averages and higher moments for the ell torsion in class groups we prove upper bounds for the average size of the ell torsion mathrm cl k ell of the class group of k as k runs through certain natural families of number fields and ell is a positive integer we refine a key argument used in almost all results of this type which links upper bounds for mathrm cl k ell to the existence of many primes splitting completely in k that are small compared to the discriminant of k our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument in conjunction with new counting results for these invariants this leads to significantly improved upper bounds for the average and sometimes even higher moments of mathrm cl k ell for many families of number fields k considered in the literature for example for the families of all degree d fields for d in 2 3 4 5 and non d 4 if d 4 as an application of the case d 2 we obtain the best upper bounds for the number of d p fields of bounded discriminant for primes p 3

phase separation and coarsening in one dimensional driven diffusive systems local dynamics leading to long range hamiltonians a driven system of three species of particles diffusing on a ring is studied in detail the dynamics is local and conserves the three densities a simple argument suggesting that the model should phase separate and break the translational symmetry is given we show that for the special case where the three densities are equal the model obeys detailed balance and the steady state distribution is governed by a hamiltonian with asymmetric long range interactions this provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension the steady state of finite size systems is studied using a generalized matrix product ansatz the coarsening process leading to phase separation is studied numerically and in a mean field model the system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size the typical domain size is shown to grow logarithmically in time generalizations to a larger number of species are discussed

universality classes in two component driven diffusive systems we study time dependent density fluctuations in the stationary state of driven diffusive systems with two conserved densities rho lambda using monte carlo simulations of two coupled single lane asymmetric simple exclusion processes we present numerical evidence for universality classes with dynamical exponents z 1 sqrt 5 2 and z 3 2 but different from the kardar parisi zhang kpz universality class which have not been reported yet for driven diffusive systems the numerical asymmetry of the dynamical structure functions converges slowly for some of the non kpz superdiffusive modes for which mode coupling theory predicts maximally asymmetric z stable levy scaling functions we show that all universality classes predicted by mode coupling theory for two conservation laws are generic they occur in two component systems with nonlinearities in the associated currents already of the minimal order rho lambda 2 rho mu the macroscopic stationary current density relation and the compressibility matrix determine completely all permissible universality classes through the mode coupling coefficients which we compute explicitly for general two component systems

convergence of the iterates of descent methods for analytic cost functions in the early eighties lojasiewicz in seminari di geometria 1982 1983 universita di bologna istituto di geometria dipartimento di matematica 1984 pp 115 117 proved that a bounded solution of a gradient flow for an analytic cost function converges to a well defined limit point in this paper we show that the iterates of numerical descent algorithms for an analytic cost function share this convergence property if they satisfy certain natural descent conditions the results obtained are applicable to a broad class of optimization schemes and strengthen classical weak convergence results for descent methods to strong limit point convergence for a large class of cost functions of practical interest the result does not require that the cost has isolated critical points and requires no assumptions on the convexity of the cost nor any nondegeneracy conditions on the hessian of the cost at critical points

asymptotic analysis of the exponential penalty trajectory in linear programming we consider the linear program min c x ax b and the associated exponential penalty functionfr x c x r exp aix bi r forr close to 0 the unconstrained minimizerx r offr admits an asymptotic expansion of the formx r x rd r wherex is a particular optimal solution of the linear program and the error term r has an exponentially fast decay using duality theory we exhibit an associated dual trajectory r which converges exponentially fast to a particular dual optimal solution these results are completed by an asymptotic analysis whenr tends to the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution

line search algorithms for locally lipschitz functions on riemannian manifolds this paper presents line search algorithms for finding extrema of locally lipschitz functions defined on riemannian manifolds to this end we generalize the so called wolfe conditions for nonsmooth functions on riemannian manifolds using varepsilon subgradient oriented descent directions and the wolfe conditions we propose a nonsmooth riemannian line search algorithm and establish the convergence of our algorithm to a stationary point moreover we extend the classical bfgs algorithm to nonsmooth functions on riemannian manifolds numerical experiments illustrate the effectiveness and efficiency of the proposed algorithm

