analytical and belief propagation studies of random constraint satisfaction problems with growing domains we study solution space structure and solution finding algorithms of a representative hard random constraint satisfaction problem with growing domains known as model rb using rigorous methods we show that solutions are grouped into disconnected clusters before the theoretical satisfiability phase transition point using the cavity method it is shown that the entropy density obtained by belief propagation bp on random model rb instances which corresponds well to the analytical results vanishes as the control parameter constraint tightness approaches the satisfiability threshold from an algorithmic point of view we find that reinforced bp which performs much better than all existing algorithms allows us to find solutions efficiently for instances in the regime that is very close to the satisfiability transition these results also can shed light on the effectiveness of bp reinforcement on problems with a large number of states

stability analysis on the finite temperature replica symmetric and first step replica symmetry broken cavity solutions of the random vertex cover problem the vertex cover problem is a prototypical hard combinatorial optimization problem it was studied in recent years by physicists using the cavity method of statistical mechanics in this paper the stability of the finite temperature replica symmetric rs and the first step replica symmetry broken 1rsb cavity solutions of the vertex cover problem on random regular graphs of finite vertex degree k are analyzed by population dynamics simulations we found that 1 the lowest temperature for the rs solution to be stable t rs k is not a monotonic function of k 2 at relatively large connectivity k and temperature t slightly below the dynamic transition temperature t d k the 1rsb solutions with small but non negative complexity values are stable and 3 the dynamical transition temperature t d and kauzmann temperature t k is equal to each other similar results are obtained on random poissonian graphs

ground state entropy of the random vertex cover problem counting the number of ground states for a spin glass or nondeterministic polynomial complete combinatorial optimization problem is even more difficult than the already hard task of finding a single ground state in this paper the entropy of minimum vertex covers of random graphs is estimated through a set of iterative equations based on the cavity method of statistical mechanics during the iteration both the cavity entropy contributions and cavity magnetizations for each vertex are updated this approach overcomes the difficulty of iterative divergence encountered in the zero temperature first step replica symmetry breaking 1rsb spin glass theory it is still applicable when the 1rsb mean field theory is no longer stable the method can be extended to compute the entropies of ground states and metastable minimal energy states for other random graph spin glass systems

t 0 mean field population dynamics approach for the random 3 satisfiability problem during the past decade phase transition phenomena in the random 3 satisfiability 3 sat problem has been intensively studied by statistical physics methods in this work we study the random 3 sat problem by the mean field first step replica symmetry broken cavity theory at the limit of temperature t 0 the reweighting parameter y of the cavity theory is allowed to approach infinity together with the inverse temperature beta with fixed ratio r y beta focusing on the system s space of satisfiable configurations we carry out extensive population dynamics simulations using the technique of importance sampling and we obtain the entropy density s r and complexity sigma r of zero energy clusters at different r values we demonstrate that the population dynamics may reach different fixed points with different types of initial conditions by knowing the trends of s r and sigma r with r we can judge whether a certain type of initial condition is appropriate at a given r value this work complements and confirms the results of several other very recent theoretical studies

quadratic integral games and causal synthesis the game problem for an input output system governed by a volterra integral equation with respect to a quadratic performance functional is an untouched open problem in this paper it is studied by a new approach called projection causality the main result is the causal synthesis which pro vides a causal feedback implementation of the optimal strategies in the saddle point sense the linear feedback operator is determined by the solution of a fredholm integral operator equation which is independent of data functions and control functions two application examples are included the first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables the second is the standard linear quadratic differential games for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential riccati operator equation

complexity of several constraint satisfaction problems using the heuristic classical algorithm walksat we determine the complexity of several constraint satisfaction problems using the heuristic algorithm walksat at large sizes n the complexity increases exponentially with n in all cases perhaps surprisingly out of all the models studied the hardest for walksat is the one for which there is a polynomial time algorithm

