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In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

This paper treats the strict semi-stability of the symmetric powers [Formula: see text] of a stable vector bundle [Formula: see text] of rank [Formula: see text] with even degree on a smooth projective curve [Formula: see text] of genus [Formula: see text]. The strict semi-stability of [Formula: see text] is equivalent to the orthogonality of [Formula: see text] or the existence of a bisection on the ruled surface [Formula: see text] whose self-intersection number is zero. A relation between the two interpretations is investigated in this paper through elementary transformations. This paper also gives a classification of [Formula: see text] with strictly semi-stable [Formula: see text]. Moreover, it is shown that when [Formula: see text] is stable, every symmetric power [Formula: see text] is stable for all but a finite number of [Formula: see text] in the moduli of stable vector bundles of rank [Formula: see text] with fixed determinant of even degree on [Formula: see text].

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

In this paper, we introduce a new functional for the conformal spectrum of the conformal Laplacian on a closed manifold [Formula: see text] of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds. The functional introduced here has been investigated in the recent work [S. Perez-Ayala, Extremal eigenvalues of the conformal Laplacian under Sire–Xu normalization, preprint (2019); arXiv:2011.06018].

In this paper, we propose and study a new algorithm for finding a common element of the sets of solutions of split equality pseudomonotone equilibrium, split equality monotone variational inclusion and fixed point problems for Bregman relatively nonexpansive mappings in [Formula: see text]-uniformly convex and uniformly smooth Banach spaces. Our iterative method uses stepsizes which do not require prior knowledge of the operator norm. We apply our result to solve split equality variational inequality and split equality convex minimization problems. The result present in this paper unifies and extends several existing results in the literature.

This paper is the study of Hankel type translation and Hankel type convolution for linear canonical Hankel type transformations. In this paper, I have studied some inequalities associated with Hankel type translation and Hankel type convolution. Also, I have studied some applications of linear canonical Hankel transformation to a canonical convolution integral equation and a generalized nonlinear parabolic equation.

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

In this paper, we study the existence and the numerical estimates of the solutions for a set of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for the exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at origin and infinity. The novelty of this paper is to use variational methods to obtain the multiplicity of solutions of boundary value problems with the nonlinearity depending on the fractional derivative.

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.

In this paper, we study split generalized mixed equilibrium problem and fixed point problem in real Hilbert spaces with a view to analyze an iterative method for approximating a common solution of split generalized mixed equilibrium problem and fixed point problem of an infinite family of a quasi-nonexpansive multi-valued mappings. The iterative algorithm introduced in this paper is designed in such a way that it does not require the knowledge of the operator norm. We state and prove a strong convergence result of the aforementioned problems and also give application of our main result to split variational inequality problem. Our result complements and extends some related results in literature.

Very recently, the notion of graphical rectangular [Formula: see text]-metric spaces has been introduced, where the weight of the symmetric edges endowing the directed graph is defined in terms of the underlying metric. Due to its intimate relationship with the directed graphs, such spaces have a wide range of applications in science and engineering. In this paper, we rectify some errors that arose in the paper [Younis et al., A novel approach of graphical rectangular [Formula: see text]-metric spaces with an application to the vibrations of a vertical heavy hanging cable, J. Fixed Point Theory Appl. 21(1) (2019) 33]. We show that the main result of the paper described above can be modified, and we present the same in this note.

In this paper, we present a new hybrid method to solve a nonlinear unconstrained optimization problem by using conjugate gradient, which is a convex combination of Liu–Storey (LS) conjugate gradient method and Hager–Zhang (HZ) conjugate gradient method. This method possesses the sufficient descent property with Strong Wolfe line search and the global convergence with the strong Wolfe line search. In the end of this paper, we illustrate our method by giving some numerical examples.

This is about the paper “Bi-interior ideals of semigroups” by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. 38 (2018) 69–78. According to Theorem 3.11 (also Theorem 3.3(8)) of the paper, the intersection of a bi-interior ideal [Formula: see text] of a semigroup [Formula: see text] and a subsemigroup [Formula: see text] of [Formula: see text] is a bi-interior ideal of [Formula: see text]. Regarding to Theorem 3.6, every bi-interior ideal of a regular semigroup is an ideal of [Formula: see text]. We give an example that the above two results are not true for semigroups. According to the same paper, if [Formula: see text] is a regular semigroup then, for every bi-interior ideal [Formula: see text], every ideal [Formula: see text] and every left ideal [Formula: see text] of [Formula: see text], we have [Formula: see text]. The proof is wrong, we provide the corrected proof. In most of the results of the paper the assumption of unity is not necessary. Care should be taken about the proofs in the paper.