dini derivative and a characterization for lipschitz and convex functions on riemannian manifolds dini derivatives in riemannian manifold settings are studied in this paper in addition a characterization for lipschitz and convex functions defined on riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the dini derivative are given

quasi tangent vectors in flow invariance and optimization problems on banach manifolds abstract the aim of this paper is threefold 1 1 to introduce the notion of quasi tangent vector to a subset s of a banach manifold m and to extend the definitions and some properties of a generalized gradients and tangent cones from a banach space e to m 2 2 to apply some of the above notions to the problem of flow invariance of a closed subset of the manifold m generalizing nagumo brezis result 3 3 to extend some fundamental results in optimization e g clarke s general lagrange multipliers rule in terms of generalized gradient from e to m points 1 and 3 are based on the results of clarke rockafellar and ursescu on tangent cones

universality class of interface growth with reflection symmetry we investigate interface dynamics in 1 1 dimensions respecting reflection symmetry in the continuum approach of kardar parisi and zhang the leading nonlinearity is then of the form ht 3 on the basis of monte carlo simulations for a driven lattice gas we argue that the nonlinearity is marginally irrelevant thus the universality class is the one of equilibrium interfaces with a purely relaxational bulk dynamics

exact scaling functions for one dimensional stationary kpz growth we determine the stationary two point correlation function of the one dimensional kpz equation through the scaling limit of a solvable microscopic model the polynuclear growth model the equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a riemann hilbert problem related to the painleve ii equation we solve these equations numerically with very high precision and compare our up to numerical rounding exact result with the prediction of colaiori and moore 1 obtained from the mode coupling approximation

convergence and efficiency of subgradient methods for quasiconvex minimization we study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint in a hilbert space our setting is very general the objective is only upper semicontinuous on its domain which need not be open and various subdifferentials may be used we extend previous results by proving convergence in objective values and to the generalized solution set for classical stepsizes t k 0 t k and weak or strong convergence of the iterates to a solution for t k l2 l1 under mild regularity conditions for bounded constraint sets and suitable stepsizes the method finds e solutions with an efficiency estimate of o e 2 thus being optimal in the sense of nemirovskii

p class groups of certain extensions of degree p kkt p be an odd prime number in this article we study the distribution of p class groups of cyclic number fields of degree p and of cyclic extensions of degree p of an imaginary quadratic field whose class number is coprime to p we formulate a heuristic principle predicting the distribution of the p class groups as galois modules which is analogous to the cohen lenstra heuristics concerning the prime to p part of the class group although in our case we have to fix the number of primes that ramify in the extensions considered using results of gerth we are able to prove part of this conjecture furthermore we present some numerical evidence for the conjecture

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

the 4 class ranks of quadratic fields let k be a quadratic extension field of the rational numbers q let c k be the 2 class group of k in the narrow sense it is a classical result that rank c k t l where t is the number of primes that ramify in k q now let c ai a e ck and let r k denote the 4 class rank of k in the narrow sense i e rk rank kc z dimv2 ttc z t 41 k kj here f 2 is the finite field with two elements and 2 4 ck c k is an elementary abelian 2 group which we are viewing as a vector space o v e r f 2 given a quadratic field k one can compute r k by computing the rank over f2 of a certain matrix of legendre symbols cf 11 now assume k is imaginary quadratic so k q 1 mm where m is a square free positive integer for each positive integer t each nonnegative integer e and each positive real number x we define

hessian riemannian gradient flows in convex programming in view of solving theoretically constrained minimization problems we investigate the properties of the gradient flows with respect to hessian riemannian metrics induced by legendre functions the first result characterizes hessian riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities giving a new motivation for the introduction of bregman type distances then the general evolution problem is introduced and global convergence is established under quasi convexity conditions with interesting refinements in the case of convex minimization some explicit examples of these gradient flows are discussed dual trajectories are identified and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints some convergence rate results are established in the case of a linear objective function several optimality characterizations of the orbits are given optimal path of viscosi

invariant monotone vector fields on riemannian manifolds abstract various concepts of invariant monotone vector fields on riemannian manifolds are introduced some examples of invariant monotone vector fields are given several notions of invexities for functions on riemannian manifolds are defined and their relations with invariant monotone vector fields are studied

