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Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

The concept of an involution or anti‐involution is a self‐inverse linear mapping that plays a prominent role in the theory of algebraic structures, particularly rings, hyperrings, ordered semigroups, and ordered semihypergroups. Nowadays, the study of involutions in ordered hyperstructures is a particular area of research in the field of hyperstructure theory, especially in ordered hyperstructures. In this research, we apply the concept of fuzzy sets to anti‐involution hyperideals of ordered anti‐involution LA‐semihypergroups. We shall propose the mathematical formulation of fuzzy anti‐involution hyperideals in ordered anti‐involution LA‐semihypergroups and provide some examples. This concept is a novel extension of fuzzy involution ordered semigroups. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g (x σ (y)) = g (x) g (y) + f (x) f (y) , x , y ∈ S , f (x σ (y)) = f (x) g (y) + f (y) g (x) , x , y ∈ S , and f (x σ (y)) = f (x) g (y) - f (y) g (x) , x , y ∈ S , where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications. [ABSTRACT FROM AUTHOR]

In this paper, we prove results on various prime ideals of seminearring, which is a generalization of nearring. We obtain the relationship among various prime ideals of S-semigroup, which is a module over seminearring. In addition, we prove one-one correspondence between the strong ideals of S-semigroup R and the strong ideals of homomorphic image of R. [ABSTRACT FROM AUTHOR]

A significant development in the field of gyrogroups was the introduction of the space of all relativistically admissible velocities, which brought gyrogroups into the mainstream. A group has various generalizations, one of which is the notion of gyrogroups. Moreover, for any pair (a; b) in this structure, there exists an automorphism gyr[a; b] that fulfills left associativity and left loop property. The motivation behind this study is to generalize gyrogroups and semigroups, which has led to the introduction of gyrosemigroups. Accordingly, in this paper, some classes of gyrosemigroups are presented. Also, all gyrosemigroups of order 2 are characterized. Furthermore, the gyrosemigroups with an identity or a zero are studied. [ABSTRACT FROM AUTHOR]

We prove an analog of the Tits alternative for endomorphisms of P1. In particular, we show that if S is a finitely generated semigroup of endomorphisms of P1 over C, then either S has polynomially bounded growth or S contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field of any characteristic and Prep.f / 6D Prep.g/, then for all sufficiently large j, the semigroup hf j; gj i is a free semigroup on two generators. [ABSTRACT FROM AUTHOR]

As concerning the views of T norms and T conorms, the intent of article is to define and probe the fuzzy semigroups, fuzzy ideals, fuzzy bi-ideals, bifuzzy subsemigroups, bifuzzy ideals, bifuzzy bi-ideals, fuzzy (1; 2)-ideals and bifuzzy (1; 2)-ideals in any given semigroup. Also, we indicate and study their basic properties of them in completely regular semigroups. Finally, we extend these concepts and so characterise (pre)image of them in semigroup homomorphisms. [ABSTRACT FROM AUTHOR]

Let R be a prime ring and Z(R) denotes the center of R. In this study, we expose the commutativity of R as a consequence of specific differential identities involving derivations acting on left ideals of R. Finally, we give examples that demonstrate the necessity of hypotheses taken in the theorems. [ABSTRACT FROM AUTHOR]

Complementary pairs of symmetric 2-designs are equivalent to coherent configurations of type (2, 2; 2). D. G. Higman studied these coherent configurations and adjacency algebras of coherent configurations over a field of characteristic zero. These are always semisimple. We investigate these algebras over fields of any characteristic prime and the structures. [ABSTRACT FROM AUTHOR]

In this paper, prime ideal bundles and semi-prime and irreducible ideal bundles of a Lie algebra bundle are defined and their relation with prime ideal bundles is studied. [ABSTRACT FROM AUTHOR]

In this paper, we introduced m-extension dual complex Fibonacci p-numbers. We established the properties of mextension dual complex Fibonacci p-numbers. They are connected to complex Fibonacci numbers, complex Fibonacci p-numbers and dual complex Fibonacci p-numbers. [ABSTRACT FROM AUTHOR]