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path [Formula: see text], if there are pegs in [Formula: see text] and [Formula: see text] and exists a hole in [Formula: see text], then [Formula: see text] can jump over [Formula: see text] into [Formula: see text], and after that, the peg in [Formula: see text] is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex [Formula: see text], starting with a hole [Formula: see text], a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.

In this paper, we are concerned with the following integral system [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] ([Formula: see text]). When [Formula: see text], such an integral system is associated with the best constants of the Hardy–Littlewood–Sobolev inequality. Chen, Li and their cooperators obtained optimal integrability intervals of the finite energy solutions by an argument of contraction and shrinking operators. This result is helpful to well understand the classification of the extremal functions of the Hardy–Littlewood–Sobolev inequality. The critical condition plays a key role in their work. In this paper, we study optimal integrability intervals when the positive solutions have some initial integrability. Now, the critical condition is not necessary, and we apply a weaker condition, the Serrin-type condition, to establish some important relations of exponents which come into play to lift the regularity. In addition, we also generalize this result to the case of [Formula: see text].

Intrinsic tame filling functions are quasi-isometry invariants that are refinements of the intrinsic diameter function of a group. The main purpose of this paper is to show that every finite presentation of a group has an intrinsic tame filling function that is equivalent to its intrinsic diameter function. We also introduce some alternative filling functions — based on concepts similar to those used to define intrinsic tame filling functions — that are potential proper refinements of the intrinsic diameter function.

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic [Formula: see text]-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic [Formula: see text]-metric spaces, where [Formula: see text] is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case [Formula: see text] taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic [Formula: see text]-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy[Formula: see text] [Formula: see text]AMS-[Formula: see text], Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for [Formula: see text]-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in [Formula: see text], Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.

The investigation of co-tame automorphisms of the affine space [Formula: see text] is helpful to understand the structure of its automorphisms group. In this paper, we show the co-tameness of several classes of automorphisms, including some 3-parabolic automorphisms, power-linear automorphisms, homogeneous automorphisms in small dimension or small transcendence degree. We also classify all additive-nilpotent automorphisms in dimension four and show that they are co-tame.

This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left [Formula: see text]-dinilpotent dimonoid ([Formula: see text]), the free abelian dimonoid of rank [Formula: see text] and the free [Formula: see text]-dimonoid to a generalized digroup. We also count the cardinalities of the free left [Formula: see text]-dinilpotent dimonoid and the free [Formula: see text]-dimonoid for a finite case.

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if [Formula: see text] is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group [Formula: see text], then [Formula: see text] is a tower over [Formula: see text]. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.

In this paper, we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.

Let [Formula: see text] be a finite group. How minimal subgroups can be embedded in [Formula: see text] is a question of particular interest in studying the structure of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for all Sylow subgroups [Formula: see text] of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] with [Formula: see text]-embedded subgroups.

Let [Formula: see text] be a graph with edge set [Formula: see text]. For an edge [Formula: see text] in [Formula: see text], we define [Formula: see text], where [Formula: see text] and [Formula: see text] are degrees of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. For [Formula: see text], the graph invariants [Formula: see text], [Formula: see text] and [Formula: see text] are defined as [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] means that the edges [Formula: see text] and [Formula: see text] are incident. In this paper, some relationship between these graph invariants and some classical topological indices were presented. Moreover, some bounds for [Formula: see text], [Formula: see text] and [Formula: see text] are obtained and trees with the first through the third smallest [Formula: see text] and [Formula: see text], as well as the trees with the first through the forth smallest [Formula: see text] are also characterized.

In this paper, we study the dynamical behavior of solutions of nonlinear Schrödinger equations with quadratic interaction and [Formula: see text]-critical growth. We give sharp conditions under which the existence of global and blow-up solutions are deduced. We also show the existence, stability, and blow-up behavior of normalized solutions of this system.

In this paper, we prove that the dominion of any full orthodox subsemigroup of a medial orthodox semigroup is described by the Isbell zigzag theorem in the category of medial orthodox semigroups. As a consequence, the dominions of any full completely regular subsemigroup of a medial completely regular semigroup as well as that of any full Clifford subsemigroup of a medial Clifford semigroup are also described by the Isbell zigzag theorem in the categories of all medial completely regular semigroups and of all medial Clifford semigroups, respectively.

