convergence rates for forward backward dynamical systems associated with strongly monotone inclusions abstract we investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and lipschitz continuous one in a real hilbert space we show that these trajectories strongly converge with exponential rate to a zero of the sum provided the latter is strongly monotone we derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper convex and lower semicontinuous function with a smooth convex one provided the objective function fulfills a strong convexity assumption in the particular case of minimizing a smooth and strongly convex function we prove that its values converge along the trajectory to its minimum value with exponential rate too

homotopy approach for solving constrained optimization problems a homotopy approach for solving constrained parameter optimization problems is examined the first order necessary conditions with the complementarity conditions represented using a technique due to mangasarian 1967 are solved the equations are augmented to avoid singularities which occur when the active constraint changes the chow yorke 1978 algorithm is used to track the homotopy path leading to the solution to the desired problem at the terminal point a simple example which illustrates the technique and an application to a fuel optimal orbital transfer problem are presented

exploring collision free path planning by using homotopy continuation methods autonomous and semi autonomous robots play significant roles in space and terrestrial exploration even more in unfavorable and dangerous environments although recent advances allow robots to evolve in many such environments one of the most important problems remains the establishment of collision free trajectories in static or partially temporal static environments this paper presents a different approach to address this problem proposing a methodology based on homotopy continuation methods hcm capable of generating collision free trajectories in two and three dimensions the basic idea behind the proposal relies on the construction of a nonlinear equation representing the map of the environment making it possible to apply hcm methods to obtain collision free paths a series of simulations are presented to show the effectiveness of the method avoiding circular semi rectangular spherical shaped and semi parallelepipeds obstacles

borcherds products and arithmetic intersection theory on hilbert modular surfaces we prove an arithmetic version of a theorem of hirzebruch and zagier saying that hirzebruch zagier divisors on a hilbert modular surface are the coecients of an elliptic modular form of weight two moreover we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its petersson metric on a regular model of a hilbert modular surface and study faltings heights of arithmetic hirzebruch zagier divisors

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria let h r be a c1 function on a real hilbert space h and let 0 be a positive damping parameter for any control function e r r which tends to zero as t we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator hbfc x t x t x t e t x t 0 we show that if e t does not tend to zero too rapidly as t then the term e t x t asymptotically acts as a tikhonov regularization which forces the trajectories to converge to a particular equilibrium indeed in the main result of this paper it is established that when is convex and s argmin under the key assumption that e is a slow control i e 0 e t dt then each trajectory of the hbfc system strongly converges as t to the element of minimal norm of the closed convex set s as an application we consider the damped wave equation with neumann boundary condition formula

homotopy perturbation method for a type of nonlinear coupled equations with parameters derivative abstract in this paper homotopy perturbation method is directly extended to investigate nonlinear coupled equations with parameters derivative and to derive their numerical solutions these nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations by choosing different values of the parameters in general formal numerical solutions as a result a very rapidly convergent series solution is obtained the efficiency and accuracy of the method are verified by using two famous examples coupled burgers and mkdv equations numerical solutions show that good results have been achieved

newton homotopy analysis method for nonlinear equations abstract in this paper we present an efficient numerical algorithm for solving nonlinear algebraic equations based on newton raphson method and homotopy analysis method also we compare homotopy analysis method with adomian s decomposition method and homotopy perturbation method some numerical illustrations are given to show the efficiency of algorithm

application of homotopy perturbation methods for solving systems of linear equations abstract in this paper homotopy perturbation methods hpms are applied to obtain the solution of linear systems and conditions are deduced to check the convergence of the homotopy series moreover we have adapted the richardson method the jacobi method and the gauss seidel method to choose the splitting matrix the numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius

theory of globally convergent probability one homotopies for nonlinear programming for many years globally convergent probability one homotopy methods have been remarkably successful on difficult realistic engineering optimization problems most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective convergence theory has been derived for a few particular problems and considerable fixed point theory exists but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking this paper derives some probability one homotopy convergence theorems for unconstrained and inequality constrained optimization for linear and nonlinear inequality constraints and with and without convexity some insight is provided into why the homotopies used in engineering practice are so successful and why this success is more than dumb luck by presenting the theory as variations on a prototype probability one homotopy convergence theorem the essence of such convergence theory is elucidated