near optimal configurations in mean field disordered systems we present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems our approach is based on a generalization of the cavity method to a system interacting with its ground state with this technique we study the random matching problem as well as the mean field diluted spin glass as a by product of this approach we calculate the de almeida thouless transition line of the spin glass on a fixed connectivity random graph

satisfiability threshold for random regular nae sat we consider the random regular k nae sat problem with n variables each appearing in exactly d clauses for all k exceeding an absolute constant it k 0 we establish explicitly the satisfiability threshold d star equiv d star k we prove that for d d star the problem is unsatisfiable with high probability if the threshold d star lands exactly on an integer we show that the problem is satisfiable with probability bounded away from both zero and one this is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by krzakala et al proc natl acad sci 104 25 10318 10323 2007 our proof verifies the one step replica symmetry breaking formalism for this model we expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs

backward doubly stochastic volterra integral equations and their applications abstract in this paper we introduce a new class of equations called backward doubly stochastic volterra integral equations bdsvies for short first the well posedness of bdsvies in the sense of introduced m solutions is established second a comparison theorem of bdsvies is proved as an application of the comparison theorem we derive the existence of solutions of bdsvies with continuous coefficients third a duality principle between linear forward doubly stochastic volterra integral equations fdsvies for short and bdsvies is obtained moreover by virtue of the duality principle a maximum principle of pontryagin type is established for an optimal control problem of fdsvies

statistical mechanics of maximal independent sets the graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory a maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints it is known that finding minimum and maximum density maximal independent sets are hard optimization problems in this paper we use cavity method of statistical physics and monte carlo simulations to study the corresponding constraint satisfaction problem on random graphs we obtain the entropy of maximal independent sets within the replica symmetric and one step replica symmetry breaking frameworks shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms this is of particular relevance for the application to the study of strategic interactions in social and economic networks where maximal independent sets correspond to pure nash equilibria of a graphical game of public goods allocation

statistical mechanics of the minimum vertex cover problem in stochastic block models the minimum vertex cover min vc problem is a well known np hard problem earlier studies illustrate that the problem defined over the erdos renyi random graph with a mean degree c exhibits computational difficulty in searching the min vc set above a critical point c e 2 718 here we address how this difficulty is influenced by the mesoscopic structures of graphs for this we evaluate the critical condition of difficulty for the stochastic block model we perform a detailed examination of the specific cases of two equal size communities characterized by in and out degrees which are denoted by c in and c out respectively our analysis based on the cavity method indicates that the solution search once becomes difficult when c in c out exceeds e from below but becomes easy again when c out is sufficiently larger than c in in the region c out e experiments based on various search algorithms support the theoretical prediction

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

extremal optimization for graph partitioning extremal optimization is a new general purpose method for approximating solutions to hard optimization problems we study the method in detail by way of the computationally hard np hard graph partitioning problem we discuss the scaling behavior of extremal optimization focusing on the convergence of the average run as a function of run time and system size the method has a single free parameter which we determine numerically and justify using a simple argument on random graphs our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes with an approximation error decreasing over run time roughly as a power law t 20 4 on geometrically structured graphs the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials but when only the best runs are considered results consistent with theoretical arguments are recovered

causal feedback optimal control for volterra integral equations the optimal control problem for volterra integral equations with respect to quadratic criteria is studied by a projection causality approach the work features a synthesis result where the optimal control is implemented via a linear causal feedback in which the feedback operator is determined by solving an independent fredholm integral operator equation

backward stochastic volterra integral equations with additive perturbations the paper discusses a large class of stochastic volterra integral equations whose coefficients additively depend on small perturbations their solutions are compared in the l2 sense with the solutions of the appropriate unperturbed equations of the equal or simpler type more precisely we prove for an arbitrary 0 that there exists an interval t t 0 t on which the l2 difference between the solutions of the perturbed and unperturbed equations is less than