Finding the solution of the absolute value equations (AVEs) has attracted much attention in recent years. In this paper, we propose and analyze two generalized accelerated overrelaxation (AOR) methods for solving AVEs [Formula: see text], where [Formula: see text] is an [Formula: see text]-matrix. Furthermore, we discuss the convergence of the methods under some suitable assumptions. Numerical results are given to verify the effectiveness of our methods.

In this paper, our objective is to study numerous fundamental properties of the algebra of generalized derivations on a multiplicative BiHom-Poisson superalgebra [Formula: see text]. First, we present a description of some generalized derivations, central derivations, quasiderivations, centroid and quasicentroid of the multiplicative BiHom-Poisson superalgebra [Formula: see text] More specifically, we obtain some important connections and properties of these derivations. We prove that in a larger multiplicative BiHom-Poisson superalgebra, [Formula: see text] can be embedded as a derivations algebra. Finally, we show that when [Formula: see text] is centerless, the derivations of the larger multiplicative BiHom-Poisson superalgebras have direct sum decompositions.

The domatic number [Formula: see text] of a graph [Formula: see text] is the maximum number of disjoint dominating sets in a dominating set partition of a graph [Formula: see text]. For any graph [Formula: see text], [Formula: see text] where [Formula: see text] is the minimum degree of [Formula: see text], and [Formula: see text] is domatically full if the equality holds, i.e., [Formula: see text]. In this paper, we characterize domatically full Cartesian products of a path of order 2 and a tree of order at least 3. Moreover, we show a characterization of the Cartesian product of a longer path and a tree of order at least 3. By using these results, we also show that for any two trees of order at least 3, the Cartesian product of them is domatically full.

In this paper, we study the solvability of a very general class of integral equations whose kernel depends on four different functions. Necessary and sufficient conditions for the unique solvability of such integral equations are obtained. To achieve such a goal, the main technique consists in introducing eight new convolutions weighted by multi-dimensional Hermite functions and using them as convolutions somehow associated with our integral equations. In this way, two Young-type inequalities are also obtained.

In this paper, we study the existence and uniqueness of solutions for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations. The investigation of the main results depends on Schauder’s fixed point theorem and Banach’s contraction principle. An illustrative example is given to show the applicability of theoretical results.

The subjectivity domain of a module allows us to evaluate how close (or far) a module is from being injective. In this paper, we consider three families of rings characterized by their RD-injective left [Formula: see text]-modules via the subinjective domain: those whose noninjective RD-injective left [Formula: see text]-modules are subinjective relatively only to divisible modules, those whose noninjective RD-injective left [Formula: see text]-modules are subinjective relatively only to fp-injective modules and those whose noninjective RD-injective left [Formula: see text]-modules are subinjective relatively only to injective modules.

Let [Formula: see text] be a lattice, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] the set of all elements of [Formula: see text] which are not prime to [Formula: see text]. In this paper, we investigate some interesting properties of [Formula: see text] and introduce the total graph of a lattice [Formula: see text] with respect to an ideal [Formula: see text]. It is the graph with all elements of [Formula: see text] as vertices and for distinct [Formula: see text], the vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We investigated the basic properties of this graph and its induced subgraphs as diameter, girth, clique number, cut vertex and independence number.

This paper deals with the Ulam–Hyers stability of the following additive-quadratic mixed type functional equation: [Formula: see text] [Formula: see text] [Formula: see text] in modular spaces using the direct method.

The aim of this paper is to introduce new notions called generalized [Formula: see text] contraction and rational type [Formula: see text] contraction for hybrid maps in complete G-metric spaces. First, using new concept of generalized [Formula: see text] contraction we establish a common fixed point theorem for a hybrid pair of weakly compatible maps. Second, we prove a common fixed point theorem for occasionally coincidentally idempotent mappings satisfying another new concept of rational type [Formula: see text] contraction under common limit range property. Our results generalize and improve several known results of the existing literature. Also, we give an example to indicate the usefulness of our main results.

In this paper, on using a left exact preradical, we define a module [Formula: see text] to be EPD (EPG) if and only if each ep-submodule [Formula: see text] of [Formula: see text] is a direct summand (there exists a direct summand [Formula: see text] such that [Formula: see text] is essential in both [Formula: see text] and [Formula: see text]). We investigate structural properties of former new classes of modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for EPD and EPG-modules. We show that the EPD property is inherited by finite direct sums however we provide algebraic topological examples which show that direct summands do not enjoy the property. Moreover, we obtain that EPG property is closed under right essential overring and rational hull. We provide examples to illustrate some of the results by making special choices of left exact preradicals.