Let R be a commutative semiring and I be a k-ideal of R. In this paper, we introduce the k-ideal-based graph of R, denoted by ΓI ∗ (R). The basic properties and possible structures of the graph are studied. [ABSTRACT FROM AUTHOR]

We define cohomology of module bundles over associative algebra bundles. We establish a one to one correspondence between the first cohomology classes and the extensions of module bundles. Using this correspondence we give sufficient condition in terms of cohomology for an algebra bundle to be semisimple. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I, J be two ideals of R, and M, N be two R-modules. Let S be a Serre subcategory of the category of R-modules. We introduce Serre cohomological dimension of N, M with respect to (I, J), as cdS(I, J, N, M) = sup{i ∈ N0 : HiI,J (N, M) 6∈ S}. We study some properties of cdS(I, J, N, M), and we get some formulas and upper bounds for it. [ABSTRACT FROM AUTHOR]

As we know that all non-trivial permutation identities are not preserved under epimorphisms of partially ordered semigroups. In this paper towards this open problem, first we show that certain non-trivial identities in conjunction with the permutation identity z1z2 ... zn = zi1 zi2 ... zin (n ≥ 2) with in 6= n [i1 6= 1] are preserved under epimorphisms of partially ordered semigroups. Further, we extend a result of Ahanger and Shah which showed that the center of a partially ordered semigroup S is closed in S and show that the normalizer of any element of a partially ordered semigroup S is closed in S. [ABSTRACT FROM AUTHOR]

The action of any group on itself by conjugation and the corresponding conjugacy relation plays an important role in group theory. Generalizing the group theoretic notion of conjugacy to semigroups is one of the interesting problems, and semigroup theorists had produced substantial amount of research in this direction. The challenge to introduce a new notion of conjugacy in semigroups is to choose the suitable set of conjugating elements. A semigroup may contain a zero, and if zero lies in the conjugating set, then the relation reduces to the universal relation as can be seen in the notions ∼l, ∼p, and ∼o. To avoid this problem, various innovative notions of conjugacy in semigroups have been considered so far, and ∼n is one of these notions. ∼n is an equivalence relation in any semigroup, coincides with the usual group theoretic notion if the underlying semigroup is a group, and does not reduce to a universal relation even if S contains a zero. In this paper, we study ∼n notion of conjugacy in some classes of epigroups. Since epigroups are generalizations of groups, our results of this paper are innovative and generalize the existing results on other notions. After proving some fundamental results, we compare our results with existing ones and prove that they are worth contribution in the study of conjugacy in epigroups. [ABSTRACT FROM AUTHOR]

This paper deals with the semigroup generation of anti‐triangular operator matrices M with unbounded entries in Hilbert space. Based on the space decomposition, some necessary and sufficient conditions are given for M to generate contraction semigroups. In addition, the anti‐triangular differential system, converted from the damping wave equation, is used to explain our work, and it is proved that the corresponding anti‐triangular operator matrix satisfies the conditions and generates a contraction semigroup. [ABSTRACT FROM AUTHOR]

In this work, we show that if F is a positive integer, then Sat (F) = { S ∣ S is a saturated numerical semigroup with Frobenius number F} is a covariety. As a consequence, we present two algorithms: one that computes Sat (F) , and another which computes all the elements of Sat (F) with a fixed genus. If X ⊆ S \ Δ (F) for some S ∈ Sat (F) , then we see that there exists the least element of Sat (F) containing X. This element is denoted by Sat (F) [ X ]. If S ∈ Sat (F) , then we define the Sat (F) -rank of S as the minimum of { c a r d i n a l i t y (X) ∣ S = Sat (F) [ X ] }. In this paper, we present an algorithm to compute all the elements of Sat (F) with a given Sat (F) -rank. [ABSTRACT FROM AUTHOR]

