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This article indicates another settheoretic formula, solely in terms of union and intersection, for the set of the limits of any given sequence (net, in general) in an arbitrary T _{1} space; this representation in particular gives a new characterization of a T _{1} space.
We give all solutions of completely multiplicative functions ƒ , g, for which the equation Ag(n + 1) = Bƒ (n) + C holds for every n ∈ ℕ. We also study the equation G(p + 1) = F(p − 1) + D and we prove some results concerning it.
We consider a graph whose vertices are legally colored using k colors and ask if the graph contains a kclique. As it turns out this very special type of kclique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.
Assume that A_{j} , j ∈ {1, … , m} are positive definite matrices of order n. In this paper we prove among others that, if 0 < l I_{n} ≤ A_{j} , j ∈ {1, … , m} in the operator order, for some positive constant l, and I_{n} is the unity matrix of order n, then
where Pk ≥ 0 for k ϵ {1, …, m} and
The evolute of a conic in the pseudoEuclidean plane is the locus of centers of all its osculating circles. It’s a curve of order six and class four in general case. In this paper we discuss and compute the order and class of evolutes of different types of conics. We will highlight those cases that have no analogy in the Euclidean plane.
Fast [12] is credited with pioneering the field of statistical convergence. This topic has been researched in many spaces such as topological spaces, cone metric spaces, and so on (see, for example [19, 21]). A cone metric space was proposed by Huang and Zhang [17]. The primary distinction between a cone metric and a metric is that a cone metric is valued in an ordered Banach space. Li et al. [21] investigated the definitions of statistical convergence and statistical boundedness of a sequence in a cone metric space. Recently, Sakaoğlu and Yurdakadim [29] have introduced the concepts of quasistatistical convergence. The notion of quasi Istatistical convergence for triple and multiple index sequences in cone metric spaces on topological vector spaces is introduced in this study, and we also examine certain theorems connected to quasi Istatistically convergent multiple sequences. Finally, we will provide some findings based on these theorems.
For a graph G, we define the lower bipartite number LB(G) as the minimum order of a maximal induced bipartite subgraph of G. We study the parameter, and the related parameter bipartite domination, providing bounds both in general graphs and in some graph families. For example, we show that there are arbitrarily large 4connected planar graphs G with LB(G) = 4 but a 5connected planar graph has linear LB(G). We also show that if G is a maximal outerplanar graph of order n, then LB(G) lies between (n + 2)/3 and 2 n/3, and these bounds are sharp.
The motions of a bar structure consisting of two congruent tetrahedra are investigated, whose edges in their basic position are the face diagonals of a rectangular parallelepiped. The constraint of the motion is the following: the originally intersecting edges have to remain coplanar. All finite motions of our bar structure are determined. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.
We study the path behavior of the symmetric walk on some special combtype subsets of ℤ^{2} which are obtained from ℤ^{2} by generalizing the comb having finitely many horizontal lines instead of one.
In a typical maximum clique search algorithm when optimality testing is inconclusive a forking takes place. The instance is divided into smaller ones. This is the branching step of the procedure. In order to ensure a balanced work load for the processors for parallel algorithms it is essential that the resulting smaller problems are do not overly vary in difficulty. The socalled splitting partitions of the nodes of the given graph were introduced earlier to meliorate this problem. The paper proposes a splitting partition of the edges for the same purpose. In the lack of available theoretical tools we assess the practical feasibility of constructing suboptimal splitting edge partitions by carrying out numerical experiments. While working with splitting partitions we have realized that they can be utilized as preconditioning tools preliminary to a large scale clique search. The paper will discuss this new found role of the splitting edge partitions as well.
We prove that for any collection F of n ≥ 2 pairwise disjoint compact convex sets in the plane there is a pair of sets A and B in F such that any line that separates A from B separates either A or B from a subcollection of F with at least n/18 sets.
In this paper, we study the existence of positive solutions for a system of nonlinear fractional differential equations. The results are based upon the fixedpoint theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, Our results generalize and include some known results.
Criteria for a diffeomorphism of a smooth manifold M to be lifted to a linear automorphism of a given real vector bundle p : V → M, are stated. Examples are included and the metric and complex vectorbundle cases are also considered.
Let X be an irreducible complex projective variety of dimension n ≥ 1. Let D be a Cartier divisor on X such that H^{i}(X, O_{X} (mD)) = 0 for m > 0 and for all i > 0, then it is not true in general that D is a nef divisor (cf. [4]). Also, in general, effective divisors on smooth surfaces are not necessarily nef (they are nef provided they are semiample). In this article, we show that, if X is a smooth surface of general type and C is a smooth hyperplane section of it, then for any nonzero effective divisor D on X satisfying H^{1}(X, O_{X} (mD)) = 0 for all m > C.K_{X} , D is a nef divisor.
In this paper, we introduce the notion of a Gel’fand Γsemiring and discuss the various characterization of simple, kideal, strong ideal, tsmall elements and additively cancellative elements of a Gel’fand Γsemiring R, and prove that the set of additively cancellative elements, set of all tsmall elements of R and set of all maximal ideal of R are strong ideals. Further, let R be a simple Gel’fand Γsemiring and 1 ≠ t ∈ R. Let M be the set of all maximal left (right) ideals of R. Then an element x of R is tsmall if and only if it belongs to every maximal one sided left (right)ideal of R containing t.