In this paper, we prove that if [Formula: see text] is a partitioning ideal of an [Formula: see text]-semiring [Formula: see text] with respect to two subsets [Formula: see text] and [Formula: see text] of [Formula: see text], then the quotient [Formula: see text]-semirings [Formula: see text] and [Formula: see text] are isomorphic. Also, the concepts of maximal homomorphism, steady homomorphism, one-one homomorphism and [Formula: see text]-homomorphism are generalized for [Formula: see text]-semirings and obtained relation between them. Finally, the fundamental theorem of semiring homomorphism is generalized for [Formula: see text]-semirings.

A collection of matrices is not always reducible, but it is beneficial for further analysis if it is. Some criteria for the algebraic reducibility have been considered in literature and, indeed, in some cases we are able to compute the reduced forms explicitly. We will review these methods in this paper. As their applications, we will consider the reductions (simplifications) of matrix continued fractions and discuss the convergence problems.

Roopaei [J. Inequal. Appl. 2020 (2020) 120] introduced the sequence space [Formula: see text] which is a matrix domain of Copson matrix in the space [Formula: see text] [Formula: see text] In this paper, we characterize the matrix classes [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text]. Moreover, we characterize compact operators associated with matrices belonging to these classes via the techniques of measures of noncompactness.

In this paper, we consider mixed approximation operators based on second-kind beta transform using Szász–Mirakjan operators. For the proposed operators, we establish some direct results, Voronovskaya-type theorem, quantitative Voronovskaya-type theorem, Grüss–Voronovskaya-type theorem, weighted approximation and functions of bounded variation.

All groups under consideration are finite. Let [Formula: see text] be some partition of the set of all primes [Formula: see text], [Formula: see text] be a group, and [Formula: see text] be a class of groups. Then, [Formula: see text] and [Formula: see text] A function [Formula: see text] of the form [Formula: see text] is called a formation [Formula: see text]-function. For any formation [Formula: see text]-function [Formula: see text] the class [Formula: see text] is defined as follows: [Formula: see text] [Formula: see text] If for some formation [Formula: see text]-function [Formula: see text] we have [Formula: see text] then the class [Formula: see text] is called [Formula: see text]-local and [Formula: see text] is called a [Formula: see text]-local definition of [Formula: see text] Every formation is called 0-multiply [Formula: see text]-local. For [Formula: see text] a formation [Formula: see text] is called [Formula: see text]-multiply [Formula: see text]-local provided either [Formula: see text] is the class of all identity groups or [Formula: see text] where [Formula: see text] is [Formula: see text]-multiply [Formula: see text]-local for all [Formula: see text] Let [Formula: see text] be a set of subgroups of [Formula: see text] such that [Formula: see text]. Then, [Formula: see text] is called a subgroup functor if for every epimorphism [Formula: see text] : [Formula: see text] and any groups [Formula: see text] and [Formula: see text] we have [Formula: see text] and [Formula: see text]. A class of groups [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] for all [Formula: see text]. In this paper, we study the properties of [Formula: see text]-closed [Formula: see text]-multiply [Formula: see text]-local formations generated by some set of groups. In particular, we obtained a description of the smallest [Formula: see text]-local definition of such formation, we also prove that [Formula: see text]-closed [Formula: see text]-multiply [Formula: see text]-local formation [Formula: see text] is one-generated if and only if [Formula: see text] is finite and [Formula: see text] have a integrated [Formula: see text]-local definition [Formula: see text] such that [Formula: see text] is a one-generated [Formula: see text]-closed [Formula: see text]-multiply [Formula: see text]-local formation for all [Formula: see text]. In addition, we prove that the class of all [Formula: see text]-closed one-generated [Formula: see text]-multiply [Formula: see text]-local formations coincides with the class of all one-generated [Formula: see text]-closed [Formula: see text]-multiply [Formula: see text]-local formations.

The irregularity of a simple undirected graph G is defined as irr(G) = Sigma(uv is an element of E(G)) vertical bar d(G)(u) - d(G) (v) vertical bar, where d(G) (u) denotes the degree of a vertex u is an element of V (G). The total irregularity of a graph as a new measure of graph irregularity is defined as irr(t) (G) = 1/2 Sigma(u,v is an element of V (G) )vertical bar d(G) (u) - d(G) (v) vertical bar. In this paper, the irregularity and the total irregularity of fractal graphs and the derived graphs from a class of fractal graphs are investigated. Exact formulae are presented for the computation of the irregularity and total irregularity of fractal-type graphs in terms of the parameters of the underlying graphs.

