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Let [Formula: see text] be a commutative ring with unity [Formula: see text] and let [Formula: see text] be an ideal expansion. In the first part of this paper, we extend the concept of [Formula: see text]-primary ideals to [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction and [Formula: see text] is an ideal expansion. We introduce some of the ideal expansion [Formula: see text] and define [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction. Also, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized [Formula: see text]-[Formula: see text]-primary ideals with respect to such an ideal expansion [Formula: see text]. In the second part of this paper we investigate 2-absorbing [Formula: see text]-primary ideals which unify 2-absorbing ideals and 2-absorbing primary ideals, where [Formula: see text] is an ideal expansion. A number of results in the two parts are given.

In an earlier paper, we studied the Hilbert quasi-polynomial functions of finitely generated bigraded modules in the general framework when the base ring is bigraded and generated by finitely many homogeneous elements of arbitrary degrees. In this paper, we introduce the concept of [Formula: see text]-good bifiltration [Formula: see text] on a finitely generated [Formula: see text]-module [Formula: see text], where [Formula: see text] and [Formula: see text] are specified noetherian filtrations on the noetherian ring [Formula: see text]. The bigraded modules associated with such bifiltrations are shown to be finitely generated under reasonable hypotheses. Their Hilbert functions are studied. The Hilbert–Samuel function of [Formula: see text] with respect to the [Formula: see text]-good bifiltration [Formula: see text] of [Formula: see text] is one of them. It is proved, among others, that this function is a quasi-polynomial function in two variables and that if [Formula: see text] is a noetherian local ring and if the filtrations [Formula: see text] and [Formula: see text] are primary filtrations, then its degree equals the Krull dimension of [Formula: see text].

The purpose of this paper is to explore the geometry of a smooth CR manifold of hypersurface type and its relationship to the higher regularity properties of the complex Green operator on [Formula: see text]-forms in the [Formula: see text]-Sobolev space [Formula: see text] for a fixed [Formula: see text] and [Formula: see text].

Let [Formula: see text] be a group and [Formula: see text] be the [Formula: see text]-absolute center of [Formula: see text], that is, the set of all elements of [Formula: see text] fixed by all class preserving automorphisms of [Formula: see text]. In this paper, we classify all finite groups [Formula: see text], whose [Formula: see text]-absolute central factors are isomorphic to the direct product of cyclic groups, [Formula: see text] and [Formula: see text]. Moreover, we consider finite groups which can be written as the union of centralizers of class preserving automorphisms and study the structure of [Formula: see text] for groups, in which the number of distinct centralizers of class preserving automorphisms is equal to 4 or 5.

In this paper, we show some [Formula: see text]-difference analogues of the second main theorems for algebraically nondegenerate meromorphic mappings over the field [Formula: see text] of zero-order meromorphic functions in [Formula: see text] satisfying [Formula: see text] intersecting hypersurfaces, located in subgeneral position in [Formula: see text], where [Formula: see text] and [Formula: see text] may be different. As an application, we give some unicity theorems for meromorphic mappings of [Formula: see text] into [Formula: see text] under the growth condition “order [Formula: see text]”, which are analogous to Picard’s theorems.

Let [Formula: see text] be a simple connected graph and [Formula: see text] a coloring of the vertices in [Formula: see text] For any [Formula: see text], let [Formula: see text] be the sum of colors of the vertices adjacent to [Formula: see text]. Then [Formula: see text] is called a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text]. The minimum number of colors needed in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text], denoted by [Formula: see text] In this paper, we prescribe a sigma coloring of the join of paths and cycles. As a consequence, we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular, let [Formula: see text], where [Formula: see text] or [Formula: see text] with [Formula: see text] If [Formula: see text], where [Formula: see text] and [Formula: see text], then [Formula: see text] if [Formula: see text] is an odd cycle, for some [Formula: see text] and [Formula: see text] otherwise.

In this paper, we consider the high-order geometric flows of a compact submanifolds [Formula: see text] in a complete Riemannian manifold [Formula: see text] with [Formula: see text], which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on [Formula: see text]. Precisely, we show that if [Formula: see text] is strictly larger than the integer part of [Formula: see text] and [Formula: see text] is an immersion for all [Formula: see text] and if [Formula: see text] is bounded by a constant which relies on the injectivity radius [Formula: see text] and sectional curvature [Formula: see text] of [Formula: see text], then [Formula: see text] must be [Formula: see text].

In this paper, we construct matrix commutators involving hyperbolic functions [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and obtain some results related to the relation between these commutators and [Formula: see text].

