near equality near convexity sums of maximally monotone operators and averages of firmly nonexpansive mappings we study nearly equal and nearly convex sets ranges of maximally monotone operators and ranges and fixed points of convex combinations of firmly nonexpansive mappings the main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings a striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular throughout examples are provided to illustrate the theory we also obtain detailed information on the domain and range of the resolvent average

on the stability of the parallel sum of maximal monotone operators identifying operators with their graphs we study the continuity of the parallel sum of two monotone operators with respect to set convergence and we examine its stability from a quantitative point of view

a weaker regularity condition for subdifferential calculus and fenchel duality in infinite dimensional spaces abstract in this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping this condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior point regularity conditions given so far in the literature moreover it provides a weak sufficient condition for fenchel duality regarding convex optimization problems in infinite dimensional spaces as an application we discuss the strong conical hull intersection property chip for a finite family of closed convex sets

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

maximal monotonicity for the sum of two subdifferential operators in lp spaces abstract this paper is devoted to providing a sufficient condition for the maximality of the sum of subdifferential operators defined on reflexive banach spaces and proving the maximal monotonicity in l p l p of the nonlinear elliptic operator u m u u with a maximal monotone graph

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

universal t matrix representations of ospq 1 2 and little q jacobi polynomials we obtain a closed form expression of the universal t matrix encapsulating the duality between the quantum superalgebra uq osp 1 2 and the corresponding supergroup ospq 1 2 the classical q 1 limit of this universal t matrix yields the group element of the undeformed ospq 1 2 supergroup the finite dimensional representations of the quantum supergroup ospq 1 2 are readily constructed employing the above mentioned universal t matrix and the known finite dimensional representations of the dually related deformed uq osp 1 2 superalgebra proceeding further we derive the product law the recurrence relations and the orthogonality of the representations of the quantum supergroup ospq 1 2 it is shown that the entries of these representation matrices are expressed in terms of the little q jacobi polynomials with q q two mutually complementary singular maps of the universal t matrix on the universal r matrix are also presented

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

a class of high order runge kutta chebyshev stability polynomials the analytic form of a new class of factorized runge kutta chebyshev frkc stability polynomials of arbitrary order n is presented roots of frkc stability polynomials of degree l m n are used to construct explicit schemes comprising l forward euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to l 2 the associated stability domain scales as m 2 along the real axis marginally stable real valued points on the interior of the stability domain are removed via a prescribed damping procedure by construction frkc schemes meet all linear order conditions for nonlinear problems at orders above 2 complex splitting or butcher series composition methods are required linear order conditions of the frkc stability polynomials are verified at orders 2 4 and 6 in numerical experiments comparative studies with existing methods show the second order unsplit frkc2 scheme and higher order 4 and 6 split frkcs schemes are efficient for large moderately stiff problems

stabilized hp finite element methods for first order hyperbolic problems we analyze the hp version of the streamline diffusion finite element method sdfem and of the discontinuous galerkin finite element method dgfem for first order linear hyperbolic problems for both methods we derive new error estimates on general finite element meshes which are sharp in the mesh width h and in the spectral order p of the method assuming that the stabilization parameter is o h p for piecewise analytic solutions exponential convergence is established on quadrilateral meshes for the dgfem we admit very general irregular meshes and for the sdfem we allow meshes which contain hanging nodes numerical experiments confirm the theoretical results

kolmogorov equations in infinite dimensions well posedness and regularity of solutions with applications to stochastic generalized burgers equations we develop a new method to uniquely solve a large class of heat equations so called kolmogorov equations in infinitely many variables the equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper lyapunov type functions this way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally lipschitz drifts apart from general analytic interest the main motivation is to apply this to uniquely solve martingale problems in the sense of stroock varadhan given by stochastic partial differential equations from hydrodynamics such as the stochastic navier stokes equations in this paper this is done in the case of the stochastic generalized burgers equation uniqueness is shown in the sense of markov flows

a new condition for maximal monotonicity via representative functions abstract in this paper we give a weaker sufficient condition for the maximal monotonicity of the operator s a t a where s x x t y y are two maximal monotone operators a x y is a linear continuous mapping and x y are reflexive banach spaces we prove that our condition is weaker than the generalized interior point conditions given so far in the literature this condition is formulated using the representative functions of the operators involved in particular we rediscover some sufficient conditions given in the past using the so called fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator respectively

