously, since the leading-order term in~\\eqref{intomega} is identical to that obtained in the special case\n $w=0$.\n\nFollowing the conventions of past work ({\\it e.g.} \\cite{El-Menoufi:2013pza}), we give the ADM tension rather than a pressure, where tension is simply minus the pressure\\footnote{This convention\nis natural in asymptotically flat static spacetimes where the gravitational tension can be shown to be positive \\cite{Traschen:2003jm}.}.\n Assembling the pieces, at late times the mass and tensions of the soliton in the metric \\eqref{latemetric} are\n\\begin{subequations}\n\\label{admcharges}\n\\bea\n{\\cal M} = {\\cal T} _j & = -{ V \\over 16 \\pi l } e^{5y_+\/l}\\left( 1 -{1\\over 2}\\ev{\\Omega} \\right) \\ , \\quad j=1,2,3, \\\\\n{\\cal T}_v &= {4V \\over 16 \\pi l }e^{5y_+\/l} \\left( 1 -{1\\over 2} \\ev{\\Omega} \\right).\n\\eea\n\\end{subequations}\nThe expressions for the ADM charges have the same structure as the components of the boundary stress tensor, relaxing to the \nequilibrium values like $\\ev{\\Omega} $. \nSince~$\\ev{\\Omega} > 0$, the mass of the AdS soliton cosmology decreases as $ \\ev{\\Omega} $ goes to zero, approaching \nits negative static value at late times, consistent with the energy bound conjectured in \\cite{Horowitz:1998ha}. The tension ${\\cal T}_v$ around the compact dimension increases to its static positive value, while the trace ${\\cal M} + {\\cal T}_v +\\Sigma _j {\\cal T}_j $ vanishes\nthroughout the relaxation process.\n\n\n\n\\section{Entanglement Entropy}\n\\label{entropy}\n\nThe new AdS cosmological solutions allow us to compute how the entanglement entropy of a region in the dual CFT approaches equilibrium. To perform the computation, we use the holographic prescription \\cite{Ryu:2006ef,Hubeny:2007xt}, which proposes that the entanglement entropy of a region~$\\mathcal{R}$ (called the entangling region) in the boundary CFT is equal to\n\\be\n\\label{sa}\nS_\\mathcal{R} = \\frac{\\mathrm{Area}\\left[\\Sigma\\right]}{4G_N},\n\\ee\nwhere~$\\Sigma$ (referred to as the entangling surface) is the minimal-area extremal surface in the bulk spacetime anchored to~$\\partial\\mathcal{R}$ and homologous to~$\\mathcal{R}$. Note that in this section we have restored Newton's constant $G_N$. We will also keep~$w$ general, though we emphasize that only the case~$w = 1$ (wherein the bulk matter is a scalar field) has a well-understood CFT dual.\n\nParametrizing~$\\Sigma$ as~$X^a(\\sigma^i)$, with~$\\sigma^i$ coordinates on~$\\Sigma$, $i=1,..., D-2$, the area functional is\n\\be\n\\label{eq:area}\nA = \\int \\sqrt{h} \\, d^{D-2} \\sigma,\n\\ee\nwhere~$h$ is the determinant of the induced metric on the surface\n\\be\n\\label{eq:inducedh}\nh_{ij} = g_{ab} \\partial_i X^a \\partial_j X^b.\n\\ee\n\nIn general, extremizing~\\eqref{eq:area} to obtain the entangling surface is difficult to accomplish analytically, and the AdS soliton cosmologies\nare no exception. However, we can make progress by working in the non-solitonized AdS cosmology~\\eqref{metric} and noting that the calculations performed there should approximate those in the AdS soliton cosmology, as long as the relevant surfaces do not extend too deeply into the spacetime. The boundary metric is then\n\\be\n\\label{eq:bndryflat}\nds^2_\\partial = -dT^2 (1-\\delta \\tilde{g}_{TT}) + dR^2 (1+\\delta \\tilde{g}_{RR}) - 2\\delta \\tilde{g}_{TR} \\, dR \\, dT + R^2 d\\Omega_{(2)}^2,\n\\ee\nand the full metric is given in (\\ref{metric}) with $\\gamma_{\\mu\\nu}$ equal to $ds^2_\\partial$.