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

probability one homotopies in computational science probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that under mild assumptions are globally convergent for a wide range of problems in science and engineering convergence theory robust numerical algorithms and production quality mathematical software exist for general nonlinear systems of equations and special cases such as brouwer fixed point problems polynomial systems and nonlinear constrained optimization using a sample of challenging scientific problems as motivation some pertinent homotopy theory and algorithms are presented the problems considered are analog circuit simulation for nonlinear systems reconfigurable space trusses for polynomial systems and fuel optimal orbital rendezvous for nonlinear constrained optimization the mathematical software packages hompack90 and polsys plp are also briefly described

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

appendix to the iwasawa conjecture for totally real fields by a wiles arithmetic minimal compactification of the hilbert blumenthal moduli spaces the arithmetic theory of toroidal compactification of the hilbert blumenthal moduli spaces was described in rapoport s thesis rap while the arithmetic theory of minimal compactification has been lacking for a long time after faltings work on the arithmetic compactification of siegel moduli spaces fa it was obvious to the experts that the same things could be done in the hilbertblumenthal case in similar but simpler ways in fact what emerged was a machinery to compactify moduli spaces coming from abelian varieties it turns out that the arithmetic minimal compactification of the hilbert blumenthal moduli spaces could have been done more than ten years ago rap contains everything that is needed so in some sense i am writing a summary and an appendix to rap the purpose of this appendix is to explain this remark and show that this theory of minimal compactification can be useful in number theory for instance one can replace for almost all primes by for all primes outside the level and the discriminant in the main result of br la two more applications are given here one about p adic monodromy the other about positivity of the hodge line bundle the sections of which are hilbert modular forms for a systematic exposition of the theory of semi abelian degeneration and compactification the reader is referred to the book fa ch where the more

proximal point algorithm controlled by a slowly vanishing term applications to hierarchical minimization let phi 0 mathbb r n to mathbb r cup infty be a closed convex function and phi 1 mathbb r n to mathbb r be a finite convex function that are bounded from below our goal is to build an algorithm which first minimizes the map phi 0 and secondly the map phi 1 over the set s 0 rm argmin phi 0 for that purpose we define the following proximal type algorithm x n 1 x n lambda n in partial eta n phi 0 varepsilon n phi 1 x n 1 leqno mathcal a 1 where lambda n is a positive step sequence eta n is a summable error sequence and varepsilon n is a control sequence tending toward 0 partial eta denotes the eta approximate subdifferential when varepsilon n is a slow control i e sum n 0 infty varepsilon n infty we prove that under adequate conditions the sequence x n defined by mathcal a 1 tends toward an element of s 1 rm argmin s 0 phi 1 more generally given finite convex functions phi 2 ldots phi n mathbb r n to mathbb r let us define the sets s i i in 1 ldots n by the recursive relation s i rm argmin s i 1 phi i we introduce an extension of algorithm mathcal a 1 to minimize hierarchically each function phi i on the set s i 1 for i in 1 ldots n

strong asymptotic convergence of evolution equations governed by maximal monotone operators with tikhonov regularization abstract we consider the tikhonov like dynamics u t a u t e t u t where a is a maximal monotone operator on a hilbert space and the parameter function e t tends to 0 as t with 0 e t d t when a is the subdifferential of a closed proper convex function f we establish strong convergence of u t towards the least norm minimizer of f in the general case we prove strong convergence towards the least norm point in a 1 0 provided that the function e t has bounded variation and provide a counterexample when this property fails

asymptotic behavior of coupled dynamical systems with multiscale aspects abstract we study the asymptotic behavior as time variable t goes to of nonautonomous dynamical systems involving multiscale features as a benchmark case given h a general hilbert space h r and h r two closed convex functions and a function of t which tends to as t goes to we consider the differential inclusion x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled systems we show several results ranging from weak ergodic to strong convergence of the trajectories as a key ingredient we assume that for every p belonging to the range of n c 0 t p t c p t d t where is the fenchel conjugate of c is the support function of c argmin and n c x is the normal cone to c at x as a by product we revisit the system x t t x t x t 0 where t tends to zero as t goes to and 0 t d t whose asymptotic behavior can be derived from the preceding one by time rescaling applications are given in game theory optimal control variational problems and pdes

asymptotic behavior of solutions to a class of nonlinear evolution equations on etudie les equations de la chaleur a controle de temperature a l interieur ou sur la frontiere du domaine pour cela on considere un type de problemes variationnels paraboliques avec obstacles

weak convergence of a relaxed and inertial hybrid projection proximal point algorithm for maximal monotone operators in hilbert space this paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a hilbert space which unifies three previously studied strategies relaxation inertial type extrapolation and projection step the first two strategies are intended to speed up the convergence of the standard proximal point algorithm while the third permits one to perform inexact proximal iterations with fixed relative error tolerance the paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters

a new class of alternating proximal minimization algorithms with costs to move given two objective functions f mathcal x mapsto r cup infty and g mathcal y mapsto r cup infty on abstract spaces mathcal x and mathcal y and a coupling function c mathcal x times mathcal y mapsto r we introduce and study alternative minimization algorithms of the following type x 0 y 0 in mathcal x times mathcal y mbox given x n y n rightarrow x n 1 y n rightarrow x n 1 y n 1 mbox as follows left begin array l x n 1 in mbox argmin f xi beta n c xi y n alpha n h x n xi xi in mathcal x y n 1 in mbox argmin g eta mu n c x n 1 eta nu n k y n eta eta in mathcal y end array right their most original feature is the introduction of the terms h mathcal x times mathcal x mapsto r and k mathcal y times mathcal y mapsto r which are costs to change or to move distance like functions relative entropies accounting for various inertial friction or anchoring effects these algorithms are studied in a general abstract framework the introduction of the costs to change h and k leads to proximal minimizations with corresponding dissipative effects as a result the algorithms enjoy nice convergent properties coefficients alpha n beta n mu n nu n are nonnegative parameters when taking alpha n nu n 0 and quadratic costs on a hilbert space one recovers the classical alternating minimization algorithm which itself is a natural extension of the alternating projection algorithm of von neumann a number of new significant results hold in general metric spaces we pay particular attention to the following cases 1 mathcal x d mathcal x and mathcal y d mathcal y are complete metric spaces and h geq d mathcal x k geq d mathcal y high local costs to move the algorithms then provide sequences that converge to nash equilibria 2 mathcal x mathcal y mathcal h is a hilbert space the costs to change are quadratic low local costs to move and the functions f g mathcal h mapsto r cup infty are closed convex proper then some of the classical convergence theorems for alternating convex minimization algorithms including those of acker and prestel are properly extended with original proofs

a homotopy approach for nonlinear control synthesis in this paper the authors propose a control method for nonlinear systems which relies on the homotopy method for solving nonlinear equations and tangent linearized control systems they present a mathematically rigorous treatment of the theoretical foundations including a constructive proof of the exponential error decay for the controlled system which contains an explicit control design instruction for illustration this design instruction is applied to an example system the behavior of the controlled system is examined by numerical simulations to demonstrate the advantages and drawbacks of the proposed method they include a short comparison with two usual nonlinear control techniques by means of the considered example system

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria a note from l 1 to non l 1 in h attouch m o czarnecki asymptotic control and stabilization of non linear oscillators with non isolated equilibria j differential equations 179 2002 278 310 we exhibited a sharp condition ensuring the efficiency of the tikhonov like control term in the hbfc system precisely let h r be a c1 function on a real hilbert space h let 0 be a positive damping parameter and let r r be a control function which decreases to zero as t in order to select particular equilibria in the important case where has non isolated equilibria we introduced in h attouch m o czarnecki j differential equations 179 2002 278 310 the following damped nonlinear oscillator and studied its asymptotic behavior hbfc x t x t x t t x t 0 we established that when is convex and s argmin under the key assumption that is a slow control i e 0 t dt then each trajectory of the hbfc system strongly converges when t to the element of minimal norm of the closed convex set s the condition on the control term is sharp indeed when 0 t dt the trajectory weakly converges but it may not strongly converge and we have no information a priori on the weak limit in this note we give an answer to the following question when does an l1 control term becomes or behave non l1 precisely take a control term l1 let x be the solution of the corresponding hbfc system take n to be a non increasing truncation of n t t for t 0 n let x be the solution of the corresponding hbfc system we show that limn xn x 0 in particular the weak limits of the trajectories xn strongly converge when n to the strong limit of the trajectory x in other words there is no loss of the information gained by the slow behavior for t n