on the existence of fields governing the 2 invariants of the classgroup of as varies this paper formulates general conjectures relating the structure of the 2 classgroup c2 dp associated to q fdp to the splitting of the ideal p in certain algebraic number fields here d i 2 mod 4 is a fixed integer and p varies over primes the conjectures assert that there exists an algebraic number field qj d such that the artin symbol q2 d q p determines the first j 2 invariants of the group c2 dp i e it determines c2 dp c2 dp 2j these conjectures imply that the set of primes p for which c2 dp has a given set of 2 invariants has a natural density which is a rational number existing results prove the conjectures wheneverj i or 2 and also for an infinite set of d withj 3 the smallest open case isj 3 d 21 this paper presents evidence concerning these conjectures for d 4 8 and 21 numerical evidence is given that q23 21 exists and that natural densities which are rational numbers exist for the sets of primes with 2 h dp for d 4 and 8 for i j 7 a search for the hypothetical field 04 4 ruled out the simplest candidate fields 04 4 is not a normal extension of q of degree 16 ramifying only at 2

surrogate dual multiplier search procedures in integer programming search procedures for optimal lagrange multipliers are highly developed and provide good bounds in branch and bound procedures that have led to the successful application of lagrangean duality in integer programming although the surrogate dual generally provides a better objective bound there has been little development of surrogate multiplier search procedures this paper develops and empirically analyzes several surrogate multiplier search procedures results indicate that the procedures can produce possibly superior bounds in an amount of time comparable to other techniques our discussion also highlights the similarity of the procedures to some well known lagrangean search techniques

on the 4 rank of class groups of quadratic number fields we prove that the 4 rank of class groups of quadratic number fields behaves as predicted in an extension due to gerth of the cohen lenstra heuristics

algorithms for the quasiconvex feasibility problem we study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x r n that satisfies the inequalities f 1 x 0 f 2 x 0 f m x 0 where all functions are continuous and quasiconvex we consider the consistent case when the solution set is nonempty since the fenchel moreau subdifferential might be empty we look at different notions of the subdifferential and determine their suitability for our problem we also determine conditions on the functions that are needed for convergence of our algorithms the quasiconvex functions on the left hand side of the inequalities need not be differentiable but have to satisfy a lipschitz or a holder condition

on the projected subgradient method for nonsmooth convex optimization in a hilbert space we consider the method for constrained convex optimization in a hilbert space consisting of a step in the direction opposite to anek subgradient of the objective at a current iterate followed by an orthogonal projection onto the feasible set the normalized stepsizesek are exogenously given satisfying k 0 k k 0 k2 0 we prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions and is unbounded otherwise among the features of our convergence analysis we mention that it covers the nonsmooth case in the sense that we make no assumption of differentiability off and much less of lipschitz continuity of its gradient also we prove weak convergence of the whole sequence rather than just boundedness of the sequence and optimality of its weak accumulation points thus improving over all previously known convergence results we present also convergence rate results 1998 the mathematical programming society inc published by elsevier science b v

proximal subgradient and a characterization of lipschitz function on riemannian manifolds a characterization of lipschitz behavior of functions defined on riemannian manifolds is given in this paper first it is extended the concept of proximal subgradient and some results of proximal analysis from hilbert space to riemannian manifold setting a technique introduced by clarke stern and wolenski f h clarke r j stern p r wolenski subgradient criteria for monotonicity the lipschitz condition and convexity canad j math 45 1993 1167 1183 for generating proximal subgradients of functions defined on a hilbert spaces is also extended to riemannian manifolds in order to provide that characterization a number of examples of lipschitz functions are presented so as to show that the lipschitz behavior of functions defined on riemannian manifolds depends on the riemannian metric

optimal control problems on manifolds a dynamic programming approach abstract dynamic programming identifies the value function of continuous time optimal control with a solution to the hamilton jacobi equation appropriately defined this relationship in turn leads to sufficient conditions of global optimality which have been widely used to confirm the optimality of putative minimisers in continuous time optimal control the dynamic programming methodology has been used for problems with state space a vector space however there are many problems of interest in which it is necessary to regard the state space as a manifold this paper extends dynamic programming to cover problems in which the state space is a general finite dimensional c manifold it shows that also in a manifold setting we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous lower bounded generalised solution of the hamilton jacobi equation the application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum

barrier operators and associated gradient like dynamical systems for constrained minimization problems we study some continuous dynamical systems associated with constrained optimization problems for that purpose we introduce the concept of elliptic barrier operators and develop a unified framework to derive and analyze the associated class of gradient like dynamical systems called a driven descent method a dm prominent methods belonging to this class include several continuous descent methods studied earlier in the literature such as steepest descent method continuous gradient projection methods and newton type methods as well as continuous interior descent methods such as lotka volterra type differential equations and riemannian gradient methods related discrete iterative methods such as proximal interior point algorithms based on bregman functions and second order homogeneous kernels can also be recovered within our framework and allow for deriving some new and interesting dynamics we prove global existence and strong viability results of the corresponding trajectories of a dm for a smooth objective function when the objective function is convex we analyze the asymptotic behavior at infinity of the trajectory produced by the proposed class of dynamical systems a dm in particular we derive a general criterion ensuring the global convergence of the trajectory of a dm to a minimizer of a convex function over a closed convex set this result is then applied to several dynamics built upon specific elliptic barrier operators throughout the paper our results are illustrated with many examples

the robustness of spontaneous symmetry breaking in a bridge model a simple two species asymmetric exclusion model in one dimension with bulk and boundary exchanges of particles is investigated for the existence of spontaneous symmetry breaking the model is a generalization of the bridge model for which earlier studies have confirmed the existence of symmetry broken phases and the motivation here is to check the robustness of the observed symmetry breaking with respect to additional dynamical moves in particular the boundary exchange of the two species of particle our analysis based on general considerations mean field approximation and numerical simulations shows that the symmetry breaking in the bridge model is sustained for a range of values of the boundary exchange rate moreover the mechanism through which symmetry is broken is similar to that in the bridge model our analysis allows us to plot the complete phase diagram of the model demarcating regions of symmetric and symmetry broken phases

approximation and regularization of lipschitz functions convergence of the gradients we examine the possible extensions to the lipschitzian setting of the classical result on c 1 convergence first approximation if a sequence f n of functions of class c 1 from r n to r converges uniformly to a function f of class c 1 then the gradient of f is a limit of gradients of f n in the sense that graph f c lim inf n graph f n second regularization the functions f n can be chosen to be of class c and c 1 converging to f in the sense that lim n f n f f n f 0 in other words the space of c functions is dense in the space of c 1 functions endowed with the c 1 pseudo norm we first deepen the properties of warga s counterexample 1981 for the extension of the approximation part to the lipschitzian setting this part cannot be extended even if one restricts the approximation schemes to the classical convolution and the lasry lions regularization we thus make more precise various results in the literature on the convergence of sub differentials we then show that the regularization part can be extended to the lipschitzian setting namely if f r n r is a locally lipschitz function we build a sequence of smooth functions f n n n such that 0 in other words the space of c functions is dense in the space of locally lipschitz functions endowed with an appropriate lipschitz pseudo distance up to now rockafellar and wets 1998 have shown that the convolution procedure permits us to have the equality lim sup n graph f n graph f which cannot provide the exactness of our result as a consequence we obtain a similar result on the regularization of epi lipschitz sets with both functional and set parts we improve previous results in the literature on the regularization of functions and sets

revisiting normalized gradient descent fast evasion of saddle points the paper considers normalized gradient descent ngd a natural modification of classical gradient descent gd in optimization problems it is shown that contrary to gd ngd escapes saddle points quickly a serious shortcoming of gd in nonconvex problems is that it can take arbitrarily long to escape from the neighborhood of a saddle point in practice this issue can significantly slow the convergence of gd particularly in high dimensional nonconvex problems the paper focuses on continuous time dynamics it is shown that 1 ngd almost never converges to saddle points and 2 the time required for ngd to escape from a ball of radius r about a saddle point x is at most 5 sqrt kappa r where kappa is the condition number of the hessian of f at x as a simple application of these results a global convergence time bound is established for ngd under mild assumptions

