layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

hom nijienhuis operators and t extensions of hom lie superalgebras abstract in this paper we study hom nijienhuis operators and t extensions of hom lie superalgebras we show that a linear map between hom lie superalgebras is a morphism if and only if its graph is a hom subalgebra and show that the infinitesimal deformation generated by a hom nijienhuis operator is trivial moreover we introduce the definition of t extensions of hom lie superalgebras and show that t extensions preserve many properties such as nilpotency solvability and decomposition in some sense in particular we discuss the equivalence of t extensions using cohomology

the generalization of bivariate mkz operators by multiple generating functions in the present paper we study approximation properties of multiple generating functions type bivariate meyer konig and zeller mkz operators with the help of volkov type theorem we compute the order of convergence of these operators by means of modulus of continuity and the elements of modified lipschitz class finally we give application to partial differential equations

integrals for monoidal hom hopf algebras and their applications integrals of monoidal hom hopf algebras are introduced and the existence and uniqueness of integrals for finite dimensional monoidal hom hopf algebras are investigated first then integrals are applied to the maschke type theorem for monoidal hom hopf algebras controlling the semisimplicity and separability of monoidal hom hopf algebras further monoidal hom algebras are characterized with additional frobenius property and the question when finite dimensional monoidal hom hopf algebras are frobenius is studied as applications of integrals the maschke type theorem for hom smash product is given and the morita context in the hom category h mk is constructed

yetter drinfeld modules for hom bialgebras the aim of this paper is to define and study yetter drinfeld modules over hom bialgebras a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism yetter drinfeld modules over a hom bialgebra with bijective structure map provide solutions of the hom yang baxter equation the category yhhd of yetter drinfeld modules with bijective structure maps over a hom bialgebra h with bijective structure map can be organized in two different ways as a quasi braided pre tensor category if h is quasitriangular respectively coquasitriangular the first respectively second quasi braided pre tensor category yhhd contains as a quasi braided pre tensor subcategory the category of modules respectively comodules with bijective structure maps over h

the hom yang baxter equation hom lie algebras and quasi triangular bialgebras we study a twisted version of the yang baxter equation called the hom yang baxter equation hybe which is motivated by hom lie algebras three classes of solutions of the hybe are constructed one from hom lie algebras and the others from drinfeld s dual quasi triangular bialgebras each solution of the hybe can be extended to operators that satisfy the braid relations assuming an invertibility condition these operators give a representation of the braid group

relative hom hopf modules and total integrals let h be a monoidal hom hopf algebra and a a right h hom comodule algebra we first investigate the criterion for the existence of a total integral of a in the setting of monoidal hom hopf algebras also we prove that there exists a total integral h a if and only if any representation of the pair h a is injective in a functorial way as a corepresentation of h which generalizes doi s result finally we define a total quantum integral h hom h a and prove the following affineness criterion if there exists a total quantum integral and the canonical map a ba a h a bb 1 a b 0 b 1 is surjective then the induction functor a b k b k ah is an equivalence of categories

radford s biproducts and yetter drinfeld modules for monoidal hom hopf algebras let h be a monoidal hom bialgebra and b be a left h hom module algebra and also a left h hom comodule coalgebra then in this paper we first introduce the notion of a hom smash coproduct which is a monoidal hom coalgebra second we find sufficient and necessary conditions for the hom smash product algebra structure and the hom smash coproduct coalgebra structure on b h to afford b h a monoidal hom bialgebra structure generalizing the well known radford s biproduct where the conditions are equivalent to that b is a bialgebra in the category of hom yetter drinfeld modules hydhh finally we illustrate the category of hom yetter drinfeld modules hydhh and prove that the category hydhh is a braided monoidal category

bivariate positive linear operators constructed by means of q lagrange polynomials abstract in this paper we propose an extension of the linear positive operators constructed by means of the q lagrange polynomials and investigate the order of convergence by means of the complete and partial moduli of continuity and the peetre s k functional we also define the associated generalized boolean sum gbs operators and investigate the rate of convergence of these operators with the aid of mixed modulus of smoothness for bogel continuous and bogel differentiable functions finally we introduce an sth s being a non negative integer order generalization of the above operators and study the degree of approximation of these operators for sth order continuously differentiable lipschitz class functions