In this paper, we explore the concept of topologically dense injectivity of monoid acts. It is shown that topologically dense injective acts constitute a class strictly larger than the class of ordinary injective ones. We determine a number of acts satisfying topologically dense injectivity. Specifically, any strongly divisible as well as strongly torsion free S -act over a monoid S is topologically dense injective if and only if S is a left reversible monoid. Furthermore, we establish a counterpart of the Skornjakov criterion and also identify a class of acts satisfying the Baer criterion for topologically dense injectivity. Lastly, some homological classifications for monoids by means of this type of injectivity of monoid acts are also provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce an iterative algorithm for the split equality problem with an equilibrium problem, a variational inequality problem, and a fixed point problem of nonexpansive semigroups. We establish a strong convergence theorem of common solutions by the uniformly continuity rather than the Lipchitz continuity of the mappings in real Hilbert spaces. The proposed algorithm only requires one projection each per iteration onto the feasible sets. We also propose a self-adaptive technique that generates non-monotonic sequence of step sizes. Finally, we present a numerical example to illustrate the significance and efficient performance of our algorithm. Our results develop and unify several optimization results in the literature. [ABSTRACT FROM AUTHOR]

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. Some applications to the basic dynamical properties of the action are indicated, covering topics as topological transitivity, limit points and periodic points. We then explore the connections of these recurrence sets with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions). [ABSTRACT FROM AUTHOR]

This article presents a geometric-flavoured proof of the diamagnetic inequality for the heat semigroup in a Hermitian bundle based on Chernoff's theorem about the approximation of contraction semigroups in Banach spaces. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, and σ: S → S an automorphism that need not be involutive. We determine the complex-valued solutions of the following functional equation f(xy) - µ(y) f(σ(y)x) = 2f(x)g(y), x,y ∈ S, where µ: S→C is a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S. This enables us to solve the functional equation f (xφ(y))-f(ψ(y)x) = 2f(x)g(y), x,y ∈ S, where φ, ψ: S → S are automorphisms such that φ is involutive and ψ is not necessarily involutive. Some consequences of these results are presented. [ABSTRACT FROM AUTHOR]

A family U of non-empty subsets of a set D is called an upfamily if for each set U E U any set F U U belongs to U. The upfamily extension υ(D) of D consists of all upfamilies on D. Any associative binary operation ∗: D×D → D can be extended to an associative binary operation ... In the paper, we show that the upfamily extension (υ(D), -|, -|) of a (strong) doppelsemigroup (D, -|, -|) is a (strong) doppelsemigroup as well and study some properties of this extension. Also we introduce the upfamily functor in the category DSG whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms. We prove that the automorphism group of the upfamily extension of a doppelsemigroup (D, -|, -|) of cardinality |D| ≥ 2 contains a subgroup, isomorphic to C2 × Aut(D, -|, -|). Also we describe the structure of upfamily extensions of all two-element doppelsemigroups and their automorphism groups. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp - Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp (Rd) and a time weighted Lr (Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space. [ABSTRACT FROM AUTHOR]

In this paper, we prove the boundedness of the Riesz potential Iα in local variable Morrey-Lorentz spaces. Also we apply our results to particular operators such as fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups in these spaces. [ABSTRACT FROM AUTHOR]

The residual finite-dimensionality of a |${\textrm{C}}^{\ast }$| -algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of when residual finite-dimensionality of an operator algebra is inherited by its maximal |${\textrm{C}}^{\ast }$| -cover, which we establish in many cases of interest. [ABSTRACT FROM AUTHOR]

Semigroups are the building blocks of algebra as they have application in automata, coding theory, formal languages, and theoretical computer science. They are also used in the solutions of graph theory and optimization theory. For the advanced study of algebraic structures and their applications, ideals are essential. The generalization of ideals in algebraic structures is necessary for further research on algebraic structures. The main purpose of this paper is to present the notion of soft intersection almost subsemigroup of a semigroup, which is a generalization of soft intersection subsemigroup and investigate its basic properties in detail. In this context, we also obtain many striking relationships between almost subsemigroups and soft intersection almost subsemigroups concerning minimality, primeness, semiprimeness and strongly primeness. [ABSTRACT FROM AUTHOR]