inexact subgradient methods for quasi convex optimization problems in this paper we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi convex constrained optimization problem the inexactness stems from computation errors and noise which come from practical considerations and applications assuming that the computational errors and noise are deterministic and bounded we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima in both cases using the constant and diminishing stepsize rules we describe convergence results in both objective values and iterates and finite convergence to approximate optimality we also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the cobb douglas production efficiency problem the numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm especially for the large scale problems

onsager relations and eulerian hydrodynamic limit for systems with several conservation laws we present the derivation of the hydrodynamic limit under eulerian scaling for a general class of one dimensional interacting particle systems with two or more conservation laws following yau s relative entropy method it turns out that in case of more than one conservation laws in order that the system exhibit hydrodynamic behaviour some particular identities reminiscent of onsager s reciprocity relations must hold we check validity of these identities whenever a stationary measure with product structure exists it also follows that as a general rule the equilibrium thermodynamic entropy as function of the densities of the conserved variables is a globally convex lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit as concrete examples we also present a number of models modeling deposition or domain growth phenomena the onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel the fact that equilibrium thermodynamic entropy is lax entropy for the arising euler equations was noticed earlier in the context of hamiltonian systems with weak noise see ref 7

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

pairwise balance and invariant measures for generalized exclusion processes we characterize the steady state of a driven diffusive lattice gas in which each site holds several particles and the dynamics is activated and asymmetric using a quantum hamiltonian formalism we show that for arbitrary transition rates the model has product invariant measure in the steady state a pairwise balance condition is shown to hold configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to pairwise balance is more general than the condition of detailed balance and holds in the non equilibrium steady state of a number of stochastic models

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

counting dihedral and quaternionic extensions we give asymptotic formulas for the number of biquadratic extensions of that admit a quadratic extension which is a galois extension of with a prescribed galois group for example with a galois group isomorphic to the quaternionic group our approach is based on a combination of the theory of quadratic equations with some analytic tools such as the siegel walfisz theorem and the double oscillations theorem

generalized gradients and characterization of epi lipschitz sets in riemannian manifolds abstract in this paper a notion of generalized gradient on riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented a characterization of the tangent cone to a nonempty subset s of a riemannian manifold m at a point x is obtained then these results are applied to characterize epi lipschitz subsets of complete riemannian manifolds

p ell torsion points in finite abelian groups and combinatorial identities abstract the main aim of this article is to compute all the moments of the number of p l torsion elements in some type of finite abelian groups the averages involved in these moments are those defined for the cohen lenstra heuristics for class groups and their adaptation for tate shafarevich groups in particular we prove that the heuristic model for tate shafarevich groups is compatible with the recent conjecture of poonen and rains about the moments of the orders of p selmer groups of elliptic curves for our purpose we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way moreover from our probabilistic model we derive combinatorial identities some of which appearing to be new the others being related to the theory of symmetric functions in some sense our method therefore gives for these identities a somehow natural algebraic context

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

pseudo jacobian and global inversion of nonsmooth mappings on riemannian manifolds abstract we introduce a notion of pseudo jacobian for continuous nonsmooth mappings between riemannian manifolds then we discuss the global inversion of continuous mappings between riemannian manifolds which may be non locally lipschitz for this purpose we obtain a version of the hadamard integral condition for invertibility by using pseudo jacobian

primal dual optimization algorithms over riemannian manifolds an iteration complexity analysis in this paper we study nonconvex and nonsmooth multi block optimization over euclidean embedded smooth riemannian submanifolds with coupled linear constraints such optimization problems naturally arise from machine learning statistical learning compressive sensing image processing and tensor pca among others by utilizing the embedding structure we develop an admm like primal dual approach based on decoupled solvable subroutines such as linearized proximal mappings where the duality is with respect to the embedded euclidean spaces first we introduce the optimality conditions for the afore mentioned optimization models then the notion of epsilon stationary solutions is introduced as a result the main part of the paper is to show that the proposed algorithms possess an iteration complexity of o 1 epsilon 2 to reach an epsilon stationary solution for prohibitively large size tensor or machine learning models we present a sampling based stochastic algorithm with the same iteration complexity bound in expectation in case the subproblems are not analytically solvable a feasible curvilinear line search variant of the algorithm based on retraction operators is proposed finally we show specifically how the algorithms can be implemented to solve a variety of practical problems such as the np hard maximum bisection problem the ell q regularized sparse tensor principal component analysis and the community detection problem our preliminary numerical results show great potentials of the proposed methods