hom l r smash products hom diagonal crossed products and the drinfeld double of a hom hopf algebra abstract we introduce the hom analogue of the l r smash product and use it to define the hom analogue of the diagonal crossed product when h is a finite dimensional hom hopf algebra with bijective antipode and bijective structure map we define the drinfeld double of h its algebra structure is a hom diagonal crossed product and it has all expected properties namely it is quasitriangular and modules over it coincide with left right yetter drinfeld modules over h

on hom algebras with surjective twisting abstract a hom associative structure is a set a together with a binary operation and a self map such that an twisted version of associativity is fulfilled in this paper we assume that is surjective we show that in this case under surprisingly weak additional conditions on the multiplication the binary operation is a twisted version of an associative operation as an application an earlier result fregier and gohr 1 on weakly unital hom algebras is recovered with a different proof in the second section consequences for the deformation theory of hom algebras with surjective twisting map are discussed

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

symmetries and the u condition in hom yetter drinfeld categories let h s be a monoidal hom hopf algebra and hydhh the hom yetter drinfeld category over h then in this paper we first find sufficient and necessary conditions for hydhh to be symmetric and pseudosymmetric respectively second we study the u condition in hydhh and show that the hom yetter drinfeld module h adjoint resp h m coadjoint satisfies the u condition if and only if s2 id finally we prove that hydhh over a triangular resp cotriangular hom hopf algebra contains a rich symmetric subcategory

chaotic orbits for systems of nonlocal equations we consider a system of nonlocal equations driven by a perturbed periodic potential we construct multibump solutions that connect one integer point to another one in a prescribed way in particular heteroclinic homoclinic and chaotic trajectories are constructed this is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework

the generalization of meyer konig and zeller operators by generating functions in this paper theorems are proved concerned with some approximation properties of generating functions type meyer konig and zeller operators with the help of a korovkin type theorem secondly we compute the rates of convergence of these operators by means of the modulus of continuity peetre s k functional and the elements of the modified lipschitz class also we introduce the rth order generalization of these operators and we obtain approximation properties of them in the last part we give some applications to the differential equations

nonlinear equations for fractional laplacians ii existence uniqueness and qualitative properties of solutions this paper which is the follow up to part i concerns the equation delta s v g v 0 in mathbb r n with s in 0 1 where delta s stands for the fractional laplacian the infinitesimal generator of a l evy process when n 1 we prove that there exists a layer solution of the equation i e an increasing solution with limits pm 1 at pm infty if and only if the potential g has only two absolute minima in 1 1 located at pm 1 and satisfying g 1 g 1 0 under the additional hypothesis g 1 0 and g 1 0 we also establish its uniqueness and asymptotic behavior at infinity furthermore we provide with a concrete almost explicit example of layer solution for n geq 1 we prove some results related to the one dimensional symmetry of certain solutions in the spirit of a well known conjecture of de giorgi for the standard laplacian

statistical weighted b summability and its applications to approximation theorems in this paper which deals essentially with various summability concepts and summability techniques and shows how these concepts and techniques lead to a number of approximation results we have used the new concept of weighted a summability proposed by mohiuddine 2016 and introduced the notions of statistically weighted b summability and weighted b statistical convergence with respect to the weighted regular method we then prove a korovkin type approximation theorem for functions of two variables and also present an example via generalized meyer konig and zeller type operator to show that our proposed method is stronger than its classical and statistical versions furthermore the rate of convergence of approximating positive linear operators are estimated by means of the modulus of continuity and some voronovskaja type results are investigated computational and geometrical approaches to illustrate some of our results are also presented

hom lie hopf algebras abstract we studied both the double cross product and the bicrossproduct constructions for the hom hopf algebras of general type this allows us to consider the universal enveloping hom hopf algebras of hom lie algebras which are of id type we show that the universal enveloping hom hopf algebras of a matched pair of hom lie algebras form a matched pair of hom hopf algebras we observe also that the semi dualization of a double cross product hom hopf algebra is a bicrossproduct hom hopf algebra in particular we apply this result to the universal enveloping hom hopf algebras of a matched pair of hom lie algebras to obtain hom lie hopf algebras