This study aims to introduce the concept of soft intersection almost weak-interior ideals of a semigroup, which extends the notion of nonnull soft intersection weak-interior ideals of a semigroup. We explore the properties of the ideal in depth. We show that soft intersection almost ideal and soft intersection almost weak-interior ideal coincide with each other when the soft set is idempotent and we also illustrate that an idempotent soft almost weak-interior ideal is a soft intersection almost subsemigroup. We also establish significant connections between almost weak-interior ideals and soft intersection weak-interior ideals of a semigroup concerning minimality, primeness, semiprimeness, and strong primeness. [ABSTRACT FROM AUTHOR]

The paper is devoted to two problems pivotal in the generation theory. The first concerns verifying whether a normalized holomorphic function is an infinitesimal generator. It is also important to trace the influence of properties of a generator on the dynamic behavior of the generated semigroup. Recently, these problems have been studied using the so-called filtrations of generators. In this paper we study three filtrations. The study of the first began earlier, the second is a new one, while the third consists of functions studied previously with no connection to complex dynamics. For all filtration classes studied in the paper, dynamic properties of the corresponding semigroups are established and estimates for the Fekete-Szegö functional are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce triple θ-centralizers and weak triple θ-centralizers on an algebra A, where θ: A → A is a triple homomorphism. Some observations concerning triple θ-centralizers, weak triple θ-centralizers and approximate weak triple θ-centralizers are given. [ABSTRACT FROM AUTHOR]

Let [0,∞) be the set of all non-negative real numbers. The set B[0,∞) = [0,∞) × [0,∞) with the following binary operation (a, b)(c, d) = (a + c - min{b, c}, b + d - min{b, c}) is a bisimple inverse semigroup. In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup B[0,∞) with an adjoined compact ideal of the following tree types. The semigroup B[0,∞) with the induced usual topology τu from R2, with the topology τL which is generated by the natural partial order on the inverse semigroup B[0,∞), and the discrete topology are denoted by B1 [0,∞), B2 [0,∞), and Bd [0,∞), respectively. We show that if SI 1 (SI 2) is a Hausdorff locally compact semitopological semigroup B1 [0,∞) (B2 [0,∞)) with an adjoined compact ideal I then either I is an open subset of SI 1 (SI 2) or the topological space SI 1 (SI 2) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on S1 0 = B1 [0,∞) ∪ {0} (resp. S2 0 = B2 [0,∞) ∪ {0}) with an adjoined zero 0 is either homeomorphic to the one-point Alexandroff compactification of the topological space B1 [0,∞) (resp. B2 [0,∞)) or zero is an isolated point of S1 0 (resp. S2 0). Also, we proved that if SId is a Hausdorff locally compact semitopological semigroup Bd [0,∞) with an adjoined compact ideal I then I is an open subset of SId. [ABSTRACT FROM AUTHOR]

This document presents a study on the power graphs of finite cyclic groups. The authors derive lower and upper bounds for the energies of these power graphs by analyzing the adjacency matrix structure. They also conduct boundary studies to compare the obtained bounds. The document includes references to related research on graph energy and power graphs of other mathematical structures. [Extracted from the article]

EL-hypergroups were defined by Chvalina 1995. Till now, no exact statistics of EL-hypergroups have been done. Moreover, there is no classification of EL-hypergroups and EL2-hypergroups even over small sets. In this paper, we classify all EL-(semi)hypergroups over sets with two elements obtained from quasi ordered semigroups. Also, we characterize all quasi ordered Hv-group and then we enumerate the number of EL2-Hv-hypergroups and EL2-hypergroups of order 2. [ABSTRACT FROM AUTHOR]