In this paper, one-dimensional time fractional telegraph equation is deduced using fractional Ohm’s law in transmission lines. The nonlocal behavior, arising from the time fractality, is discussed via fractional calculus. Here, the time fractional differential equation related to electrical GLRC circuit is proposed in terms of the Liouville–Caputo fractional derivative. To keep the dimensionality of the physical parameters [Formula: see text] the new parameter [Formula: see text] is introduced. This parameter ([Formula: see text]) characterizes the existence of fractional structure in the system. The analytical solution of the proposed equation is obtained by using Laplace transform method in terms of Mittag-Leffler function which describes physical system with memory. The order of the time derivative being considered is [Formula: see text]. The result of an example shows that the behavior of voltage depends on the value of the fractional order of the proposed equation.

Various ample divisors on smooth del Pezzo surfaces of degree [Formula: see text] have been verified to allow polar cylinders in [I. Cheltsov, J. Park and J. Won, Cylinders in del Pezzo surfaces, Int. Math. Res. Not. 2017(4) (2015) 1179–1230]. We show that affine cones over smooth del Pezzo surfaces of degree [Formula: see text] polarized by such ample divisors are flexible in codimension one. All varieties in this paper are assumed to be defined over an algebraically closed field of characteristic zero.

This paper studies the bi-compact-[Formula: see text]-open topology on the family [Formula: see text] of all continuous functions from a completely regular space [Formula: see text] to the space [Formula: see text]. We compare this topology with open-point topology, bi-point-open topology and bi-compact-open topology and mention an example of a space asked in [P. Garg and S. Kundu, The compact-[Formula: see text]-open topology on [Formula: see text], Topology Appl. 159(8) (2012) 2082–2089] which satisfies [Formula: see text]. We also investigate submertrizability, mertrizability, separability, [Formula: see text]ech-completeness and Baire property of bi-compact-[Formula: see text]-open topology on [Formula: see text].

Denote $M_k$ the set of complex $k$ by $k$ matrices. We will analyze here quantum channels $\phi_L$ of the following kind: given a measurable function $L:M_k\to M_k$ and the measure $\mu$ on $M_k$ we define the linear operator $\phi_L:M_k \to M_k$, via the expression $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho {L(v)}^\dagger \, \dm(v)$. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where $L$ was the identity. Under some mild assumptions on the quantum channel $\phi_L$ we analyze the eigenvalue property for $\phi_L$ and we define entropy for such channel. For a fixed $\mu$ (the \textit{a priori} measure) and for a given a Hamiltonian $H: M_k \to M_k$ we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such $H$) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed $\mu$ (with more than one point in the support) the set of $L$ such that it is $\phi$-Erg (also irreducible) for $\mu$ is a generic set. We describe a related process $X_n$, $n\in \mathbb{N}$, taking values on the projective space $ P(\C^k)$ and analyze the question of the existence of invariant probabilities. We also consider an associated process $\rho_n$, $n\in \mathbb{N}$, with values on $\mathcal{D}_k$ ($\mathcal{D}_k$ is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator which is fixed for $\phi_L$., Comment: We correct some mistakes in some proofs and we add a new section on genericty of ergodicity. We change the title

The aim of this paper is to explain how to parametrize the equations of the facets of Kirwan polytopes using the notion of Ressayre’s pairs.

Let [Formula: see text] be a minimal parabolic subgroup of a real reductive Lie group [Formula: see text] and [Formula: see text] a closed subgroup of [Formula: see text]. Then it is proved by Kobayashi and Oshima that the regular representation [Formula: see text] contains each irreducible representation of [Formula: see text] at most finitely many times if the number of [Formula: see text]-orbits on [Formula: see text] is finite. Moreover, they also proved that the multiplicities are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite, where [Formula: see text] are complexifications of [Formula: see text], respectively, and [Formula: see text] is a Borel subgroup of [Formula: see text]. In this paper, we prove that the multiplicities of the representations of [Formula: see text] induced from a parabolic subgroup [Formula: see text] in the regular representation on [Formula: see text] are uniformly bounded if the number of [Formula: see text]-orbits on [Formula: see text] is finite. For the proof of this claim, we also show the uniform boundedness of the dimensions of the spaces of group invariant hyperfunctions using the theory of holonomic [Formula: see text]-modules.