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

minimal vertex covers on finite connectivity random graphs a hard sphere lattice gas picture the minimal vertex cover or maximal independent set problem is studied on random graphs of finite connectivity analytical results are obtained by a mapping to a lattice gas of hard spheres of chemical radius 1 and they are found to be in excellent agreement with numerical simulations we give a detailed description of the replica symmetric phase including the size and entropy of the minimal vertex covers and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1 43 the replica symmetric solution breaks down at c e approximately 2 72 we give a simple one step replica symmetry broken solution and discuss the problems in the interpretation and generalization of this solution

paradox of enrichment a fractional differential approach with memory abstract the paradox of enrichment poe proposed by rosenzweig m rosenzweig the paradox of enrichment science 171 1971 385 387 is still a fundamental problem in ecology most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels poe resolution via these two organizational levels can be interpreted as a microscopic and macroscopic level solution fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes the derivatives can be physically interpreted via two time scales that are considered simultaneously the ideal equably flowing homogeneous local time and the cosmic inhomogeneous non local time several mechanisms and theories have been proposed to resolve the poe problem but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non local effects which capture memory here we formulate the fractional counterpart of the rosenzweig model and analyze the stability behavior of a system we conclude that there is a threshold for the memory effect parameter beyond which the rosenzweig model is stable and may be used as a potential agent to resolve poe from a new perspective via fractional differential equations

clustering of neural code words revealed by a first order phase transition a network of neurons in the central nervous system collectively represents information by its spiking activity states typically observed states i e code words occupy only a limited portion of the state space due to constraints imposed by network interactions geometrical organization of code words in the state space critical for neural information processing is poorly understood due to its high dimensionality here we explore the organization of neural code words using retinal data by computing the entropy of code words as a function of hamming distance from a particular reference codeword specifically we report that the retinal code words in the state space are divided into multiple distinct clusters separated by entropy gaps and that this structure is shared with well known associative memory networks in a recallable phase our analysis also elucidates a special nature of the all silent state the all silent state is surrounded by the densest cluster of code words and located within a reachable distance from most code words this code word space structure quantitatively predicts typical deviation of a state trajectory from its initial state altogether our findings reveal a non trivial heterogeneous structure of the code word space that shapes information representation in a biological network

clustering analysis of the ground state structure of the vertex cover problem vertex cover is one of the classical np complete problems in theoretical computer science a vertex cover of a graph is a subset of vertices such that for each edge at least one of the two endpoints is contained in the subset when studied on erd os r enyi random graphs with connectivity c one observes a threshold behavior in the thermodynamic limit the size of the minimal vertex cover is independent of the specific graph recent analytical studies show that on the phase boundary for small connectivities cle the system is replica symmetric while for larger connectivities replica symmetry breaking occurs this change coincides with a change of the typical running time of algorithms from polynomial to exponential to understand the reasons for this behavior and to compare with the analytical results we numerically analyze the structure of the solution landscape for this purpose we have also developed an algorithm which allows the calculation of the backbone without the need to enumerate all solutions we study exact solutions found with a branch and bound algorithm as well as configurations obtained via a monte carlo simulation we analyze the cluster structure of the solution landscape by direct clustering of the states by analyzing the eigenvalue spectrum of correlation matrices and by using a hierarchical clustering method all results are compatible with a change at c e for small connectivities the solutions are collected in a finite small number of clusters while the number of clusters diverges slowly with system size for larger connectivities and replica symmetry breaking but not one step replica symmetry breaking 1 rsb occurs

number of guards needed by a museum a phase transition in vertex covering of random graphs in this letter we study the np complete vertex cover problem on finite connectivity random graphs when the allowed size of the cover set is decreased a discontinuous transition in solvability and typical case complexity occurs this transition is characterized by means of exact numerical simulations as well as by analytical replica calculations the replica symmetric phase diagram is in excellent agreement with numerical findings up to average connectivity e where replica symmetry becomes locally unstable