This paper studies the state estimation problem for memristor-based stochastic neural networks (MSNNs) with mixed variable delays. A new Lyapunov-Krasovskii functional (LKF) with quadruple integral terms is incorporated. Then, asymptotic stability conditions are estab- lished for the error system using a linear matrix inequality technique. The estimator gain can be obtained by solving the linear matrix inequalities. Numerical simulations are given to demon- strate the effectiveness and superiority of the new scheme. [ABSTRACT FROM AUTHOR]

We study the problem of finding the common element of the set of fixed points of two nonexpansive mappings in Hilbert spaces. Some previous attempts in this direction make some restrictive assumptions which may be difficult to check in practice on the generated sequence. In this paper, we introduce a viscosity S-iteration scheme for finding the common fixed point of two nonexpansive mappings. Under some very mild conditions, we obtain a strong convergence theorem for the sequence generated by our algorithm. We apply our main result to approximating common fixed points of nonexpansive semigroups. We also provide numerical examples to support our main results and illustrate the efficiency and effectiveness of our algorithm by comparing with some existing algorithms in literature. This work generalizes and improves some existing works in the literature in this direction. [ABSTRACT FROM AUTHOR]

For a µ;n dichotomic systems generalization of Palmer's lemma was proved. Necessary and sufficient conditions of the existence of bounded on the whole axis solutions and quasisolutions that minimize the residual norm were obtained. Index of the corresponding operator was found. [ABSTRACT FROM AUTHOR]

In 2004, Patrício and Puystjens characterized the relation between Drazin invertible elements (resp., Moore-Penrose invertible elements) of two semigroups pRp and pRp+1 p of a ring R for some idempotent (resp., projection) p ∈ R. In this paper, we consider the relevant result for pseudo core invertible elements of such two semigroups of a ring for some projection, which is then applied to characterize the relation between pseudo core invertible elements of the matrix semigroup AA†RmxmAA†+Im AA† and the matrix semigroup A†ARnxnA†A+In A†A, where A ∈ Rmxn with A† existing. Also, similar equivalence involving DMP invertible elements is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we have proved certain kinds of Tauberian theorems for Nörlund-Cesáro summability methods in m-normed space X in statistically sense. We give necessary and sufficient Tauberian condition for this method of summability, in statistically sense. Also, we prove that Tauberian theorems under these summability methods are valid under oscillating slow condition of Schmidt-type and Hardy-type conditions. [ABSTRACT FROM AUTHOR]

This paper discusses the S-asymptotically periodic problem of fractional evolution equation with delay. By introducing a new noncompact measure theory involving infinite interval, we study the existence of S-asymptotically periodic mild solutions under the situation that the relevant semigroup is noncompact and the nonlinear term satisfies more general growth conditions instead of Lipschitz-type conditions. Moreover, by establishing a new Gronwall-type integral inequality corresponding to fractional differential equation with delay, we consider the global asymptotic behavior of S-asymptotically periodic mild solutions, which will make up for the blank of this field. [ABSTRACT FROM AUTHOR]

It is well known that all subvarieties of the variety of all semigroups are not absolutely closed. So, it is worth to find subvarieties of the variety of all semigroups that are closed in itself or closed in the containing varieties of semigroups. We have gone through this open problem and able to determine that the varieties of right [left] normal bands and left [right] quasinormal bands are closed in the varieties of semigroups defined by the identities axy = xany [axy = aynx], axy = xnay [axy = ayxn] (n > 1); and axy = axnay [axy = ayxny] (n > 1), axy = anxary [axy = ayrxyn] (n, r ∊ N), respectively. [ABSTRACT FROM AUTHOR]

We present a new method to obtain weighted L¹-estimates of global solutions to the Cauchy problem for the semilinear heat equation with a simple power of super-critical Fujita exponent. Our approach is based on direct and explicit computations of commutation relations between the heat semigroup and monomial weights in Rn, while it is independent of the standard parabolic arguments which rely on the comparison principle or some compactness arguments. We also give explicit asymptotic profiles with parabolic self-similarity of the global solutions. [ABSTRACT FROM AUTHOR]