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

high order splitting schemes with complex timesteps and their application in mathematical finance high order splitting schemes with complex timesteps are applied to kolmogorov backward equations stemming from stochastic differential equations in the stratonovich form in the setting of weighted spaces the necessary analyticity of the split semigroups can easily be proved a numerical example from interest rate theory the cir2 model is considered the numerical results are robust for drift dominated problems and confirm our theoretical results

letter to the editor on borwein s paper adjoint process duality we give counterexamples for some statements of borwein and sufficient conditions for the validity of these

enlargements of positive sets in this paper we introduce the notion of enlargement of a positive set in ssd spaces to a maximally positive set a we associate a family of enlargements e a and characterize the smallest and biggest element in this family with respect to the inclusion relation we also emphasize the existence of a bijection between the subfamily of closed enlargements of e a and the family of so called representative functions of a we show that the extremal elements of the latter family are two functions recently introduced and studied by stephen simons in this way we extend to ssd spaces some former results given for monotone and maximally monotone sets in banach spaces

twist deformation of the rank one lie superalgebra the drinfeld twist is applied to deform the rank one orthosymplectic lie superalgebra the twist element is the same as for the sl 2 lie algebra due to the embedding of the sl 2 into the superalgebra the r matrix has the direct sum structure in the irreducible representations of the dual quantum group is defined using the frt formalism it includes the jordanian quantum group as subalgebra and grassmann generators as well

boundary half strips and the strong chip when the subdifferential sum rule formula holds for the indicator functions iota c and iota d of two closed convex sets c and d of a locally convex space x the pair c d is said to have the strong conical hull intersection property the strong chip the specification of a well known theorem due to moreau to the case of the support functionals sigma c and sigma d subsumes the fact that the pair c d has the strong chip whenever the inf convolution of sigma c and sigma d is exact in this article we prove in the setting of euclidean spaces that if the pair c d has the strong chip while the boundary of c does not contain any half strip then the inf convolution of sigma c and sigma d is exact moreover when the boundary of a closed and convex set c does contain a half strip it is possible to find a closed and convex set d such that the pair c d has the strong chip while the inf convolution of sigma c and sigma d is not exact the validity of the converse of moreau s theorem in euclidean spaces is thus associated with the absence of half strips within the boundary of concerned convex sets

convex analysis can be helpful for the asymptotic analysis of monotone operators asymptotic analysis of monotone operators we use representations of maximal monotone operators for studying recession or asymptotic operators associated to maximal monotone operators such a concept is useful for dealing with unboundedness

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

semigroup splitting and cubature approximations for the stochastic navier stokes equations approximation of the marginal distribution of the solution of the stochastic navier stokes equations on the two dimensional torus by high order numerical methods is considered the corresponding rates of convergence are obtained for a splitting scheme and the method of cubature on wiener space applied to a spectral galerkin discretization of degree n while the estimates exhibit a strong n dependence convergence is obtained for appropriately chosen time step sizes results of numerical simulations are provided and confirm the applicability of the methods

on the necessity of the moreau rockafellar robinson qualification condition in banach spaces as well known the moreau rockafellar robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended real valued convex lower semi continuous functions defined on a locally convex space is exact and that the subdifferential of the sum of these functions is the sum of their subdifferentials this note is devoted to proving that this condition is in a certain sense also necessary provided the underlying space is a banach space our result is based upon the existence of a non supporting weak closed hyperplane to any weak closed and convex unbounded linearly bounded subset of the topological dual of a banach space