exact shock measures and steady state selection in a driven diffusive system with two conserved densities we study driven 1d lattice gas models with two types of particles and nearest neighbor hopping we find the most general case when there is a shock solution with a product measure which has a density profile of a step function for both densities the position of the shock performs a biased random walk we calculate the microscopic hopping rates of the shock we also construct the hydrodynamic limit of the model and solve the resulting hyperbolic system of conservation laws in case of open boundaries the selected steady state is given in terms of the boundary densities

fibonacci family of dynamical universality classes universality is a well established central concept of equilibrium physics however in systems far away from equilibrium a deeper understanding of its underlying principles is still lacking up to now a few classes have been identified besides the diffusive universality class with dynamical exponent formula see text another prominent example is the superdiffusive kardar parisi zhang kpz class with formula see text it appears e g in low dimensional dynamical phenomena far from thermal equilibrium that exhibit some conservation law here we show that both classes are only part of an infinite discrete family of nonequilibrium universality classes remarkably their dynamical exponents formula see text are given by ratios of neighboring fibonacci numbers starting with either formula see text if a kpz mode exist or formula see text if a diffusive mode is present if neither a diffusive nor a kpz mode is present all dynamical modes have the golden mean formula see text as dynamical exponent the universal scaling functions of these fibonacci modes are asymmetric levy distributions that are completely fixed by the macroscopic current density relation and compressibility matrix of the system and hence accessible to experimental measurement

local convergence of the proximal point method for a special class of nonconvex functions on hadamard manifolds abstract local convergence analysis of the proximal point method for a special class of nonconvex functions on hadamard manifold is presented in this paper the well definedness of the sequence generated by the proximal point method is guaranteed moreover it is proved that each cluster point of this sequence satisfies the necessary optimality conditions and under additional assumptions its convergence for a minimizer is obtained

finite time consensus on strongly convex balls of riemannian manifolds with switching directed communication topologies abstract it is known that a consensus problem on any connected complete riemannian manifold can be transformed into the one on its strongly convex balls via the compression decompression along geodesics from the viewpoint of interior metrics this paper mainly provides a consensus protocol for strongly convex geodesic balls in which the communication can be switching and directed with the aid of nonsmooth analysis tools on riemannian manifolds our analysis shows that all dynamical points involved can achieve consensus in finite time meanwhile the corresponding global algorithm is given with its application to the consensus problem of rotation attitudes as well as a case simulation to demonstrate and verify our proposed techniques

on quasi convex duality the familiar fenchel moreau rockafellar duality scheme deduced from a conjugation is shown to be applicable to quasi convex problems here quasi affine functions take the place of affine functions the links with other quasi convex dualities are examined

phase transition in the abc model recent studies have shown that one dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules the mathrm abc model which comprises three particle species that diffuse asymmetrically around a ring shows anomalous coarsening into a phase separated steady state in the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one no phase separation occurs and the steady state of the system is disordered in the present work we consider the weak asymmetry regime q mathrm exp ensuremath ensuremath beta n where n is the system size and study how the disordered state is approached in the case of equal densities we find that the system exhibits a second order phase transition at some nonzero ensuremath beta c the value of ensuremath beta c 2 ensuremath pi sqrt 3 and the optimal profiles can be obtained by writing the exact large deviation functional for nonequal densities we write down mean field equations and analyze some of their predictions

applications of proximal calculus to fixed point theory on riemannian manifolds abstract we prove a general form of a fixed point theorem for mappings from a riemannian manifold into itself which are obtained as perturbations of a given mapping by means of general operations which in particular include the cases of sum when a lie group structure is given on the manifold and composition in order to prove our main result we develop a theory of proximal calculus in the setting of riemannian manifolds

