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+{"text":"\\section{Introduction}\n\nIn mechanical and structural systems the knowledge of all possible\nsolutions is crucial for safety and reliability. In devices modelled\nby linear ordinary differential equations we can predict the existing\nsolutions using analytical methods \\cite{rao1995mechanical,nayfeh2008linear}.\nHowever, in case of complex, nonlinear systems analytical methods\ndo not give the full view of system's dynamics \\cite{warminski2003approximate,he2004homotopy,belendez2006analytical,nayfeh2011introduction}.\nDue to nonlinearity, for the same set of parameters more then one\nstable solution may exist \\cite{feudel1998dynamical,ChudzikPSK11,Yanchuk2011,gerson2012design,brzeski2012dynamics,menck2013basin,Stability_threshold}.\nThis phenomenon is called multistability and has been widely investigated\nin all types of dynamical systems (mechanical, electrical, biological,\nneurobiological, climate and many more). The number of coexisting\nsolutions strongly depends on the type of nonlinearity, the number of degrees\nof freedom and the type of coupling between the subsystems. Hence, usually\nthe number of solutions vary strongly when values of system's parameters changes. \n\nAs an example, we point out the classical tuned mass absorber\n\\cite{arnoldfr19955,Cartmell1994173,Alsuwaiyan2002791,fischer2007wind,BVG2008,Ikeda2010,Chung2012,Brzeski2014298}.\nThis device is well known and widely used to absorb energy and mitigate\nunwanted vibrations. However, the best damping ability is achieved\nin the neighbourhood of the multistability zone \\cite{brzeski2012dynamics}.\nAmong all coexisting solutions only one mitigates oscillations effectively.\nOther solutions may even amplify an amplitude of the base system.\nSo, it is clear that only by analyzing all possible solutions we can\nmake the device robust. \n\nSimilarly, in systems with impacts one solution can ensure correct\noperation of a machine, while others may lead to damage or destruction\n\\cite{Brzeski_bells,blazejczyk1998co,de2001basins,qun2003coexisting,de2004controlling}.\nThe same phenomena is present in multi-degree of freedom systems where\ninteractions between modes and internal resonances play an important\nrole \\cite{bux1986non,haquang1987non,cartmell1988simultaneous,orlando2013influence}. \n\nPractically, in nonlinear dynamical systems with more then one degree\nof freedom it is impossible to find all existing solutions without\nhuge effort and using classical methods of analytical and numerical\ninvestigation (path-following, numerical integration, basins of attractions),\nespecially in cases when we analyse a wider range of system's parameters and we\ncannot precisely predict the initial conditions. Moreover, solutions\nobtained by integration may have meager basins of attraction and it\ncould be hard or even impossible to achieve them in reality. That is why\nwe propose here a new method basing on the idea of basin stability \\cite{menck2013basin}.\nThe classical basin stability method is based on the idea of Bernoulli\ntrials, i.e., equations of system's motion are integrated $N$ times\nfor randomly chosen initial conditions (in each trial they are different).\nAnalyzing the results we asses the stability of each solution. If\nthere exist only one solution the result of all trials is the same.\nBut, if more attractors coexist we can estimate the probability of their\noccurrence for a chosen set of initial conditions. In mechanical and\nstructural systems we want to be sure that a presumed solution is stable\nand has the dominant basin of attraction in a given range of system's parameters.\nTherefore, we build up a basin stability method by drawing values of\nsystem's parameters. We take into account the fact that values of\nparameters are measured or estimated with some finite precision and\nalso that they can slightly vary during normal operation. \n\nThe paper is organized as follows. In Section 2 we introduce simple\nmodels which we use to demonstrate the main idea of our approach.\nIn the next section we present and describe the proposed method. Section\n4 includes numerical examples for systems described in Section 2.\nFinally, in Section 5 our conclusions are given.\n\n\n\\section{Model of systems\\label{sec:Model-of-systems}}\n\nIn this section we present systems that we use to present our method.\nTwo models are taken from our previous papers \\cite{brzeski2012dynamics,czolczynski2012synchronization}\nand the third one was described by Pavlovskaia et. al.\n\\cite{pavlovskaia2010complex}. We deliberately picked models whose\ndynamics is well described because we can easily evaluate the correctness\nand efficiency of the method we propose. \n\n\n\\subsection{Tuned mass absorber coupled to a Duffing oscillator}\n\nThe first example is a system with a Duffing oscillator and a tuned\nmass absorber. It was investigated in \\cite{brzeski2012dynamics}\nand is shown in Figure \\ref{fig:Duffing1}. The main body consists\nof mass $M$ fixed to the ground with nonlinear spring (hardening\ncharacteristic $k_{1}+k_{2}y^{2}$) and a viscous damper (damping coefficient\n$c_{1}$). The main mass is forced externally by a harmonic excitation\nwith amplitude $F$ and frequency $\\omega$. The absorber is modelled\nas a mathematical pendulum with length $l$ and mass $m$. A small\nviscous damping is present in the pivot of the pendulum. \n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure1}\n\\par\\end{centering}\n\n\\caption{The model of the first considered system. Externally forced Duffing\noscillator with attached pendulum (tuned mass absorber). \\label{fig:Duffing1}}\n\\end{figure}\n\n\nThe equations of the system's motion are derived in \\cite{brzeski2012dynamics},\nhence we do not present their dimension form. Based on the following\ntransformation of coordinates and parameters we reach the dimensionless\nform: \\foreignlanguage{english}{$\\omega_{1}^{2}=\\frac{k_{1}}{M+m}$,\n$\\omega_{2}^{2}=\\frac{g}{l}$, $a=\\frac{m}{M+m}$, $b=\\left(\\frac{\\omega_{2}}{\\omega_{1}}\\right)^{2}$,\n$\\alpha=\\frac{k_{2}l^{2}}{(M+m)\\omega_{1}^{2}}$, $f=\\frac{F}{(M+m)l\\omega_{1}^{2}},$\n$d_{1}=\\frac{c_{x}}{(M+m)\\omega_{1}}$, $d_{2}=\\frac{c_{\\varphi}}{ml^{2}\\omega_{2}}$,\n$\\mu=\\frac{\\omega}{\\omega_{1}}$, $\\tau=t\\omega_{1}$, $x=\\frac{y}{l}$,\n$\\dot{x}=\\frac{\\dot{y}}{\\omega_{1}l}$, $\\ddot{x}=\\frac{\\ddot{y}}{\\omega_{1}^{2}l}$,\n$\\gamma=\\varphi,$ $\\dot{\\gamma}=\\frac{\\dot{\\varphi}}{\\omega_{2}},$\n$\\gamma=\\frac{\\ddot{\\varphi}}{\\omega_{2}^{2}}$.} \n\nThe dimensionless equations are as follows:\n\n\\begin{equation}\n\\begin{array}{c}\n\\ddot{x}-ab\\ddot{\\gamma}\\sin\\gamma-ab\\dot{\\gamma}^{2}\\cos\\gamma+x+\\alpha x^{3}+d_{1}\\dot{x}=f\\cos\\mu\\tau,\\\\\n\\\\\n\\ddot{\\gamma}-\\frac{1}{b}\\ddot{x}\\sin\\gamma+\\sin\\gamma+d_{2}\\dot{\\gamma}=0,\n\\end{array}\\label{eq:row bez}\n\\end{equation}\n where $\\mu$ is the frequency of the external forcing and we consider it\nas controlling parameter. The dimensionless parameters have the following\nvalues: $f=0.5$, $a=0.091$, $b=3.33$, $\\alpha=0.031$, $d_{1}=0.132$\nand $d_{2}=0.02$. Both subsystems (Duffing oscillator and the pendulum)\nhave a linear resonance for $\\mu=1.0$. \n\n\n\\subsection{System with impacts}\n\nAs the next example we analyse a system with impacts \\cite{pavlovskaia2010complex}.\nIt is shown in Figure \\ref{fig:Impact} and consists of mass\n$M$ suspended by a linear spring with stiffness $k_{1}$ and a viscous\ndamper with the damping coefficient $c$ to harmonically moving frame.\nThe frame oscillates with amplitude $A$ and frequency $\\Omega$.\nWhen amplitude of mass $M$ motion reaches the value $g$, we observe soft\nimpacts (spring $k_{2}$ is much stiffer than spring $k_{1}$). \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure2}\n\\par\\end{centering}\n\n\\caption{The model of the second considered system. Externally forced oscillator with\nimpacts. \\label{fig:Impact}}\n\\end{figure}\n\n\nThe dimensionless equation of motion is as follow (for derivation\nsee \\cite{pavlovskaia2010complex}) :\n\n\\[\n\\ddot{x}+2\\xi\\dot{x}+x+\\beta\\left(x-e\\right)\\mathrm{H}\\left(x-e\\right)=a\\omega^{2}\\sin\\left(\\omega\\tau\\right)\n\\]\n\n\nwhere $x=\\frac{y}{y_{0}}$ is the dimensionless vertical displacement\nof mass $M$, $\\tau=\\omega_{n}t$ is the dimensionless time, $\\omega_{n}=\\frac{k_{1}}{M}$,\n$\\beta=\\frac{k_{2}}{k_{1}}$ the stiffness ratio, $e=\\frac{g}{y_{0}}$\n the dimensionless gap between equilibrium of mass $M$ and the\nstop suspended on the spring $k_{2}$, $a=\\frac{A}{y_{0}}$ and $\\omega=\\frac{\\Omega}{\\omega_{n}}$\nare dimensionless amplitude and frequency of excitation, $\\xi=\\frac{c}{2m\\omega_{n}}$\nis the damping ratio, $y_{0}=1.0\\:[\\mathrm{mm}]$ and $\\mathrm{H}(\\cdot)$\n the Heaviside function. In our calculations we take the following\nvalues of system's parameters: $a=0.7$, $\\xi=0.01$, $\\beta=29$,\n$e=1.26$. As a controlling parameter we use the frequency of excitation\n$\\omega$. \n\n\n\\subsection{Beam with suspended rotating pendula}\n\nThe last considered system consists of a beam which can move in\nthe horizontal direction and $n$ rotating pendula. The beam has the mass\n$M$ and supports $n$ rotating, excited pendula. Each pendulum has\nthe same length $l$ and masses $m_{i}$ $(i=1,\\:2,\\ldots,\\:n)$.\nWe show the system in Figure \\ref{fig:Beam_model} \\cite{czolczynski2012synchronization}.\nThe rotation of the $i$-th pendulum is given by the variable $\\varphi_{i}$\nand its motion is damped by the viscous friction described by the damping\ncoefficient $c_{\\varphi}$. The forces of inertia of each pendulum\nacts on the beam causing its motion in the horizontal direction (described\nby the coordinate $x$). The beam is considered as a rigid body, so\nwe do not consider the elastic waves along it. We describe the phenomena\nwhich take place far below the resonances for longitudinal oscillations\nof the beam. The beam is connected to a stationary base by a light\nspring with the stiffness coefficient $k_{x}$ and viscous damper with\na damping coefficient $c_{x}$. The pendula are excited by external\ntorques proportional to their velocities: $N_{0}-\\dot{\\varphi}_{i}N_{1}$,\nwhere $N_{0}$ and $N_{1}$ are constants. If no other external forces\nact on the pendulum, it rotates with the constant velocity $\\omega=N_{0}\/N_{1}$.\nIf the system is in a gravitational field (where $g=9.81\\:[\\mathrm{m\/s^{2}}]$\nis the acceleration due to gravity), the weight of the pendulum causes\nthe unevenness of its rotation velocity, i.e., the pendulum slows\ndown when the centre of its mass goes up and accelerates when the\ncentre of its mass goes down. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure3}\n\\par\\end{centering}\n\n\\caption{The model of the third considered system. Horizontally moving beam\nwith attached pendulums. \\label{fig:Beam_model}}\n\\end{figure}\n\n\nThe system is described by the following set of dimensionless equations:\n\n\\begin{equation}\nm_{i}l^{2}\\ddot{\\varphi}_{i}+m_{i}\\ddot{x}l\\cos{\\varphi_{i}}+c_{\\varphi}\\dot{\\varphi}_{i}+m_{i}gl\\sin{\\varphi_{i}}=N_{0}-\\dot{\\varphi}_{i}N_{1}\\label{eq:Beam1}\n\\end{equation}\n\n\n\\begin{equation}\n\\left({M+\\sum\\limits _{i=1}^{n}{m_{i}}}\\right)\\ddot{x}+c_{x}\\dot{x}+k_{x}x=\\sum\\limits _{i=1}^{n}{m_{i}l\\left({-\\ddot{\\varphi}_{i}\\cos\\varphi_{i}+\\dot{\\varphi}_{i}^{2}\\sin\\varphi_{i}}\\right)}\\label{eq:Beam2}\n\\end{equation}\nIn our investigation we analyze two cases: a system with two pendula\n(where $n=2$ and $i=1,\\:2$) and with $20$ pendula ($n=20$ $i=1,\\:2,\\:...,n$)..\nThe values of the parameters are as follows: $m_{i}=\\frac{2.00}{n}$,\n$l=0.25$, $c_{\\varphi}=\\frac{0.02}{n}$, $N_{0}=5.0$0, $N_{1}=0.50$,\n$M=6.00$, $g=9.81$, $c_{x}=\\frac{\\ln\\left(1.5\\right)}{\\pi}\\sqrt{k_{x}\\left(M+\\sum\\limits _{i=1}^{n}{m_{i}}\\right)}$and\n$k_{x}$ is a controlling parameter. The derivation of the system's equations\ncan be found in \\cite{czolczynski2012synchronization}. We present the\ntransformation to a dimensionless form in Appendix A. \n\n\n\\section{Methodology}\n\nIn \\cite{menck2013basin} Authors present a ``basin stability''\nmethod which let us estimate the stability and number of solutions\nfor given values of system parameters. The idea behind basin stability\nis simple, but it is a powerful tool to assess the size of complex\nbasins of attraction in multidimensional systems. For fixed values\nof system's parameters, $N$ sets of random initial conditions are\ntaken. For each set we check the type of final attractor. Based on\nthis we calculate the chance to reach a given solution and determine\nthe distribution of the probability for all coexisting solutions. This\ngives us information about the number of stable solutions and the\nsizes of their basins of attraction.\n\nWe consider the dynamical system $\\dot{\\mathbf{x}}=f\\,(\\mathbf{x},\\,\\omega)$,\nwhere $\\mathbf{x}\\in\\mathtt{\\mathbb{R}^{n}}$ and $\\omega\\in\\mathbb{R}$\nis the system's parameter. Let ${\\cal B\\subset\\mathbb{\\mathtt{\\mathbb{R}^{n}}}}$\nbe a set of all possible initial conditions and ${\\cal C}\\subset\\mathbb{\\mathtt{\\mathbb{R}}}$\na set of accessible values of system's parameter. Let us assume that an\nattractor ${\\cal A}$ exists for $\\omega\\in{\\cal C_{A}}\\subset{\\cal C}$\nand has a basin of attraction $\\beta({\\cal A})$. Assuming random\ninitial conditions the probability that the system will reach attractor\n${\\cal A}$ is given by $p\\left({\\cal A}\\right)$. If this probability\nis equal to $p\\left({\\cal A}\\right)=1.0$ this means that the considered\nsolution is the only one in the taken range of initial conditions\nand given values of parameters. Otherwise other attractors coexist.\nThe initial conditions of the system are random from set ${\\cal B_{A}}\\subset{\\cal B}$.\nWe can consider two possible ways to select this set. \n\n\\begin{description}\n  \\item[I] The first ensures that set the ${\\cal B_{A}}$ includes values of initial \nconditions leading to all possible solutions. This approach is appropriate if \nwe want to get a general overview of the system's dynamics. \n  \\item[II] In the second approach we use a narrowed set of initial conditions that \ncorresponds to practically accessible initial states.\\ldots\n\\end{description}\n\nIn our method we chose the second approach because it let us take into \naccount constrains imposed on the system and because in engineering we \nusually know or expect the initial state of the system with some finite \nprecision. \n\nIn the classical approach of Menck et. al. \\cite{menck2013basin} the\nvalues of system's parameters are fixed and do not change during calculations.\nThe novelty of our method is that we not only draw initial conditions\nbut also values of some selected parameters of the system. We assume\nthat the initial conditions and some of the system's parameters are chosen\nrandomly. Then using $N$ trials of numerical simulations we estimate\nthe probability that the system will reach a given attractor ${\\cal A}$\n($p\\left({\\cal A}\\right)$). The idea is to take into consideration\nthe fact that the values of system's parameters are measured or estimated\nwith some finite accuracy which is often hard to determine. Moreover\nvalues of parameters can vary during normal operation. Therefore drawing\nvalus of parameters we can describe how a mismatch in their values influences\nthe dynamics of the system and estimate the risk of failure. In many\npractical applications one is interested in reaching only one presumed\nsolution ${\\cal A}$, and the precise description of other coexisting\nattractors is not necessary. We usually want to know the probability\nof reaching the expected solution $p\\left({\\cal A}\\right)$ and the chance\nthat the system behave differently. If $p\\left({\\cal A}\\right)$ is\nsufficiently large, we can treat the other attractors as an element\nof failure risk. \n\nIn our approach we perform the following steps:\n\\begin{description}\n\t\\item[I] We pick values of system's parameters from the set\n\t\t${\\cal C_{A}}\\subset{\\cal C}$. \n\t\\item[II]We select the set ${\\cal C_{A}}$ so that\n\t\tit consists of all practically accessible values of system's parameters\n\t\t$\\omega$ . This let us ensure that a given solution indeed exists in a practically\n\t\taccessible range (taking into account the mismatch in parameters).\n\t\\item[III] We subdivide the set $C_{A}$ in to $m=1,2,\\dots M$ equally spaced\n\t\tsubsets. The subsets ${\\cal C}_{A}^{m}$ do not overlap and the relation\n\t\t$\\bigcup_{m=1\\dots M}{\\cal C}_{A}^{m}=C_{A}$ is always fulfilled.\n\t\\item[IV] Then for each subset ${\\cal C}_{A}^{m}$ we randomly pick $N$ sets\n\t\tof initial conditions and value of the considered parameter. For each\n\t\tset we check the final attractor of the system. \n\t\\item[V] After a suficient number\n\t\tof trials we calculate the probability of reaching a presumed solution\n\t\tor solutions. \n\t\\item[VI]Finally we describe the relation between the\n\t\tvalue of the system's parameter and the ``basin stability'' of reachable\n\t\tsolutions.\\ldots\n\\end{description}\n\nIn our calculations for each range of parameter values\n(subset ${\\cal C}_{A}^{m}$) we draw from $N=100$ up to $N=1000$\nsets of initial conditions and parameter. The value of $N$ strongly depends\non the complexity of the analysed system. Also the computation time for a\nsingle trial should be adjusted for each system independently\nsuch that it can reach the final attractor. In general, we recommend that\nin most cases $N$ should be at least 100.\n\n\n\\section{Numerical results}\n\n\n\\subsection{Tuned mass absorber coupled to a Duffing oscillator}\n\nAt the beginning we want to recall the results we present in our previous\npaper \\cite{brzeski2012dynamics}. As a a summary we show Figure \\ref{fig:Two-paramters_colour}\nwith a two dimensional bifurcation diagram obtained by the path-following\nmethod. It gives bifurcations for varying amplitude $f$ and frequency\n$\\mu$ of the external excitation (see Eq. \\ref{eq:row bez}). Lines shown\nin the plot correspond to different types of bifurcations (period\ndoubling, symmetry breaking, Neimark-Sacker and resonance tongues).\nWe present these lines in one style because the structure is too complex\nto follow bifurcation scenarios and we do not need that data (details\nare shown in \\cite{brzeski2012dynamics}). We mark areas where we\nobserve the existence of one solution (black colour), or the coexistence of two (grey) and three\n(hatched area) stable solutions. The remaining part of the diagram\n(white area) corresponds to situations where there are four or more\nsolutions. Additionally, by white colour we also mark areas where\nonly the Duffing system is oscillating in 1:1 resonance with the frequency\nof excitation and the pendulum is in a stable equilibrium position,\ni.e., HDP (hanging down pendulum) state. In this case the dynamics\nof the system is reduced to the oscillations of summary mass ($M+m$).\n\nThe detailed analysis of system \\ref{eq:row bez} is time consuming\nand creation of Figure \\ref{fig:Two-paramters_colour} was preceded\nby complex analysis done with large computational effort. Additionally,\nthe obtained results give us no information about the size of the basins\nof attraction of each solution - which practically means that some\nof the solutions may occur only very rarely in the real system (i.e. due to not accessible\ninitial conditions). Nevertheless, such analysis gives us an in-depth\nknowledge about the bifurcation structure of the system. As we can see, the\nrange where less then three solutions exist is rather small, especially\nfor $\\mu<2.0$. To illustrate our method of analysis, we focus on three\nsolutions: $2:1$ oscillating resonance, HDP and $1:1$ rotating resonance\nassuming that only they have some practical meaning.\n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure4}\n\\par\\end{centering}\n\n\\caption{Two-parameter bifurcations diagram of the system (1) in the plane $\\left(f,\\,\\mu\\right)$\nshowing periodic oscillations and rotations of the pendulum. Black colour\nindicates one attractor, grey colour shows two coexisting attractors\n(the same as for black but with a coexisting stable steady state of\nthe pendulum). In the hatched area we observe the coexistence of stable\nrotations and a stable steady state of the pendulum. A detailed analysis\nis presented in \\cite{brzeski2012dynamics}. \\label{fig:Two-paramters_colour}}\n\\end{figure}\n\n\nTo show our results obtained with integration, we compute bifurcation diagrams\nfor $f=0.5$ in the range $\\mu\\in[0.1,\\:3.0]$ (see Figure \\ref{fig:tma_bif}).