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

on various new concepts of statistical convergence for sequences of random variables via deferred cesaro mean abstract the paper aims to introduce and investigate various aspects of statistical convergence for sequences of random variables as well as for sequences of real numbers via deferred cesaro summability mean we first established several new theorems presenting the connection among them by using the fundamental limit concepts of sequences of random variables as application of the findings we have proposed different new korovkin type approximation results with algebraic test functions for a sequence of random variables on a banach space and also demonstrated the effectiveness of the findings further several examples have been incorporated to illustrate the findings

homogenization and orowan s law for anisotropic fractional operators of any order abstract we consider an anisotropic levy operator i s of any order s 0 1 and we consider the homogenization properties of an evolution equation the scaling properties and the effective hamiltonian that we obtain are different according to the cases s 1 2 and s 1 2 in the isotropic one dimensional case we also prove a statement related to the so called orowan s law that is an appropriate scaling of the effective hamiltonian presents a linear behavior

the category of yetter drinfel d hom modules and the quantum hom yang baxter equation in this paper we introduce the category of yetter drinfel d hom modules which is a braided monoidal category and show that the category of yetter drinfel d hom modules is a full monoidal subcategory of the left center of left hom module category also we study the equivalent relationship between the category of yetter drinfel d hom modules and the category of hom modules over the drinfel d double finally the faddeev reshetikhin takhtajan frt type theorem for the quantum hom yang baxter equation is investigated

hom novikov algebras we study a twisted generalization of novikov algebras called hom novikov algebras in which the two defining identities are twisted by a linear map it is shown that hom novikov algebras can be obtained from novikov algebras by twisting along any algebra endomorphism all algebra endomorphisms on complex novikov algebras of dimensions 2 or 3 are computed and their associated hom novikov algebras are described explicitly another class of hom novikov algebras is constructed from hom commutative algebras together with a derivation generalizing a construction due to dorfman and gel fand two other classes of hom novikov algebras are constructed from hom lie algebras together with a suitable linear endomorphism generalizing a construction due to bai and meng

on quantum galois theory the goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the operator content of orbifold models to explain what this might mean and to clarify the title of the paper we will assume that the reader is familiar with the algebraic formulation of 2 dimensional cft in the guise of vertex operator algebras voa see b flm and dm for more information on this point in the paper dvvv several ideas are proposed concerning the structure of a holomorphic orbifold in other words if v is a holomorphic voa and if g is a finite group of automorphisms of v then the sub voa v g of g invariants is itself a voa and the subject of dvvv is very much concerned with speculation on the nature of the v modules it turns out to be more useful at least for purpose of inductive proofs to take v to be a simple voa we will then see that v g is also simple whenever g is a finite group of automorphisms of v one consequence of our main results is the following

integrals quantum galois extensions and the affineness criterion for quantum yetter drinfel d modules abstract in this paper we shall generalize the notion of an integral on a hopf algebra introduced by sweedler by defining the more general concept of an integral of a threetuple h a c where h is a hopf algebra coacting on an algebra a and acting on a coalgebra c we prove that there exists a total integral c hom c a of h a c if and only if any representation of h a c is injective in a functorial way as a corepresentation of c in particular the quantum integrals associated to yetter drinfel d modules are defined let now a be an h bicomodule algebra h y d a the category of quantum yetter drinfel d modules and b a a s 1 a 1 a 1 a 0 1h a the subalgebra of coinvariants of the verma structure a h y d a we shall prove the following affineness criterion if there exists h hom h a a total quantum integral and the canonical map a b a h a a b b s 1 b 1 b 1 ab 0 is surjective i e a b is a quantum homogeneous space then the induction functor b a m b h y d a is an equivalence of categories the affineness criteria proven by cline parshall and scott and independently by oberst for affine algebraic groups schemes and schneider in the noncommutative case are recovered as special cases