the spinor genus of the integral trace let k be a number field of degree at least 3 in this article we show that the genus of the integral trace form of k contains only one spinor genus additionally we show that exactly 43 resp 29 resp 58 of quadratic resp real quadratic resp imaginary quadratic fields have the same property introduction let k be a number field the rational quadratic form x 7 trk q x has been extensively studied by several authors see for example ba bale c p ep ga mau mire and s for arithmetic purposes a finer invariant of a number field is its integral trace form i e the integral quadratic form obtained by restricting trk q x to the maximal order in k recent applications of the integral trace form can be found in the work of bhargava and shnidman see bhsh where they count cubic orders using the shape an invariant closely related to the integral trace form some other applications of the integral trace on cubic fields can be found in man and man2 given a non degenerate integral quadratic form q it is of great interest to study the number of spinor classes on its genus this is explained by the famous result of eichler eich which says that for an indefinite form q of dimension of at least 3 the spinor genus and the isometry class coincide in this article we analyze the spinor genus of the integral trace form of a number field our main theorem is the following theorem cf theorem 2 12 let k be a non quadratic number field then the genus of the integral trace form of k contains only one proper spinor genus in the case of quadratic fields it is not necessary true that the genus and the proper spinor genus of the integral trace coincide however they agree and differ

the parity of the period of the continued fraction of d we call a positive square free integer d special if d is not divisible by primes congruent to 3 mod 4 we show that the period of the expansion of ad in continued fractions is asymptotically more often odd than even when we restrict to special integers we note that this period is always even for a non special square free integer d it is well known that the above period is odd if and only if the negative pell equation x2 dy2 1 is solvable the latter problem is solvable if and only if the narrow and the ordinary class groups of q ad are equal in a prior work we fully described the asymptotics of the 4 ranks of those class groups here we get the first non trivial results about the asymptotic behavior of the 8 rank of the narrow class group for example we show that more than 76 of the quadratic fields q ad where d is special have the property that the 8 rank of the narrow class group is zero

viscosity solutions of minimization problems viscosity methods for minimization problems are revisited from some modern perspectives in variational analysis variational convergences for sequences of functions epi convergence gamma convergence mosco convergence and for sequences of operators graph convergence provide a flexible tool for such questions it is proved in a rather large setting that the solutions of the approximate problems converge to a viscosity solution of the original problem that is a solution that is minimal among all the solutions with respect to some viscosity criteria various examples coming from mathematical programming calculus of variations semicoercive elliptic equations phase transition theory hamilton jacobi equations singular perturbations and optimal control theory are considered

proximal point method for minimizing quasiconvex locally lipschitz functions on hadamard manifolds abstract in this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex locally lipschitz objective functions on hadamard manifolds to reach this goal we use the concept of clarke subdifferential on hadamard manifolds and assuming that the function is bounded from below we prove the global convergence of the sequence generated by the method to a critical point of the function

nonsmooth analysis on smooth manifolds we study infinitesimal properties of nonsmooth nondifferentiable functions on smooth manifolds the eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function a subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems nonclassical problems of calculus of variations and generalized solutions of first order partial differential equations on manifolds we also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions

on the negative pell equation we give asymptotic upper and lower bounds for the number of squarefree d 0 d x such that the equation x 2 dy 2 1 is solvable these estimates as usual can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm 1 and give strong evidence in the direction of a conjecture due to p stevenhagen

nonlinear fluctuating hydrodynamics in one dimension the case of two conserved fields we study the bs model which is a one dimensional lattice field theory taking real values its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges the bs model has two locally conserved fields the peak structure of their steady state space time correlations is determined through numerical simulations and compared with nonlinear fluctuating hydrodynamics which predicts a traveling peak with kpz scaling function and a standing peak with a scaling function given by the maximally asymmetric levy distribution with parameter alpha 5 3 as a by product we completely classify the universality classes for two coupled stochastic burgers equations with arbitrary coupling coefficients

a subgradient method for vector optimization problems vector optimization problems are a significant extension of scalar optimization and have many real life applications we consider an extension of the projected subgradient method to convex vector optimization which works directly with vector valued functions without using scalar valued objectives we eliminate the scalarization approach a popular strategy for solving vector optimization problems exploring strongly the structure of these kinds of problems under suitable assumptions we show that the sequence generated by the algorithm converges to a weakly efficient optimum point