\nIn Figure \\ref{fig:tma_bif}(a) we increase $\\mu$ from $0.1$ to\n$3.0$ and in Figure \\ref{fig:tma_bif}(b) we decrease $\\mu$ from $3.0$\nto $0.1$. As the initial conditions we take the equilibrium position\n($x_{0}=\\dot{x}_{0}=0.0$ and $\\gamma_{0}=\\dot{\\gamma}_{0}=0.0$).\nIn both panels we plot the amplitude of the pendulum $\\gamma$. Ranges where the \ndiagrams differ we mark by grey rectangles. It is easy to see that there\nare two dominating solutions: HDP and $2:1$ internal resonance. Near\n$\\mu=1.0$ we observe a narrow range of $1:1$ and $9:9$ resonances\nand chaotic motion (for details see Figure 6 in \\cite{brzeski2012dynamics}).\nBased on previous results we know that we detected all solutions existing\nin the considered range, however we do not have information about the size of\ntheir basins of attraction and coexistence. Hence the analysis with the \nproposed method should give us new important information about the system's\ndynamics. Contrary to the bifurcation diagram obtained by path-following\nin Figure \\ref{fig:tma_bif}, we do not observe rotating solutions\n(the other set of initial conditions should be taken).\n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure5}\n\\par\\end{centering}\n\n\\caption{Bifurcation diagram showing the behaviour of the pendulum suspended\non the Duffing oscillator. For subplot (a) the value of the bifurcation parameter\n$\\mu$ was increased, while for subplot (b) we decreased the value of $\\mu$.\nGray rectangles mark the range of the bifurcation parameter $\\mu$ for\nwhich different attractors coexist. A detailed analysis is presented\nin \\cite{brzeski2012dynamics}. \\label{fig:tma_bif}}\n\\end{figure}\n\n\nIn Figure \\ref{fig:Duff_prob} we show the probability of reaching the\nthree aforementioned solutions obtained using the proposed method.\nThe initial conditions are random numbers drawn from the following\nranges: $x_{0}\\in[-2,\\:2]$, $\\dot{x}_{0}\\in[-2,\\:2]$, $\\gamma_{0}\\in[-\\pi,\\:\\pi]$\nand $\\dot{\\gamma}_{0}\\in[-2.0,\\:2.0]$ (ranges there selected basing\non the results from \\cite{brzeski2012dynamics}). The frequency of excitation\nis within a range $\\mu\\in[0,\\:3.0]$ (Figure \\ref{fig:Duff_prob}(a,c)\n), then we refine it to $\\mu\\in[1.25,\\:2.75]$ (Figure \\ref{fig:Duff_prob}(b,d)\n). In both cases we take $15$ equally spaced subsets of $\\mu$ and\nin each subset we calculate the probability of reaching a given solution.\nFor each subset we calculate $100$ trials each time drawing initial\nconditions of the system and a value of $\\mu$ from the appropriate\nrange. Then we plot the dot in the middle of the subset which indicate\nthe probability of reaching a given solution in each considered range.\nLines that connect the dots are shown just to ephasize the tendency. For\neach range we take $N=1000$ because we want to estimate the probability\nof a solution with small a basin of stability (1:1 rotating periodic solution).\n\nAs we can see in Figure \\ref{fig:Two-paramters_colour}, the $2:1$\nresonance solution exists in the area marked by black colour around\n$\\mu=2.0$ and coexists with HDP in the neighbouring grey zone. In Figure\n\\ref{fig:Duff_prob} we mark the probability of reaching the $2:1$ resonance\nusing blue dots. As we expected, for $\\mu<1.4$ and $\\mu>2.2$ the solution\ndoes not exist. In the range $\\mu\\in[1.4,\\:2.2]$ the maximum value of\nprobability $p(2:1)=0.971$ is reached in the subset $\\mu\\in[1.8,\\:2.0]$\nand outside that range the probability decreases. To check if we can\nreach $p(2:1)=1.0$, we decrease the range of parameter's values to\n$\\mu\\in[1.25,\\:2.75]$ and the size of subset to $\\Delta\\mu=0.1$\n(we still have 15 equally spaced subsets). The results are shown in\nFigure \\ref{fig:Duff_prob}(b) similarly in blue colour. In the range\n$\\mu\\in[1.95,\\:2.05]$ the probability $p(2:1)$ is equal to unity\nand in the range $\\mu\\in[1.85,\\:1.95]$ it is slightly smaller $p(2:1)=0.992$.\nHence, for both subsets we can be nearly sure that the system reaches the $2:1$\nsolution. This gives us indication of how precise we have to set the\nvalue of $\\mu$ to be sure that the system will behave in a presumed\nway.\n\nA similar analysis is performed for HDP. The values of probability\nis indicated by the red dots. As one can see for $\\mu<0.8$, $\\mu\\in[1.2,\\:1.4]$\nand $\\mu\\in[2.6,\\:2.8]$, the HDP is the only existing solution. The\nrapid decrease close to $\\mu\\approx1.0$ indicates the $1:1$ resonance\nand the presence of other coexisting solutions in this range (see\n\\cite{brzeski2012dynamics}). In the range $\\mu\\in[1.2,\\:1.4]$ the probability\n$p(\\mathrm{HDP})=1.0$ which corresponds to a border between solutions\nborn from $1:1$ and $2:1$ resonance. Hence, up to $\\mu=2.0$ the\nprobability of the HDP solution is a mirror refection of $p(2:1)$. The\nsame tendency is observed in the narrowed range as presented in Figure\n\\ref{fig:Duff_prob}(b). Finally, for $\\mu>2.0$ the third considered\nsolution comes in and we start to observe an increase of probability\nof the rotating solution $S(\\mu,\\:\\mathrm{HDP})$ as shown in Figure \\ref{fig:Duff_prob}(c).\nHowever, the chance of reaching the rotating solution remains small and never\nexceeds $p(\\mathrm{1:1})=8\\times10^{-3}$. We also plot the probability\nof reaching the rotating solution in the narrower range of $\\mu$ in Figure\n\\ref{fig:Duff_prob}(d). The probability is similar to the one presented\nin Figure \\ref{fig:Duff_prob}(c) - it is low and does not exceed $p(\\mathrm{1:1})=8\\times10^{-3}$.\nNote that the results presented in Figure \\ref{fig:Duff_prob}(a,b) and\nFigure \\ref{fig:Duff_prob}(c,d) are computed for different sets of\nrandom initial conditions and parameter values; hence the obtained\nprobability can be slightly different. \n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure6}\n\\par\\end{centering}\n\n\\caption{Probability of reaching given solutions in (1) system with tuned mass\nabsorber. Subplots (a,b) present solutions with $2:1$ periodic oscillations\n(blue) and without motion of the pendulum (red). Subplots (c,d) present the\nprobability of reaching $1:1$ rotations (black). (Please note that\nin both cases (a,b) and (c,d) the initial conditions and parameter\nare somehow random, hence the results may slightly differ). \\label{fig:Duff_prob}}\n\\end{figure}\n\n\n\n\\subsection{System with impacts}\n\nIn this subsection we present our analysis of different periodic solutions\nin the system with impacts. A discontinuity usually increases the number\nof coexisting solutions. Hence, in the considered system we observe\na large number of different stable orbits and their classification\nis necessary. In Figure \\ref{fig:ImpactBif} we show two bifurcation\ndiagrams with $\\omega$ as controlling parameter. Both of them start\nwith initial conditions $x_{0}=0.0$ and $\\dot{x}_{0}=0.0$. In panel\n(a) we increase $\\omega$ from $0.801$ to $0.8075$; while in panel\n(b) we decrease $\\omega$ in the same range. We select the range of\n$\\omega$ basing on the results presented in \\cite{pavlovskaia2010complex}.\nAs one can see, both diagrams differ in two zones marked by grey colour.\nHence, we observe a coexistence of different solutions, i.e., in the range\n$\\omega\\in[0.8033,\\:0.8044]$ solutions with period-3 and -2 are present,\nwhile in the range $\\omega\\in[0.8068,\\:0.8075]$ we detected solutions\nwith period-2 and -5. As presented in \\cite{pavlovskaia2010complex}\nsome solutions appear from a saddle-node bifurcation and we are not\nable to detect them with the classical bifurcation diagram. The proposed\nmethod solves this problem and shows all existing solutions in\nthe considered range of excitation frequency. \n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure7}\n\\par\\end{centering}\n\n\\caption{Bifurcation diagram showing the behaviour of impacting oscillator (2).\nFor subplot (a) the value of the bifurcation parameter $\\omega$ was increased\nwhile for subplot (b) we decreased the value of $\\omega$. Grey rectangles\nmark the range of the bifurcation parameter $\\omega$ for which different\nattractors coexist. Further analysis can be found in \\cite{pavlovskaia2010complex}.\n\\label{fig:ImpactBif}}\n\\end{figure}\n\n\nWe focus on periodic solutions with periods that are not longer than\neight periods of excitation. We observe periodic solutions with higher\nperiods in the narrow range of $\\omega$ but the probability that\nthey will occur is very small and we can neglect them. All non-periodic\nsolutions are chaotic (quasiperiodic solutions are not present in\nthis system). The results of our calculations are shown in Figure\n\\ref{fig:Impact_prob}(a,b). We take initial conditions from the following\nranges $x_{0}\\in[-2,\\:2]$, $\\dot{x}_{0}\\in[-2,\\:2]$. The controlling\nparameter $\\omega$ is changed from $0.801$ to $0.8075$ with step\n$\\Delta\\omega=0.0005$ in Figure \\ref{fig:Impact_prob}(a) and from\n$0.806$ to $0.8075$ with the step $\\Delta\\omega=0.0001$ in Figure \\ref{fig:Impact_prob}(b)\n(in each subrange of excitation's frequency we pick the exact value of\n$\\omega$ randomly from this subset). The probability of periodic\nsolutions is plotted by lines with different colours and markers.\nWe detect the following solutions: period-1, -2, -3, -5 (two different\nattractors with large and small amplitude), -6 and -8. The dot lines\nindicate the sum of all periodic solutions' probability (also with\nperiod higher then eight). Hence, when its value is below $1$, chaotic solution exist. Dots are drawn for mean value i.e, middle\nof the subset. For each range we take $N=200$ and we increase the calculation\ntime because the transient time is sufficiently larger than in the\nprevious example due to the piecewise smooth characteristic of spring's\nstiffness.\n\nAs we can see, the chance of reaching a given solution strongly depends\non $\\omega$. Hence, in the sense of basin stability we can say that\nstability of solutions rely upon the $\\omega$ value. In Figure \\ref{fig:Impact_prob}(a)\nthe probability of a single solution is always smaller than one. Nevertheless,\nwe observe two dominant solutions: period-5 with large amplitude in\nthe first half of the considered $\\omega$ range and period-2 in the second\nhalf of the range. The maximum registered value of probability is $p(\\mathrm{period-2})=0.92$\nand it refers to the period-2 solution for $\\omega\\approx0.80675$. To\ncheck if we can achieve even higher probability we analyse a narrower\nrange of $\\omega$ and decrease the step (from $\\Delta\\omega=0.0005$\nto $\\Delta\\omega=0.0001$). In Figure \\ref{fig:Impact_prob}(b) we\nsee that in range $\\omega\\in[0.8069,\\:0.807]$ the probability of reaching\nthe period-2 solution is equal to $1$. Hence, in the sense of basin stability\nit is the only stable solution. Also in the range $\\omega\\in[0.8065,\\:0.8072]$\nthe probability of reaching this solution is higher then $0.9$ and\nwe can say that its basin of attraction is strongly dominant. \n\nOther periodic solutions presented in Figure \\ref{fig:Impact_prob}(a)\nare: period-1 is present in the range $\\omega\\in[0.801,\\:0.8025]$ with the\nhighest probability $p(\\mathrm{period-1})=0.4$, period-3 exists in the\nrange $\\omega\\in[0.803,\\:0.805]$ with the maximum probability $p(\\mathrm{period-3})=0.36$,\nperiod-2 is observed in two ranges $\\omega\\in[0.8025,\\:0.8035]$ and\n$\\omega\\in[0.804,\\:0.8045]$ with the highest probability equal to $0.18$\nand $0.12$ respectively. Solution with period-5 (small amplitude's\nattractor) exists also in two ranges $\\omega\\in[0.8055,\\:0.8065]$\nand $\\omega\\in[0.807,\\:0.8075]$ with the highest probability equal to\n$0.14$ and $0.4$3 respectively.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure8}\n\\par\\end{centering}\n\n\\caption{Probability of reaching given solutions in the impacting system. Subplots\n(a,b) present different periodic solutions and the summary probability\nof reaching any periodic solution. In Subplot (a) we analyze $\\omega\\in[0.801,\\:0.8075]$\nwith the step $\\Delta\\omega=0.0005$, and in subplot (b) we narrow\nthe range $\\omega\\in[0.806,\\:0.8075]$ and decrease the step size\n$\\Delta\\omega=0.0001$. (Please note that in cases (a) and (b) the\ninitial conditions and parameter are somehow random, hence the results\nmay slightly differ).\\label{fig:Impact_prob}}\n\\end{figure}\n\n\n\n\\subsection{Beam with suspended rotating pendula}\n\nThe third considered system consists of a beam that can move horizontally\nwith two ($n=2$) or twenty ($n=20$) pendula suspended on it. As\na control parameter we use $k_{x}$ which describes the stiffness\nof the beam's support. For the considered range of $k_{x}\\in[100,\\,5000]$\ntwo stable periodic attractors exist in that system. One corresponds\nto complete synchronization of the rotating pendula. The second one is called anti-phase synchronization and refers to the state when\nthe pendula rotate in the same direction but are shifted in phase by $\\pi$.\n\nIn Figure \\ref{fig:pendulaBif} we show four bifurcation diagrams\nwith $k_{x}$ as the controlling parameter and a Pioncare map of rotational\nspeed of the pendula. The subplots (a,b) refer to the system with two\npendula ($n=2$). We start with zero initial conditions: $x_{0}=0.0$,\n$\\dot{x}_{0}=0.0$, $\\varphi_{10}=0.0$, $\\dot{\\varphi}_{10}=0.0$,\n$\\varphi_{20}=0.0$, $\\dot{\\varphi}_{20}=0.0$ and take $k_{x}\\in[100,\\,5000]$.\nThe parameter $k_{x}$ is increasing in subplot (a) and decreasing in\n(b). We see that in the range marked by grey rectangle both complete\nand anti-phase synchronization coexist. In subplots (c,d) we present\nresults for twenty pendula ($n=20$). We start the integration from initial\nconditions that refer to anti-phase synchronization (two clusters\nof 10 pendula shifted by $\\pi$) i.e. $x_{0}=0.1$, $\\dot{x}_{0}=0.00057$,\n$\\varphi_{k0}=0.0$, $\\dot{\\varphi}_{k0}=9.81$, $\\varphi_{j0}=3.09$,\n$\\dot{\\varphi}_{j0}=9.784$ where: $k=1,2,\\ldots10$ and $j=11,12,\\ldots20$.\nThe value of $k_{x}$ is increasing in subplot (c) and decreasing in (d).\nSimilarly as in the two pendula case, we observe the region ($k_{x}\\in[100,\\,750]$)\nwhere two solutions coexist: anti-phase synchronization and non-synchronous\nstate. To further analyse multistability in that system we use \nproposed method.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{Figure9}\n\\par\\end{centering}\n\n\\caption{Bifurctaion diagram showing the behaviour of two (a,b) and twenty\n(c,d) pendula suspended on the moving beam. For subplots (a,c) the value\nof the bifurcation parameter $k_{x}$ was increased, while for subplots\n(b,d) we decreased the value of $k_{x}$. Grey rectangles mark the\nranges of the bifurcation parameter $k_{x}$ for which different attractors\ncoexist. Further analysis of number of solutions can be found in \\cite{czolczynski2012synchronization}.\n\\label{fig:pendulaBif}}\n\\end{figure}\n\n\nIn Figure \\ref{fig:Pendula_prob} we present how the probability of\nreaching a given solution depends on the parameter $k_{x}$ . In subplot\n(a) we show the results for the system with 2 pendula, while in subplot\n(b) results obtained for the system with 20 pendula suspended\non the beam are given. In both cases we consider $k_{x}\\in[0,\\:5000]$ and assume\nthe following ranges of initial conditions: $x_{0}\\in[-0.15,\\:0.15]$,\n$\\dot{x}_{0}\\in[-0.1,\\:0.1]$, $\\varphi_{i0}\\in[-\\pi,\\:\\pi]$, $\\varphi_{20}\\in[-\\pi,\\:\\pi]$,\n$\\dot{\\varphi}_{10}\\in[-3.0,\\:3.0]$ and $\\dot{\\varphi}_{20}\\in[-3.0,\\:3.0]$\nin Figure \\ref{fig:Pendula_prob}(a) and $x_{0}\\in[-0.15,\\:0.15]$,\n$\\dot{x}_{0}\\in[-0.1,\\:0.1]$, $\\varphi_{i0}\\in[-\\pi,\\:\\pi]$, $\\dot{\\varphi}_{i0}\\in[-\\pi,\\:\\pi]$\nwhere $i=1\\dots\\:20$ in Figure \\ref{fig:Pendula_prob}(b). We take\n$20$ subsets of parameter $k_{x}$ values with the step equal to\n$\\Delta k_{x}=250$ and mark their borders with vertical lines. For\neach set we run $N=100$ simulations; each one with random initial\nconditions and $k_{x}$value drawn from the respective subset. Then,\nwe estimate the probability of reaching given solution. The dots in\nFigure \\ref{fig:Pendula_prob} indicate the probability of reaching a\ngiven solution in the considered range (dots are drawn for mean value,\ni.e, middle of subset). Contrary to both already presented systems,\nthis one has a much larger dimension of phase space (six and forty two),\nhence we decide to decrease number of the trials to $N=100$ in order\nto minimise the time of calculations. \n\nIn Figure \\ref{fig:Pendula_prob}(a) we show the results for 2 pendula.\nWhen $k_{x}\\in[0,\\:250]$ only anti-phase synchronization is possible.\nThen, with the increase of $k_{x}$ we observe a sudden change\nin the probability and for $k_{x}\\in[750,\\:1750]$ only complete synchronization\nexists. For $k_{x}>2000$ a probability of reaching both solutions fluctuates\naround $p(\\mathrm{complete})=0.7$ for complete and $p(\\mathrm{\\mathrm{anti-phas}e})=0.3$\nfor anti-phase synchronization. Further increase of $k_{x}$\ndoes not introduce any significant changes. \n\nIn Figure \\ref{fig:Pendula_prob}(b) we show the results for twenty\npendula. For $k_{x}\\in[0,\\:250]$ the system reaches solutions different\nfrom the two analysed (usually chaotic). Then, the probability of reaching\ncomplete synchronization drastically increases and for $k_{x}\\in[750,\\:5000]$\nit is equal to $p(complete)=1.0$ which means that the pendula always\nsynchronize completely. We also present the magnification of the plot\nwhere we see that in fact for $k_{x}\\in[715,\\:5000]$ we will always\nobserve complete synchronization of the pendula. Please note that\nfor calculating both plots we use random initial conditions and $k_{x}$\nvalue hence, the results for a narrower range may differ. Anti-phase\nsynchronization was never achieved with randomly chosen initial conditions.\nThis means that even though this solution is stable for $k_{x}\\in[100,\\:750]$\n(see Figure \\ref{fig:pendulaBif}(c)) it has a much smaller basin of attraction\nand is extremely hard to obtain in reality. The results presented in Figure\\ref{fig:Pendula_prob}\nprove that by proper tuning of the parameter $k_{x}$ we can control the\nsystems behaviour even if we can only fix the $k_{x}$ value with finite\nprecision. \n\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics{Figure10}\n\\par\\end{centering}\n\n\\caption{Probability of reaching given solutions in the system with rotating\npendula. Subplot (a) refers to the case with two pendula and (b) with\ntwenty pendula. (Please note that on plot (b) and its magnification\nthe initial conditions and parameter are somehow random, hence the\nresults may slightly differ). \\label{fig:Pendula_prob}}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nIn this paper we propose a new method of detection of solutions' in \nnon-linear mechanical or structural systems. The method allows\nto get a general view of the system's dynamics and estimate the risk that\nthe system will behave behave differently than assumed. To achieve\nthis goal we extend the method of basin stability \\cite{menck2013basin}.\nWe build up the classical algorithm and draw not only initial conditions\nbut also values of system's parameters. We take this into account\nbecause the identification of parameters' values is quite often not\nvery precise. Moreover values of parameters often slowly vary during\noperation. Whereas in practical applications we usually need certainty\nthat the presumed solution is stable and its basin of stability is large\nenough to ensure its robustness. Hence, there is a need to describe\nhow small changes of parameters' values influence the behaviour of\nthe system. Our method provides such a description and allows us to estimate\nthe required accuracy of parameters values and the risk of unwanted\nphenomena. Moreover it is relatively time efficient and does not require\nhigh computational power.\n\nWe show three examples, each for a different class of systems: a tuned\nmass absorber, a piecewise smooth oscillator and a multi-degree of freedom\nsystem. Using the proposed method we can estimate the number of existing\nsolutions, classify them and predict their probability of appearance.