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

In this paper, we study the existence and concentration behaviors of positive solutions to the following fractional nonlinear Schrödinger equation: [Formula: see text] where [Formula: see text] is a positive parameter, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is the fractional Laplacian. When the potential [Formula: see text] decays to zero like [Formula: see text], [Formula: see text], and [Formula: see text] like [Formula: see text] with [Formula: see text], we will show that the existence of ground states [Formula: see text] belonging to [Formula: see text], which concentrates at a minimum point of the auxiliary function [Formula: see text].

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

In this paper, we introduce the concept of an [Formula: see text]-pseudo complementation [Formula: see text] on an Almost Distributive Lattice [Formula: see text] for an arbitrary fixed element [Formula: see text] and any [Formula: see text] of [Formula: see text] and discuss several properties of this. Here, we obtain a bijection correspondence between the [Formula: see text]-pseudo complementations on [Formula: see text] and maximal elements of [Formula: see text] We prove that the set [Formula: see text] is a Boolean algebra which is independent(upto isomorphism) of the [Formula: see text]-pseudo complementation [Formula: see text].

For positive integer [Formula: see text], a graph [Formula: see text] is said to be [Formula: see text]-magic if the edges of [Formula: see text] can be labeled with the nonzero elements of Abelian group [Formula: see text], where [Formula: see text] (the set of integers) and [Formula: see text] is the group of integers mod [Formula: see text], so that the sum of the labels of the edges incident to any vertex of [Formula: see text] is the same. When this constant sum is [Formula: see text], we say that [Formula: see text] is a zero-sum [Formula: see text]-magic graph. The set of all [Formula: see text] for which [Formula: see text] is a zero-sum [Formula: see text]-magic graph is the null set of [Formula: see text]. In this paper, we will completely determine the null set of the join of a finite number of paths.

Let [Formula: see text] be the semigroup of all transformations on a set [Formula: see text]. For an arbitrary equivalence relation [Formula: see text] on [Formula: see text] and a cross-section [Formula: see text] of the partition [Formula: see text] induced by [Formula: see text], let [Formula: see text] [Formula: see text] Then [Formula: see text] and [Formula: see text] are subsemigroups of [Formula: see text]. In this paper, we characterize left regular, right regular and completely regular elements of [Formula: see text] and [Formula: see text]. We also investigate conditions for which of these semigroups to be left regular, right regular and completely regular semigroups.

Let [Formula: see text] be a connected finite graph and [Formula: see text] be the set of functions defined on [Formula: see text]. Let [Formula: see text] be the discrete [Formula: see text]-Laplacian on [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is positive everywhere. Consider the operator [Formula: see text]. We prove that [Formula: see text] is one-to-one, onto and preserves order. So it implies that there exists a unique solution to the equation [Formula: see text] for any given [Formula: see text]. We also prove that the equation [Formula: see text] has a solution which is unique up to a constant, where [Formula: see text] is the average of [Formula: see text]. With the help of these results, we finally give various conditions such that the [Formula: see text]th Kazdan–Warner equation [Formula: see text] has a solution on [Formula: see text] for given [Formula: see text] and [Formula: see text]. Thus we generalize Grigor’yan, Lin and Yang’s work Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations 55(4) (2016) Paper No. 92, 13 pp. for [Formula: see text] to any [Formula: see text].

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

A bounded operator [Formula: see text] in a Banach space [Formula: see text] is said to satisfy the descent spectrum equality, if the descent spectrum of [Formula: see text] as an operator on [Formula: see text] coincides with the descent spectrum of [Formula: see text] as an element of the algebra [Formula: see text] of all bounded linear operators on [Formula: see text]. In this paper, we give some conditions under which the equality [Formula: see text] holds for a single operator [Formula: see text].

In this paper, the classes of groups with given systems of [Formula: see text]-subnormal subgroups are studied. In particular, it is showed that if [Formula: see text] and [Formula: see text] are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose [Formula: see text]-subgroups are all [Formula: see text]-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with [Formula: see text]-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong [Formula: see text], where [Formula: see text] is a hereditary saturated formation, is obtained.

Let [Formula: see text] be a graph and [Formula: see text] A dominating set [Formula: see text] is a set of vertices such that each vertex of [Formula: see text] is either in [Formula: see text] or has at least one neighbor in [Formula: see text]. The minimum cardinality of such a set is called the domination number of [Formula: see text], [Formula: see text] [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text] A set [Formula: see text] is a strong-dominating set, shortly sd-set, (weak-dominating set, shortly wd-set) of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number [Formula: see text] of [Formula: see text] is the minimum cardinality of an sd-set (wd-set). In this paper, we present weak and strong domination numbers of thorn graphs.