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

homotopy analysis method for limit cycle flutter of airfoils an analytical approximate method for non linear problems namely the homotopy analysis method is for the first time employed to propose a new approach for the flutter system of a two dimensional airfoil with a cubic structural nonlinearity the frequencies and amplitudes of limit cycle flutters are described as maclaurin series of an embedding parameter a series of algebraic equations in the priori unknown coefficients of the series are derived all the equations are linear except the first one this provides us with a simple iteration scheme to seek the solutions to any desired accuracy numerical examples are presented to illustrate the validity and efficiency of the proposed approach the approximations of the frequencies and amplitudes of the limit cycles are obtained very accurately furthermore it is proved the frequencies are independent of the coefficient of the cubic nonlinearity while the amplitudes are in inverse proportion to the square root of the coefficient

e adic and x adic representations associated to abelian varieties defined over number fields 0 introduction let a be an abelian variety over a number field k of dimension n 1 and let g gal kik for each prime number let vi a be the qe adic tate module of a one general problem is to study the image of the e adic representation vr a of g let ge be the e adic lie algebra of the image of this f adic representation fix an embedding cr k c c and denote by a c the abelian variety a x c in 12 mumford and tate conjectured that ge m 0q qe where mn is the lie algebra of the mumford tate group mt a c cf 9 section 3 as is now well known in general g is contained in m 0q qe cf 9 16 various results toward this general problem have been obtained for some classes of abelian varieties cf 22 section 4 more recently some important general results on e adic representations attached to abelian varieties have been established especially one has the following results i the rank of g is independent of c serre 24 zarhin 34 ii geis algebraic and contains the homotheties bogomolov 1 iii g is reductive and end ve a endk a 0 qe faltings 10 these make the determination of g possible for some other classes of abelian varieties for example when d dim a is odd and endt a z j p serre has proved that ge m 0q qe sp 2d qe 3 qe id where id is the 2d x 2d identity matrix cf 25 the purpose of this article is to extend serre s method to some other cases as a result ge can be determined for the following types of simple abelian varieties

approaching the solving of constrained variational inequalities via penalty term based dynamical systems abstract we investigate the existence and uniqueness of locally absolutely continuous trajectories of a penalty term based dynamical system associated to a constrained variational inequality expressed as a monotone inclusion problem relying on lyapunov analysis and on the ergodic continuous version of the celebrated opial lemma we prove weak ergodic convergence of the orbits to a solution of the constrained variational inequality under investigation if one of the operators involved satisfies stronger monotonicity properties then strong convergence of the trajectories can be shown

probability one homotopy methods for constrained clustering abstract many algorithms for constrained clustering have been developed in the literature that aim to balance vector quantization requirements of cluster prototypes against the discrete satisfaction requirements of constraint must link or must not link sets a significant amount of research has been devoted to designing new algorithms for constrained clustering and understanding when constraints help clustering however no method exists to systematically characterize solution sets as constraints are gently introduced and how to assist practitioners in choosing a sweet spot between vector quantization and constraint satisfaction a homotopy method is presented that can smoothly track solutions from unconstrained to constrained formulations of clustering beginning the homotopy zero curve tracking where the solution is fairly well understood the curve can then be tracked into regions where there is only a qualitative understanding of the solution set finding multiple local solutions along the way experiments demonstrate how the new homotopy method helps identify better tradeoffs and reveals insight into the structure of solution sets not obtainable using pointwise exploration of parameters

penalization in non classical convex programming via variational convergence a penalty method for convex functions which cannot necessarily be extended outside their effective domains by an everywhere finite convex function is proposed and combined with the proximal method proofs of convergence rely on variational convergence theory

alternating proximal algorithms for linearly constrained variational inequalities application to domain decomposition for pde s abstract let x y z be real hilbert spaces let f x r g y r be closed convex functions and let a x z b y z be linear continuous operators let us consider the constrained minimization problem p min f x g y a x b y given a sequence n which tends toward 0 as n we study the following alternating proximal algorithm a x n 1 argmin n 1 f 1 2 a b y n z 2 2 x n x 2 x y n 1 argmin n 1 g 1 2 a x n 1 b z 2 2 y n y 2 y where and are positive parameters it is shown that if the sequence n tends moderately slowly toward 0 then the iterates of a weakly converge toward a solution of p the study is extended to the setting of maximal monotone operators for which a general ergodic convergence result is obtained applications are given in the area of domain decomposition for pde s

stabilization of oscillators subject to dry friction finite time convergence versus exponential decay results we investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion s x t x t f x t 0 t 0 where f r n r is a smooth potential and r n r is a convex function the friction is modelized by the subdifferential term x when 0 int 0 dry friction condition it was shown by adly attouch and cabot 2006 that the unique solution to s converges in a finite time toward an equilibrium state too provided that f x int 0 in this paper we study the delicate case where the vector f x belongs to the boundary of the set 0 we prove that either the solution converges in a finite time or the speed of convergence is exponential when a b 2 2 a 0 b 0 we obtain the existence of a critical coefficient be 0 below which every solution stabilizes in a finite time it is also shown that the geometry of the set 0 plays a central role in the analysis

