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A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G + uv) < k for any pair of non-adjacent vertices u and v of G. The problem that arises is that of characterizing k-i critical graphs. In this paper, we characterize connected 3-i-critical graphs with minimum vertex cutset of size 2. More specifically, we show that if G is a connected 3-i-critical graph with minimum vertex cutset S of size 2 and the number of components of G−S is exactly two, then G is isomorphic to a graph in one of nine classes of connected 3-i-critical graphs. The results in this paper together with results in [1] and [2] provide a complete characterization of connected 3-i-critical graphs with a minimum cutset of size at most 3. [ABSTRACT FROM AUTHOR]

A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The total domination number, γt(G), is the minimum cardinality of a total dominating set of G. Chellali and Haynes [J. Combin. Math. Combin. Comput.58 (2006), 189–193] showed that if T is a nontrivial tree of order n, with ℓ leaves, then γt(T) ≥ (n − ℓ + 2)/2. In this paper, we first characterize all trees T of order n with ℓ leaves satisfying γt(T) = ⌈(n−ℓ+2)/2⌉. We then generalize this result to connected graphs and show that if G is a connected graph of order n ≥ 2 with k ≥ 0 cycles and ℓ leaves, then γt(G) ≥ ⌈(n − ℓ + 2)/2⌉ − k. We also characterize the graphs G achieving equality for this new bound. [ABSTRACT FROM AUTHOR]

Let C(X) denote the ring of all real-valued continuous functions on a topological space X; and C∞(X) be the subring of all functions C(X) which vanish at infinity. In [2], the paper "Rings of continuous functions vanishing at infinity," (Comment. Math. Univ. Carolin.45(3) (2004), 519–533), by A.R. Aliabad, F. Azarpanah, and M. Namdari, it is shown that for every completely regular Hausdorff space X, whenever C∞(X) ≠ (0), then there exists a locally compact space Y such that C∞(X) ≅ C∞(Y). In fact, the space Y may be considered as a nonempty open locally compact subspace of X. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames. [ABSTRACT FROM AUTHOR]

In a metric space M = (X, d), a line induced by two distinct points x, x′ ∈ X, denoted by , is the set of points given by A line is universal whenever. Chen and Chvátal [Disc. Appl. Math. 156 (2008), 2101-2108.] conjectured that in any finite metric space M = (X, d) either there is a universal line, or there are at least |X| different (nonuniversal) lines. A particular problem derived from this conjecture consists of investigating the properties of M that determine the existence of a universal line, and the problem remains interesting even if we can check that M has at least |X| different lines. Since the vertex set of any connected graph, equipped with the shortest path distance, is a metric space, the problem automatically becomes of interest in graph theory. In this paper, we address the problem of characterizing graphs that have universal lines. We consider several scenarios in which the study can be approached by analysing the existence of such lines in primary subgraphs. We first discuss the wide class of separable graphs, and then describe some particular cases, including those of block graphs, rooted product graphs and corona graphs. We also discuss important classes of nonseparable graphs, including Cartesian product graphs, join graphs and lexicographic product graphs. [ABSTRACT FROM AUTHOR]

Group coextensions of monoids, which generalise Schreier-type extensions of groups, have originally been defined by P.A. Grillet and J. Leech. The present paper deals with pomonoids, that is, monoids that are endowed with a compatible partial order. Following the lines of the unordered case, we define pogroup coextensions of pomonoids. We furthermore generalise the construction to the case that pomonoids instead of pogroups are used as the extending structures. The intended application lies in fuzzy logic, where triangular norms are those binary operations that are commonly used to interpret the conjunction. We present conditions under which the coextension of a finite totally ordered monoid leads to a triangular norm. Triangular norms of a certain type can therefore be classified on the basis of the presented results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a generalized Wintgen-type inequality for submanifolds in generalized (κ,µ)-space forms. Further, we discuss the inequality for bi-slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds and slant submanifolds in various ambient spaces. In particular, we generalize some recent results of Mihai, Tohoku Math. J.69 (2017), and Aquib et al., Tamkang J. Math.50 (2019) and Mathematics7(12) (2019). [ABSTRACT FROM AUTHOR]

Let (tn(t)) be the sequence of trigonometric functions on [0, 1] with real scalars. We prove a generalization of the likely known result: - for every p ⩾ 1 the series Σ (∣tn(t)∣p)/n is unbounded on every set A with positive measure. This provides an alternate proof of a result in a recent paper of the author; the methods used are classical. [ABSTRACT FROM AUTHOR]

In this paper, for 𝒞 a strict braided monoidal category with tensor product ⊗, we improve the definition of Hom-associative algebra by removing the multiplicativity condition of the automorphismα. After that we state the close connection between the classical notions of (co)algebra, (co)module and Hopf algebra and the corresponding ones in the Hom-world. As a consequence we can study the questions about the Hom-world by passing to the classical one, and the main contribution of this paper is to illustrate the suitable procedure to move from one side to the other. Finally, we apply our techniques to get in the Hom setting some classical results about cleft and Galois extensions. [ABSTRACT FROM PUBLISHER]