mean field linear quadratic gaussian lqg games for stochastic integral systems in this technical note we formulate and investigate a class of mean field linear quadratic gaussian lqg games for stochastic integral systems unlike other literature on mean field games where the individual states follow the controlled stochastic differential equations sdes the individual states in our large population system are characterized by a class of stochastic volterra type integral equations we obtain the nash certainty equivalence nce equation and hence derive the set of associated decentralized strategies the epsilon nash equilibrium properties are also verified due to the intrinsic integral structure the techniques and estimates applied here are significantly different from those existing results in mean field lqg games for stochastic differential systems for example some fredholm equation in the mean field setup is introduced for the first time as for applications two types of stochastic delayed systems are formulated as the special cases of our stochastic integral system and relevant mean field lqg games are discussed

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

optimal transportation network with concave cost functions loop analysis and algorithms transportation networks play a vital role in modern societies structural optimization of a transportation system under a given set of constraints is an issue of great practical importance for a general transportation system whose total cost c is determined by c sigma c i j ij i ij with c ij i ij being the cost of the flow i ij between node i and node j banavar and co workers phys rev lett 84 4745 2000 proved that the optimal network topology is a tree if c ij proportional to parallel to i ij parallel to gamma with 0 1 the same conclusion also holds in the more general case where all the flow costs are strictly concave functions of the flow i ij to further understand the qualitative difference between systems with concave and convex cost functions a loop analysis of transportation cost is performed in the present paper and an alternative mathematical proof of the optimality of tree formed networks is given the simple intuitive picture of this proof then leads to an efficient global algorithm for the searching of optimal structures for a given transportation system with concave cost functions

a positivstellensatz for non commutative polynomials a non commutative polynomial which is positive on a bounded semi algebraic set of operators has a weighted sum of squares representation this positivstellensatz parallels similar results in the commutative case a broader issue is to what extent does real semi algebraic geometry extend to non commutative polynomials our strict positivstellensatz is positive news on the opposite extreme from strict positivity would be a real nullstellensatz we give an example which shows that there is no non commutative real nullstellensatz along certain lines however we include a successful type of non commutative nullstellensatz proved by george bergman

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

optimal control of impulsive volterra equations with variable impulse times we obtain necessary conditions of optimality for impulsive volterra integral equations with switching and impulsive controls with variable impulse time instants the present work continues and complements our previous work on impulsive volterra control with fixed impulse times

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

maximally flexible solutions of a random k satisfiability formula random k satisfiability k sat is a paradigmatic model system for studying phase transitions in constraint satisfaction problems and for developing empirical algorithms the statistical properties of the random k sat solution space have been extensively investigated but most earlier efforts focused on solutions that are typical here we consider maximally flexible solutions which satisfy all the constraints only using the minimum number of variables such atypical solutions have high internal entropy because they contain a maximum number of null variables which are completely free to choose their states each maximally flexible solution indicates a dense region of the solution space we estimate the maximum fraction of null variables by the replica symmetric cavity method and implement message passing algorithms to construct maximally flexible solutions for single k sat instances

detecting the solution space of vertex cover by mutual determinations and backbones to solve the combinatorial optimization problems especially the minimal vertex cover problem with high efficiency is a significant task in theoretical computer science and many other subjects aiming at detecting the solution space of vertex cover a new structure named mutual determination is defined and discovered for vertex cover on general graphs which results in the emergence of strong correlations among the unfrozen nodes based on the backbones and mutual determinations with node ranks by leaf removal we propose a mutual determination and backbone evolution algorithm to achieve the reduced solution graph which provides a graphical expression of the solution space of vertex cover by this algorithm the whole solution space and detailed structures such as backbones can be obtained strictly when there is no leaf removal core on the given graph compared with the current algorithms the mutual determination and backbone evolution algorithm performs as well as the replica symmetry one in a certain interval but has a small gap higher than the replica symmetric breaking one and has a relatively small error for the exact results the algorithm with the mutual determination provides a new viewpoint to solve vertex cover and understand the organizations of the solution spaces and the reduced solution graph gives an alternative way to catch detailed information of the ground steady states