Let [Formula: see text] be a quasi-compact scheme and [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the set of closed points of [Formula: see text] and the closure of the subset [Formula: see text]. Let [Formula: see text] be a nonempty subset of [Formula: see text]. We define the [Formula: see text]-Zariski topology graph on the scheme [Formula: see text], denoted by [Formula: see text], as an undirected graph whose vertex set is the set [Formula: see text], for two distinct vertices [Formula: see text] and [Formula: see text], there is an arc from [Formula: see text] to [Formula: see text], denoted by [Formula: see text], whenever [Formula: see text]. In this paper, we study the connectivity properties of the graph [Formula: see text], we establish the relationship between the connectivity of the graph [Formula: see text] and the structure of irreducible components of the scheme [Formula: see text]. Also, we characterize when the complement graph of the Zariski topology graph [Formula: see text] is a complete multipartite graph.

An edge irregular total [Formula: see text]-labeling on simple and undirected graph [Formula: see text] is a map [Formula: see text] such that for any different edge [Formula: see text] and [Formula: see text] their weights [Formula: see text] and [Formula: see text] are distinct. The minimum [Formula: see text] for which the graph [Formula: see text] has an edge irregular total [Formula: see text]-labeling is called the total edge irregularity strength of [Formula: see text] and is denoted by tes[Formula: see text]. In this paper, we determine the exact value of the total edge irregularity strength of families of ladder-related graphs, namely triangular ladder graphs, diagonal ladder graphs and circular triangular ladder graphs.

In this paper, we investigate spreading properties of the solutions of the Kolmogorov-Petrovsky-Piskunov-type, (to be simple,KPP-type) lattice system \begin{equation}\label{firstequation}\overset{.}u_{i}(t) =d^{\prime}_{i}(u_{i+1}(t)-u_{i}(t))+d_{i}(u_{i-1}(t)-u_{i}(t))+f(i,u_{i}).\end{equation} we develop some new discrete Harnack-type estimates and homogenization techniques for this lattice system to construct two speeds $\overline\omega \leq \underline \omega$ such that $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{i\geq\omega t}|u_i(t)|=0$ for any $\omega>\overline{\omega}$, and $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{0\leq i\leq\omega t}|u_i(t)-1|=0$ for any $\omega

Let [Formula: see text] be a fixed integer, we define the multiplicative function [Formula: see text], where [Formula: see text] is the Piltz divisor function and [Formula: see text] is the unitary analogue function of [Formula: see text]. The main purpose of this paper to use elementary methods to study the mean value of the function [Formula: see text].

A module which is invariant under automorphisms of its injective envelope is called an automorphism-invariant module. The class of automorphism-invariant modules was introduced and investigated by Lee and Zhou in 2013. In this paper, we study the class of modules which are invariant under all nilpotent endomorphisms of their injective envelopes of index two, such as modules are called 2-nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that a nonsingular module [Formula: see text] is a weak duo 2-nilpotent-invariant module if and only if [Formula: see text] is a strongly regular ring. For the ring [Formula: see text] satisfying every cyclic right [Formula: see text]-module is 2-nilpotent-invariant, we prove that [Formula: see text], where [Formula: see text] are rings which satisfy [Formula: see text] is a semi-simple Artinian ring and [Formula: see text] is square-free as a right [Formula: see text]-module and all idempotents of [Formula: see text] is central.

Let [Formula: see text] be a ring and [Formula: see text] be a right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text] for some essential submodule [Formula: see text] of [Formula: see text] of singular elements of[Formula: see text] with respect to[Formula: see text] , and we investigate the properties of it. For example, it is shown that [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text]. Also if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text], where [Formula: see text] is the right singular ideal of [Formula: see text]. We prove that if [Formula: see text] is a semisimple module or a prime module, then [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we have [Formula: see text] and if [Formula: see text], then [Formula: see text]. We show that [Formula: see text] and [Formula: see text]. In the end, the singular elements of some rings with respect to the formal triangular matrix ring are investigated.

In this paper, we gives lower and upper bounds on the covering radius of codes over [Formula: see text], where [Formula: see text] is a prime integer with respect to Lee distance. We also determine the covering radius of various Repetition codes over [Formula: see text], where [Formula: see text] is a prime integer.

Let [Formula: see text] be a finite group and [Formula: see text] be the set of nonlinear irreducible character degrees of [Formula: see text]. Suppose that [Formula: see text] is the set of primes dividing some elements of [Formula: see text]. The bipartite divisor graph for [Formula: see text], [Formula: see text], is a graph whose vertices are the disjoint union of [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is connected to a vertex [Formula: see text] if and only if [Formula: see text]. In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.