\nWorking in pure AdS has the significant advantage that the extremal surface is known for a spherical entangling region on the boundary \\cite{Ryu:2006ef}. This allows us to use perturbative techniques to compute the time dependent correction to the area as the metric approaches the static AdS spacetime in the future.\n\nIn order to compute the late-time behavior of the entanglement entropy we work to first order in powers of $R\/T$ in~$\\delta \\tilde{g}_{TT}$,~$\\delta \\tilde{g}_{RR}$, and~$\\delta \\tilde{g}_{TR}$, given in equations \\eqref{fndef} and \\eqref{lateomega}. We take the boundary of the entangling region to be a sphere of radius~$R_0$ at some time~$T_0$; the corresponding entangling surface~$\\Sigma$ in pure AdS was found in \\cite{Ryu:2006ef}. We may then perturb off of this solution to compute the leading correction to the area. There are two natural options for how this sphere should evolve in time: (i) the sphere can be of fixed proper size in the asymptotically static coordinates so that~$R_0$ is held constant as~$T_0$ advances; or (ii) the sphere can be comoving, so\nthat fluid elements on the boundary of the sphere follow geodesics, and~$R_0$ grows like~$a(t)$. We will discuss both choices below.\n\n\n\\subsection{Zeroth Order Solutions}\n\nAt zeroth order, the boundary metric~\\eqref{eq:bndryflat} is just Minkowski space. Parametrizing the surface by~$z \\equiv l e^{-y\/l}$ and the coordinates on the sphere, the area functional~\\eqref{eq:area} is\n\\be\nA = 4\\pi l^3 \\int _\\epsilon ^1 dx \\, {(1-x^2 )^{1\/2} \\over x^3 },\n\\ee\nwhere $\\epsilon = z_\\mathrm{cut} \/ R_0 $ and $z_\\mathrm{cut}$ is a UV cutoff to regulate the integral. The corresponding entangling surfaces were calculated in~\\cite{Ryu:2006ef} and are given by \n\\be\n\\label{zerosurf}\n\\Sigma_{0}: \\quad z^2 + R^2 = R_0 ^2 \\ , \\quad T = T_0,\n\\ee\nwith area \n\\be\n\\label{zeroarea}\nA^{(0)} = l^3 \\left[ {A_\\mathrm{static} \\over 2 z_\\mathrm{cut} ^2} - \\pi \\ln\\left( \\frac{A_\\mathrm{static} }{ \\pi z_\\mathrm{cut}^2 }\\right)-\\pi \\right],\n\\ee\nwhere~$A_\\mathrm{static} = 4\\pi R_0 ^2$ is the area of~$\\partial\\Sigma _0 $. The first term in the above expression denotes the usual area law growth of the entanglement entropy, while the coefficient of the logarithmically divergent term provides a UV-independent measure of the entanglement entropy.\n\n\n\\subsection{First Order Corrections: Approach to Equilibrium}\n\nNow, consider corrections to~\\eqref{zeroarea} which arise both from perturbations to the metric and to the surface~$X^a$. Write each as a zeroth order piece plus a perturbation,\n\\be\n\\label{realdeal}\ng_{ab} = g_{ab}^{(0)} + \\delta g_{ab} \\ , \\quad \\ X^a (\\sigma_i ) = X^a _{(0)} + \\delta X^a .\n\\ee\nTo first order the volume element on the surface becomes\n\\be\n\\label{deltaa}\nh = h^{(0)} \\left( 1 + \\mathrm{Tr}\\left[ \\delta g_{ab}\\partial_i X_{(0)}^a \\partial_j X_{(0)}^b + 2 g_{ab} ^{(0)} \\partial_i \\delta X^a \\partial_j X_{(0)}^b\\right]\\right).\n\\ee\nHowever, the second term in the trace is a variation of the surface in the background metric, and so this integrates to zero since the background surface is extremal. Thus the first-order change in the area is governed by the perturbation to the metric:\n\\be\n\\label{deltaatwo}\n\\delta A ={1\\over 2} \\int \\sqrt{h^{(0)} } \\, \\mathrm{Tr}\\left[ \\delta g_{ab}\\partial_i X_{(0)}^a \\partial_j X_{(0)}^b\\right] \\, d^{d-1} \\sigma.