In this article, we establish two new and different versions of fractional HermiteHadamard type inequality for the convex functions with respect to a pair of functions. Moreover, we establish a new Simpson’s type inequalities for differentiable convex functions with respect to a pair of functions. We also prove two more Simpson’s type inequalities for differentiable convex functions with respect to a pair of functions using the power mean and Hölder’s inequalities. It is also shown that the newly established inequalities are the extension of some existing results. Finally, we add some mathematical examples and their graphs to show the validity of newly established results. [ABSTRACT FROM AUTHOR]

A new approach for constructing vector Lyapunov function for nonlinear nonautonomous large-scale systems is proposed. It is assumed that independent subsystems are linear periodic systems. The components of the vector Lyapunov function are chosen as a quadratic form with a variable matrix. This matrix is an approximate solution of the Lyapunov matrix differential equation. This solution is constructed using the discretization method and the representation of the evolution operator proposed by Magnus. Sufficient conditions for the asymptotic stability of a trivial solution of a nonlinear large-scale system are established. The effectiveness of obtained results are illustrated by the example of stability investigation for coupled systems. [ABSTRACT FROM AUTHOR]

A newly-disclosed non-standard finite difference method has been used to discretize a prey-predator model to investigate the critical normal form coefficients of bifurcations for both one-parameter and two-parameter bifurcations. The discrete-time prey-predator model exhibits variety of local bifurcations such as period-doubling, Neimark-Sacker, and strong resonances. Critical normal form coefficients are determined to reveal dynamical scenario corresponding to each bifurcation point bifurcation. We also investigates the complex dynamics of the model numerically using by MATLAB package MATCONTM based on numerical continuation technique. [ABSTRACT FROM AUTHOR]

In this article, we deal with a logarithmic wave equation with distributed delay. Firstly, we establish the well-posedness by utilizing the semigroup theory. Later, we obtain the global existence of solutions by using the well-depth method. Moreover, under appropriate assumptions on the weight of the distributed delay and that of strong damping, we get the exponential decay results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a new faster iterative method, called the MNiteration process, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the MN-iteration of continuous functions on an arbitrary interval is established. We also compare the rate of convergence between the proposed iteration and some other iteration processes in the literature. Specifically, our main result shows that MN-iteration converges faster than NSP-iteration to the fixed point. We finally give numerical examples to compare the result with Mann, Ishikawa, Noor, SP and NSP iterations. Our findings improve corresponding results in the contemporary literature. [ABSTRACT FROM AUTHOR]

In this article, inspired by Berinde (Approximating fixed points of enriched nonexpansive mappings by Krasnoselskii iteration in Hilbert spaces, Carpathian J. Math. 35(3), 2019, 293-304), we define and study a new enriched class of mappings which includes many other contractive type mappings. We prove an existence result for the fixed point of newly introduced mapping. We also estimate fixed points of the proposed mapping via newly modified Mann iteration. In the process, some convergence results are also obtained for the proposed class of mappings in Uniformly convex Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce new class of 2D-k-Laguerre-Konhauser polynomials, δL k,n(α,β) (x, y), which generalizes the 2D-Laguerre-Konhauser polynomials (see [18]). Then, we define a new family of bivariate k-Mittag-Leffler functions E k,α,β,δ(γ)(x, y) and establish the k-Riemann-Liouville double fractional integral and derivative of the functions E(γ) k,α,β,δ(x, y). Moreover, we introduce an integral operator kε(γ) α,β,δ;ω1,ω2;a+,c+ which contains the bivariate k-Mittag-Leffler functions E(γ) k,α,β,δ(x, y) in the kernel and investigate the semigroup property of this operator. Finally, the left inverse operator of the integral operator kε(γ) α,β,δ;ω1,ω2;a+,c+ is constructed. [ABSTRACT FROM AUTHOR]

We give a logarithmic estimate of an elliptic semi-group generated by a big-order generator by using the Malliavin Calculus of Bismut type and large deviation estimates. [ABSTRACT FROM AUTHOR]