In this paper, the minimization problem [Formula: see text] where [Formula: see text] is studied when [Formula: see text] ([Formula: see text]) is an open, bounded, convex domain with smooth boundary and [Formula: see text]. We show that [Formula: see text] is either zero, when the maximum of the distance function to the boundary of [Formula: see text] is greater than [Formula: see text], or it is a positive real number, when the maximum of the distance function to the boundary of [Formula: see text] belongs to the interval [Formula: see text]. In the latter case, we provide estimates for [Formula: see text] and show that for [Formula: see text] sufficiently large [Formula: see text] coincides with the principal frequency of the [Formula: see text]-Laplacian in [Formula: see text]. Some particular cases and related problems are also discussed.

In this paper, we introduce the concept of algebra cone generalized [Formula: see text]-metric space over Banach algebra by replacing the constant [Formula: see text] with [Formula: see text] where [Formula: see text] is the spectral radius of [Formula: see text], with this modification we shall prove Banach and Kannan fixed point results for contractive generalized Lipschitz mappings in such a space. Moreover, we present one example in support of our results.

Let [Formula: see text] be a simple graph. A hop dominating set [Formula: see text] is called a connected hop dominating set of [Formula: see text] if the induced subgraph [Formula: see text] of [Formula: see text] is connected. The smallest cardinality of a connected hop dominating set of [Formula: see text], denoted by [Formula: see text], is called the connected hop domination number of [Formula: see text]. In this paper, we characterize the connected hop dominating sets in the join, corona and lexicographic product of graphs and determine the corresponding connected hop domination number of these graphs. The study of these concepts is motivated with a social network application.

Let [Formula: see text] be a simple connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. The eccentricity [Formula: see text] of a vertex [Formula: see text] in [Formula: see text] is the largest distance between [Formula: see text] and any other vertex of [Formula: see text]. The eccentric adjacency index (also known as Ediz eccentric connectivity index) of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the sum of degrees of neighbors of the vertex [Formula: see text]. In this paper, we determine the unicyclic graphs with largest eccentric adjacency index among all [Formula: see text]-vertex unicyclic graphs with a given girth. In addition, we find the tree with largest eccentric adjacency index among all the [Formula: see text]-vertex trees with a fixed diameter.

In this paper, we show some Second Main Theorems for zero-order meromorphic mappings intersecting slowly moving targets in [Formula: see text] by considering their [Formula: see text]-Casorati determinant. Our results are [Formula: see text]-difference analogues of Cartan’s Second Main Theorem for moving targets. As an application, we give an unicity theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] under the growth condition “order [Formula: see text]”.

In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] is a nonnegative potential function with nonempty zero set [Formula: see text], [Formula: see text] is a positive function such that the operator [Formula: see text] is indefinite and non-degenerate for [Formula: see text] large. We show that both in subcritical and critical cases, equation [Formula: see text] admits a least energy solution which for [Formula: see text] large localized near the zero set [Formula: see text].

Let [Formula: see text] and [Formula: see text] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [Formula: see text] and their dual spaces. More precisely, the authors prove that [Formula: see text] and, for any [Formula: see text], [Formula: see text], where [Formula: see text] denotes the local BMO space, [Formula: see text], for any [Formula: see text] and [Formula: see text], the inhomogeneous Lipschitz space and [Formula: see text] a variant of the local Orlicz–Hardy space related to the Orlicz function [Formula: see text] for any [Formula: see text] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

In this paper, we study the existence of positive multi-peak solutions to the fractional Schrödinger–Poisson system [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text] is a positive function, [Formula: see text] and [Formula: see text] Under some given conditions which are given in Sec. ??, we prove the existence of a positive solution with m-peaks and concentrating near a given local maximum point of [Formula: see text]

In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].

In this paper, we consider the following singular elliptic problem involving an exponential nonlinearity in two dimensions: [Formula: see text] [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show the existence and multiplicity of positive solutions globally with respect to the bifurcation parameter [Formula: see text].

In this paper, we introduce the concept of quasi-convex functions by considering a subclass [Formula: see text] of normalized analytic functions in the unit disk [Formula: see text] for some convex function [Formula: see text] with fixed second coefficient of order [Formula: see text], such that [Formula: see text] We obtain results on growth and distortion theorems and radius of convexity.

Given a group [Formula: see text], we study right and left zeros, idempotents, the minimal ideal, left cancelable and right cancelable elements of the semigroup [Formula: see text] of [Formula: see text]-linked upfamilies and characterize groups [Formula: see text] whose extensions [Formula: see text] are commutative. We finish the paper with the complete description of the structure of the semigroups [Formula: see text] for all groups [Formula: see text] of cardinality [Formula: see text].