exceptional splitting of reductions of abelian surfaces heuristics based on the sato tate conjecture and the lang trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction we prove this result for abelian surfaces with real multiplication as in previous work by charles and elkies this shows that a density 0 set of primes pertaining to the reduction of abelian varieties is infinite the proof relies on the arakelov intersection theory on hilbert modular surfaces

adaptive multiprecision path tracking this article treats numerical methods for tracking an implicitly defined path the numerical precision required to successfully track such a path is difficult to predict a priori and indeed it may change dramatically through the course of the path in current practice one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved to avoid unnecessary computational cost it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path we present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates we then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations

convergence of a relaxed inertial proximal algorithm for maximally monotone operators in a hilbert space mathcal h given a mathcal h rightarrow 2 mathcal h a maximally monotone operator we study the convergence properties of a general class of relaxed inertial proximal algorithms this study aims to extend to the case of the general monotone inclusion ax ni 0 the acceleration techniques initially introduced by nesterov in the case of convex minimization the relaxed form of the proximal algorithms plays a central role it comes naturally with the regularization of the operator a by its yosida approximation with a variable parameter a technique recently introduced by attouch peypouquet math program ser b 2018 https doi org 10 1007 s10107 018 1252 x for a particular class of inertial proximal algorithms our study provides an algorithmic version of the convergence results obtained by attouch cabot j differ equ 264 7138 7182 2018 in the case of continuous dynamical systems

a convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming abstract we present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u t t u t where t t 0 is a family of closed proper convex functions the result is used to study the flow generated by the family t x f x r t where f x r ctx r exp aix bi r is the exponential penalty approximation of the linear program min ctx ax b and r t is a positive function tending to 0 when t we prove that the trajectory u t converges to an optimal solution u of the linear program and we give conditions for the convergence of an associated dual trajectory t toward an optimal solution of the dual program

the inverse kinematics problem of spatial 4p3r robot manipulator by the homotopy continuation method with an adjustable auxiliary homotopy function abstract in this paper the concept of an adjustable auxiliary homotopy function for the homotopy continuation method is presented by means of adjusting the auxiliary function we can solve non linear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the newton raphson method and so on

asymptotic behavior of gradient like dynamical systems involving inertia and multiscale aspects abstract in a hilbert space h we study the asymptotic behavior as time variable t goes to of nonautonomous gradient like dynamical systems involving inertia and multiscale features given h r and h r two convex differentiable functions a positive damping parameter and t a function of t which tends to zero as t goes to we consider the second order differential equation x t x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled nonlinear oscillators assuming that t tends to zero moderately slowly as t goes to infinity we show that the trajectories converge weakly in h the limiting equilibria are solutions of the hierarchical minimization problem which consists in minimizing over the set c of minimizers of as key assumptions we suppose that 0 t d t and that for every p belonging to a convex cone c depending on the data and 0 t p c t p d t where is the fenchel conjugate of and c is the support function of c an application is given to coupled oscillators

an arithmetic intersection formula on hilbert modular surfaces in this paper we obtain an explicit arithmetic intersection formula on a hilbert modular surface between the diagonal embedding of the modular curve and a cm cycle associated to a non biquadratic cm quartic field this confirms a special case of the author s conjecture with j bruinier in by and is a generalization of the beautiful factorization formula of gross and zagier on singular moduli as an application we proved the first non trivial non abelian chowla selberg formula a special case of colmez conjecture

solving the nonlinear equations by the newton homotopy continuation method with adjustable auxiliary homotopy function in this paper the concept of adjustable auxiliary homotopy function for the newton homotopy continuation method is presented by means of adjusting the auxiliary function we can solve the nonlinear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the newton raphson method and so on

threading homotopies and dc operating points of nonlinear circuits this paper studies continuation methods for finding isolated zeros of nonlinear functions given a nonlinear function f bbb r n rightarrow bbb r n a em threading homotopy is a function h mathbf x lambda bbb r n 1 rightarrow bbb r n with h mathbf x 0 equiv f mathbf x such that the zero set of h is a single connected curve containing all zeros of f mathbf x for a c 1 function f a necessary condition for the existence of a nondegenerate c 1 threading homotopy is that the topological degree of f mathbf x be 1 0 or 1 for c 2 mappings in all dimensions except possibly n 2 this condition is also a sufficient condition for existence of a c 2 threading homotopy which is weakly proper over 0 a homotopy h is em weakly proper over 0 if for every interval a b the set h 1 bf 0 cap bbb r n times a b is compact this condition rules out any part of the zero set escaping to infinity at a finite value of the homotopy parameter threading homotopies are potentially applicable in continuation methods for finding all dc operating points of nonlinear circuits we show that most transistor circuits have dc operating point equations f mathbf x bf 0 with deg f pm 1 so that threading homotopies exist in principle for such operating point equations the explicit construction of such threading homotopies remains an open problem