\nNevertheless, in many cases it is not necessary to distinct all solutions\nexisting in a system but it is enough to focus on an expected solution,\nwhile usually other periodic, quasi-periodic and chaotic solutions\nare classified as undesirable. Such a strategy simplifies the analysis and\nreduces the computational effort. We can focus only on probable solutions\nand reduce the number of trials omitting a precise description of solutions\nwith low probability. \n\nThe proposed method is robust and can be used not only for mechanical\nand structural systems but also for any system given by differential\nequations where the knowledge about existing solutions is crucial. \n\n\n\\section*{Acknowledgement}\n\n\nThis work is funded by the National Science Center Poland based on\nthe decision number DEC-2015\/16\/T\/ST8\/00516. PB\nis supported by the Foundation for Polish Science (FNP).\n\n\n\\section*{Appendix A}\n\nThe motion of the system presented in Figure \\ref{fig:Beam_model}\nis described by the following set of two second order ODEs:\n\n\\begin{equation}\nm_{iD}l_{D}^{2}\\ddot{\\varphi'}_{i}+m_{iD}\\ddot{x'}l_{D}\\cos{\\varphi'_{i}}+c_{\\varphi D}\\dot{\\varphi'}_{i}+m_{iD}g_{D}l_{D}\\sin{\\varphi'_{i}}=N_{0D}-\\dot{\\varphi'}_{i}N_{1D}\\label{eq:Beam1-1-1}\n\\end{equation}\n\n\n\\begin{equation}\n\\left({M_{D}+\\sum\\limits _{i=1}^{n}{m_{iD}}}\\right)\\ddot{x'}+c_{xD}\\dot{x'}+k_{xD}x'=\\sum\\limits _{i=1}^{n}{m_{iD}l_{D}\\left({-\\ddot{\\varphi'}_{i}\\cos\\varphi'_{i}+\\dot{\\varphi'}_{i}^{2}\\sin\\varphi'_{i}}\\right)}\\label{eq:Beam2-1-1}\n\\end{equation}\n\n\nThe values of parameters and their dimensions are as follow: $m_{iD}=\\frac{2.00}{n}\\,[kg]$,\n$l_{D}=0.25\\,[m]$, $c_{\\varphi D}=\\frac{0.02}{n}\\,[Nms]$, $N_{0D}=5.00\\,[Nm]$,\n$N_{1D}=0.50\\,[Nms]$, $M_{D}=6.00\\,[kg]$, $g_{D}=9.81\\,[\\frac{m}{s^{2}}]$,\n$c_{x_{D}}=\\frac{\\ln\\left(1.5\\right)}{\\pi}\\sqrt{k_{x}\\left(M+\\sum\\limits _{i=1}^{n}{m_{i}}\\right)}\\,[\\frac{Ns}{m}]$\nand $k_{xD}\\,[\\frac{N}{m}]$ is controlling parameter. The derivation\nof the above equations can be found in \\cite{czolczynski2012synchronization}.\n\\foreignlanguage{english}{We perform a transformation to a dimensionless\nform in a way that enables us to hold parameters' values. It is because\nwe want to present new results in a way that thay can be easily compared\nto results of the investigation presented in }\\cite{czolczynski2012synchronization}\\foreignlanguage{english}{.\nWe introduce dimensionless time $\\tau=t\\omega_{0}$, where $\\omega_{0}=1\\,\\mathrm{[Hz]}$,\nand unit parameters $m_{0}=1.0\\,[kg]$, }$l_{0}=1.0\\,[m]$\\foreignlanguage{english}{\nand reach the dimensionless equations:}\n\n\\begin{equation}\nm_{i}l^{2}\\ddot{\\varphi}_{i}+m_{i}\\ddot{x}l\\cos{\\varphi_{i}}+c_{\\varphi}\\dot{\\varphi}_{i}+m_{i}gl\\sin{\\varphi_{i}}=N_{0}-\\dot{\\varphi}_{i}N_{1}\\label{eq:Beam1-1}\n\\end{equation}\n\n\n\\begin{equation}\n\\left({M+\\sum\\limits _{i=1}^{n}{m_{i}}}\\right)\\ddot{x}+c_{x}\\dot{x}+k_{x}x=\\sum\\limits _{i=1}^{n}{m_{i}l\\left({-\\ddot{\\varphi}_{i}\\cos\\varphi_{i}+\\dot{\\varphi}_{i}^{2}\\sin\\varphi_{i}}\\right)}\\label{eq:Beam2-1}\n\\end{equation}\n\n\n\\selectlanguage{english}%\nwhere: $x=\\frac{x'}{l_{0}}$, $\\dot{x}=\\frac{\\dot{x'}}{l_{0}\\omega_{0}}$,\n$\\ddot{x}=\\frac{\\ddot{x'}}{l_{0}\\omega_{0}^{2}}$, $\\varphi_{i}=\\varphi'_{i}$,\n$\\dot{\\varphi}_{i}=\\frac{\\dot{\\varphi'}_{i}}{\\omega_{0}}$, $\\ddot{\\varphi}_{i}=\\frac{\\ddot{\\varphi'}_{i}}{\\omega_{0}^{2}}$,\n$m_{i}=\\frac{m_{iD}}{m_{0}}$, $l=\\frac{l_{D}}{l_{0}}$, $c_{\\varphi}=\\frac{c_{\\varphi D}}{m_{0}l_{o}^{2}\\omega_{0}}$,\n$N_{0}=\\frac{N_{0D}}{m_{0}l_{o}^{2}\\omega_{0}^{2}}$, $N_{1}=\\frac{N_{1D}}{m_{0}l_{o}^{2}\\omega_{0}}$,\n$M=\\frac{M_{D}}{m_{0}}$, $g=\\frac{g_{D}}{l_{o}\\omega_{0}^{2}}$,\n$c_{x}=\\frac{c_{xD}}{m_{0}\\omega_{0}}$ and dimensionless control\nparameter $k_{x}=\\frac{k_{xD}}{m_{0}\\omega_{0}^{2}}$. Dimensionless\nparameters have the following values: {$m_{i}=\\frac{2.0}{n}$,\n$l=0.25$, $c_{\\varphi}=\\frac{0.02}{n}$, $N_{0}=5.0$, $N_{1}=0.5$,\n$M=6.0$, $g=9.81$.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\nThough up to now there is no experimental indication why quantum evolution may be nonlinear, it has been traditionally considered both as a possible way out to the measurement problem\\cite{Wig62a} or as matter of theoretical considerations to be contrasted with high-finesse experiments\\cite{Wei89a}. One of the most remarkable consequences of these considerations\\cite{Gis90a} was the possibility, under the nonlinearity assumption, of supraluminal communication between two spatially separated parties. This soon led some authors to conclude that any nonlinear quantum evolution would necessarily entail the possibility of such a communication\\cite{Gis89a,GisRig95a} and even to consider the relativistic postulate of 'no-faster-than-light' phenomena as the theoretical basis for the quantum evolution to be linear\\cite{SimBuzGis01a}. Recently\\cite{FerSalSan03b} we have proven that this implicaction is not strict, i.e.\\ that there exist possible nonlinear quantum evolutions not implying this fatal supraluminal communication.\\\\\n\nHere we extend our previous result to finite quantum systems of arbitrary dimensions. We formulate the 'no-signaling' condition for these systems and show a full-flegded infinity of examples fulfilling this condition. Everything is expressed in Bloch space language\\cite{Kim03a,ByrKha03a}, i.e.\\ the states of quantum systems are expressed as \n\n\\begin{equation}\n\\rho(t)=\\frac{1}{N}\\left(\\mathbb{I}_{N}+\\mathbf{r}(t)\\cdot\\mathbf{\\sigma}\\right)\n\\end{equation}\n\n\\noindent and orthogonal projectors as\n\n\\begin{equation}\nP=P_{0}\\mathbb{I}_{N}+\\mathbf{P}\\cdot\\mathbf{\\sigma}\n\\end{equation}\n\\noindent where $\\mathbf{r}(t)$ is a time-dependent so-called Bloch vector belonging to a particular convex subset of $\\mathbb{R}^{N^{2}-1}$, $\\mathbf{\\sigma}\\equiv\\left(\\sigma_{1},\\dots,\\sigma_{N^{2}-1}\\right)$ are the traceless orthogonal generators of $SU(N)$ and $(P_{0},\\mathbf{P})\\equiv(P_{0},P_{1},\\dots,P_{N^{2}-1})$ are real numbers subjected to certain restrictions (cf.\\ \\cite{Kim03a} for the details).\n\n\n\\section{The 'no-signaling' condition}\n\n\nAs remarked in \\cite{FerSalSan03b}, the impossibility of communication through the projection postulate, i.e.\\ at a speed faster than that of light, is obtained only after imposing that \\emph{the \\textbf{probability distribution} of any observable of one subsystem \\textbf{only} depends on its own reduced state}. The mathematical translation of this criterion is straightforward provided one is familiar with the preceding language. Let us consider a two-partite system of subsystems $1$ and $2$, which have dimensions $N_{1}$ and $N_2$, respectively. Their common density matrix, using a tensor product basis, will be given by\n\n\\begin{equation}\n\\rho_{12}=\\frac{1}{N_{1}N_{2}}\\left(\\mathbb{I}_{N_{1}N_{2}}+\\mathbf{r}^{(1)}\\cdot\\mathbf{\\sigma}\\otimes\\mathbb{I}_{N_{2}}+\\mathbb{I}_{N_{1}}\\otimes\\mathbf{r}^{(2)}\\cdot\\lambda+\\sum_{ij}r_{ij}^{(12)}\\sigma_{i}\\otimes\\lambda_{j}\\right)\n\\end{equation}\n\n\\noindent and an orthogonal projector for each of them by\n\n\\begin{equation}\nP^{(1)}=P_{0}^{(1)}\\mathbb{I}_{N_{1}}+\\mathbf{P}^{(1)}\\cdot\\sigma\\quad P^{(2)}=P_{0}^{(2)}\\mathbb{I}_{N_{2}}+\\mathbf{P}^{(2)}\\cdot\\lambda\n\\end{equation}\n\n\\noindent respectively, where $\\sigma$ ($\\lambda$) stands for the traceless orthogonal generators of $SU(N_{1})$ ($SU(N_{2})$) and $\\mathbf{P}^{(1)}$ ($\\mathbf{P}^{(2)})$ is a $(N^{2}_{1}-1)$($(N^{2}_{2}-1)$)-dimensional vector restricted to some given subset\\footnote{Namely, $P_{0}=P_{0}^{2}+\\mathbf{P}\\cdot\\mathbf{P}$ and $2P_{0}P_{n}+z_{ijn}P_{i}P_{j}=P_{n}$, where $z_{ijk}\\equiv g_{ijk}+if_{ijk}$, the latter denoting the completely symmetric and antisymmetric tensors of the Lie algebra $\\mathfrak{su}(N_{j})$, respectively.}.\\\\\nSuppose now that an orthogonal projector $(u_{0},\\mathbf{u})$ is measured upon subsystem $2$. Then $N_{2}$ possible outcomes $(u_{0}^{(k)},\\mathbf{u}^{(k)})$ will result with probabilites $p_{k}=u_{0}^{(k)}+\\mathbf{u}^{(k)}\\cdot\\mathbf{r}^{(2)}$ given by the trace rule. Also, the projection postulate allows us to conclude that after such a measurement, the reduced density operator for its partner, subsystem $1$ will be given by\n\n\\begin{equation}\n\\rho_{k}^{(1)}(0)=\\frac{1}{N_{1}}\\left(\\mathbb{I}_{N_{1}}+\\mathbf{r}^{(1;k)}\\cdot\\sigma\\right)\n\\end{equation} \n\n\\noindent where $\\mathbf{r}^{(1;k)}$ is an $(N^{2}_{1}-1)$-dimensional vector ($k=1,\\dots,N_{2}$ possible outcomes) dependent on the joint state $\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r_{ij}^{(12)}$ and on the measured observable $(u_{0},\\mathbf{u})$:\n\n\\begin{equation}\n\\mathbf{r}^{(1;k)}_{j}=\\frac{u_{0}^{(k)}r_{j}^{(1)}+\\sum_{n=1}^{N_{2}^{2}-1}r_{jn}^{(12)}u_{n}^{(k)}}{u_{0}^{(k)}+\\mathbf{u}^{(k)}\\cdot\\mathbf{r}^{(2)}}\\equiv r^{(1;k)}_{j}(0)\n\\end{equation}\n\nIn these conditions, the probability distribution $\\mathbb{P}$ of an arbitrary orthogonal projector $(v_{0},\\mathbf{v})$ with $p=1,\\dots,N_{1}$ possible outcomes $(v_{0}^{(p)},\\mathbf{v}^{(p)})$ at time $t$ of subsystem $1$ will be given by \n\n\\begin{equation}\n\\mathbb{P}^{(1)}(t;v^{(p)})=\\sum_{k=1}^{N_{2}}(u_{0}^{(k)}+\\mathbf{r}^{(2)}\\cdot\\mathbf{u}^{(k)})(v_{0}^{(p)}+\\mathbf{v}^{(p)}\\cdot\\mathbf{r}^{(1)}(t;\\mathbf{r}^{(1;k)}(0))\n\\end{equation}\n\n\\noindent where $\\mathbf{r}^{(1)}(t;\\mathbf{r}^{(1;k)}(0))$ denotes the Bloch vector of subsystem $1$ at time $t$ with initial condition $\\mathbf{r}^{(1;k)}(0)$.\\\\\n \nThe 'no-signaling' condition can then be easily formulated. The independece with respect to other partners' reduced state and their mutual correlations will be expressed as\n\n\\begin{eqnarray}\\label{NoSig1}\n\\frac{\\partial\\mathbb{P}^{(1)}(t;v^{(p)})}{\\partial r_{k}^{(2)}}&=&0\\\\\n\\label{NoSig2}\\frac{\\partial\\mathbb{P}^{(1)}(t;v^{(p)})}{\\partial r_{ij}^{(12)}}&=&0\n\\end{eqnarray}\n\n\\noindent Finally, the independence with respect to observables to be measured in spatially separated subsystems will be expressed as\n\n\\begin{equation}\\label{NoSig3}\n\\frac{\\partial\\mathbb{P}^{(1)}(t;v^{(p)})}{\\partial u^{(k)}_{\\mu}}=0\\quad\\mu=0,1,\\dots,N_{1}^{2}-1\n\\end{equation}\n\nThese three conditions are the mathematical translation of the previously formulated 'no-signaling' condition. The reader may check for himself that, as expected, the usual linear quantum evolution fulfills each of them (see also below).\n\n\\section{Consequences}\n\nOne of the main consequences of eqs.\\ (\\ref{NoSig1}), (\\ref{NoSig2}) and (\\ref{NoSig3}) arises after noticing that they must be valid for any particular value of the parameters involved, which implies $\\mathbf{r}^{(i)}(t;\\mathbf{r}_{k})=A^{(i)}(t)\\mathbf{r}_{k}$, where $A^{(i)}(t)$ is a time-dependent matrix. In other words, the reduced dynamics in absence of interactions (spatial separation) must be linear. Note that this does not exhaust the possibility of having nonlinear joint evolution. Indeed reduced linearity in absence of interactions entails neither joint linearity nor even reduced unitarity. Expressing this in Bloch vector language, if $(\\mathbf{r}^{(1)}(t),\\mathbf{r}^{(2)}(t),r_{ij}^{(12)}(t))$ denotes the Bloch vector of a two-partite system and if $H=H_{0}\\mathbb{I}_{N_{1}N_{2}}+\\mathbf{H}\\cdot\\sigma_{12}$ ($\\mathbf{H}=(\\mathbf{H}^{(1)},\\mathbf{H}^{(2)},H^{(12)})$ and $\\sigma_{12}=(\\sigma\\otimes\\mathbb{I}_{N_{2}},\\mathbb{I}_{N_{1}}\\otimes\\lambda,\\sigma\\otimes\\lambda)$) denotes its joint Hamiltonian, then any evolution given by\n\n \\begin{eqnarray}\n\\mathbf{r}^{(1)}(t)&=&\\mathbf{F}_{1}(t;H,\\mathbf{r}(0))\\\\\n\\mathbf{r}^{(2)}(t)&=&\\mathbf{F}_{2}(t;H,\\mathbf{r}(0))\\\\\nr^{(12)}(t)&=&F_{12}(t;H,\\mathbf{r}(0))\n\\end{eqnarray}\n\n\\noindent such that in absence of interactions ($H^{(12)}=0$) satisfies\n\n\\begin{eqnarray}\n\\label{RedLin1}\\mathbf{r}^{(1)}(t)&=&M^{(1)}(t;\\mathbf{H}^{(1)})\\mathbf{r}^{(1)}(0)\\\\\n\\label{RedLin2}\\mathbf{r}^{(2)}(t)&=&M^{(2)}(t;\\mathbf{H}^{(2)})\\mathbf{r}^{(2)}(0)\n\\end{eqnarray}\n\n\n\\noindent where $M^{(k)}(t;\\mathbf{H}^{(k)})$ denotes a time-dependent matrix depending only on the Hamiltonian of the $k$th subsystem, is free of supraluminal communication.\\\\\n\nIt should be clear that this nonlinearity only affects the evolution and never the static structure of the theory, i.e.\\ the principle of superposition of quantum states at a given instant of time is still valid, only the evolution of these states is affected.\\\\\n\nAlternatively, one can express these nonlinearities through the evolution equations:\n\n\\begin{eqnarray}\n\\frac{dr^{(1)}_{i}}{dt}&=&\\left(\\sum_{m,n=1}^{N_{1}^{2}-1}f_{imn}^{(1)}H^{(1)}_{m}r^{(1)}_{n}+\\sum_{j,m,n=1}^{N_{1}^{2}-1}f_{ijm}^{(1)}H^{(12)}_{jn}r^{(12)}_{mn}\\xi_{i;jn}^{(1)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})\\right)\\nonumber\\\\\n&&\\\\\n\\frac{dr^{(2)}_{i}}{dt}&=&\\left(\\sum_{m,n=1}^{N_{2}^{2}-1}f^{(2)}_{imn}H^{(2)}_{m}r^{(2)}_{n}+\\sum_{j,m,n=1}^{N_{2}^{2}-1}f^{(2)}_{ijm}H^{(12)}_{jn}r^{(12)}_{nm}\\xi_{i;jn}^{(2)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})\\right)\\\\\n\\frac{dr^{(12)}_{pq}}{dt}&=&2\\left(\\sum_{i,j=1}^{N_{1}^{2}-1}f^{(1)}_{jip}H^{(1)}_{j}r^{(12)}_{iq}+\\sum_{i,j=1}^{N_{2}^{2}-1}f^{(2)}_{jip}H^{(1)}_{j}r^{(12)}_{qi}+\\right.\\nonumber\\\\\n&+&\\sum_{i,j=1}^{N_{1}^{2}-1}\\sum_{m,n=1}^{N_{2}^{2}-1}\\textrm{Im}\\left[z_{ijp}^{(1)}z_{mnq}^{(2)}\\right]H_{im}^{(12)}r_{jn}^{(12)}\\xi_{pq;im}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})+\\nonumber\\\\\n&+&\\left.\\sum_{i,j=1}^{N_{1}^{2}-1}f_{ijp}^{(1)}H^{(12)}_{iq}r^{(1)}_{j}\\xi_{pq;iq}^{(12)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})+\\sum_{i,j=1}^{N_{2}^{2}-1}f_{ijp}^{(1)}H^{(12)}_{qi}r^{(2)}_{j}\\xi_{pq;iq}^{(12)}(\\mathbf{r}^{(1)},\\mathbf{r}^{(2)},r^{(12)})\\right)\\nonumber\\\\\n\\end{eqnarray}\n\n\\noindent where the functions $\\xi$ are completely arbitrary. Notice that in absence of interactions ($H^{(12)}=0$), one recovers the usual well-known quantum evolution.\n\n\\section{Conclusions\n\nThe main two conclusions to be drawn are that (i) nonlinear evolution does not necessarily imply the possibility of supraluminal communication between two arbitrary finite quantum systems, and (ii) non linear terms, in order to fulfill the no-signaling condition, must be necessarily associated to interactions.\\\\\n\nThis reopens a door, originally suggested by Wigner, to explore possible solutions to the measurement problem without contradicting other well contrasted theories.\\\\\n\nA third generalization of this approach can be undertaken by focusing on non-projective measurements, but on generalized measurements, i.e.\\ on POVM's \\cite{Per93a}.\n\n\n\\section*{Acknowledgements}\nWe acknowlegde financial support from Spanish Ministry of Science and Techmology through project no.\\ FIS2004-01576. M.F.\\ also acknowledges financial support from Oviedo University (ref.\\ no.\\ MB-04-514).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\nWe propose a novel FL method (CoFED) that is simultaneously compatible with heterogeneous tasks, heterogeneous models, and heterogeneous training processes. Compared with the traditional method, CoFED is more suitable for CS-FL settings with fewer participants but higher heterogeneity. CoFED decouples the models and training processes of different participants, thereby enabling each participant to train its independently designed model for its unique task via its optimal training methods. In addition, CoFED protects private data, models and training methods of all participants under FL settings. CoFED enables participants to share multiparty knowledge to increase their local model performance. The method produces promising results under non-IID data settings for models with heterogeneous architectures, which is more practical but is usually difficult to handle in existing FL methods. Moreover, the CoFED method is efficient since training can be performed in only one communication round.\n\nThe CoFED method may be limited by the availability of public unlabeled datasets. Although we conduct numerous experiments to demonstrate that CoFED has low requirements for public datasets and that the use of irrelevant or randomly generated datasets is still effective, some failure scenarios may still occur; this is a problem that we hope to address in the future.\n\\section{Experiments}\n\nIn this section, we execute the CoFED method under different FL settings to explore the impacts of different conditions and compare it with existing FL methods. The source code can be found at https:\/\/github.com\/flcao\/CoFED.\n\n\n\n\n\n\n\\subsection{Model Settings}\nThe CoFED method enables participants to independently design different models. For example, some participants use CNNs as classifiers, while others use support vector machine (SVM) classifiers. We use randomly generated 2-layer or 3-layer CNNs with different architectures as the models for the different participants in image classification tasks to illustrate that CoFED is compatible with heterogeneous models. Ten of the 100 employed model architectures are shown in Table \\ref{netStruct}.\n\n\\begin{table}[ht]\n\\caption{Network Architectures}\n\\label{netStruct}\n\\centering\n\\begin{tabular}{cccc}\n\\toprule\nModel & \\begin{tabular}[c]{@{}c@{}}1st \\\\ conv layer\\end{tabular} & \\ \\begin{tabular}[c]{@{}c@{}}2nd \\\\ conv layer\\end{tabular} & \\ \\begin{tabular}[c]{@{}c@{}}3rd \\\\ conv layer\\end{tabular} \\\\ \\midrule\n1 & 24 3x3 & 40 3x3 & none \\\\\n2 & 24 3x3 & 32 3x3 & 56 3x3 \\\\ \n3 & 20 3x3 & 32 3x3 & none \\\\ \n4 & 24 3x3 & 40 3x3 & 56 3x3 \\\\ \n5 & 20 3x3 & 32 3x3 & 80 3x3 \\\\ \n6 & 24 3x3 & 32 3x3 & 80 3x3 \\\\ \n7 & 32 3x3 & 32 3x3 & none \\\\ \n8 & 40 3x3 & 56 3x3 & none \\\\ \n9 & 32 3x3 & 48 3x3 & none \\\\ \n10 & 48 3x3 & 56 3x3 & 96 3x3 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo demonstrate that CoFED is applicable to broader ranges of models and task types than other approaches, we choose four types of classifiers as the participant models in the Adult dataset experiment: decision trees, SVMs, generalized additive models, and shallow neural networks.\n\n\\subsection{CIFAR Dataset Experiment}\nWe set the number of participants to 100 in our CIFAR experiments. For each participant, we randomly select a few of the 20 superclasses of CIFAR100 \\cite{krizhevsky2009learning} as its label space. Since we try to study the effect of the proposed method on heterogeneous tasks, the label spaces of different participants are generally different, but some overlap may occur. The local dataset of each participant consists of the samples in its label space.\n\nThe distributions of each superclass possessed by the different participants who own this superclass sample may encounter two situations. In the first case, we assume that the samples of each participant with the superclass are uniformly and randomly sampled from all samples of the superclass in CIFAR100 (that is, the IID setting). In this case, each participant usually has samples belonging to all 5 subclasses of each superclass in its label space. In the second case, we assume that each participant who owns the superclass only has samples belonging to some of the subclasses of its superclass (in our experiment, 1 or 2 subclasses), that is, the non-IID setting. The details of the experimental data settings are as follows.\n\n\\textbf{IID Data Setting}: Each participant is randomly assigned 6 to 8 superclasses in CIFAR100 as its label space. For the local training sets, each participant has 50 instances from each superclass in the CIFAR100 training set, and these 50 samples are evenly sampled from the samples of this superclass in the training set of CIFAR100. No overlap occurs between the training sets of any participants. For the test sets, all instances in the test set of CIFAR100 are used, whereby each participant's test set has 500 instances for each superclass.\n\n\\textbf{Non-IID Data Setting}: This setting is almost the same as the IID setting; the difference is that the sample of a superclass of each participant is only randomly acquired from 1 to 2 subclasses of the superclass in the CIFAR100 training set. The configuration of the test set is exactly the same as that used with the IID setting. The non-IID data setting is often regarded as more difficult than the IID one since a model that learns only 1 or 2 subclasses of a superclass during the training process is required to recognize all 5 subclasses included in the test set.\n\nIn this experiment, the public unlabeled dataset used in CoFED is the training set of CIFAR10 \\cite{krizhevsky2009learning}. Since CoFED is compatible with heterogeneous training processes, we conduct a grid search to determine the optimal training parameter settings for each participant task. The training configuration optimized for each participant (including the trainer, learning rate, and learning rate decay) is used in the initial local training phase, and the update training settings in the final step are adjusted based on these parameters. We always use a minibatch size of 50 for the local training process and 1000 for the update training process.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/iidcifar_cifar10.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/noniidcifar_cifar10.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{Results of the CIFAR experiment. The X-axis value is the relative test accuracy, which is the ratio of the CoFED test accuracy to the local test accuracy.}\n  \\label{cifar_cifar10}\n\\end{figure}\n\n\nWe test our proposed CoFED method separately under the IID setting and non-IID setting, and the hyperparameter $\\alpha$ is set to 0.3 for both settings. We compare the test classification accuracies of the models trained by CoFED with those of the models utilizing local training, and the results are shown in Fig. \\ref{cifar_cifar10}(a) and Fig. \\ref{cifar_cifar10}(b). Under the IID setting, CoFED method improves the relative test accuracy of each participant model by 10\\%-32\\% with an average of 17.3\\%. This demonstrates that CoFED can increase participant model performance even when the divergences of models are not large (such as those under the IID data setting). For the non-IID setting, CoFED can lead to greater model performance gains due to the greater divergence of data distributions that make locally trained models more divergent. In this experiment, CoFED achieves a relative average test accuracy improvement of 35.6\\%, and for each participant, the improvement ranges from 14\\% to 67\\%. The performance boost under the Non-IID setting is better than the IID setting, which indicates that CoFED suffers less from statistical heterogeneity.\n\n\n\\subsection{FEMNIST Dataset Experiment}\nFEMNIST dataset is a handwritten character dataset of LEAF \\cite{caldas2018leaf} which is a benchmark framework for FL. It consists of samples of handwritten characters from different users, and we select the 100 users with the most samples of FEMNIST as participants. Forty percent of selected samples are used as training set, and the rest are test set. \n\nThe architectures of participant models are the same as those in the CIFAR experiment, and the local training hyperparameters are tuned in similarly to those in the CIFAR experiment. In this experiment, we use random crops of the images in the Chars74k-Kannada dataset \\cite{de2009character} to construct an unlabeled public dataset with nearly 50,000 items. Chars74k-Kannada contains handwritten character images of English and Kannada, and only the handwritten Kannada characters are used as unlabeled public dataset in our experiment. The hyperparameter $\\alpha$ is set to 0.01, and the results are shown in Fig. \\ref{fem_kan}. CoFED improves the relative test accuracies of the models for almost all participants, with an average improvement of 15.6\\%.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Figures\/fem_kan.eps}\n\\caption{Results of the FEMNIST experiment.}\n\\label{fem_kan}\n\\end{figure}\n\n\n\\subsection{Public Unlabeled Dataset Availability}\nOne of the major concerns regarding the CoFED method concerns whether a public dataset is available. The role of the public dataset is to express and share the knowledge that the participant models need to learn. Generally, the participants in FL cannot learn knowledge from samples generated entirely by random values, but this does not mean that we must use samples that are highly related to the participants' local data to construct a public dataset.\n\nFor example, in the CIFAR experiment, we use the samples of CIFAR100 to construct the local data of the participants, but we use CIFAR10 as the public dataset. The categories of the data samples contained in CIFAR10 and CIFAR100 only overlap slightly. Therefore, we can regard CIFAR10 as a dataset composed of pictures randomly obtained from the Internet without considering the similarity between its contents and the samples of participants (from CIFAR100). A large morphology difference between English characters and Kannada characters is also observed in the FEMNIST experiment. However, CoFED can effectively improve the performance of almost all participant models in both experiments, which makes us want to know how different participant models use public datasets that are not relevant to them to share knowledge. Therefore, we review the results of pseudolabel aggregation and check how these models classify the irrelevant images. Fig. \\ref{pseudo} shows a partial example of the pseudolabel aggregation results of the 10 participant models (out of 100 participants).\n\n\\begin{figure*}[ht]\n\\includegraphics[width=\\linewidth]{Figures\/aggregating.eps}\n\\caption{The results of pseudolabel aggregation. The images obtained from the training set of CIFAR10 are scattered across the superclasses of CIFAR100. Ten images per superclass are randomly selected.}\n\\label{pseudo}\n\\end{figure*}\n\nFirst, we notice that some trucks and automobiles are correctly classified into the vehicles\\_1 category. This indicates that the unlabeled instances whose categories are contained in the label spaces of the participant models are more likely to be assigned to the correct category. However, exceptions occur; some automobiles are assigned to the {\\it{flowers}} and {\\it{fruit and vegetables}} categories. A common feature possessed by these automobile images is that they contain many red parts, which may be regarded as a distinctive feature of {\\it{flowers}} or {\\it{fruit and vegetables}} by the classifiers. In addition, the unlabeled samples that are not included in the label spaces of any classifier are also classified into the groups that match their visual characteristics. For example, the corresponding instances of aquatic\\_mammals and fish usually have blue backgrounds, which resemble water. Another interesting example is the people category. Although almost no human instances are contained in the training set of CIFAR10, some of the closest instances are still given, including the person on a horse.\n\nMoreover, we also try to replace the public dataset in the CIFAR experiment with the ImageNet dataset \\cite{van2016pixel} with almost the same size as that of CIFAR10. The CoFED method can still achieve considerable performance improvements, as shown in Fig. \\ref{cifar_imagenet}(a) and Fig. \\ref{cifar_imagenet}(b).\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/iidcifar_imagenet.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/noniidcifar_imagenet.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{Results of the CIFAR experiment obtained by using the ImageNet dataset as an unlabeled public dataset.}\n  \\label{cifar_imagenet}\n\\end{figure}\n\n\n\\subsection{Unlabeled Dataset Size}\nAccording to (\\ref{13}), (\\ref{14}) and (\\ref{15}), when an existing classifier has a higher generalization accuracy and a larger labeled training set, a larger unlabeled dataset is needed to improve its accuracy. This suggests that in the CoFED method, a larger unlabeled dataset can produce a more obvious performance improvement for a given group of participant models. We redo the CIFAR experiment with 10 participants and vary the size of the unlabeled dataset between 500 and 50,000 samples in the training set of CIFAR10. The experimental results are shown in Fig. \\ref{pubsize}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figures\/pubsize.eps}\n\\caption{Size of the unlabeled public dataset vs. the mean relative test accuracy.}\n\\label{pubsize}\n\\end{figure}\n\nThe experimental results are consistent with the theoretical analysis. This shows that the strategy of increasing the size of the unlabeled dataset can be used to boost the performance improvements exhibited by all participant models. Considering that the difficulty of collecting unlabeled data is much lower than that of collecting labeled data in general, this strategy is feasible in many practical application scenarios.\n\n\n\\subsection{Hyperparameter $\\alpha$}\nFrom the theoretical analysis, increasing the reliability of the pseudolabeling process is likely to bring more significant performance improvements, which is also very intuitive. Therefore, we use a hyperparameter $\\alpha$ to improve the reliability of pseudolabel aggregation. A larger $\\alpha$ requires a higher percentage of participants to agree to increase the reliability of the pseudolabels, but this may also reduce the number of available pseudolabel instances. In particular, when the participants disagree greatly, requiring excessive consistency across the results of different participant models may stop the spread of multiparty knowledge.\n\nIn this section, we repeat the CIFAR experiment for 10 participants with different $\\alpha$ and record the changes in the total number of samples generated by pseudolabel aggregation when different $\\alpha$ values are taken in Fig. \\ref{idxSiz}. The changes in the test accuracies of all participant models in the CoFED method are shown in Fig. \\ref{alphasize}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figures\/idxSize.eps}\n\\caption{$\\alpha$ vs. the size of the pseudolabel aggregation results. For $\\alpha=1$, the total numbers of IID and non-IID data are both 0, which cannot be shown with the log scale.}\n\\label{idxSiz}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{Figures\/alphasize.eps}\n\\caption{$\\alpha$ vs. the mean relative test accuracy.}\n\\label{alphasize}\n\\end{figure}\n\nThe results show that when the value of $\\alpha$ changes, its impact on CoFED is not monotonic. Although an excessively large value of $\\alpha$ can increase the reliability of the generated pseudolabels, this also greatly reduces the number of samples in the pseudolabel aggregation results, which may degrade the training results. In the FEMNIST experiment, we find that a larger $\\alpha$ value may greatly reduce the number of samples in the pseudolabel aggregation results, so we set the $\\alpha$ value to 0.01.\n\nAt the same time, an excessively large $\\alpha$ value makes the pseudolabel aggregation process more inclined to choose the samples that most participants agree on. Since these sample are approved by most participants, it is almost impossible to bring new knowledge to these participants. In addition, we find that a larger $\\alpha$ value has a more severe impact on the non-IID data setting, where performance degradation is more significant than in the IID cases. This is because the differences between the models trained on the non-IID data are greater, and the number of samples that most participants agree on is smaller. Therefore, when $\\alpha$ increases, the number of available samples decreases faster than in the IID case, as shown in Fig. \\ref{idxSiz}, which causes greater performance degradation under the non-IID setting.\n\nOn the other hand, an $\\alpha$ that is too small decreases the reliability of the pseudolabel aggregation results, which may introduce more mislabeled samples, making the CoFED method less effective. In addition, an excessively small $\\alpha$ value may cause a large increase in the number pseudolabel aggregation samples, resulting in increased computation and communication overheads.\n\nIn summary, the effectiveness of the CoFED method can be affected by the value of the hyperparameter $\\alpha$, and adopting an appropriate $\\alpha$ value can yield greater performance improvements and avoid unnecessary computation and communication costs.\n\n\\subsection{Adult Dataset Experiment}\nAdult dataset\\cite{kohavi1996scaling} is a census income dataset to be used to classify a person's yearly salary based on their demographic data. We set the number of participants to 100 in our Adult experiments. For each participant, we randomly select 200 samples from the training set of the Adult dataset without replacement. Since we try to study the effect of the proposed method on heterogeneous models, 4 types of classifiers are chosen: decision trees, SVMs, generalized additive models, and shallow neural networks. The number of utilized classifiers of each type is 25. The size of unlabeled public dataset in this experiment is 5000, and each sample is randomly generated with valid values for the input properties used by the classifiers. The Adult test set is used to measure the performance of each participant's classifier.\n\nThe results are shown in Fig. \\ref{4c}. The proposed method improves the test accuracies of most classifiers, with an average of 8.5\\%, and the performance of the classifier that benefits the most increases by more than 25\\%. This demonstrates that the proposed method is applicable not only to image classification tasks involving CNN models but also to nonimage classification tasks with traditional machine learning models.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{Figures\/4classifiers.eps}\n\\caption{Results of the Adult experiment.}\n\\label{4c}\n\\end{figure}\n\n\\subsection{Comparison with Other FL Methods}\nTo the best of our knowledge, CoFED is the first FL method that tries to be compatible with heterogeneous models, tasks, and training processes simultaneously. Therefore, FL methods are available for comparison under HFMTL settings. We use two well-known comparison methods. One is the personalized FedAvg method, which represents the classic parameter aggregation-based FL strategy that is not compatible with heterogeneous models with different architectures; the other is the FedMD method \\cite{li2019fedmd}, which supports different neural network architectures and is used to evaluate the performance of CoFED under heterogeneous models.\n\nTo enable the personalized FedAvg and FedMD methods to handle heterogeneous classification tasks, we treat each participant's local label space $\\mathcal{Y}_i$ as the union of the spaces $\\mathcal{Y}$ in (\\ref{union}). In this way, we can handle heterogeneous tasks with the idea of personalized FedAvg and FedMD. The main steps are as follows.\n\\begin{enumerate}\n    \\item Use the FedAvg or FedMD algorithm to train a global model whose label space is the union of the label spaces of all participants.\n    \\item The global model is fine-tuned on each participant's local dataset for a personalized model for their local task..\n\\end{enumerate}\n\nIn this comparison experiment, we use the data configuration presented in Section 4.3. Considering that the personalized FedAvg method can not be used for models with different architectures, 100 participants share the same architecture neural network model. In the FedMD comparison experiment, we use the same setup as that in Section 4.3; that is, we select 100 participants with different neural network architectures.\n\nWe try a variety of different hyperparameter settings to achieve better performance in the personalized FedAvg experiment. We use $E=20$ as the number of local training rounds in each communication iteration, $B=50$ is set as the local minibatch size used for the local updates, and participants also perform local transfer learning to train their personalized models in each communication iteration. In the FedMD experiment, We use the similar settings of \\cite{li2019fedmd}, and it should be noted that FedMD uses the labeled data of CIFAR10 as the public dataset, which is different from the unlabeled data used in CoFED.\n\n\\begin{table}[]\n\\caption{Comparison Results}\n\\label{iidtb}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{@{}c@{}c@{}ccc}\n\\hline\n                         &          & \\begin{tabular}[c]{@{}c@{}}Personalized\\\\ FedAvg\\end{tabular} & FedMD & CoFED \\\\ \\hline\n\\multirow{2}{*}{IID}     & Accuracy & \\textbf{1.06}                                                          & 0.87  & 1.00\\tnote{*}  \\\\\n                         & Rounds   & 100                                                           & 8\\tnote{\\dag}     & \\textbf{1}     \\\\ \\hline\n\\multirow{2}{*}{Non-IID} & Accuracy & 0.94                                                          & 0.94  & \\textbf{1.00}\\tnote{*}  \\\\\n                         & Rounds   & 150                                                           & 17\\tnote{\\dag}    & \\textbf{1}     \\\\ \\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\item[*] Reference value.\n\\item[\\dag] The round with the best performance.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedavg_iid.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedavg_noiid.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{Personalized FedAvg method vs. CoFED. To facilitate the comparison, the test accuracy of the CoFED model is used as the reference accuracy, and the value of the Y-axis is the ratio of the comparison algorithm's test accuracy to the reference accuracy. Since each participant has a corresponding ratio, the blue line represents the average of the ratios of all participants corresponding to the given number of iterations.}\n  \\label{fedavg}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedmd_iid.eps}\n    \\caption{IID setting}\n  \\end{subfigure}%\n \n  \\begin{subfigure}{.5\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fedmd_noiid.eps}\n    \\caption{Non-IID setting}\n  \\end{subfigure}%\n  \\caption{FedMD vs. CoFED.}\n  \\label{fedmd}\n\\end{figure}\n\nThe results of the comparison between FedAvg and CoFED are shown in Fig. \\ref{fedavg}. The relative test accuracy is calculated as the average of the ratios of all participants to the CoFED test accuracy. Under the IID setting, FedAvg reaches the performance level of CoFED after 42 communication rounds and is finally 6\\% ahead of COFED in Fig. \\ref{fedavg}(a). Under the non-IID setting, FedAvg stabilizes after 150 communication rounds. At this time, CoFED still leads FedAvg by 6\\% in Fig. \\ref{fedavg}(b). The comparing results of personalized FedAvg and CoFED are shown in Fig. \\ref{fedmd}. For both data settings, CoFED outperforms FedMD, and CoFED leads by 14\\% under the IID setting and 35\\% under the non-IID one.\n\nIn terms of communication cost, if we do not consider the communication overhead required to initially construct the public dataset, CoFED achieves better performance with lower communication overheads in all cases because CoFED only needs to pass through the label data (not the sample itself) during the training process, and iterating for multiple rounds is not required. This assumption is not unrealistic because the construction of public datasets may not require the central server to distribute data to the participants; the participants can instead obtain data from a third party, which does not incur communication costs between the participants and the central server. Even if we include that paradigm, CoFED still achieves better performance with lower communication costs except under the IID and identical architecture model settings. In fact, the IID setting used in the FedAvg comparison experiment is not the scenario that is considered most by CoFED because the model architectures are the same and can be shared under that setting.\n\\section{Introduction}\nFederated learning (FL) allows different participants to collaborate in solving machine learning problems under the supervision of a center without disclosing participants' private data \\cite{kairouz2021advances}. The main purpose of FL is to achieve improved model quality by leveraging the multiparty knowledge from participants' private data without the disclosure of data themselves.\n\nFL was originally proposed for training a machine learning model across a large number of users' mobile devices without logging their private data to a data center \\cite{konevcny2015federated}. In this scenario, a center orchestrates edge devices to train a global model that serves a global task. However, some new FL settings have emerged in many fields, including medicine \\cite{rieke2020future}, \\cite{xiao2021federated}, \\cite{raza2022designing}, \\cite{courtiol2019deep}, \\cite{adnan2022federated}, finance \\cite{li2020preserving}, \\cite{gu2021privacy}, and network \\cite{zhang2021survey}, \\cite{nguyen2021federated}, \\cite{ghimire2022recent}, \\cite{regan2022federated}, where the participants are likely companies or organizations. Generally, the terms \\textit{cross-device federated learning} (CD-FL) and \\textit{cross-silo federated learning} (CS-FL) can refer to the above two FL settings \\cite{kairouz2021advances}. However, the majority of these studies' contributions concern training reward mechanisms \\cite{tang2021incentive}, topology designs \\cite{marfoq2020throughput}, data protection optimization approaches \\cite{zhang2020batchcrypt}, etc., and for their core CS-FL algorithms, they simply follow the idea of gradient aggregation used in CD-FL. Such studies ignore the fact that organizations or companies, as participants, may be more heterogeneous than device participants.\n\nOne of the most important heterogeneities to address under the CS-FL setting is model heterogeneity; that is, models may have different architecture designs. Different devices in CD-FL typically share the same model architecture, which is given by a center. As a result, the models obtained through local data training on different devices differ only in terms of their model parameters, allowing the center to directly aggregate the gradients or model parameters uploaded by the participating devices. However, participants in a CS-FL scenario are usually independent companies or organizations, and they are capable of designing unique models. They prefer to use their own independently designed model architectures rather than sharing the same model architecture with others. At this time, the strategy used in CD-FL cannot be applied to CS-FL models with different architectures. Furthermore, model architectures may also be intellectual properties that need to be protected, and companies or organizations that own these properties do not want them to be exposed to anyone else, which makes model architecture sharing hard for CS-FL participants to accept. An ideal CS-FL method should treat each participant's model as a black box, without the need for its parameters or architecture.\n\nThe heterogeneity among models results in the need for training heterogeneity. Under the CD-FL setting, participants usually train their models according to the configuration of a center, which may include training algorithms (such as stochastic gradient descent) and parameter settings (such as the learning rate and minibatch size). However, when participants like companies or organizations use their independently designed model architectures in CS-FL, they need to choose different local training algorithms that are suitable for their models and exert full control over their local training processes. Decoupling the local training processes of different participants not only enables them to choose suitable algorithms for their models but also prevents the leakage of their training strategies, which may be their intellectual properties.\n\nIn addition, CS-FL is more likely to face task heterogeneity than CD-FL. In CD-FL scenarios, all devices usually share the same target task. In terms of classification tasks, the task output categories of all devices are exactly the same. Under the CS-FL setting, because the participating companies or organizations are independent of each other and have different business needs, their tasks may be different. Of course, we must assume that there are still similarities between these tasks. In terms of classification tasks, the task output categories of different participants in CS-FL may be different. For example, an autonomous driving technology company and a home video surveillance system company both need to complete their individual machine learning classification tasks. Although both of them need to recognize pedestrians, the former must also recognize vehicles, whereas the latter must recognize indoor fires. Therefore, the task output categories of the autonomous driving technology company include pedestrians and vehicles without indoor fires, whereas the task output categories of the home video surveillance system company include pedestrians and indoor fires without vehicles. It is easy to see that, in contrast to the complete task consistency of CD-FL, CS-FL participation by independent companies or organizations is more likely to encounter situations in which the different participants possess heterogeneous tasks.\n\n\\begin{figure}[t]\n  \\centering\n  \\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fig0.eps}\n    \\caption{CS-FL setting without heterogeneity}\n  \\end{subfigure}%\n  \\par\\bigskip\n  \\begin{subfigure}{\\columnwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Figures\/fig1.eps}\n    \\caption{CS-FL setting with heterogeneity (HFMTL)}\n  \\end{subfigure}%\n  \\caption{(b) is an example of an HFMTL setting with 3 participants. A central server needs to coordinate 3 participants to solve their classification tasks via federated training. Compared with (a), which has no heterogeneity, (b) also contains different machine learning model architectures and training optimization algorithms, and the tasks of participants are distinct from one another(i.e., different label spaces).}\n  \\label{fig1}\n\\end{figure}\n\nAlthough the number of participants in CS-FL is much smaller than that in CD-FL in general, the heterogeneity of the participants, including model heterogeneity, training heterogeneity and task heterogeneity, may bring more challenges. Overall, we use the term \\textit{heterogeneous federated multitask learning} (HFMTL) to refer to the FL settings that contain the above three heterogeneity requirements, and Fig. \\ref{fig1}(b) shows an example of the HFMTL setting with three participants. Different participants may have different label spaces. For example, Participant $B$ has a label space $\\left\\{0, 1, 2, 3\\right\\}$. This space can be different from those of other participants, e.g., Participant $C$ has $\\left\\{1, 4, 5\\right\\}$. The classification task of Participant $B$ is to classify inputs with the labels in $\\left\\{0, 1, 2, 3\\right\\}$, while Participant $C$ aims to classify inputs with the labels in $\\left\\{1, 4, 5\\right\\}$. Therefore, they have different tasks and usually have different categories of local training data.\n\nOur main motivation in this paper is to address the needs of \\textit{model heterogeneity}, \\textit{training heterogeneity}, and \\textit{task heterogeneity} in CS-FL settings. Aiming at the HFMTL scenario with these heterogeneities, we propose a novel FL method. The main contributions of this paper are as follows.\n\n\\begin{itemize}\n\\item We propose an FL method that is simultaneously compatible with heterogeneous tasks, models, and training processes, and it boost the performance of each participant's model.\n\\item Compared with the existing FL methods, the proposed method not only protects private local data but also protects the architectures of private models and private local training methods.\n\\item Compared with these other methods, the proposed method achieves more performance improvements for the non-independent and identically distributed (non-IID) data settings in FL and can be completed in one round.\n\\item We conduct comprehensive experiments to corroborate the theoretical analysis conclusions and the impacts of different settings on the suggested method.\n\\end{itemize}\n\nThe rest of this paper is structured as follows. Section 2 provides a brief overview of relevant works. Section 3 contains the preliminaries of our method. Section 4 describes the proposed method in detail. Section 5 offers the experimental results and analysis. Section 6 summarizes the paper.\n\\section*{Acknowledgment}\nThis research was partially supported by the National Key Research and Development Program of China (2019YFB1802800), PCL Future Greater-Bay Area Network Facilities for Large-scale Experiments and Applications (PCL2018KP001).\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Methodology}\n\\subsection{The Overall Steps of CoFED}\nThe main incentive of FL is that it improves the performance of participant models. The poor performance of locally trained models is mainly due to insufficient local training data, so these models fail to learn enough task expertise. Hence, to increase the participant model performance, it is vital to allow them to learn from other participants. The most popular and straightforward methods of sharing knowledge are shared models or data, however both are forbidden by FL settings, thus we must devise alternative methods.\n\nA research \\cite{wang2007analyzing} discovered that a sufficiently enough amount of variety across the models is necessary to increase classification model performance. Generally, it is difficult to generate models with large divergences under single-view settings. However, under FL settings, The architectures of the participant models may be highly varied, and they may have been trained on distinct local datasets that are very likely to be distributed differently. All of these may increase the diversity of different participants' models. Therefore, if enough unlabeled data are obtained, we can regard the federated classification problem as a semisupervised learning problem, and it is very suitable to adopt cotraining-like techniques due to the high diversity between different participant models. We provide the overall steps of CoFED as follows.\n\n\\begin{enumerate}\n    \\item \\textbf{Local training}: Each participant independently trains a local model on its private dataset.\n    \\item \\textbf{Pseudolabeling}: Each participant pseudolabels an unlabeled dataset, which is public to all participants, with its locally trained model.\n    \\item \\textbf{Pseudolabel aggregation}: Each participant uploads its pseudolabeling results to a central server, and the center votes for the pseudolabeled dataset with high confidence for each category based on the category overlap statuses of different participants and the pseudolabeling results. After that, the center sends all pseudolabels to each participant.\n    \\item \\textbf{Update training}: Each participant trains its local model on the new dataset created by combining the local dataset with the received pseudolabeled dataset.\n\\end{enumerate}\n\nIt can be seen that there are some differences between CoFED and the cotraining process under single-view settings. First, in cotraining, the training sets used by different classifiers are the same, while the training sets of the different classifiers in CoFED come from different participants, so they are usually different. Second, the target tasks of different classifiers in cotraining are the same; that is, they have the same label space. However, in CoFED, the label spaces of different classifiers are different, and the pseudolabeling of unlabeled samples need to be performed according to the pseudolabel aggregation results of the overlapping classification process. Furthermore, cotraining completes training by repeating the above process, while the process is performed only once in CoFED.\n\n\\subsection{Analysis}\n\nSuppose that we are given two binary classification models $f$ and $g$ from a hypothesis space $\\mathcal{H}: \\mathcal{X} \\rightarrow \\mathcal{Y}, |\\mathcal{H}| < \\infty$, and an oracle model $h \\in \\mathcal{H}$ whose generalization error is zero. We can define the generalization disagreement between $h_1 \\in \\mathcal{H}$ and $h_2 \\in \\mathcal{H}$ as:\n\\begin{equation}\n    \\begin{aligned}\n     d(h_1,h_2)&=d(h_1,h_2|\\mathcal{X}) \\\\\n               &=\\mathbf{Pr}(h_1(x) \\ne h_2(x) | x \\in \\mathcal{X})\n    \\end{aligned}\n\\end{equation}\n\nTherefore, the generalization errors of $f$ and $g$ can be computed as $d(f,h)$ and $d(g,h)$, respectively. Let $\\varepsilon$ bound the generalization error of a model, and let $\\delta>0$; a learning process generates an approximate model $h'$ for $h$ with respect to $\\varepsilon$ and $\\delta$ if and only if:\n\\begin{equation}\n    \\mathbf{Pr}(d(h', h) \\ge \\varepsilon) \\le \\delta\n\\end{equation}\n\nSince we usually have only a training dataset $X \\subset \\mathcal{X}$ containing finite samples, the training process minimizes the disagreement over $X$:\n\\begin{equation}\n    \\mathbf{Pr}(d(f,h|X) \\ge \\varepsilon) \\le \\delta\n\\end{equation}\n\n\\textbf{Theorem 1.} \\textit{\nGiven that $f$ is a probably approximately correct (PAC) learnable model trained on $L \\subset \\mathcal{D}$, $g$ is a PAC model trained on $L_{g} \\subset \\mathcal{D}$, and $\\varepsilon_f < \\frac{1}{2}$, $\\varepsilon_g < \\frac{1}{2}$. $f$ and $g$ satisfy that the following:\n\\begin{align}\n    \\label{9}\n    & \\mathbf{Pr}(d(f, h) \\ge \\varepsilon_f) \\le \\delta \\\\\n    \\label{10}\n    & \\mathbf{Pr}(d(g, h) \\ge \\varepsilon_g) \\le \\delta\n\\end{align}\nIf we use $g$ to pseudolabel an unlabeled dataset $X_u \\subset \\mathcal{X}$, we generate a pseudolabeled dataset\n\\begin{equation}\n    P=\\left \\{ (x,y)|x \\in X_u, y=g(x) \\right \\}\n\\end{equation}\nand combine $P$ and $L$ into a new training dataset $C$. After that, $f'$ is trained on $C$ by minimizing the empirical risk. Moreover,}\n\\begin{align}\n    \\label{13}\n    & |L| \\varepsilon_f < \\sqrt[|P|\\varepsilon_g]{(|P|\\varepsilon_g)!} \\hspace{1mm} e-|P|\\varepsilon_g \\\\\n    \\label{14}\n    & \\varepsilon_{f'} = \\max \\left \\{\\varepsilon_f + \\frac{|P|}{|L|}(\\varepsilon_g-d(g,f')), 0 \\right \\}\n\\end{align}\n\\textit{where $e$ is the base for natural logarithms; then,}\n\\begin{equation}\n    \\label{15}\n    \\mathbf{Pr}(d(f',h) \\ge \\varepsilon_{f'}) \\le \\delta\n\\end{equation}\n\nTheorem 1 has been proven in \\cite{wang2007analyzing}. Assume that $f$ and $g$ are 2 models from different participants that satisfy (\\ref{9}) and (\\ref{10}). The right side of (\\ref{13}) monotonically increases as $|P|\\varepsilon_g \\in (0, \\infty)$, which indicates that a larger pseudolabeled dataset $P$ enables a larger upper bound of $|L| \\varepsilon_f$. That is, if $f$ is a model trained on a larger training dataset with higher generalization accuracy, a larger unlabeled dataset is required to further improve the generalization accuracy of $f$.It can be seen from (\\ref{14}) and (\\ref{15}) that when $d(g,f')$ is larger, the lower bound of the generalization error of $f'$ under the same confidence is smaller, which is because that $f'$ is trained on $L$ and $P$, and $P$ is generated by $g$, $d(g,f')$ mainly depends on  the degree of divergence between $f$ and $g$, i.e., the difference between the training dataset of $f$ and $g$. In FL settings, substantial diversity across different local training datasets is fairly prevalent, hence the performance improvement requirement is generally satisfied. The same conclusion applies to the boost version $g'$ obtained by switching $f$ and $g$ due to symmetry.\n\nOn the other hand, if $|P|$ is sufficiently large since $P$ is generated by $g$, the $f'$ trained on $C$ can be treated as proximal to $g$. That is, if we repeat the above process on $f'$ and $g$, $f'$ and $g$ may be too similar to improve them. Therefore, we utilize a large unlabeled dataset and only perform the above process once instead of iterating for multiple rounds; this technique can also avoid the computational cost increase caused by multiple training iterations.\n\nAn intuitive explanation for the CoFED method is that when the different participant models have great diversity, the differences in the knowledge possessed by the models are greater. As a result, the knowledge acquired by each participant model from others contains more knowledge that is unknown to itself. Therefore, more distinctive knowledge leads to greater performance gains. In addition, when mutual learning between different models is sufficient, the knowledge difference between them will almost disappear, and it is difficult for mutual learning to provide any participant's model with distinctive knowledge.\n\n\\subsection{Pseudolabel Aggregation}\n\nAfter each participant uploads its pseudolabeling results to the public unlabeled dataset, the center exploits their outputs to pseudolabel the unlabeled dataset. In this subsection, we explain the implementation details of this step.\n\nAssume that the union of the label spaces of all participants' tasks is\n\\begin{equation}\n\\label{union}\n\\mathcal{Y}=\\bigcup_{i=1}^N{\\mathcal{Y}_i=\\left\\{ c_k \\right\\} , k=1,2,\\cdots,n_c}\n\\end{equation}\nwhere $n_c$ is the number of elements in the whole label space $\\mathcal{Y}$, and each category $c_k$ exists in the label space of one or multiple participants' label spaces:\n\\begin{equation}\n    c_k\\in \\bigcap_{j=1}^{m_k}{\\mathcal{Y}_{i_j}, 1\\leq i_1 < i_2 < \\cdots< i_{m_k}\\leq N}\n\\end{equation}\n$m_k$ is the number of participants who possess category $c_k$. At the same time, we define the pseudolabeled dataset $P_k$ for category $c_k$, and $P_k$ is used to store the indices of the instances in the public dataset corresponding to each category $c_k$ after pseudolabel aggregation.\n\nFor each category $c_k$ existing in $\\mathcal{Y}_{i_j}$, the model $f_{i_j}$ classifies the instances in the public unlabeled dataset $D^{pub}$ as belonging to category $c_k$, and the set of these instances can be defined as\n\\begin{equation}\n    S_j=\\left\\{ x|f_{i_j}\\left( x \\right) \\equiv c_k,x\\in D^{pub} \\right\\} , j=1,2,\\cdots,m_k\n\\end{equation}\n\nFor an instance $x\\in D^{pub}$, if we regard the outputs of different participant models on $x$  for category $c_k$ as the outputs of a two-class (belonging to category $c_k$ or not) ensemble classifier $g$, since (\\ref{14}) suggests that a lower generalization error bound $\\varepsilon_g$ is helpful, we can set a hyperparameter $\\alpha$ to make the results more reliable. That is, if\n\\begin{equation}\n    \\frac{|\\left\\{ S_j|x\\in S_j \\right\\} |}{m_k} > \\alpha, 0 \\le \\alpha \\le 1,\n\\end{equation}\nan $\\alpha$ value of 0 means that whether $x$ is marked as belonging to category $c_k$ requires only one participant to agree, while an $\\alpha$ of 1 means that the consent of all participants is required. After that, we can put the index of $x$ into $P_k$. After all $P_k$ are generated, the central server sends the corresponding pseudolabeled dataset to each participant's task based on its label space $\\mathcal{Y}_i$. For the participants whose label space is $\\mathcal{Y}_i$, the corresponding pseudolabeled dataset received from the center is:\n\\begin{equation}\n    R_i = \\{(P_k, c_k)|c_k \\in \\mathcal{Y}_i\\}\n\\end{equation}\nAssuming that $C_i[index]$ stores the results of $f_i(D^{pub}[index])$, $\\mathcal{P}=\\{P_k|c_k \\in \\mathcal{Y}\\}$ and $M=|D^{pub}|$, Algorithm 1 describes the above process.\n\\begin{algorithm}[!t]\n\\DontPrintSemicolon\n \\caption{Pseudolabel aggregation}\n \\KwIn{$\\{C_i\\}$, $\\mathcal{Y}_i$, $\\mathcal{Y}$, $\\alpha$, $M$}\n \\KwOut{$\\{P_k\\}$}\n \\SetKwBlock{Begin}{function}{end function}\n \\Begin(\\text{Aggregation} {$(\\{C_i\\}$, $\\mathcal{Y}_i$, $\\mathcal{Y}$, $\\alpha$, $M)$})\n {\n  \\tcp*[l]{Initialization}\n  $TOTAL$ = empty dict\\;\n  $\\mathcal{P}$ = empty set\\;\n  \n  \\tcp*[l]{Counting $c_k$}\n  \\ForAll {$c_k \\in \\mathcal{Y}$}\n  {\n    $TOTAL[c_k] = |\\{i|c_k \\in \\mathcal{Y}_i\\}|$\\;\n    $\\mathcal{P}[c_k]$ = empty set\\;\n  }\n  \n  \\tcp*[l]{Label aggregating}\n  \\ForAll {$index = 1$ \\textbf{to} $M$}\n  {\n    $COUNT$ = empty dict with default value 0\\;\n    \\ForAll {$i = 1$ \\textbf{to} $N$}\n    {\n        $COUNT[C_i[index]] = COUNT[C_i[index]] + 1$\\;\n    }\n    \\ForAll {$c_k \\in keys(COUNT)$}\n    {\n        \\uIf {$\\frac{COUNT(c_k)}{TOTAL(c_k)}>\\alpha$}\n        {\n            add $index$ to $P_k$\\;\n        }\n        \\Else\n        {\n           continue\\;\n        }\n        \n    }\n  }\n\n \\Return{$\\{P_k\\}$}\n }\n \\end{algorithm}\n\nIt should be pointed out that some indices of $x \\in D^{pub}$ may exist in multiple $P_k$ because all $\\mathcal{Y}_i$ are different from each other. Therefore, the different $P_k$ in $R_i$ may overlap, resulting in contradictory labels. To build a compatible pseudolabeled dataset $R_i$, the indices contained by different $P_k$ should be removed from $R_i$.