An involuted semilattice [Formula: see text] is a semilattice [Formula: see text] with an identity element [Formula: see text] and with an involution [Formula: see text] satisfying [Formula: see text] and [Formula: see text]. We consider involuted semilattices [Formula: see text] with an identity [Formula: see text] such that there is a subsemilattice [Formula: see text] without [Formula: see text] with the property that any [Formula: see text] belongs to exactly one of the following four sets : [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. In this paper, we introduce an associative binary operation [Formula: see text] on [Formula: see text] in the following quite natural way: [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] and characterize all endomorphisms of the orthodox semigroup [Formula: see text].

In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a [Formula: see text] bounded domain [Formula: see text] or [Formula: see text], if the initial data [Formula: see text] with [Formula: see text] satisfies [Formula: see text].

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: [Formula: see text] Here [Formula: see text] is a domain (bounded or unbounded) in [Formula: see text] which is convex in [Formula: see text]-direction. [Formula: see text] is the nonlocal fractional Laplacian operator which is defined as [Formula: see text] Under various conditions on [Formula: see text] and on a solution [Formula: see text] it is shown that [Formula: see text] is strictly increasing in [Formula: see text] in the left half of [Formula: see text], or in [Formula: see text]. Symmetry (in [Formula: see text]) of some solutions is proved.

Let [Formula: see text], [Formula: see text]. Then we have the algebraic curve [Formula: see text], and its coordinate algebras (the Riemann surfaces) [Formula: see text] and [Formula: see text] The Lie algebras [Formula: see text] and [Formula: see text] are called the [Formula: see text]th superelliptic Lie algebras associated to [Formula: see text]. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form [Formula: see text].

In this paper, we define a sub-class of analytic bi-univalent functions associated with a certain differential operator [Formula: see text]. Bounds for the general Taylor–Maclaurin coefficients [Formula: see text] for the functions in this class are obtained. Estimates for the coefficient [Formula: see text] and the estimate for the functional [Formula: see text] for any real [Formula: see text], are also found. Results for the specific values of the parameters [Formula: see text], are also given mentioning some of the results obtained earlier.

In this paper, we are concerned with the zero Dirichlet boundary value problem associated to the quasilinear elliptic equation [Formula: see text] where [Formula: see text] is an open and bounded set in [Formula: see text] ([Formula: see text]), [Formula: see text] is a continuously differentiable real function in [Formula: see text], [Formula: see text] is an elliptic, bounded and symmetric matrix, [Formula: see text] is non-negative and may be singular at zero and [Formula: see text]. We give sufficient conditions on [Formula: see text], [Formula: see text] and [Formula: see text] in order to have a comparison principle and, as a consequence, uniqueness of positive solutions being continuous up to the boundary.

Let [Formula: see text] be a symplectic diffeomorphism on a closed [Formula: see text][Formula: see text]-dimensional Riemannian manifold [Formula: see text]. In this paper, we show that [Formula: see text] is Anosov if any of the following statements holds: [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property.

The intersecting kernel of a Heegaard splitting [Formula: see text] for a compact orientable 3-manifold [Formula: see text] is the subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is the homomorphism induced by the inclusion [Formula: see text], [Formula: see text]. In the paper, we obtain some invariants of 3-manifolds [Formula: see text] from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.

In this paper, we study [Formula: see text]-expansive homeomorphisms on surfaces. We prove that when [Formula: see text] is a [Formula: see text]-expansive homeomorphism defined on a compact boundaryless surface [Formula: see text] with non-wandering set [Formula: see text] being the whole of [Formula: see text] then [Formula: see text] is expansive. This condition on the non-wandering set cannot be relaxed: we present an example of a [Formula: see text]-expansive homeomorphisms on a surface with genus [Formula: see text] with wandering points that is not expansive.

Let [Formula: see text] be a [Formula: see text]-dimensional ([Formula: see text]), compact smooth Riemannian manifold endowed with a symplectic form [Formula: see text]. In this paper, we show that, if a symplectic diffeomorphism [Formula: see text] is [Formula: see text]-robustly measure expansive, then it is Anosov and a [Formula: see text] generic measure expansive symplectic diffeomorphism [Formula: see text] is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system [Formula: see text] is robustly measure expansive, then [Formula: see text] is Anosov.

In a graph [Formula: see text], a non-empty set [Formula: see text] is said to be an open packing set if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text] Let [Formula: see text] and let [Formula: see text] denote the maximum cardinality of an open packing set in [Formula: see text] which contains [Formula: see text]. Then [Formula: see text] is called the open packing saturation number of [Formula: see text]. In this paper, we initiate a study on this parameter.