a modified formula of ancient chinese algorithm by the homotopy continuation technique a modified formula iteration representation of ancient chinese algorithm by combining the homotopy continuation technique is presented considering the convergence the new iteration formula and the technique for the choice of the auxiliary homotopy function are developed once we have the condition f x 1 f x 2 or f x is an even function we will diverge in traditional ancient chinese algorithm the paper provides some useful rules to overcome this defect

second order forward backward dynamical systems for monotone inclusion problems we begin by considering second order dynamical systems of the from ddot x t gamma t dot x t lambda t b x t 0 where b cal h rightarrow cal h is a cocoercive operator defined on a real hilbert space cal h lambda 0 infty rightarrow 0 infty is a relaxation function and gamma 0 infty rightarrow 0 infty is a damping function both depending on time for the generated trajectories we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator b the framework allows us to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one this captures as a particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one furthermore we prove that when b is the gradient of a smooth convex function the value of the latter converges along the ergo

a dynamic gradient approach to pareto optimization with nonsmooth convex objective functions in a general hilbert framework we consider continuous gradient like dynamical systems for constrained multiobjective optimization involving non smooth convex objective functions our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions based on the yosida regularization of the subdi erential operators involved in the system we obtain the existence of strong global trajectories we prove a descent property for each objective function and the convergence of trajectories to weak pareto minima this approach provides a dynamical endogenous weighting of the objective functions applications are given to cooperative games inverse problems and numerical multiobjective optimization

generalized solutions for the sum of two maximally monotone operators a common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions a classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations in this paper we introduce a normal problem associated with finding a zero of the sum of two maximally monotone operators if the original problem admits solutions then the normal problem returns this same set of solutions the normal problem may yield solutions when the original problem does not admit any furthermore it has attractive variational and duality properties several examples illustrate our theory

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

convergence of damped inertial dynamics governed by regularized maximally monotone operators in a hilbert space setting we study the asymptotic behavior as time t goes to infinity of the trajectories of a second order differential equation governed by the yosida regularization of a maximally monotone operator with time varying positive index lambda t the dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient gamma t a suitable tuning of the parameters gamma t and lambda t makes it possible to prove the weak convergence of the trajectories towards zeros of the operator when the operator is the subdifferential of a closed convex proper function we estimate the rate of convergence of the values these results are in line with the recent articles by attouch cabot and attouch peypouquet in this last paper the authors considered the case gamma t frac alpha t which is naturally linked to nesterov s accelerated method we unify and often improve the results already present in the literature

compact structures in a class of nonlinearly dispersive equations with time fractional derivatives in this work variants of the kdv equation with time fractional derivatives which exhibits compactons solitons with compact support are investigated new solitary wave solutions with compact support are developed the homotopy perturbation method is employed to derive compact solutions for these variants the chosen initial solution or trial function plays a major role in changing the physical structure of the solution the proposed scheme can be used to construct compact and noncompact solutions for a wide class of nonlinear fractional evolution equations

errata computing all solutions to polynomial systems using homotopy continuation in a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems a major feature of this new approach is that the start system for the homotopy need not be a random or generic system also homotopy paths are strictly increasing in the homotopy parameter in this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies a feature of our implementation is that we eliminate divergent paths entirely we include performance statistics for homotopies derived from more traditional approaches for comparison generally the new approach is faster and more reliable

asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces abstract by suitably modifying the methods used by b djafari rouhani ergodic theorems for non expansive sequences in hilbert spaces and related problems part i pp 1 76 thesis yale university and a new proof of the weak convergence theorems for non expansive sequences and curves in hilbert spaces preprint 1984 and by b djafari rouhani and s kakutani ergodic theorems for non expansive non linear operators in a hilbert space preprint 1984 we study the asymptotic behaviour of the quasi autonomous dissipative system du dt au where a is a monotone operator in a hilbert space h and l 1 0 h