\n\n\\subsection{Unlabeled Dataset}\nTo perform the CoFED method, we need to build a public unlabeled dataset for all participants. Although an unlabeled dataset that is highly relevant to the classification tasks is preferred, we find that even less relevant datasets can yield sufficient results in our experiments. For the image classification tasks that we focus on, the almost unlimited image resources that are publicly accessible on the Internet can be built into an unlabeled dataset. Another benefit of utilizing resources that each participant can independently obtain is that this strategy can prevent the distribution of unlabeled datasets by a central server, thereby saving the limited communication resources in an FL scenario.\n\nThe reason why less relevant datasets work is that even though different objects may have commonality, we can use this commonality as the target of the tasks involving different objects. For example, if someone asks you what an apple looks like when you have nothing else but a pear, you might tell the person that it looks similar to a pear. This may give him or her some incorrect perceptions about apples, but it is better than nothing, and this person will at least be less likely to recognize a banana as an apple. That is, although you have not been able to tell him or her exactly what an apple looks like, the process still improves his or her ability to recognize apples. For a similar reason, even if a less relevant unlabeled dataset is used to transfer knowledge, it can also yield improved model performance in FL scenarios.\n\n\\subsection{Training Process}\nIn the CoFED method, each participant needs to train its model twice, i.e., local training and update training. Both training processes are performed locally, where no exchange of any data with other participants or the central server is necessary. The benefits of this approach are as follows.\n\\begin{enumerate}\n    \\item It prevents the leakage of participant data, including the local data and models.\n    \\item It avoids the loss of stability and performance through communication.\n    \\item It decouples the training processes of different participants so that they can independently choose the training algorithms and training configurations that are most suitable for their models.\n\\end{enumerate}\n\n\\subsection{Different Participant Credibility Levels}\nIn practical applications, the problem of different participant credibility levels may be encountered, resulting in uneven pseudolabel quality for different participants. Credibility weights can be assigned to the pseudolabels provided by different participants. Accordingly, Algorithm 1 can be modified to calculate $TOTAL(c_k)$ and $COUNT(c_k)$ by adding the weights of different participant, and the unmodified version of Algorithm 1 is equivalent to the case in which the weight of each participant is 1.\n\nDifferent bases can be used for setting the weights. For example, since the quality of the model trained on a larger training set is generally higher, weights based on the size of the utilized local dataset may be helpful for the unbalanced data problem. The test accuracy can also be used as a basis for the weights, but the participants may be reluctant to provide this information. Therefore, we can make decisions according to the actual scenario.\n\n\\subsection{Non-IID Data Settings for Heterogeneous Tasks}\nData can be non-IID in different ways. We have pointed out that the heterogeneous task setting itself is an extreme non-IID case of the personalized task setting. However, under the heterogeneous task setting, the instance distributions of a single category in the local datasets of different participants can still be IID or non-IID. The IID case means that the instances of this category owned by different participants have the same probability distribution, while in the extreme non-IID case, each participant may only have the instances of one subclass of this category, and different participants have different subclasses. A non-IID example is a case in which pets are contained in the label spaces of two participants, but all pet instances in the local training set of one participant are dogs, while the other participant only has cat images in its local dataset.\n\nThe existing FL methods based on parameter aggregation usually work well for IID data but suffer when encountering non-IID data. Zhao \\textit{et al.} \\cite{zhao2018federated} showed that the accuracy of convolutional neural networks (CNNs) trained with the FedAvg algorithm could be significantly reduced, by up to 55\\%, with highly skewed non-IID data. Since non-IID data cannot be prevented in practice, addressing them has always been regarded as an open challenge in FL \\cite{kairouz2021advances}, \\cite{li2020federated}. Fortunately, in the CoFED method, where model diversity is helpful for improving performance, a non-IID data setting is usually beneficial. This is because models trained on non-IID data generally have more divergences than models trained on IID data.\n\\section{Related Work}\nThe federated average (FedAvg) algorithm was originally proposed by McMahan \\textit{et al.} \\cite{mcmahan2017communication} to solve machine learning federated optimization problems on mobile devices. The core idea of FedAvg method is to pass model parameters instead of private data from different data sources, and use a weighted average of model parameters from different data sources as a global model. In each round of communication in the FedAvg method, the central server broadcasts the global model to participants, and then the participants who received the global model continue to train the global model using local private data. Following the local training phase, participants submit trained models to the center which utilizes the weighted average of different participant models as the global model for the next round of communication.\n\n\nInspired by the FedAvg algorithm, many FL methods \\cite{wang2019adaptive}, \\cite{li2020federatedhn} based on the aggregation of model parameters have been proposed. These methods are mainly suitable for federated training under the CD-FL setting where the tasks and model architectures are usually published by a central server, and all participants (i.e. devices) sharing the same model architecture makes aggregation of model parameters an effective knowledge sharing strategy. In addition, under the CD-FL setting, it is also practical for the central server to control the local training process and parameters (such as learning rate, epoch number, and mini-batch size, etc.). However, under the CS-FL setting where the model architectures and tasks of different participants may be different, and the training process and parameters are reluctant to be controlled by the center, these FL methods based on aggregation of model parameters are generally unable to cope with these heterogeneity challenges. FedProx proposed by Li \\textit{et al.} \\cite{li2020federatedhn} introduced a proximal term to overcome statistical heterogeneity and systematic heterogeneity (i.e., stragglers), but FedProx is unable to cope with heterogeneous model architectures due to model parameter aggregation.\n\nIn recent years, personalized federated learning has been proposed to address the personalized needs of participants. Smith \\textit{et al.} \\cite{smith2017federated} proposes a multitask FL framework MOCHA which clusters different tasks according to their relevance by an estimated matrix. Khodak \\textit{et al.} \\cite{khodak2019adaptive} proposed an adaptive meta-learning framework utilizing online learning ARUBA, which has the advantage of eliminating the need for hyperparameter optimization during personalized FL. Lin \\textit{et al.} \\cite{lin2020meta} and Fallah \\textit{et al.} \\cite{fallah2020personalized} proposed a model-agnostic meta-learning method and their variants to achieve personalized federated learning. Dinh \\textit{et al.} \\cite{dinh2020personalized} proposed a personalized FL method based on meta-learning by Moreau envelope, which leverages the $l2$-norm regularization loss to balance the personalization performance and generalization performance.\n\n\nIt should be pointed out that personalized tasks are different from heterogeneous tasks. Tasks with personalized settings always have the same label spaces, while tasks with heterogeneous settings have different label spaces. A typical example of personalized tasks that can be presented from \\cite{smith2017federated} is to classify users' activities by using their mobile phone accelerometer and gyroscope data. For each user, the model needs to provide outputs from the same range of activity categories, but the classification for each user is regarded as a personalized task due to the differences in their heights, weights, and personal habits. Despite these differences, from the data distribution perspective, the data of heterogeneous classification tasks can be regarded as non-IID sampling results from an input space $\\mathcal{X}$, which contains all instances of all labels in a label space $\\mathcal{Y}$, and $\\mathcal{Y}$ is the union of the label spaces of all participants. Each participant only has instances of some categories in $\\mathcal{Y}$ since its label space is a subset of $\\mathcal{Y}$. Therefore, an FL method that is compatible with personalized tasks can also be used for heterogeneous tasks.\n\nHowever, the core idea of all these methods is model parameter aggregation, which requires the model architectures of all participants to be consistent or only partially different. However, under CS-FL settings, participants as companies or organizations usually need to use their own independently designed model architectures. Li \\textit{et al.} \\cite{li2019fedmd} proposed a FL method FedMD leveraging model distillation to transfer the knowledge of different participant's model by aligning the output of neural networks models on a public dataset. Although FedMD participants are allowed to use neural network models of different architectures, they must be consistent in the logit output layer, and FedMD is not compatible with participants using non-neural network models. Also, FedMD needs a large number of labeled data for participant model alignment, which raises the bar for its application.\n\nModel parameter aggregation also leads to the leakage of participant models. Many studies try to overcome model leakage using various secure computing methods including differential privacy \\cite{wei2020federated}, \\cite{hu2020personalized}, \\cite{adnan2022federated}, secure multiparty computing \\cite{yin2020fdc}, \\cite{liu2020secure}, homomorphic encryption \\cite{jia2021blockchain}, \\cite{fang2021privacy}, \\cite{zhang2020batchcrypt}, blockchain \\cite{li2022blockchain}, \\cite{otoum2022federated}, and trusted execution environments \\cite{mo2021ppfl}, \\cite{chen2020training}, However, these technologies still have certain drawbacks, such as high computational costs or hardware specific.\n\\section{Preliminaries}\nWe first formulate the HFMTL problem and introduce the cotraining method that inspires us to propose the communication-efficient FL (CoFED) method.\n\\subsection{HFMTL}\nAn FL setting contains $N$ participants, and each of them has its own classification task $T_i$, input space $\\mathcal{X}_i$, output space $\\mathcal{Y}_i$, and $\\mathcal{D}_i$ consisting of all valid pairs $(\\boldsymbol{x}, y)$, where $\\boldsymbol{x} \\in \\mathcal{X}_i, y \\in \\mathcal{Y}_i$. Since we indicate that $T_i$ is a classification task, $\\mathcal{Y}_i$ is the label space of $T_i$. Each participant trains its machine learning model with supervised learning to perform a classification task, so each participant has its own local data $D_i \\subset \\mathcal{D}_i$.\n\nSince the HFMTL settings contain heterogeneous tasks and heterogeneous models $f$, we can assume that for the general $i \\ne j$:\n\\begin{equation}\n\\mathcal{Y}_i\\ne \\mathcal{Y}_j, f_i\\ne f_j\n\\end{equation}\nOn the other hand, the classification tasks of each participant should have commonality with the tasks of other participants. In our setting, this commonality manifests as the overlap between the label spaces. This means that:\n\\begin{equation}\n    \\forall \\mathcal{Y}_i\\rightarrow \\left\\{ j|\\mathcal{Y}_i\\cap \\mathcal{Y}_j\\ne \\oslash , i\\ne j \\right\\} \\ne \\oslash \n\\end{equation}\n\nAssuming that a model $f_i$ has been trained for $T_i$, its generalization accuracy can be defined as:\n\\begin{equation}\n    GA(f_i) = \\mathbf{E}_{(\\boldsymbol{x}, y) \\sim \\mathcal{D}_i}[\\mathbf{I}\\left( f_i\\left( \\boldsymbol{x} \\right) \\equiv y \\right)]\n\\end{equation}\nwhere $\\mathbf{E}[\\cdot]$ is the expectation and $\\mathbf{I}(\\cdot)$ is an indicator function: $\\mathbf{I}(\\text{TRUE}) \\equiv 1 $ and $\\mathbf{I}(\\text{FALSE}) \\equiv 0 $.\n\nThe goal of this paper is to propose an HFMTL method that is compatible with heterogeneous tasks, models, and training processes. This method should help to improve the performance of each participant model without sharing the local private datasets $D_i$ and private models $f_i$ of these participants:\n\\begin{equation}\n    f_i^{fed} = \\underset{f_i}{arg\\max}\\,\\,GA\\left( f_i \\right) \n\\end{equation}\nIn addition, assuming that the model locally trained by the participant is $f_{i}^{loc}$, we expect that the model $f_{i}^{fed}$ trained with our FL method yields better performance for each participant:\n\\begin{equation}\n    GA(f_{i}^{fed}) > GA(f_{i}^{loc}), i=1,2,\\cdots,N\n\\end{equation}\n\n\\subsection{Cotraining}\nIn HFMTL settings, many tricky problems come from the differences between participant models and tasks. However, cotraining is an effective method for training different models. Therefore, ideas that are similar to cotraining can be used to solve problems under HFMTL settings.\n\nCotraining is a semi-supervised learning technique proposed by Blum \\textit{et al.} \\cite{blum1998combining} that utilizes unlabeled data from different views. The \\textit{view} means a subset of instance attributes, and two classifiers that can be obtained by training on different view data. Cotraining lets the two classifiers jointly give some pseudo-labels for unlabeled instances, and then retraining the two classifiers on the original training set and the pseudo-labeled dataset can improve the classifier performance. Wang \\textit{et al.} \\cite{wang2007analyzing} pointed out that cotraining is still effective for a single view, and applying cotraining can bring greater classifier performance improvement when  the divergence of different classifiers is larger. In FL, local models of different participants are trained from their local data. Under the HFMTL setting, due to differences in model architecture and data distribution, models of different parties are likely to have large divergences, which is beneficial for applying Cotraining.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\n\n\tWe studied the properties of the MSSM Higgs bosons, h and H through the decay into b-quarks in associated production with a top-quark pair. There was used the tree-level Higgs sector described by two parameters M$_A$ and tan$\\beta$  and found their optimal values according to experimental data of ATLAS detector. Using the restricted parameter space we calculated cross sections of associated $t\\bar{t}h(H)$ production at 13 and 14 TeV, the corresponding kinematical cuts, mass distributions and Branching Ratios of h and H decays into $b\\bar{b}$ quark pair. \n\n\\section{Introduction}\n\n    Supersymmetry searches are the most attractive in the aspect of searching for new physics beyond the standard model. The search for an extended sector of Higgs bosons is especially urgent, since they are the lightest candidates for supersymmetric particles and information on their production cross sections and decay widths provides additional knowledge about the Yukawa coupling constants. The dependences on these couplings of the cross section and Higgs branching ratios have been studied intensively and it was found, that there are indirect constraints from experimental data on the scalar and \npseudoscalar H-top couplings k$_t$ and $\\tilde{k}$, and these constraints are relatively weak, \\cite {1.}  \n\t\n\tOne of the important channels of such searches is the $t\\bar{t}H$ Higgs boson production channel. The search strategy for the $t\\bar{t}H$ process has been studied in various Higgs decay modes: $b\\bar{b}$, \\cite{2.}, $\\tau\\bar{\\tau}$, \\cite{3.} and WW$^*$, \\cite{4.}.  Furthermore, from experimental point of view the $H\\rightarrow b\\bar{b}$ decay mode is more prefferable due to the possibility of the reconstruction of the Higgs boson kinematics, which allows to extract the information about the top\u2013Higgs interaction. \n    As the decay of Higgs boson into two b-quarks ($H\\rightarrow b\\bar{b}$) is the most probable, \\cite{5.} our paper is devoted to the consideration of this decay channel. Higgs boson decay into b-quarks in associated production with a top-quark pair is connected with\ntesting the predictions of the Standard Model (SM) and\nvery sensitive to effects of physics beyond the SM (BSM). So,\nwe used Minimal Supersymmetric Standard Model (MSSM) as base theory for futher calculations. Our purpose was to calculate production cross sections $\\sigma(t\\bar{t}h),\\ \\sigma(t\\bar{t}H)$ at the centre-of-mass energy of $\\sqrt{s}$ = 13 TeV and to compare obtained data with experimental data. We also found p$_T$ and rapidity distributions, parameter space and mass distributions, which corresponds to the best fit with experimental data.\n\n\\section{Search channels and parameter cuts of Higgs boson production}\n\tATLAS \\cite{6.} and CMS \\cite{7.} have searched for the process of  Higgs boson production ($t\\bar{t}H(b\\bar{b})$)  intensively using the 8 TeV data set. Later new data collected in proton\u2013proton collisions at the LHC between 2015 and 2018 at a centre-of-mass energy of $\\sqrt{s}$ = 13 TeV were analysed, corresponding to an integrated luminosity of 139 fb$^{-1}$, \\cite{8.}. The measured signal strength, defined as the ratio of the measured signal yield to that predicted by the SM, \n\\[ \\mu = 0.35\\pm0.20 (stat.)^{+0.30}_{-0.28}(syst.)=0.35^{+0.36}_{-0.34} \\ ,\\]\ncorresponds to an observed (expected) significance of 1.0 (2.7) standard deviations. The measured 95$\\%$ confidence level (CL) cross-section upper limits in each bin for simplified template cross-sections (STXS) formalism are shown in Fig.1\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{1.eps}\\\\\n\\emph{\\textbf{Fig.1}} {\\emph{The measured 95$\\%$ CL cross-section upper limits with the theoretical uncertainty \nconnected with signal scale and PDF uncertainties.}}\n\\end{center}\n\n        \n    The Higgs sector of the Minimal Supersymmetric extension of the SM \\cite{9.} consists of five physical Higgs bosons, two neutral CP-even bosons, h, H, one neutral CP-odd boson, A, and a charged Higgs pair, H$^{\\pm}$. \nAs there is the experimental deviation from the SM, we used BSM model - MSSM and studied the properties of two Higgs bosons: the lightest Higgs boson, h and CP-even Higgs boson, H. Our analysis of  Higgs boson production in association with a pair of top quarks and decaying into a pair of b-quarks ($t\\bar{t}H(b\\bar{b})$) is presented by Feynman diagrams in Fig.2.\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{2.eps}\\\\\n\\emph{\\textbf{Fig.2}} {\\emph{Representative tree-level Feynman diagrams for the production of a Higgs boson in association with a\ntop-quark pair ($t\\bar{t}H$) in (a) the t-channel and (b) the s-channel and the subsequent decay of the Higgs boson into $b\\bar{b}$, from \\cite{8.}.\n}}\n\\end{center}\nThe tree-level Higgs sector, can be described by two parameters, the mass of the CP-odd Higgs boson, M$_A$, and the ratio of the two vacuum expectation values of the two Higgs doublets, tan$\\beta = v_2\/v_1$. So, our purpose was to chooose the optimal value of the correspomding parameters M$_A$ and tan$\\beta$. We calculated Branching ratio (BR) of h and H using FeynHiggs program \\cite{10.} as the function of M$_A$ at 13 TeV (Fig. 3) as well as production cross section as the function of tan$\\beta$ at M$_A$=200 GeV (Fig.4).\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{3.eps}\\\\\n\\emph{\\textbf{Fig.3}} {\\emph{Branching ratio (BR) of h and H decays using FeynHiggs program as the function of M$_A$ at 13 TeV.}}\n\\end{center}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{4.eps}\\\\\n\\emph{\\textbf{Fig.4}} {\\emph{Production cross section of Higgs bosons h and H for two search channels as the function of tan$\\beta$ at M$_A$=200 GeV at 13 TeV.}}\n\\end{center}\n    \nFrom the obtained data we came to the conclusion about the optimum parameters of M$_A$=200 GeV and tan$\\beta$ =2.\n\n\\section{Results of calculations}\n    As gluon-gluon and quark-antiquark processes are most preferable for the Higgs boson production we have considered the following search channels using Pythia program \\cite{11.}, presented in Table 1 and Table 2\n\n\\begin{center}\n{\\it\\normalsize Table 1. Search channels and production cross sections of h and H bosons at the centre-of-mass energy of $\\sqrt{s}$ = 13 TeV at M$_A$ = 200 GeV and tan$\\beta$ =2}\\\\\n\\vspace*{3mm}\n\\begin{tabular}{|c|c|} \\hline \nSearch channels& Production cross sections (pb) \\\\ \n& (with stat. err.) \\\\ \\hline \\hline\ngg $\\rightarrow$  $ht\\bar{t}$ &   2.609e-01 +\/- 5.084e-03   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $ht\\bar{t}$   &   1.034e-01 +\/- 1.287e-03\n   \\\\ \\hline\ngg $\\rightarrow$  $Ht\\bar{t}$\u00a0\u00a0\u00a0  &   2.017e-02 +\/- 5.618e-04   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$ $Ht\\bar{t}$   &    7.524e-03 +\/- 3.862e-04\n   \\\\ \\hline\n     gg $\\rightarrow$  $Ht\\bar{t}$ (SM)\u00a0\u00a0  &    2.530e-1\u00a0+\/- 2.704e-3  \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $Ht\\bar{t}$ (SM)\u00a0\u00a0\u00a0   &    1.041e-1 +\/-\u00a0 2.572e-3\n   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n{\\it\\normalsize Table 2. Search channels and production cross sections of h and H bosons at the centre-of-mass energy of $\\sqrt{s}$ = 14 TeV at M$_A$ = 200 GeV and tan$\\beta$ =2}\\\\\n\\vspace*{3mm}\n\\begin{tabular}{|c|c|} \\hline \nSearch channels& Production cross sections (pb) \\\\ \n& (with stat. err.)\\\\ \\hline \\hline\ngg $\\rightarrow$ $ht\\bar{t}$\u00a0     &  3.134e-01 +\/- 4.095e-04   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $ht\\bar{t}$   &  1.171e-01 +\/- 1.660e-04  \\\\ \\hline\ngg $\\rightarrow$ $Ht\\bar{t}$\u00a0\u00a0\u00a0   &  2.514e-02 +\/- 6.099e-05   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $Ht\\bar{t}$ &  9.221e-03 +\/- 5.913e-05 \\\\ \\hline\n     gg $\\rightarrow$ $Ht\\bar{t}$ (SM)\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0   &  2.530e-1\u00a0+\/- 2.704e-3   \\\\ \\hline\n$q\\bar{q}$ $\\rightarrow$  $Ht\\bar{t}$ (SM)\u00a0\u00a0\u00a0   &  1.041e-1\u00a0+\/- 2.572e-3  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\nThe  kinematical cuts on h boson corresponding to the production cross section are presented in Fig. 5.\n\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{5.eps}\\\\\n\\emph{\\textbf{Fig.5}} {\\emph{Transverse momentum (up) and rapidity distributions (down) of h boson at the energy of 13 TeV.}}\n\\end{center}\nThe  kinematical cuts on H boson corresponding to the production cross section are presented in Fig. 6.\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{6.eps}\\\\\n\\emph{\\textbf{Fig.6}} {\\emph{Transverse momentum (up) and rapidity distributions (down) of H boson at the energy of 13 TeV.}}\n\\end{center}\nFrom the obtained data we came to the conclusion about the most suitable range of transverse momentum variation of h boson (50; 150) GeV and (100; 300) GeV of H boson. As for rapidity distributions of both Higgs bosons, their maximum region is  (-1,1), which signals about the angle range along the longitudinal (beam) direction ~ 45-90$^{\\circ}$ of the Higgs bosons. \n\t\n    As for the mass detemination of both Higgs bosons, we received the mass distributions, presented in Fig. 7\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{7.eps}\\\\\n\\emph{\\textbf{Fig.7}} {\\emph{Mass distributions of h (up) and H (down) Higgs bosons obtained at 13 TeV.}}\n\\end{center}\nFrom Fig.7 we see, that the mass of h boson is about 126 GeV, which coincides with the mass of the SM Higgs boson. As for the H boson, it's mass is approximately 330 GeV. We also calculated kinematical and mass distributions of both Higgs bosons at 14 TeV and didn't find a significant difference with previous calculations at 13 TeV.\n \\section{Conclusions}\n\nThe study of the properties of the Higgs boson is an urgent task, as evidenced by the experimental data of the ATLAS and CMS collaborations. We have presented the actual experimental data of the cross sections of Higgs boson decay into b-quarks  in associated production with a top-quark pair in  pp collisions at $\\sqrt{s}$ = 13 TeV and an integrated luminosity of 139 fb$^{-1}$ with the ATLAS detector. This result corresponds to an observed (expected) significance of 1.0 (2.7) standard deviations. To clarify the ambiguities, we decided to consider the minimal extension of SM \u2013 MSSM model. We used the tree-level Higgs sector which can be described by two parameters M$_A$ and tan$\\beta$. The calculation of BR($h\\rightarrow b\\bar{b}$) as the function of M$_A$ and production cross section of both Higgs bosons as the function of tan$\\beta$ gives us the possibility to choose the optimal parameter space corresponding to the maximum values of BR and the production cross section. Using received parameters (M$_A$ = 200, tan$\\beta$=2) we calculated cross sections of associated $t\\bar{t}h(H)$ production at 13 and 14 TeV and the corresponding kinematical cuts on transverse momentum and rapidity. We found out the value of the mass of h (126 GeV) and H (330 GeV) at the chosen parameters from the constructed mass distribution. The values of BR($h\\rightarrow b\\bar{b}$) and BR($H\\rightarrow b\\bar{b}$)  are equal to 0.85 and 0.05 correspondingly.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nIt is well known that pulsars have considerably steeper spectral indices than the background population of radio sources. Their flux density (S$_\\nu$) can be described by a single power-law with slope $\\alpha$ (i.e. S$_\\nu\\propto{\\nu}^\\alpha$).  Observationally-derived spectral indices have been determined variously to be in the range of $-1.6\\pm{0.3}$ \\citep{lylg95} to  $-1.8\\pm{0.2}$ \\citep{mkkw00}. \\citet{blv13} attempted to remove pulsar survey biases to derive an intrinsic spectral index of $-1.4\\pm{1.0}$. There have been claims that millisecond pulsars have more shallow spectral indices on average than ``normal\" (i.e. non recycled)  pulsars \\citep{kll+99}, but this may be due to an observational bias \\citep{blv13}. \n\nDeviations from this pure power-law behavior have been seen at both high and low frequencies, with some fraction (10\\%) having evidence for flat spectra and spectral steepening above several GHz \\citep{mkkw00}. Low frequency turnovers first seen by \\cite{sie73}, have now been measured for both normal and millisecond (MSP) pulsars, typically below 100 MHz \\citep[e.g.][]{drt+13,kvl+15}. External free-free absorption, either in the immediate environment of the pulsar or along the line of sight, gives a good explanation for the origin of the high frequency turnover pulsars \\citep{lrkm15,rla16}.  The origin of the low frequency turnovers, is not so clear. They could be telling us something fundamental about the energy distribution of the coherent emitting electrons, or the turnover could be due to absorption, occurring either within the pulsar magnetosphere or along the line of sight.\n\nProgress in understanding these low frequency behaviors and their dependence (if any) on known pulsar parameters has been slow, owing to a shortage of flux density measurements below 1 GHz \\citep{mgj+94}. The best efforts to date are those of \\citet{mms00} who made measurements of 235 pulsars at 102.5 MHz, while 30 MSPs were observed at 102 and  110 MHz  \\citep{kl01}.  Fortunately, the situation is changing with new instruments such as the LOw-Frequency ARray (LOFAR)  and the Long Wavelength Array (LWA). There are more recent LWA measurements of 44 pulsars from 10-88 MHz \\citep{srb+15} and LOFAR has now observed large samples of normal and MSPs at 110-188 MHz and \\citep{bkk+15, kvh+16}. \n \nIn this paper we use the recently completed GMRT Sky Survey (TGSS ADR) to study known radio and gamma-ray pulsar populations at 150 MHz. This paper is arranged as follows. In \\S\\ref{survey} we briefly describe the TGSS ADR survey while in \\S\\ref{method} we outline our search methods for both the radio-loud and radio-quiet samples.  The results are discussed in \\S\\ref{results} where we derive estimates of the spectral index distribution of the TGSS ADR pulsars, and we compare the derived flux densities and the detection statistics with previously published samples. Our conclusions and suggestions for future work are given in \\S\\ref{sec:conclude}.\n\n\\section{The TGSS ADR Survey}\\label{survey}\n\nThe Giant Metrewave Radio Telescope (GMRT) was used to carry out a radio continuum survey at a frequency of 150 MHz using a total of 2000 hrs of observations. The entire sky was surveyed in over 5000 partially overlapping pointings from $-55$\\degr\\, declination to the northern polar cap covering 37,000 deg${^2}$.\n\nThe entirety of these data have recently been re-processed \\citep[TGSS ADR;][]{int16} creating high-quality images of approximately 90\\% of the entire sky. The TGSS ADR achieves a median rms noise level of \\mjybeam{3.5} and an angular resolution of \\asec{25} for Dec.$>19^{\\circ}$, and \n\\asec{25}$\\times$\\asec{25}\/cos(DEC-19$^\\circ$) for more southern declinations. In the final catalog there are some 0.62 million radio sources down to the 7$\\sigma$ level. Compared to existing meter-wavelength surveys \\citep{lcv+14,hpo+15,wlb+15},  the TGSS ADR represents a significant improvement in terms of number of radio sources, sensitivity and angular resolution. The improved angular resolution in particular allows accurate matching of radio sources with counterparts at other wavelengths. The capabilities of the TGSS ADR are well-matched to existing surveys \\citep{bwh95,ccg+98,bls+99} and provide a large frequency leverage arm for spectral index measurements. For more details on this survey and how to obtain the publicly available mosaic images and source catalog, see \\citet{int16}.\n\nThe observing bandwidth and integration time of the survey are especially relevant to the detection of the phase-averaged emission from pulsars. The original data were recorded with 256 frequency channels across 16.7 MHz of total bandwidth centered on 147.5 MHz. \nThe GMRT visibility data from the archive were saved as 16.1 s averages. Typically each (pointing) direction on the sky was observed as a series of short snapshots 3 to 5 times over the course of a single night's observing. The total integration time was 15 minutes per pointing on average.\nDuring imaging of the pointings, these data were combined in time and frequency to create a single Stokes-I image.\nThe full duration spent on any given point on the sky is more difficult to quantify. The final data products used for the analysis are 25 deg$^2$ image mosaics, formed by combining overlapping 7.6 deg$^2$ pointing images.\nNote that some pointings were observed repeatedly during multiple observing sessions, imaged separately, and combined in creating the mosaics.\nAs a result, many sources have been observed more than once, sometimes separated by days or even months.\nAs the survey's pointing centers followed the FIRST survey hexagonal grid strategy \\citep{bwh95}, the TGSS ADR will have similar duration statistics \\citep[see][]{thw+11}.\n\n\\section{Methods\\label{method}}\n\n\\subsection{Radio Loud Pulsars}\\label{sec:loud}\n\nWhile pulsars are typically detected by their pulsed, periodic emissions, they can also be identified in interferometric images as phase-averaged continuum point sources. \\citet{kcac98} were the first to employ a wide-field radio survey for this purpose. They used the 1.4 GHz NRAO VLA Sky Survey \\citep[NVSS;][]{ccg+98} to identify 79 known pulsars from the total intensity alone, while \\citet{ht99} added in the polarized intensity to identify 97 pulsars from the same survey. The 325 MHz Westerbork Northern Sky Survey  \\citep[WENSS;][]{rtb+97} was used by \\citet{kou00} to find radio emission toward 25 known pulsars. For this project we employ the TGSS ADR at 150 MHz, using a version of the source catalog that was formed by running the source extraction algorithm PyBDSM \\citep{mr15}\\footnote{http:\/\/www.astron.nl\/citt\/pybdsm\/} with its default parameters searching the mosaicked images down to a 5$\\sigma$ detection threshold. For our list of known pulsars we used the HEASARC 27 December 2015 version of the ATNF Pulsar Catalog \\citep{mhth05}. A total of 1238 pulsars were selected with dec.$\\geq-52^\\circ$ and having known positions with $\\Delta$dec.$<\\pm{3^{\\prime\\prime}}$ or $\\Delta$R.A.$<0.35^s$. {A history file listing the contents and any changes to the ATNF database as of December 2015 is at their web site\\footnote{http:\/\/www.atnf.csiro.au\/research\/pulsar\/psrcat\/catalogueHistory.html}. Up to approximately November 2015, our sample included all well-localized normal pulsars from the \\emph{Fermi}  sample at Stanford University as well as the MSP sample at the University of West Virginia.} The pulsar positions were corrected for proper motion using the mean epoch of the TGSS ADR of 11 January 2011 (MJD\\,55579.0). \n\nFollowing \\citet{hwb15}, we searched for matches between the PSR and TGSS ADR catalogs out to a radius of 30\\% of the FWHM of the 25$^{\\prime\\prime}$ beam, or 7.5$^{\\prime\\prime}$. We find 200 known pulsars, or 16\\% of the sample, are associated with TGSS ADR sources.  We list these detections in Table \\ref{tab:bright} along with some basic pulsar parameters (period and dispersion measure) along with the total flux density and peak flux from the TGSS ADR.  {The ratio of the total flux density and the peak flux can be a useful proxy in helping decide whether the radio emission is extended or unresolved, and therefore likely phase-averaged pulsar emission. We return to this point, as well as discussing the rates of false positives in \\S\\ref{ids}.} For all matches, we derived two-point spectral indices between the TGSS ADR total flux densities at 150 MHz and the 1400 MHz values from the pulsar catalog. The 400 MHz flux density was used in those few cases where the 1400 MHz values were missing. When no values were provided in the pulsar catalog database, we obtained flux densities from the original literature.\n\nThe search method above was supplemented with an image-based approach for pulsars below the 5$\\sigma$ limit of the catalog. For all  of the original 1238 well-localized pulsars, we measured the peak flux in the TGSS ADR images at the pixel corresponding to the pulsar position, along with an estimate of the rms noise in immediate vicinity. We did not attempt to search over some radius and fit a Gaussian since below 5$\\sigma$ such a approach would likely lead to many more false positives. The positive identifications are defined as having S$_p$\/$\\sigma_{rms}\\geq$2.5. Our justification for this choice of threshold is shown in  Fig. \\ref{fig:noise}. We make an estimate of the shape of the signal-to-noise distribution as an estimate of the blank sky in the vicinity of these pulsars (solid line). The positive S\/N peaks are strongly skewed above that expected from Gaussian noise. From the ratio of the levels of the positive and negative 2.5$\\sigma$ bins we estimate that 4\\% of the detections at this level will be false positives, or about 2 sources.\n\nWith this image-based approach we find significant emission towards another 88 pulsars. For each of these we inspected the mosaic images to verify that the emission was coming from a point centered on the pulsar position and was not due to a nearby extended source or an image artifact. Table \\ref{tab:faint} lists the peak flux and rms toward all 88 pulsars, along with some basic pulsar parameters identical to those in Table \\ref{tab:bright}.  Given the lower significance, we are not as confident in these identifications as we are with those in Table \\ref{tab:bright}.\n\n\n\n\\subsection{Radio Quiet Pulsars}\\label{quiet}\n\nMotivated by recent claims of the detection of pulsed radio emission from the ``radio-quiet'' PSR\\,J1732$-$3131 \\citep{mad12}, we carried out a search for emission at 150 MHz. In the Second \\emph{Fermi}  Large Area Telescope (LAT) Pulsar Catalog \\citep[2PC;][]{aaa+13} there are 35 PSRs that are radio quiet, defined as having a phase-averaged flux density S(1.4 GHz)$\\leq 30$ $\\mu$Jy. In the meantime the sample has grown; an updated list of radio quiet pulsars is available on the LAT team web site\\footnote{\\url{https:\/\/confluence.slac.stanford.edu\/display\/GLAMCOG\/Public+List+of+LAT-Detected+Gamma-Ray+Pulsars}}. For accurate pulsar positions we began with the recent compilation of \\citet{krj+15}, which has a table of positions for normal pulsars and MSPs obtained from both timing and multi-wavelength observations. All positions from \\citet{krj+15} are computed at the epoch MJD 55555 (25 Dec 2010). This is useful when comparing to the TGSS ADR which was observed around the same epoch. Additional X-ray positions are taken from \\citet{mmd+15}. \n\nThe final list consisted of 30 radio quiet pulsars in the declination range of TGSS ADR with localizations of an arcsecond or better (Table \\ref{tab:unknown}). For all pulsars we extracted image cutouts and looked for faint point sources at the pulsar position. We then made a final stacked image of all 30 pointings weighted by the inverse square of the local rms noise for each image. As the image pixel size is the same \\asec{6.2} in all images, this ensures accurate image stacking.  No source is detected. The rms noise is \\mjybeam{0.7} and the max\/min on the image is  approximately $\\pm$\\mjybeam{2.3}. \n\n\\section{Results and Discussion\\label{results}}\n\n\\subsection{Radio Loud Pulsars}\n\n\\subsubsection{Identifications}\\label{ids}\n\nThe majority of the emission that was detected at 150 MHz is likely due to phase-averaged pulsar emission. In support of this we note that the distribution of PSR-TGSS ADR offsets follows the expected Rayleigh distribution, with 95\\% of the identifications matched within a \\asec{4.5} radius. This is consistent with the astrometric accuracy (68\\% confidence) derived for the TGSS ADR of \\asec{1.55} \\citep{int16}. Given the source density of the TGSS ADR at the completeness limit of 17.6 source\/deg$^2$, and a search of 1238 positions each of radius \\asec{7.5} \n(see \\S\\ref{sec:loud}), we expect less than one false positive (i.e. a background radio source not associated with a pulsar).\n\nFurther support for pulsar identifications comes from Fig.\\ref{fig:ff}, in which we the show all pulsars (crosses) with published 400 MHz and 1400 MHz flux densities in the ATNF catalog. Those pulsars with TGSS ADR detections are indicated by circles. This figure shows that the TGSS ADR associations are well-correlated with the brightest pulsars, and thus the number of false positives are likely to be low. Furthermore, the number of associations drop off sharply for S$_{1400}<0.6$ mJy and S$_{400}<5$ mJy, as would be expected for steep spectrum pulsars given that the median noise of the TGSS ADR is \\mjybeam{3.5}. The two outliers in the bottom left corner of Fig.\\ref{fig:ff} are PSR\\, J2229+6114 and the LOFAR-identified PSR\\, J0613+3731. Their flux densities at 150 MHz appear to be dominated a pulsar wind nebula called `The Boomerang\" \\citep{kru06} in the first case and some unidentified extended emission in the second case. \n\nAs the above example illustrates, not all the matches in the TGSS ADR are from phase-averaged pulsar emission. Some radio emission is due an associated nebula (e.g. Crab), or is an from an ensemble of pulsars in a globular cluster.  {In \\citet{int16} we derive an empirical formula to help decide when a radio source is unresolved. In the high signal--to-noise case this reduces to S$_t\/$S$_p\\leq 1.13$. However, some caution is needed in applying this criteria to pulsars since they can show strong time-variability during an integration time, violating one of the central assumptions of the van Cittert-Zernike theorem upon which radio interferometric imaging is based. This can lead to deviations in the Gaussian fitted beam, or in especially strong cases, diffraction spikes around the pulsar. A visual inspection of the images is required to be sure since this same condition is likely met by strongly scintillating pulsars like PSR\\,B1937+21. We examined the images of {\\it all} of the detections in Table \\ref{tab:bright} and find that likely non-pulsar candidates are those entries for which the total flux density exceeds the peak flux (i.e. S$_t>$S$_p$) by more than 50\\%.