\n\\ee\nThe final step is to substitute the expressions for the metric perturbations (\\ref{fndef}) into the metric equations (\\ref{metric}), (\\ref{eq:bndryflat}).\n Using $R^\\prime (z) = -z\/R $ on\nthe zeroth order surface \\eqref{zerosurf}, the induced metric in the perturbed spacetime is given by\n\\be\n\\label{latethree}\n(g_{ab} ^{(0)} +\\delta g_{ab} ) \\partial_i X_{(0)}^a \\partial_j X_{(0)}^b d\\sigma^i d\\sigma^j|_{\\Sigma_0} \n= {l^2 \\over z^2 } \\left\\{ \\left( {R_0^2 \\over R^2 } + {z^2\\Omega \\over T_0^2} \\right) dz^2\n+ R^2 d\\Omega_{(2)} ^2 \\right\\},\n\\ee\nwhere $R=\\sqrt{ R_0 ^2 -z^2 }$. Using this expression in (\\ref{deltaatwo}) and substituting $\\Omega$ from (\\ref{lateomega}) gives\n the first-order correction to the area of the entangling surface\n\\bea\n\\label{latearea}\n\\delta A &= { 4\\pi l^3 \\bar{\\Omega}_* \\over ( H_* T_0 ) ^{3w+1} } \\left( { R_0 ^2 \\over 2T_0^2} \\right) \\int _\\epsilon ^1 dx {(1-x^2 )^{3\/2} \\over x } \\\\\n\t\t &= { l^3 \\bar{\\Omega}_* \\over 4 (H_* T_0 ) ^{3w+3}} H_*^2 A_\\mathrm{static} \\left( \\ln\\left( {A_\\mathrm{static}\\over \\pi z_\\mathrm{cut} ^2}\\right) -{4\\over 3} \\right).\n\\eea\nThis result is valid at sufficiently late times such that $H_* T_0 \\gg 1$ and $T_0 \\gg R_0 $. \n\nThe entanglement entropy, including the leading late time contribution, follows from substituting\n(\\ref{latearea}) and (\\ref{zeroarea}) into the entropy-area relation in equation (\\ref{sa}). \n The conversion from area to entropy contains the prefactor $l^3\/G^{(5)}_N$, which can be translated into the parameters of the dual CFT. According to the AdS\/CFT correspondence, the solutions~\\eqref{metric} in~$D = 5$ are dual to an~$\\mathcal{N} = 4$ supersymmetric Yang-Mills theory on the FRW spacetime \\eqref{cosmometric}. Following the discussion in \\cite{Ryu:2006ef}, we consider ${\\cal N} =4 \\ SU(N)$ SYM theory on AdS$_5 \\times S^5$, \n in which case the AdS radius, the ten dimensional Newton's constant, and the five dimensional Newton's constant are identified with the string coupling, string tension, and $N$ according to $l^4 = 4\\pi g_s (\\alpha^\\prime)^2 N$, $G^{(10)}_N = 8\\pi^6 g_s^2 (\\alpha^\\prime)^4 $, and \n$G^{(5)}_N =G_N^{(10)} \/ l^5$. This gives~$l^3\/G^{(5)}_N = 2N^2\/\\pi $, and the entanglement entropy is then\n\\be\\label{eq:Stot}\nS = \\frac{N^2 }{2\\pi }\\left[\\frac{A_\\mathrm{static}}{2z_\\mathrm{cut}^2} - \n\\pi \\ln \\left( {A_\\mathrm{static}\\over \\pi z_\\mathrm{cut} ^2}\\right) -\\pi\n+ \\frac{ \\bar{\\Omega}_* H_*^2 A_\\mathrm{static}}{4( H_* T )^{3w+3} }\\left( \\ln \\left( { 2A_\\mathrm{static}\\over \\pi z_\\mathrm{cut} ^2}\\right) - {4\\over 3} \\right) + \\cdots\n\\right],\n\\ee\nwhere $\\cdots$ denotes terms that are subleading at late time, and the subscript on $T$ has been dropped for simplicity. The coefficient of the logarithmic term is invariant under rescalings of the cutoff, so it serves as a regularized measure~$S_\\mathrm{ren}$ of the entanglement entropy. One finds\n\\be\n\\label{eq:Sren}\n\\delta S_\\mathrm{ren} \\simeq \\frac{N^2 }{8\\pi} \\frac{ \\bar{\\Omega}_* H_*^2 A_\\mathrm{static}}{( H_* T )^{3w+3}}.