positive semidefinite quadratic forms on unitary matrices abstract let f x 1 x n i j 1 n ij x i x j a ij a ji r be a real quadratic form such that the trace of the hermitian matrix f v 1 v n i j 1 n ij v i v j is nonnegative for all unitary 2 n 2 n matrices v 1 v n we prove that f u 1 u n is positive semidefinite for all unitary matrices u 1 u n of arbitrary size m m

clustering of solutions in the random satisfiability problem using elementary rigorous methods we prove the existence of a clustered phase in the random k sat problem for k geq 8 in this phase the solutions are grouped into clusters which are far away from each other the results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks it can be generalized to other systems of both physical and computational interest

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

coloring random graphs and maximizing local diversity we study a variation of the graph coloring problem on random graphs of finite average connectivity given the number of colors we aim to maximize the number of different colors at neighboring vertices i e one edge distance of any vertex two efficient algorithms belief propagation and walksat are adapted to carry out this task we present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution the problem and the suggested algorithms have practical relevance since various applications such as distributed storage can be mapped onto this problem

phase transitions in the q coloring of random hypergraphs we study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied hypergraph coloring is a constraint satisfaction problem where each constraint includes k variables that must be assigned one out of q colors in such a way that there are no monochromatic constraints i e there are at least two distinct colors in the set of variables belonging to every constraint this problem generalizes naturally coloring of random graphs k 2 and bicoloring of random hypergraphs q 2 both of which were extensively studied in past works the study of random hypergraph coloring gives us access to a case where both the size q of the domain of the variables and the arity k of the constraints can be varied at will our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring among other cases we revisit the hypergraph bicoloring problem q 2 where we find that for k 3 and k 4 the colorability threshold is not given by the one step replica symmetry breaking analysis as the latter is unstable towards more levels of replica symmetry breaking we also unveil and discuss the coexistence of two different 1rsb solutions in the case of q 2 k 4 finally we present asymptotic expansions for the density of constraints at which various phase transitions occur in the limit where q and or k diverge

message passing for vertex covers constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints in this paper we develop and analyze message passing techniques namely warning and survey propagation which serve as efficient heuristic algorithms for solving these computational hard problems we show also how previously obtained results on the typical case behavior of vertex covers of random graphs can be recovered starting from the message passing equations and how they can be extended

symmetrical martingale solutions of backward doubly stochastic volterra integral equations abstract this paper aims to study a new class of integral equations called backward doubly stochastic volterra integral equations bdsvies for short the notion of symmetrical martingale solutions sm solutions for short is introduced for bdsvies and the existence and uniqueness theorem for bdsvies in the sense of sm solutions is established

a reduction method for optimal control of volterra integral equations we present and analyze a novel method for solving optimal control problems for volterra integral equations based on approximating the controlled volterra integral equations by a sequence of systems of controlled ordinary differential equations the resulting approximating problems can then be solved by dynamic programming methods for ode controlled systems we also derive the connection between our version of dynamic programming and the hamiltonian equations for volterra controlled systems

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

threshold values stability analysis and high q asymptotics for the coloring problem on random graphs we consider the problem of coloring erdos renyi and regular random graphs of finite connectivity using q colors it has been studied so far using the cavity approach within the so called one step replica symmetry breaking 1rsb ansatz we derive a general criterion for the validity of this ansatz and applying it to the ground state we provide evidence that the 1rsb solution gives exact threshold values c q for the transition from the colorable to the uncolorable phase with q colors we also study the asymptotic thresholds for q 1 finding c q 2q ln q ln q 1 o 1 in perfect agreement with rigorous mathematical bounds as well as the nature of excited states and give a global phase diagram of the problem