}  For those small number of TGSS ADR detections (11) that we suspect are contaminated in this way, we add a comment in Table \\ref{tab:bright} and we do not derive a spectral index.\n\n\\subsubsection{Spectral Index Distribution}\n\nThe distribution of the two-point spectral indices of the TGSS ADR sample from Table \\ref{tab:bright}  is shown in Fig. \\ref{fig:spec_histo}. For comparison we have plotted the more comprehensive sample of 329 pulsars from the ATNF Pulsar catalog with non-zero spectral indices. As expected, the two histograms are in reasonable agreement with each other, both in terms of the width and the median of the two distributions. \n\nIf we order the spectral index values by pulsar period (Fig. \\ref{fig:alpha}) an unusual feature of our 150 MHz sample appears. The steep spectrum tail of the $\\alpha$ distribution measured at low frequencies is dominated by short period pulsars. This effect is not seen in the ATNF pulsar catalog.  We have detected many of the fastest rotating MSPs at 150 MHz, and these pulsars show a marked preference for steeper spectral index values. Of the 16 pulsars with $\\alpha<-2.5$, all but four are MSPs. Of these MSPs, all except one has been detected by the \\emph{Fermi}  gamma-rays mission including several eclipsing MSPs such as PSR\\,J1816+4510, with the steepest spectral index in Fig. \\ref{fig:alpha}. The 18 MSPs in Table \\ref{tab:faint} do not have ultra-steep spectra. \\citet{kvl+15} were the first to note a tendency for the gamma-ray MSPs to be steep-spectrum outliers based on a smaller sample. \n\nSince the values in Table \\ref{tab:bright} and Fig. \\ref{fig:alpha} are two-point values, we suspected measurement error as the source of these large values. As a first step we re-calculated the spectral index of all pulsars with $\\alpha<-2.5$ using the flux density and observing frequency taken from the original references. If no rms noise was given we assumed a fractional error of 50\\% for the flux density when estimating the uncertainty on $\\alpha$.\n\nThere are several useful compilations of flux density measurements and spectral indices we can use to cross check our measurements \\citep{tbm+98,kxl+98,kvl+15}. We find reasonable agreement in the $\\alpha$ values for all of the MSPs within the errors. For PSR\\,J1816+4510, the pulsar with the steepest 2-point spectral index in our sample, we re-fit our 150 MHz measurement along with a value at 74 MHz \\citep{kvl+15} and flux densities at 350 and 820 MHz \\citep{slr+14}. The latter two measurements were estimated from the radiometer equation so we have taken typical errors of $\\pm$50\\% on these two values. The mean spectral index is $-3.46\\pm0.10$ in agreement with a preliminary value from \\citet{kvl+15}.\n\nSince there is no evidence that the distribution of MSP spectral indices is steeper than the general population \\citep{tbm+98,kll+99}, we suspect this trend is the result of some low frequency bias.  Most of the steep spectrum MSPs in Table \\ref{tab:bright}, were discovered in low frequency searches \\citep[e.g.][]{fst+88,bhl+94,hrm+11,slr+14}. Two pulsars (B1937+21 and J0218+4232) had such steep spectral indices that they were initially identified in imaging data \\cite[e.g.][]{nbf+95}. Thus it is reasonable to expect that the TGSS ADR survey at 150 MHz would be sensitive to steep-spectrum radio sources, with a similar bias as low-frequency searches for pulsations \\citep{blv13}. This explanation, however, does not account for the preponderance of gamma-ray pulsars among our sample, nor for the unusually large fraction of (eclipsing) binaries. We know of no intrinsic property of the MSP population that would produce such an effect. {\\citet{ckr+15} noted that the nearby MSPs were susceptible to deep flux density variations at decimeter wavelengths, with strong expoential statistics such that the measured median flux density is less than the mean, skewing the spectral index to steeper values.} Since many of these systems have been found within the error ellipses of \\emph{Fermi}  unassociated sources \\citep[e.g.][]{krj+15}, a more prosaic explanation may be that the \\emph{Fermi}  mission has been such a prolific source of MSPs that they are over-represented in any sample. \n\n\\subsubsection{Comparison with the LOFAR Sample}\n\nIt is illustrative to compare the TGSS ADR and LOFAR samples. While both surveys were undertaken at the same frequency, they were observed in very different ways.  Thus a comparison could give us some insight into the different biases of each survey. LOFAR has carried out a search for pulsed emission from all northern radio pulsars \\citep{bkk+15,kvh+16}. This census was primarily  conducted with the LOFAR high-band antennas (HBA) between 110 and 188 MHz, with 400 channels each of 0.195 MHz in width, or a bandwidth of 78 MHz. Each pulsar was observed once for at least 20 minutes, although long period (P$>3$ s) normal pulsars and faint MSPs were observed up to 60 minutes in duration. Pulsed emission from a  total of 158 normal pulsars and 48 MSPs were detected. The GMRT observing method is summarized in \\S\\ref{survey} and the pulsar yield is given in \\S\\ref{ids}. We find 92 pulsars commonly detected in both the  LOFAR and TGSS ADR surveys (Tables \\ref{tab:bright} and \\ref{tab:faint}).\n\nFigure \\ref{fig:compare} (left) is a flux-flux plot of LOFAR and TGSS measured flux densities, while the same figure (right) shows a flux ratio plot of the same sample. The flux densities of the LOFAR and TGSS ADR pulsars do not agree. On average, the LOFAR pulsars are about two times brighter than the the TGSS ADR. The result persists even if we use only the bright pulsars in common (i.e. Table \\ref{tab:bright}). There are some significant outliers, dominated by bright, scintillating MSPs such as PSR\\,B1937+21 and PSR\\,J0218+4232, but the overall trend is clear. \n\nWe can immediately rule out frequency-dependent effects for this difference in the flux density scales since the surveys were performed at similar frequencies. Spectral curvature was the most likely explanation offered by \\citet{kvh+16} for why the LOFAR flux densities for one third of their MSPs are {\\it lower} than the predicted values based on an extrapolation from higher frequencies. Diffractive scintillation, while clearly important for the outliers, is not the likely origin for the systematic difference. The large observing bandwidths and the long integration times relative to the scintillation values for both the LOFAR and GMRT observations (\\S\\ref{survey}) suggest modulation of the flux density is not widespread; see \\S\\ref{sec:missing} and Appendix A of \\citet{bkk+15}. There is one important difference: the typical 20-min LOFAR integration time is a single integration, while the 15-min GMRT observations are typically subdivided into 3--5 short observations taken over a night of observing. The later is a more optimal detection strategy when there are intensity variations caused by the phase fluctuations in the interstellar medium \\citep{cl91}. If this effect is important, however, it would result in the LOFAR flux densities being {\\it lower} on average that the GMRT values, the opposite of what is seen. Temporal scattering can also reduce the measured flux density for pulsed surveys but as an imaging survey, the TGSS ADR is not sensitive to pulse smearing caused by interstellar scattering. While the LOFAR surveys are sensitive to such effects, they would also act to {\\it lower} the measured flux density. \n\nWe are left with instrumental effects associated with gain calibration. The TGSS ADR flux density scale is good to about 10\\% over the full sky.\nTaken in interferometric imaging mode, the data each day were calibrated back to several low frequency primary flux density calibrators (3C\\,48, 3C\\,147, 3C\\,286 and 3C\\,468.1). After calibration of the full survey, the accuracy of the flux density scale was cross-checked against other sky surveys such as 7C \\citep{hrw+07} and the LOFAR Multi-frequency Snapshot Survey \\citep[MSSS;][]{hpo+15} and they were found to agree at the $\\sim$5\\% level. On the other hand, the flux density calibration for the LOFAR pulsar survey was done directly using the radiometer equation for direction-dependent estimates of the antenna gain and the sky system temperature. The calibration was cross-checked with regular observations of a sample of normal pulsars and MSPs with well-determined spectra. Variations at a level 2--4 times larger than expected from scintillation alone were seen to occur and thus the resulting flux density scale was quoted with errors of $\\pm$50\\%. \n\nWe tentatively suggest that our TGSS ADR pulsar sample shows that there remains an unaccounted gain error in the LOFAR pulsar observing system that results in an overestimate of the flux density scale by about a factor of two.\n\n\\subsubsection{The Missing Pulsars}\\label{sec:missing}\n\nDespite the high yield, there are also a number pulsars in Fig. \\ref{fig:ff}  with large {decimeter} flux densities but with no TGSS ADR counterpart in Table \\ref{tab:bright}. Likewise, we failed to detect several bright pulsars which had been found in previous low frequency pulsation surveys \\citep[e.g.][]{kl01,bkk+15}. To investigate the origin of these missing pulsars, we defined a radio-bright sample from the original 1238 well-localized pulsars in \\S\\ref{sec:loud} as having 400 and 1400 MHz flux densities greater than 21 mJy and 1.8 mJy, respectively.  For a canonical pulsar spectral index these flux densities extrapolate at 150 MHz to the completeness limit of the TGSS ADR \\citep{int16}.  There are 232 such pulsars. Of this sample, 70\\% are detected and are listed in Tables \\ref{tab:bright} and \\ref{tab:faint}.\n\nWe can identify three possible reasons that about one third of this radio-bright sample of pulsars would not be detected in the TGSS ADR. The local rms noise may be too high, the pulsar spectrum may be flat or turn over at 150 MHz, or the signal may be reduced due to interstellar scintillation. It may be possible that one of these effects dominant or they are working in tandem. We will look at each of these in turn.\n\nAt low radio frequencies the synchrotron and thermal emission from the Galactic plane makes a non-negligible contribution to the system temperature of the receivers. The frequency dependence of the brightness temperature goes approximately at T$_b\\propto\\nu^{-2.6}$ \\citep{hks+81} so unless the pulsar spectrum is steeper than this value, they become increasingly more difficult to detect. While the increased brightness temperature affects pulsed and imaging searches equally, the later also suffers from increased rms due to confusion and reduced image fidelity in the presence of bright Galactic HII regions or supernova remnants.  The pulsar B\\,2319+60 is a good example of a bright pulsar confused by nearby bright, extended emission. We looked at the rms noise statistics of the  detected and non-detected samples, following up the large rms cases with a visual inspection of the TGSS ADR image data at the PSR positions. We find evidence that the rms noise of the images has some influence on the detectability of the pulsars. The median rms noise for the detections is \\mjybeam{3.5}. while for the non-detections it is nearby twice this value (\\mjybeam{6.9}).\n\nThe intrinsic spectral shape of the pulsar emission will also affect the detectability at low frequencies. The mean pulsar spectral index, while steep, has a wide scatter (\\S\\ref{sec:intro}). Likewise, for approximately 10\\% of known pulsars there is evidence of a low-frequency spectral turnover, typically around 100 MHz (\\S\\ref{sec:intro}). Our 150 MHz sample has a number of pulsars with known spectral turnovers including PSR\\,J2145-0750 \\citep{drt+13}. We lack a large public database of accurate pulsar flux densities that would be sufficient to look for a turnover frequency for our non-detections, but fortunately most of them have single power-law measurements in the ATNF pulsar catalog. The median spectral index for the detections is $\\alpha=-1.9$ and there are no pulsars in this sample as shallow as $\\alpha\\leq-0.5$. The non-detections have a much flatter median spectral index of $\\alpha=-1.3$. At least one third of our non-detections have spectral indices that are so flat that we do not expect to detect them at 150 MHz based on an extrapolation of their 400 or 1400 MHz catalog flux densities. \n\nDensity fluctuations in the ionized interstellar medium of our Galaxy can induce intensity fluctuations that may depress the flux density of a pulsar during an integration time. The characteristic time and frequency scale depends on many factors including the distance of the pulsar, the turbulent properties of the gas along the line of sight, and the relative velocities of the pulsar and the ionized gas \\citep{cwf+91}. To estimate the magnitude of strong scattering on the phase-averaged pulsar flux densities we followed the method of \\citet{kcac98}. We first estimated the scattering bandwidth and scattering time at 150 MHz for each pulsar using the NE2001 model of \\citet{cl02}. Typical scintillation timescales and bandwidths at these frequencies are small, of order 1 minute and below 1 MHz, respectively. Our values are similar to the values estimated at the same frequency by \\citet{bkk+15}. We then estimate the number of ``scintles'' that are averaged over the observed bandwidth and the duration of the observation. The intensity modulation is equal to the square root of this value. The observed bandwidth is given in \\S\\ref{survey} as 16.7 MHz. The duration of the GMRT observations are more difficult to estimate. The total integration on source is 15 minutes but it is split into 3--5 short snapshots spaced over a full night's observing. As an added complication, the image mosaics are additions of many overlapping fields and so it is possible that a single pixel may contain observations from more than one night. This sampling has the effect of smoothing out any large intensity modulations, so as a (pessimistic) estimate we take the duration as 15 min but we recognize that there may be additional temporal smoothing. Our results by and large suggest that the TGSS ADR pulsar flux densities are only being weakly modulated by scintillation in most cases. There are pulsars that are predicted to be undergoing strong diffractive scintillation at this frequency (e.g. PSR\\,B0950+08 and PSR\\,1929+10) and there are diffraction spikes centered on the MSP PSR\\,B1937+12, likely caused by intensity variations on timescales comparable to the dump time. However, we can find no systematic trend for the non-detected pulsars to have greater predicted modulations from scattering.\n\nSummarizing, we find that the bright cataloged pulsars with no TGSS ADR counterpart may be due to a combination of effects. There is evidence that the non-detections at 150 MHz have more shallow spectral indices than average, and that some of the non-detections are caused by high rms and confusion in the image plane. Strong intensity variations by interstellar scintillation is undoubtedly occurring for some pulsars but we cannot show that the non-detections differ from the detections in their scattering properties. The difficulty in estimating the true GMRT integration time for each pulsar may be masking this effect. \n\n\n\n\\subsection{Radio Quiet Pulsars}\n\nOur search did not find any significant radio emission at 150 MHz toward individual radio quiet pulsars, nor in a weighted stack of all 30 pulsars. The peak of the stacked image in Fig. \\ref{fig:stack} is 0.1$\\pm$0.7 mJy beam$^{-1}$, with upper limit (peak + 2$\\sigma$) of $<$1.5 mJy beam$^{-1}$. Recall from \\S\\ref{quiet} that ``radio-quiet\" pulsars are observationally defined as having a phase-averaged flux density at 1.4 GHz S$_\\nu<$30 $\\mu$Jy. The simplest hypothesis is that radio quiet pulsars are beamed away from the line of sight \\citep{ckr+12}. \n\nRadio quiet gamma-ray pulsars, with Geminga as the prototype, are expected, given what we know about the structure of neutron star magnetospheres \\citep{car14}. The radio emission is thought to originate further down the poles than the gamma-rays, and thus the radio will be beamed into a narrowing opening angle, increasing the probability that the beam sweeps out away from the observer's line of sight. However, it is well-known that both the radio pulse width and the component separation are frequency dependent \\citep{thor91,mr02}. As noted by \\citet{mad12}, this widening of radio beams at low frequencies might be used to detect radio quiet, gamma-ray pulsars. Such pulsars would be recognized in the image plane as having a spectral index that is much steeper than the canonical value. PSR\\,B1943+10 may be thought of the prototype of such systems, bright at 400 MHz and below but weak at 1.4 GHz, with a  spectral index $\\alpha$ steeper than $-3.0$ \\citep[see Table \\ref{tab:bright};][]{wcl+99}. The lower limit estimate on spectral index that we derive from the weighted stack at 150 MHz, assuming the defining radio-quiet 1.4 GHz flux density of 30 $\\mu$Jy, gives $\\alpha>-1.75\\pm0.20$.  This is a spectral index limit that is well within the canonical value for normal pulsars. Thus we find no evidence that these gamma-ray pulsars have radio beams that sweep close to our line of sight.\n\n\n\n\n\n\\section{Conclusion}\\label{sec:conclude}\n\nWe have identified nearly 300 pulsars at 150 MHz based in their phase-averaged emission on all-sky images. This imaging approach is complementary to pulsation studies since it is not affected by {pulse scatter} broadening or dispersion, making it sensitive to both normal and millisecond pulsars equally. Our sample includes many southern pulsars which are being detected at low radio frequencies for the first time. We anticipate that these 150 MHz flux densities will be used to study large numbers of pulsar over a wider frequency range than has hitherto been possible, and to addresses questions about the incidence and origins of low-frequency spectral turnovers. Accurate calibration between telescopes remains an important issue and we have identified a discrepancy between the flux densities of pulsars in common between GMRT and LOFAR. We suggest that the LOFAR sample may be overestimating the flux density scale by about a factor of two. It should be straightforward to test this hypothesis by observing a sample of pulsars with LOFAR in both imaging and phase-binning modes, calibrating the interferometric data in the standard way to allow proper comparison with each other and with the GMRT.\n\nWe have carried out a preliminary spectral index study of our sample. Generally there is good agreement with past work, except that we find a curious preponderance of gamma-ray MSPs with unusually steep spectral indices ($\\alpha\\leq-2.5$). Regardless of its origins, this suggests a possible way to identify new MSP candidates in \\emph{Fermi}  unassociated sources on the basis of their unusually steep spectrum at low radio frequencies. Such pulsars may have been missed in radio pulsation searches due to propagation effects caused by the interstellar medium or they may be in binary systems and thus more difficult to discover. In such cases, imaging \\emph{Fermi}  error regions with LOFAR and the GMRT could provide accurate enough positions to enable blind gamma-ray searches for pulsations.\n\n\\acknowledgments\n\nThis research has made use of data and\/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA\/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. DAF thanks T. Readhead and S. Kulkarni  for their hospitality at Caltech while this work was being written up.\nHTI acknowledges financial support through the NL-SKA roadmap project funded by the NWO. We thank A. Bilous and J. Hessels for sharing their knowledge of LOFAR pulsar flux density calibration.\n\n{\\it Facilities:} \\facility{GMRT}, \\facility{Fermi (LAT)}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}