\n\\ee\nNote that this is positive, which means that~$S$ \\textit{decreases} to its equilibrium value. This behavior differs markedly from that of quenches in CFTs~\\cite{Calabrese:2004eu,AbajoArrastia:2010yt,Albash:2010mv,\nBalasubramanian:2010ce,Caceres:2012em,\nAlishahiha:2014cwa,Alishahiha:2014jxa}, wherein the entanglement entropy grows until is saturates. There is a temptingly simple and compelling physical reason for the decrease of $S$ found in this calculation: in the Cartesian coordinates~$T,R$ there is a nonzero radial flux proportional to $g_{TR}$, so the decrease of the entropy in the ball $R \\leq R_0$ can be interpreted as due to an energy flow out of the ball. In particular, in the quasi-particle picture of entanglement entropy propagation~\\cite{Calabrese:2004eu}, entanglement is carried by entangled particle pairs; a new flow of such particles out of the entangling ball~$R \\leq R_0$ leading to a decrease in entanglement entropy is consistent with this picture. Alternatively, note that at late time, our bulk solution approach the Minkowski vacuum, and therefore the CFT evolves from an excited state to the zero-temperature vacuum state. We would therefore naturally expect probes of correlation (such as entanglement entropy) to decay in the late time limit\\footnote{We thank Juan Pedraza for this observation.}.\n\nThe time dependent contribution decays as a power law, and the time scale for the decay is set by the Hubble parameter $H_*$. The power depends on the equation of state. For example, for dust the correction goes to zero like $T^{-3} $, and for a free massless scalar field like $T^{-6} $. The time dependence in $\\delta S$ is analogous to the result of \\cite{Engelhardt:2013jda}, in which the entropy of a strip in a vacuum-Kasner AdS spacetime was found to have a power law behavior, in both cases a reflection of the time evolution of the cosmology.\n\nTurning to the amplitude of $\\delta S_\\mathrm{ren}$, we see that this is set by an interesting combination of factors. At sufficiently late times $t_* \\Omega_* \\ll 1$, so that $H_* ^2 \\bar{\\Omega}_* \\simeq 8 \\pi G_N ^{(5)}\\hat\\rho_* \/ 3 $.\nHence the dimensionless combination $H_* ^2 \\bar{\\Omega}_* A_\\mathrm{static}$ has the interpretation of the non-vacuum energy, measured in Planck units, that is\ncontained in a shell of width the Planck length that surrounds the sphere. That is, the entangling modes of the perturbation \nact like they are concentrated on the boundary of the sphere. This is a reflection of the fact that the change in the area of the \n extremal surface comes from the metric perturbations near the surface.\n\n \n\\subsection{The Cosmological View}\n\nAn alternative way to interpret the time evolution of the entanglement entropy is to take the boundary sphere to be comoving, so that\npoints on the boundary sphere follow geodesics. \n In the cosmological coordinates \\eqref{openfrw} this means that the sphere is at a fixed coordinate $\\chi= \\chi _0$. The extremal surface $\\Sigma_0$ does not lie within a slice of constant cosmological time, but it does intersect the boundary at a constant time, as can be seen by \nevaluating \\eqref{coordtrans} at $z=0$. \nTransforming the zeroth order surface (\\ref{zerosurf}) to the cosmological coordinates gives\n %\n \\be\\label{cosmosurf}\n \\Sigma_0 : \\quad a(t) \\cosh \\chi = a( t_b ) \\cosh \\chi_0 \\ , \\quad z^2 + \\cosh^2 \\chi_0 \\tanh ^2 \\chi = a(t_b )^2 \\sinh ^2 \\chi _0.\n \\ee\nLet\n\\be\nA_\\mathrm{geod}(t_b ) = 4\\pi a( t_b )^2 \\sinh ^2 \\chi _0\n\\ee\nbe the proper area of the comoving sphere on the boundary at $t_b$.\n Then in terms of the cosmological coordinates the zeroth order area (\\ref{zeroarea}) becomes\n\\be\\label{areacosmo}\nA^{(0)} (t_b ) = l^3 \\left[ { A_\\mathrm{geod}\\over 2 z_\\mathrm{cut} ^2} - \\pi \\ln \\left({ A_\\mathrm{geod} \\over \\pi z_\\mathrm{cut}^2 } \\right) -\\pi \n \\right] \n\\ee\nand the time dependent correction (\\ref{latearea}) is\n\\be\\label{dacosmo}\n\\delta A = l^3 \\bar{\\Omega}_* H_*^2 (4\\pi a_*^2 \\sinh ^2 \\chi _0 ) \\left( {a_* \\over a(t_b ) } \\right)^{3w+1} \n \\left( \\ln\\left( { A_\\mathrm{geod} \\over 2\\pi z_\\mathrm{cut}^2 }\\right) -{2\\over 3} \\right).\n\\ee\nHence the time dependent piece redshifts to zero as \n $(1+z_\\mathrm{b} )^{-(3w+1)} $ where $1+z_b = a(t_b ) \/ a_* $ is the cosmological redshift.\nThe power in the decaying term is different than for the static sphere (\\ref{latearea}) because $A_\\mathrm{geod} $ increases\n as $a^2$. \n \n So far the expressions for the area (\\ref{areacosmo}) and (\\ref{dacosmo}) are just translations from\nthe asymptotically static coordinates to the cosmological coordinates. \nThe difference \nfrom the previous section, in which the area of the boundary sphere is held constant,\n comes when one follows the time evolution by considering increasing values of the boundary time $t_b$.\nThe area of the boundary co-moving sphere increases like $a^2 (t_b )$, so \n although (the UV-independent part of) $\\delta A$ is positive and decreasing to zero, the total entropy increases with $t_b$. This brings up the important issue of \n the range of validity of the expressions in cosmological time. As discussed in\n section \\ref{entropy}, if the results are to be good approximations to the results in a solitonized spacetime, one needs to\n restrict to surfaces that do not penetrate too deeply into the bulk, which precludes taking~$R \\sim t_b \\sinh \\chi _0$ too large. This means\n that the validity of (\\ref{latearea}) is restricted to times that are not so large that the proper radius of the boundary sphere\n approaches the length scale set by the soliton, that is, we need $ t_b \\sinh \\chi _0 \\ll l e^{-y_+ \/l}$. The situation here\n is similar to that in \\cite{Engelhardt:2014mea}.\n \n \n\n\\section{Discussion}\n\\label{conclusion}\n\nIn this paper, we have shown how to construct AdS cosmologies that satisfy the Einstein equations with a nonzero stress tensor and negative cosmological constant. Our solutions were built as foliations of lower-dimensional solutions; the induced metric on each of these hypersurfaces itself satisfies Einstein's equations with a nonzero stress tensor. This construction has the advantage that the boundary metric of the AlAdS solution is just (conformal to) the induced metric on each hypersurface. Therefore, this construction offers us significant freedom in constructing AlAdS spacetimes with a boundary metric of our choosing.\n\nThe particular AdS cosmologies that we have constructed take the stress tensor to be that of a perfect fluid obeying the strong energy condition; for the equation of state~$w = 1$, the fluid is sourced by a massless noninteracting scalar field. Moreover, we have focused on the specific case in which the spatial slices of the FRW cosmologies are negatively curved, as such FRW cosmologies then approach the Milne patch of Minkowski space at late times. The AdS cosmology constructed from these slices therefore approaches the Poincar\\'e patch of AdS at late times, while the AdS soliton cosmology approaches the static AdS soliton.