gravity in non commutative geometry we study general relativity in the framework of non commutative differential geometry as a prerequisite we develop the basic notions of non commutative riemannian geometry including analogues of riemannian metric curvature and scalar curvature this enables us to introduce a generalized einstein hilbert action for non commutative riemannian spaces as an example we study a space time which is the product of a four dimensional manifold by a two point space using the tools of non commutative riemannian geometry and derive its generalized einstein hilbert action in the simplest situation where the riemannian metric is taken to be the same on the two copies of the manifold one obtains a model of a scalar field coupled to einstein gravity this field is geometrically interpreted as describing the distance between the two points in the internal space

on curvature in noncommutative geometry a general definition of a bimodule connection in noncommutative geometry has been recently proposed for a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra the corresponding curvatures are also compared

a priori estimates for the fitzpatrick function abstract new perspectives proofs and some extensions of known results are presented concerning the behavior of the fitzpatrick function of a monotone type operator in the general context of a locally convex space

noncommutative geometry of super jordanian osph 2 1 covariant quantum space extending a recently proposed procedure of construction of various elements of differential geometry on noncommutative algebras we obtain these structures on noncommutative superalgebras as an example a quantum superspace covariant under the action of super jordanian osph 2 1 is studied it is shown that there exist a two parameter family of torsionless connections and the curvature computed from this family of connections is bilinear it is also shown that the connections are not compatible with the metric

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

super jordanian algebra as a contraction limit we develop a generic representation independent contraction procedure for obtaining for instance matrices and l operators of arbitrary dimensions for the quantized super jordanian algebra from the pertinent quantities of the standard q deformed algebra

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

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

the brezis browder theorem in a general banach space during the 1970s brezis and browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces in this paper we extend and refine their result to a general banach space we also provide an affirmative answer to a problem posed by phelps and simons

differential calculus on the h superplane a noncommutative differential calculus on the h superplane is presented via a contraction of the q superplane an r matrix which satisfies both ungraded and graded yang baxter equations is obtained and a new deformation of the 1 1 dimensional classical phase space the super heisenberg algebra is introduced

exponential splitting for unbounded operators we present a convergence analysis for exponential splitting methods applied to linear evolution equations our main result states that the classical order of the splitting method is retained in a setting of unbounded operators without requiring any additional order condition this is achieved by basing the analysis on the abstract framework of semi groups the convergence analysis also includes generalizations to splittings consisting of more then two operators and to variable time steps we conclude by illustrating that the abstract results are applicable in the context of the schrodinger equation with an external magnetic field or with an unbounded potential

on the generalized parallel sum of two maximal monotone operators of gossez type d abstract the generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping in this article by assuming that the operators are maximal monotone of gossez type d we provide sufficient conditions of both interiority and closedness type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of gossez type d too this result will follow as a particular instance of a more general one concerning the maximal monotonicity of gossez type d of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals

adjoint process duality we study convex processes between topological vector spaces with particular emphasis on their adjoints this study is then applied to produce general duality results for classes of convex programs involving processes the use of processes allows one to exploit the symmetry of linear programming and to obtain significantly broader and stronger results

the parallel sum of nonlinear monotone operators on definit la somme parallele d operateurs non lineaires dans un espace de hilbert a l aide de leurs graphes

duality for constrained robust sum optimization problems given an infinite family of extended real valued functions f i i in i and a family mathcal h of nonempty finite subsets of i the mathcal h partial robust sum of f i i in i is the supremum for j in mathcal h of the finite sums sum j in j f j these infinite sums arise in a natural way in location problems as well as in functional approximation problems and include as particular cases the well known sup function and the so called robust sum function corresponding to the set mathcal h of all nonempty finite subsets of i whose unconstrained minimization was analyzed in previous papers of three of the authors https doi org 10 1007 s11228 019 00515 2 and https doi org 10 1007 s00245 019 09596 9 in this paper we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given mathcal h partial robust sum under constraints as well as closedness and convex criteria for the formulas on the subdifferential of the sup function