We study the problem of characterizing which topologies on a nonempty set are generated by a binary relation by means of its lower and upper contour sets. In this direction we consider different classical categories of topological spaces whose topology is defined by at least one binary relation. Given a topology defined by some binary relation on a set, we also analyze if the binary relation could belong to some particular category. We furnish some examples of topological spaces whose topology cannot be induced by any binary relation. We analyze some of these items in the context of pointfree topology and extend these questions to the study of bitopological spaces. [ABSTRACT FROM PUBLISHER]

This paper is devoted to derive some properties and expansions associated with the q-digamma function. The Newton series which is consisting of terms of forward difference operator, is established for the q-digamma function. The maltiplication formula of the q-gamma function is used to present some recurrence relations for the q-digamma function. The q-analogue of well-known results in the theory of the digamma function are investigated. One of the most important identities of q-beta integrals is used to introduce some integral representations of the q-digamma function. [ABSTRACT FROM AUTHOR]

Andrews, Knopfmacher and Knopfmacher have used the Schur polynomials to consider the celebrated Rogers-Ramanujan identities in the context of q-Engel expansions. We extend this view using similar polynomials, provided by Sills, in the context of Slater's list of 130 Rogers-Ramanujan type identities. [ABSTRACT FROM AUTHOR]

The notion of radial Blaschke-Minkowski homomorphisms was presented by Schuster. Afterwards, Wang et al. introduced Lp radial Blaschke-Minkowski homomorphisms. In this paper, associated with Lp-dual affine surface areas, we establish some inequalities including the Brunn-Minkowski type inequality, cyclic inequality and monotonic inequalities, and give an affirmative answer and a negative answer of Busemann-Petty problem for the Lp radial Blaschke-Minkowski homomorphisms. [ABSTRACT FROM AUTHOR]

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that if then there is a linear isometry U : X → Y so that where is defined by Representation properties of coarse isometries in free ultrafilter limits on are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we shall define the special-valued multiple Hurwitz zeta functions, namely the multiple t-values t(α) and define similarly the multiple star t-values as t⋆(α). Then we consider the sum of all such multiple (star) t-values of fixed depth and weight with even argument and prove that such a sum can be evaluated when the evaluations of t({2m}n) and t*({2m}n) are clear. We give the evaluations of them in terms of the classical Euler numbers through their generating functions. [ABSTRACT FROM AUTHOR]

The paper investigates the structure of the highly composite multicomposition function gm(n) which enumerates the number of m-compositions of n into relatively prime positive summands. Among other results, we generalize a known result by establishing that the lcm (m(m+2), (m+1) n/P1P2…Pr-1, (m+1) ø(P2α2) -1, …, (m+1) ø(Prαr)-1) divides gm(n); ∀n ≥ 3, where n = Πi=1r piαi and lcm(a, b) denotes the least common multiple of a and b. [ABSTRACT FROM AUTHOR]

The following problem is considered in this paper: Suppose k strings of unit length are to be distributed amongst x hats. If cuts in the strings are allowed, how should the string (parts) be distributed amongst the hats so that, if the shortest string is removed from each hat, the remaining (combined) string length in the hat with the most string is as small as possible? We solve this problem for a number of special cases (i.e. for certain values of x and k) and provide good bounds on the solution for the remaining cases. [ABSTRACT FROM AUTHOR]

In this paper, for a prime p, we propose some plausible definitions for the notion of Sylow fuzzy p-subgroup of a finite group. We derive a number of results for finite fuzzy groups using one of the proposed definitions. We also discuss some of the relationships between various proposed definitions for suitability, including the crisp case, with some examples. [ABSTRACT FROM AUTHOR]

This paper inductively investigates an inverse problem for difference boundary value problems with boundary conditions that depend quadratically on the eigenparameter. In particular, given the eigenvalues and the weights, we provide an algorithm to uniquely reconstruct the potential. [ABSTRACT FROM AUTHOR]

The fractional calculus includes concepts of integrals and derivatives of any complex or real order. The fractional calculus is as old as the usual calculus. Recently, many scientists have been studying on this field to provide the development and applicability to various areas of mathematics, physics, engineering and other sciences. The discrete fractional calculus has an important position in this field. Some ordinary diffrential equations can be solved by means of nabla (∇) discrete fractional calculus operator. In this paper, we aim to apply this operator to the radial Schrödinger equation for some physical potentials such as pseudoharmonic and Mie-type potentials. [ABSTRACT FROM AUTHOR]

Sibling curves were demonstrated in [1, 2] as a novel way to visualize the zeroes of real valued functions. In [3] it was shown that a polynomial of degreenhasnsibling curves. This paper focuses on the algebraic and geometric properites of the sibling curves of real and complex quadratic polynomials. [ABSTRACT FROM AUTHOR]

Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary differential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebrasl(3,IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integralsI1,I2and their quotientI2/I1are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebrasl(4,IR) of the free particle system. It is further shown that the (n+ 2)-dimensional algebras of then+ 2 first integrals of the system ofnfree particle equations are isomorphic to each other and generate the full symmetry algebrasl(n+ 2,IR) of the free particle system. [ABSTRACT FROM AUTHOR]

In this paper, for a frameL, we characterize modified sobriety in stratifiedL-topological spaces and strongL-topological spaces internally using certain classes of stratifiedL-filters. While the first characterization using completely primeL-filters is trivial and applies to all stratifiedL-topological spaces, the other one using irreducibleL-filters generalizes an approach of R.E. Hoffmann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strongL-topological spaces. [ABSTRACT FROM AUTHOR]

Straight spaces are spaces for which a continuous map defined on the space which is uniformly continuous on each set of a finite closed cover is then uniformly continuous on the whole space. Previously, straight spaces have been studied in the setting of metric spaces. In this paper, we present a study of straight spaces in the more general setting of nearness spaces. In a subcategory of nearness spaces somewhat more general than uniform spaces, we relate straightness to uniform local connectedness. We investigate category theoretic situations involving straight spaces. We prove that straightness is preserved by final sinks, in particular by sums and by quotients, and also by completions. [ABSTRACT FROM AUTHOR]

A subsetXof the vertex set of a graphGis asecure dominating setofGifXis a dominating set ofGand if, for each vertexunot inX, there is a neighbouring vertexvofuinXsuch that the swap set (X/{v}) ᑌ{u}is again a dominating set ofG, in which casevis called adefender. Thesecure domination numberofGis the cardinality of a smallest secure dominating set ofG. In this paper, we show that every graph of minimum degree at least 2 possesses a minimum secure dominating set in whichallvertices are defenders. We also characterise the classes of graphs that have secure domination numbers 1, 2 and 3. [ABSTRACT FROM AUTHOR]

In every finite poset (X,≤) we assign the so called order-matrix, whereαij∈ {−2, 0, 1, 2}. Using this matrix, we characterize the order dimension of an arbitrary finite poset. [ABSTRACT FROM AUTHOR]

A vertexυin a setSis said to becost effectiveif it is adjacent to at least as many vertices inV\Sas it is inSand is very cost effective if it is adjacent to more vertices inV\Sthan to vertices inS. A dominating setSis (very) cost effective if every vertex inSis (very) cost effective. The minimum cardinality of a (very) cost effective dominating set ofGis the (very) cost effective domination number ofG. Our main results include a quadratic upper bound on the very cost effective domination number of a graph in terms of its domination number. The proof of this result gives a linear upper bound for hereditarily sparse graphs which include trees. We show that no such linear bound exists for graphs in general, even when restricted to bipartite graphs. Further, we characterize the extremal trees attaining the bound. Noting that the very cost effective domination number is bounded below by the domination number, we show that every value of the very cost effective domination number between these lower and upper bounds for trees is realizable. Similar results are given for the cost effective domination number. [ABSTRACT FROM AUTHOR]

We investigate the interlacing of the zeros of linear combinations pn+aqm with the zeros of the components pn and qm, where {pn}∞n=0 and {qm}∞m=0 are different sequences of Jacobi polynomials. The results we prove hold when pn and qm are Jacobi polynomials Pn(α,β) n (x) and Pm(α′,β′) m (x) for certain values of α′ and β′ with m = n or m = n - 1. Numerical counterexamples are given in situations where interlacing fails to occur. We also show that the zeros of the linear combination pn + aqm interlace with the zeros of some Jacobi polynomials besides the components of the linear combination. [ABSTRACT FROM AUTHOR]

Two properties each of which is equivalent to separability for regular spaces and is weaker than separability in general are considered as well as the corresponding cardinal functions. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the scalar Kurzweil-Henstock integrability and the Kurzweil-Henstock-Pettis integrability of functions f : [1,∞) → X defined as f = Σ ∞ n=1 xnχ[n,n+1). Also the variational Henstock integrability is considered. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a graph and let S ⊑ V . The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a secure dominating set of G if for each u ϵ V \ S, there exists a vertex v ϵ S such that uv ϵ E and (S \ {v})∪{u} is a dominating set of G. The minimum cardinality of a secure dominating set in G is the secure domination number γs(G) of G. We show that if G is a connected graph of order n with minimum degree at least two that is not a 5-cycle, then γs(G) ≤ n/2 and this bound is sharp. Our proof uses a covering of a subset of V (G) by vertex-disjoint copies of subgraphs each of which is isomorphic to K2 or to an odd cycle. [ABSTRACT FROM AUTHOR]

The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of ω1 subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied. [ABSTRACT FROM AUTHOR]

For a unital Banach algebra A over ℂ, we give improvements on the well-known multiplicative spectral characterization of the Jacobson radical, (1), in terms of several spectral parameters. In particular, we show that every non-invertible element a ϵ A for which the number of elements in the spectrum of ax is less or equal to the number of elements in the spectrum of x, for all x in an arbitrary small neighborhood of the unit, must belong to the Jacobson radical. [ABSTRACT FROM AUTHOR]