\n\nSuch solutions are especially interesting because they allow us to perturbatively calculate the behavior of physically relevant quantities at late times. For instance, we have calculated the late-time perturbation to the ADM mass of the AdS soliton cosmology, and have found this perturbation to be\n\\be\n\\label{deltam}\n\\delta{\\cal M} = -{ \\Omega {\\cal M}\\over 2},\n\\ee\nwhere~${\\cal M}$ is the unperturbed mass and~$\\Omega$ is the dimensionless density parameter of the FRW cosmology, which goes to zero at late times in the solutions we are interested in. Since ${\\cal M}$ is negative for the soliton this implies that the mass \\textit{decreases} to the mass of the static AdS soliton. Hence this result is consistent with the energy conjecture of \\cite{Horowitz:1998ha} that the AdS soliton is the lowest energy spacetime with the prescribed asymptotic structure. We found the ADM tensions to be modified in a similar manner to~${\\cal M}$.\n\nMoreover, our solutions also have immediate applicability to large-$N$, strongly coupled CFTs via the AdS\/CFT correspondence. Indeed, our AdS cosmologies are dual to CFTs living on an FRW cosmology, while the massless, noninteracting scalar field in the bulk is dual to a scalar operator in the CFT with zero expectation value but nonzero source. This atypical behavior can be tied to the fact that our AdS cosmologies are singular at the Poincar\\'e horizon. This singularity is removed by ``solitonizing'': introducing a compactified direction in the bulk that caps off the geometry. This cap amounts to putting the CFT in a confined phase. \n\nAs a probe of the behavior of the CFT on these FRW spacetimes, we study the entanglement entropy~$S$ of a sphere of constant radius. Using perturbative techniques to find the leading time dependent correction to the entropy, we find that the regulated entanglement entropy $S_\\mathrm{ren}$ decays as a power law to its equilibrium value. The power depends on the equation of state of the fluid. Note that this decay to equilibrium is starkly different from the behavior of entanglement entropy after a quench, when the entanglement entropy \\textit{grows} to its equilibrium (thermal) value. However, note that our solutions at late time approach Poincar\\'e AdS, which has zero temperature; this is drastically different from the end state of a quench, which in the bulk is usually modelled by the injection of energy, forming a black hole of finite temperature.\n\nSeveral issues and questions are raised by these examples. First, can these solutions be generalized to include planar black holes in the bulk spacetimes? Such solutions could conceivably be used to model the approach of a CFT on an FRW cosmology to thermal equilibrium with a nonzero temperature~$T$, much as here we have modeled the approach to equilibrium at~$T = 0$.\nSecond, for reasons of tractability our entropy calculations have been perturbative analyses in the AdS cosmology. It would also be interesting to study the entanglement entropy of the CFT at all times, in both the AdS cosmologies and especially in the AdS soliton cosmologies, to see if there is any behavior that was not captured by our perturbative methods. Such calculations would most likely be numerical, so we leave them for the future. \n\n\n\n\\section*{Acknowledgements}\n\nThe authors with to thank Netta Engelhardt and Don Marolf for useful discussions. We also thank Juan Pedraza for useful comments on an earlier version of this paper. This project was supported in part by the National Science Foundation under Grant No PHY11-25915, by FQXi grant FRP3-1338, and by funds from the University of California.\n\n\n\n\n\\bibliographystyle{JHEP}\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}