spin glass phase transitions and minimum energy of the random feedback vertex set problem a feedback vertex set fvs of an undirected graph contains vertices from every cycle of this graph constructing a fvs of sufficiently small cardinality is very difficult in the worst cases but for random graphs this problem can be efficiently solved by converting it into an appropriate spin glass model h j zhou eur phys j b 86 455 2013 in the present work we study the spin glass phase transitions and the minimum energy density of the random fvs problem by the first step replica symmetry breaking 1rsb mean field theory for both regular random graphs and erdos renyi graphs we determine the inverse temperature beta 1 at which the replica symmetric mean field theory loses its local stability the inverse temperature beta d of the dynamical clustering phase transition and the inverse temperature beta s of the static condensation phase transition these critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way and beta d is distinct from beta s for regular random graphs of vertex degrees k 60 while beta d are identical to beta s for erdos renyi graphs at least up to mean vertex degree c 512 we then derive the zero temperature limit of the 1rsb theory and use it to compute the minimum fvs cardinality

optimal control of volterra equations with impulses we consider an optimal control problem for a system governed by a volterra integral equation with impulsive terms the impulses act on both the state and the control the control consists of switchings at discrete times the cost functional includes both an integrated cost rate continuous part and switching costs at the discrete impulse times discrete part we prove necessary optimality conditions of a form analogous to a discrete maximum principle for the particular case of a system governed by impulsive ordinary differential equations we obtain an impulsive maximum principle as a special case of the necessary optimality conditions for impulsive volterra equations

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

optimal processes governed by integral equations abstract we consider a control process governed by an integral equation with an integral constraint we derive optimality conditions using relaxed controls existence of optimal pairs is automatic furthermore for problems with linear and certain quadratic cost functions with linear integral equations and linear integral constraint the optimal controls are actually ordinary controls in the case of the linear problem considered if the control set is a convex polyhedron the control is piecewise continuous with values in the vertices of the polyhedron control processes governed by integral equations have been considered in the work of v r vinokurov siam 7 324 355 1969 however in op cit existence is not shown for the general problem and the method is rather complicated furthermore the work in op cit has some errors as pointed out in neustadt and warga s comments on vinokurov siam 8 572 1970

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

entropy landscape and non gibbs solutions in constraint satisfaction problems we study the entropy landscape of solutions for the bicoloring problem in random graphs a representative difficult constraint satisfaction problem our goal is to classify which types of clusters of solutions are addressed by different algorithms in the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem e g dynamical rigidity and satisfiability unsatisfiability sat unsat transitions in the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem for instance we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen these nonequilibrium solutions belong to the most probable unfrozen clusters

the network source location problem ground state energy entropy and effects of freezing ground state entropy of the network source location problem is evaluated at both the replica symmetric level and one step replica symmetry breaking level using the entropic cavity method the regime that is a focus of this study is closely related to the vertex cover problem with randomly quenched covered nodes the resulting entropic message passing inspired decimation and reinforcement algorithms are used to identify the optimal location of sources in single instances of transportation networks the conventional belief propagation without taking the entropic effect into account is also compared we find that in the glassy phase the entropic message passing inspired decimation yields a lower ground state energy compared to the belief propagation without taking the entropic effect using the extremal optimization algorithm we study the ground state energy and the fraction of frozen hubs and extend the algorithm to collect statistics of the entropy the theoretical results are compared with the extremal optimization results

extending positive definiteness the main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of semigroups to positive definite ones by specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems like complex two sided complex and multidimensional trigonometric moment problems in addition unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme

structure and entropy for positive definite toeplitz kernels on free semigroups abstract in this article we study the structure of positive definite toeplitz kernels on free semigroups called also multi toeplitz and its implications in noncommutative dilation theory harmonic analysis on fock spaces prediction and interpolation theory for stationary stochastic processes a parametrization of positive definite multi toeplitz kernels in terms of generalized schur sequences of contractions is obtained this leads to explicit minimal naimark dilations cholesky factorizations szego type limit theorems and entropy for positive definite multi toeplitz kernels maximal outer factors are associated with positive definite multi toeplitz kernels and are used to obtain inner outer factorization for operators on fock spaces with coefficients in hilbert spaces the kolmogorov wiener prediction problem for stochastic processes having as covariance kernels positive definite multi toeplitz kernels is considered the predication error operator is calculated in terms of schur parameters resp maximal outer factor associated with the covariance kernel of the process and a connection with a szego type infimum problem is established we solve the caratheodory interpolation problem for positive definite multi toeplitz kernels we obtain a parametrization of all solutions in terms of schur sequences and we find the maximal entropy solution the results of this article can be used to develop a theory of stochastic n linear systems

giant strongly connected component of directed networks we describe how to calculate the sizes of all giant connected components of a directed graph including the strongly connected one in particular the world wide web is a directed network the results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in and out degree distribution p k i k o we show that if p k i k o does not factorize the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in and out components the calculations of the relative sizes of all the giant components are demonstrated using the simplest examples we explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one

well posedness and regularity of backward stochastic volterra integral equations backward stochastic volterra integral equations bsvies for short are studied notion of adapted m solution is introduced well posedness of bsvies is established and some regularity results are proved for the adapted m solutions via malliavin calculus a pontryagin type maximum principle is presented for optimal controls of stochastic volterra integral equations

entropy landscape of solutions in the binary perceptron problem the statistical picture of the solution space for a binary perceptron is studied the binary perceptron learns a random classification of input random patterns by a set of binary synaptic weights the learning of this network is difficult especially when the pattern constraint density is close to the capacity which is supposed to be intimately related to the structure of the solution space the geometrical organization is elucidated by the entropy landscape from a reference configuration and of solution pairs separated by a given hamming distance in the solution space we evaluate the entropy at the annealed level as well as replica symmetric level and the mean field result is confirmed by the numerical simulations on single instances using the proposed message passing algorithms from the first landscape a random configuration as a reference we see clearly how the solution space shrinks as more constraints are added from the second landscape of solution pairs we deduce the coexistence of clustering and freezing in the solution space

schur analysis of some completion problems abstract the complete schur analysis for contractive block matrices is given together with its connections with positiveness these are used for reobtaining some results on completion problems sometimes in a more general setting these completion problems include positive completions contractive and isometric completions and the realizability of upper triangular contractions

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

volterra equations driven by semimartingales existence and uniqueness of solutions is established for stochastic volterra integral equations driven by right continuous semimartingales this resolves in the affirmative a conjecture of m berger and v mizel

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

extremal optimization at the phase transition of the three coloring problem we investigate the phase transition in vertex coloring on random graphs using the extremal optimization heuristic three coloring is among the hardest combinatorial optimization problems and is equivalent to a 3 state anti ferromagnetic potts model like many other such optimization problems it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter the graph s mean vertex degree this phase transition is often associated with the instances of highest complexity we use extremal optimization to measure the ground state cost and the backbone an order parameter related to ground state overlap averaged over a large number of instances near the transition for random graphs of size n up to 512 for these graphs benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about osn 3 5 d update steps finite size scaling yields a critical mean degree value ac 4 703s28d furthermore the exploration of the degenerate ground states indicates that the backbone order parameter measuring the constrainedness of the problem exhibits a first order phase transition

completions of matrices and the commutant lifting theorem abstract completion results are obtained for partially defined matrices with an infinite number of entries as a corollary the commutant lifting theorem of sz nagy and foia is proved by solving a completion problem

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials

a new method for optimal control of volterra integral equations abstract we formulate and analyze a new method for solving optimal control problems for systems governed by volterra integral equations our method utilizes discretization of the original volterra controlled system and a novel type of dynamic programming in which the hamilton jacobi function is parametrized by the control function rather than the state as in the case of ordinary dynamic programming we also derive estimates for the computational cost of our method

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb