the rogers ramanujan continual fraction abstract a survey of many theorems on the rogers ramanujan continued fraction is provided emphasis is given to results from ramanujan s lost notebook that have only recently been proved

dynkin s isomorphism theorem and the ray knight theorems in the case of diffusions we show that the isomorphism theorem of dynkin and the ray knight theorems can be derived from each other our proof uses additivity properties of squared bessel processes and an absolute continuity relation between squared bessel processes of dimensions one and three

solvability theory for a class of hemivariational inequalities involving copositive plus matrices applications in robotics the study of the equilibrium of an object robotic hand system including nonmonotone adhesive effects and nonclassical friction effects leads to new inequality methods in robotics the aim of this paper is to describe these inequality methods and provide a corresponding suitable mathematical theory

duality for e variational inequalities via the subdifferential calculus abstract based on the properties of the convex e subdifferential calculus we introduce to a general e variational inequality formulated with the help of a set valued operator and a perturbation function a dual one expressed by making use of the fenchel conjugate of the perturbation function under convexity hypotheses we show that the fulfillment of a regularity condition guarantees that the primal e variational inequality is solvable if and only if its dual one is solvable by particularizing the perturbation function we obtain several dual statements and we succeed to generalize and improve a duality scheme recently given by kum kim and lee an example justifying this generalization is also provided among the special instances of the general result we rediscover also the duality scheme concerning variational inequalities due to mosco

on reduced convex qp formulations of monotone lcps techniques for transforming convex quadratic programs qps into monotone linear complementarity problems lcps and vice versa are well known we describe a class of lcps for which a reduced qp formulation one that has fewer constraints than the standard qp formulation is available we mention several instances of this class including the known case in which the coefficient matrix in the lcp is symmetric

general over relaxed proximal point algorithm involving a maximal relaxed monotone mappings with applications abstract a general framework for the over relaxed proximal point algorithm based on the notion of a maximal relaxed monotonicity is developed and then the convergence analysis in the context of solving a general class of nonlinear implicit inclusion problems is explored furthermore some auxiliary results of interest involving a maximal relaxed monotone mappings in a hilbert space setting are included

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

approximation solvability of a class of nonlinear set valued variational inclusions involving a η monotone mappings abstract a new class of nonlinear set valued variational inclusions involving a monotone mappings in a hilbert space setting is introduced and then based on the generalized resolvent operator technique associated with a monotonicity the existence and approximation solvability of solutions using an iterative algorithm is investigated

convergence rates in forward backward splitting forward backward splitting methods provide a range of approaches to solving large scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out efforts at evaluating the convergence potential of such methods have so far relied on lipschitz properties and strong monotonicity or inverse strong monotonicity of the mapping involved in the forward step the perspective mainly being that of projection algorithms here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well for the first time in such a general setting global and local contraction rates are derived moreover they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected

variational composition of a monotone operator and a linear mapping with applications to elliptic pdes with singular coefficients abstract this paper proposes a regularized notion of a composition of a monotone operator with a linear mapping this new concept called variational composition can be shown to be maximal monotone in many cases where the usual composition is not the two notions coincide however whenever the latter is maximal monotone the utility of the variational composition is demonstrated by applications to subdifferential calculus theory of measurable multifunctions and elliptic pdes with singular coefficients

on a new system of nonlinear a monotone multivalued variational inclusions in this paper we introduce and study a new system of nonlinear a monotone multivalued variational inclusions in hilbert spaces by using the concept and properties of a monotone mappings and the resolvent operator technique associated with a monotone mappings due to verma we construct a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with a monotone mappings in hilbert spaces we also prove the existence of solutions for the nonlinear multivalued variational inclusions and the convergence of iterative sequences generated by the algorithm our results improve and generalize many known corresponding results

a framework of rogers ramanujan identities and their arithmetic properties the two rogers ramanujan q series where 0 1 play many roles in mathematics and physics by the rogers ramanujan identities they are essentially modular functions their quotient the rogers ramanujan continued fraction has the special property that its singular values are algebraic integral units we find a framework which extends the rogers ramanujan identities to doubly infinite families of q series identities if a 1 2 and m n 1 then we have infinite product modular function where the p x x q are hall littlewood polynomials these q series are specialized characters of affine kac moody algebras generalizing the rogers ramanujan continued fraction we prove in the case of a that the relevant q series quotients are integral units

a new relaxed proximal point procedure and applications to nonlinear variational inclusions a new relaxed algorithmic procedure based on the notion of a maximal relaxed monotonicity is developed and then the convergence analysis in the context of solving a general class of nonlinear inclusion problems is examined furthermore some results involving a maximal relaxed monotone mappings in a hilbert space setting are included

new approach to the η proximal point algorithm and nonlinear variational inclusion problems abstract a new approach using the over relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal monotonicity is developed and examined convergence analysis seems to be reasonable and finally some specializations are included furthermore the model developed in this communication seems to be appropriate to the yosida approximation in the sense that it can be applied to first order evolution equations inclusions as well

a ray knight theorem for symmetric markov processes let x be a strongly symmetric recurrent markov process with state space s and let l t x denote the local time of x at x in s for a fixed element 0 in the state space s let tau t inf s l 0 s t the 0 potential density u 0 x y of the process x killed at t 0 inf s x s 0 is symmetric and positive definite let eta eta x x in s be a mean zero gaussian process with covariance e eta eta x eta y u 0 x y the main result of this paper is the following generalization of the classical second ray knight theorem for any b in r and t 0 l x tau t frac 1 2 eta x b 2 x in s frac 1 2 eta x sqrt 2 b 2 2 x in s text in law a version of this theorem is also given when x is transient

general implicit variational inclusion problems based on a maximal m relaxed monotonicity amrm frameworks a general framework for an algorithmic procedure based on the variational convergence of operator sequences involving a maximal m relaxed monotone amrm mappings in a hilbert space setting is developed and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving a maximal m relaxed monotone mappings furthermore some specializations of interest on existence theorems and corresponding approximation solvability theorems on h maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature

mixed equilibrium problems sensitivity analysis and algorithmic aspect abstract the aim of this paper is twofold first it is to extend the sensitivity analysis framework developed recently for variational inequalities to mixed equilibrium problems the second is to propose iterative methods for solving this kind of problems in the process we establish an equivalence between an extended version of wiener hopf equations and the given problems relying on a generalization of the yosida approximation notion our results generalize results obtained for optimization variational inequalities complementarity problems and problems of nash equilibria

some theorems on the rogers ramanujan continued fraction in ramanujan s lost notebook in his first two letters to g h hardy and in his notebooks ramanujan recorded many theorems about the rogers ramanujan continued fraction in his lost notebook he offered several further assertions the purpose of this paper is to provide proofs for many of the claims about the rogers ramanujan and generalized rogers ramanujan continued fractions found in the lost notebook these theorems involve among other things modular equations transformations zeros and class invariants

sensitivity analysis for generalized nonlinear parametric a η m maximal monotone operator inclusion systems with relaxed cocoercive type operators abstract by using lim s inequalities nadler s results the new parametric resolvent operator technique associated with a m maximal monotone operators in this paper the existence theorem for a new class of generalized nonlinear parametric a m maximal monotone operator inclusion systems with relaxed cocoercive type operators in hilbert spaces is analyzed and established our results generalize sensitivity analysis results of other recent works on strongly monotone quasi variational inclusions nonlinear implicit quasi variational inclusions and nonlinear mixed quasi variational inclusion systems in hilbert spaces

a new system of variational inclusions with h η monotone operators in hilbert spaces in this paper we introduce and study a new system of variational inclusions involving h h monotone operators in hilbert space using the resolvent operator associated with h h monotone operators we prove the existence and uniqueness of solutions for this new system of variational inclusions we also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm

the primal douglas rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem we apply the douglas rachford splitting algorithm to a class of multi valued equations consisting of the sum of two monotone mappings compared with the dual application of the same algorithm which is known as the alternating direction method of multipliers the primal application yields algorithms that seem somewhat involved however the resulting algorithms may be applied effectively to problems with certain special structure in particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem

a new method for solving a system of generalized nonlinear variational inequalities in banach spaces abstract the purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem under suitable conditions some existence and strong convergence theorems are established in uniformly smooth and strictly convex banach spaces the results presented in the paper improve and extend some recent results

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

modularity of a ramanujan selberg continued fraction abstract we study a ramanujan selberg continued fraction s by employing the modular function theory we first find modular equations of s of level n for every positive integer n by using affine models of modular curves this is an extension of baruah saikia s results for level n 3 5 and 7 we further show that the ray class field modulo 4 over an imaginary quadratic field k is obtained by the value of s 2 and we prove the integrality of 1 s to find its class polynomial for k with k h where h is the complex upper half plane

warped proximal iterations for monotone inclusions abstract resolvents of set valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions we propose a generalization of this notion called warped resolvent which is constructed with the help of an auxiliary operator the properties of warped resolvents are investigated and connections are made with existing notions abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems

on the divergence of the rogers ramanujan continued fraction on the unit circle this paper studies ordinary and general convergence of the rogers ramanujan continued fraction let the continued fraction expansion of any irrational number t 0 1 be denoted by 0 e 1 t e 2 t and let the i th convergent of this continued fraction expansion be denoted by c i t d i t let s t 0 1 e i 1 t d i t infinitely often where 5 1 2 let y s exp 2 it t e s it is shown that if y e y s then the rogers ramanujan continued fraction r y diverges at y s is an uncountable set of measure zero it is also shown that there is an uncountable set of points g c y s such that if y g then r y does not converge generally it is further shown that r y does not converge generally for y 1 however we show that r y does converge generally if y is a primitive 5m th root of unity for some m e n combining this result with a theorem of i schur then gives that the continued fraction converges generally at all roots of unity

a characterization of boundary conditions yielding maximal monotone operators abstract we provide a characterization for maximal monotone realizations for a certain class of nonlinear operators in terms of their corresponding boundary data spaces the operators under consideration naturally arise in the study of evolutionary problems in mathematical physics we apply our abstract characterization result to port hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco elasticity

a monotone skew splitting model for composite monotone inclusions in duality the principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew adjoint operator an algorithmic framework is developed for solving this generic problem in a hilbert space setting new primal dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators and convergence results are established these algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration comparisons with existing methods are made and applications to composite variational problems are demonstrated

on the range of the douglas rachford operator the problem of finding a minimizer of the sum of two convex functions or more generally that of finding a zero of the sum of two maximally monotone operators is of central importance in variational analysis perhaps the most popular method of solving this problem is the douglas rachford splitting method surprisingly little is known about the range of the douglas rachford operator in this paper we set out to study this range systematically we prove that for 3 monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators a similar formula holds for the range of the corresponding displacement mapping we discuss applications to subdifferential operators to the infimal displacement vector and to firmly nonexpansive mappings various examples and counterexamples are presented including some concerning the celebrated brezis haraux theorem

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

a three term theta function identity and its applications in this paper we establish a three term theta function identity using the complex variable theory of elliptic functions this simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities from which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived we also give a new proof of the addition theorem for the weierstrass elliptic function an identity involving the products of four theta functions is given and from which one theta function identity by mccullough and shen is derived the quintuple product identity is used to derive some eisenstein series identities found in ramanujan s lost notebook and our approach is different from that of berndt and yee the proofs are self contained and elementary

generalized eckstein bertsekas proximal point algorithm based on a maximal monotonicity design a general design for the eckstein bertsekas proximal point algorithm using the notion of the a maximal monotonicity is developed convergence analysis for the generalized eckstein bertsekas proximal point algorithm in the context of solving a class of nonlinear inclusion problems is explored some auxiliary results of interest involving a maximal monotone mappings are also included the obtained results generalize investigations on general maximal monotonicity and beyond

mann and ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi variational inclusions abstract in this paper we introduce a new class of generalized nonlinear implicit quasi variational inclusions and prove its equivalence with a class of fixed point problems by making use of the properties of maximal monotone we also prove the existence of solutions for this generalized nonlinear implicit quasi variational inclusions and the convergence of iterative sequences generated by the perturbed algorithms

new perspectives on ray s theorem for the local times of diffusions a new global isomorphism theorem is obtained that expresses the local times of transient regular diffusions under px y in terms of related gaussian processes this theorem immediately gives an explicit description of the local times of diffusions in terms of 0th order squared bessel processes similar to that of eisenbaum and ray s classical description in terms of certain randomized fourth order squared bessel processes the proofs given are very simple they depend on a new version of kac s lemma for h transformed markov processes and employ little more than standard linear algebra the global isomorphism theorem leads to an elementary proof of the markov property of the local times of diffusions and to other recent results about the local times of general strongly symmetric markov processes the new version of kac s lemma gives simple short proofs of dynkin s isomorphism theorem and an unconditioned isomorphism theorem due to eisenbaum

the over relaxed proximal point algorithm based on h maximal monotonicity design and applications first the general framework for a generalized over relaxed proximal point algorithm using the notion of h maximal monotonicity also referred to as h monotonicity is developed and then the convergence analysis for this algorithm in the context of solving a general class of nonlinear inclusion problems is examined along with some auxiliary results on the resolvent operators corresponding to h maximal monotonicity

composition duality and maximal monotonicity this paper shows how the attouch thera duality principle for operator inclusions can be extended to compositions of multifunctions so that the primal and dual inclusions may involve operators between different pairs of spaces we first present the extension and give an example from computational economics to demonstrate that it has practical utility then we consider a particular case often found in applications involving real hilbert spaces and maximal monotone operators we show that in this case our duality framework includes those previously developed by rockafellar mosco and gabay finally we demonstrate that in this case under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved this proof uses an apparently new criterion for maximal monotonicity of operators of the form l tl where t is maximal monotone and l is linear and continuous with adjoint l

convergence analysis of new over relaxed proximal point algorithm frameworks with errors and applications to general a monotone nonlinear inclusion forms abstract the purpose of this paper is to introduce and study a new class of over relaxed proximal point algorithm frameworks with errors based on general a monotonicity further by using alber s inequalities the definition of normalized duality mapping on the dual spaces of banach spaces and the new proximal mapping technique associated with the general a monotone operators we discuss the approximation solvability of general a monotone nonlinear inclusion forms in banach spaces and prove the convergence analysis of iterative sequences generated by the algorithm frameworks via applying the lipschitz continuity of m 1 that is the inverse of multi valued operator m and the lipschitz continuity of proximal mapping associated with the general a monotone operators respectively finally some applications are given to show that the results presented in this paper improve generalize and unify the corresponding results of recent works

the explicit formulas and evaluations of ramanujan s theta function ψ abstract we define two quotients of theta function depending on two positive real parameters we then show how they are connected with two parameters of dedekind eta function theta function and the ramanujan weber class invariants explicit formulas for determining values of the theta function are derived and several examples will be given in addition we give some applications of these parameters for the famous rogers ramanujan continued fraction r q ramanujan s cubic continued fraction g q and the modular j invariant