For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following integral transform
where the integral is assumed to exist for t > 0.
We show among others that D(w, μ) is operator convex on (0, ∞). From this we derive that, if f : [0, ∞) → R is an operator monotone function on [0, ∞), then the function [f(0) f(t)] t
^{1} is operator convex on (0, ∞). Also, if f : [0, ∞) → R is an operator convex function on [0, ∞), then the function
under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.
Problem 2 of Welsh’s 1976 text Matroid Theory, asking for criteria telling when two families of sets have a common transversal, is solved.
Another unsolved problem in the text Matroid Theory, on whether the “join” of two nondecreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.
With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.
This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the FaedoGalerkin method. Exponential stability is proved by applying Nakao’s theorem.
We consider hypersphere x = x(u, v, w) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second LaplaceBeltrami operator the hypersphere satisfying Δ^{II}x = Ax, where A ϵ Mat (4,4).
In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.
Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width D. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.
Let P be a set of n points in general position in the plane. Let R be a set of points disjoint from P such that for every x, y € P the line through x and y contains a point in R. We show that if
We use the same approach to show a similar result in the case where each of B and G is a set of n points in general position in the plane and every line through a point in B and a point in G passes through a point in R. This provides a partial answer to a problem of Karasev.
The bound
Let E, G be Fréchet spaces and F be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator T on E into F factoring through G causes the existence of a common nuclear Köthe subspace of the triple (E, G, F). If, in addition, F has the property (y), then (E, G, F) has a common nuclear Köthe quotient.
In this paper we study the sum
In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.
The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolatefree graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4colorable, or has maximum degree at most 5.
This study proposes a new family of continuous distributions, called the Gudermannian generated family of distributions, based on the Gudermannian function. The statistical properties, including moments, incomplete moments and generating functions, are studied in detail. Simulation studies are performed to discuss and evaluate the maximum likelihood estimations of the model parameters. The regression model of the proposed family considering the heteroscedastic structure of the scale parameter is defined. Three applications on real data sets are implemented to convince the readers in favour of the proposed models.
Let [ · ] be the fioor function. In this paper, we show that when 1 < c < 37/36, then every sufficiently large positive integer N can be represented in the form
where p_{1}, p_{2}, p_{3} are primes close to squares.
In this article, we study a family of subgraphs of the Farey graph, denoted as ℱ_{N} for every N ∈ ℕ. We show that ℱ_{N} is connected if and only if N is either equal to one or a prime power. We introduce a class of continued fractions referred to as ℱ_{N} continued fractions for each N > 1. We establish a relation between ℱ_{N} continued fractions and certain paths from infinity in the graph ℱ_{N} . Using this correspondence, we discuss the existence and uniqueness of ℱ_{N} continued fraction expansions of real numbers.
Given a finite point set P in the plane, a subset S⊆P is called an island in P if conv(S) ⋂ P = S. We say that S ⊂ P is a visible island if the points in S are pairwise visible and S is an island in P. The famous Bigline Bigclique Conjecture states that for any k ≥ 3 and l ≥ 4, there is an integer n = n(k, l), such that every finite set of at least n points in the plane contains l collinear points or k pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13.
In this article, we define the notion of a generalized open book of a nmanifold over the k−sphere S^{k} , k < n. We discuss Lefschetz open book embeddings of Lefschetz open books of closed oriented 4manifolds into the trivial open book over S^{2} of the 7−sphere S^{7} . If X is the double of a bounded achiral Lefschetz fibration over D^{2} , then X naturally admits a Lefschetz open book given by the bounded achiral Lefschetz fibration. We show that this natural Lefschetz open book of X admits a Lefschetz open book embedding into the trivial open book over S^{2} of the 7−sphere S^{7} .
We show that if a nondegenerate PL map f : N → M lifts to a topological embedding in
This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.
For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is joinclosed, meetclosed, and whenever {a, x, b} ⊆ S, y ∈ L, x ∧ y = a, and x ∨ y = b, then y ∈ S. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.
In this paper, centralizing (semicentralizing) and commuting (semicommuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semicentralizing) derivations of prime semirings. Further, we observe that if there exists a skewcommuting (skewcentralizing) derivation D of S, then D = 0. It is also proved that for any two derivations d _{1} and d _{2} of a prime semiring S with char S ≠ 2 and x ^{ d 1 } x ^{ d 2 } = 0, for all x ∈ S implies either d _{1} = 0 or d _{2} = 0.
We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.
A leaf of a tree is a vertex of degree one and a branch vertex of a tree is a vertex of degree at least three. In this paper, we show a degree condition for a clawfree graph to have spanning trees with at most five branch vertices and leaves in total. Moreover, the degree sum condition is best possible.
We prove that the number of unit distances among n planar points is at most 1.94 • n ^{4/3}, improving on the previous best bound of 8 n ^{4/3}. We also give better upper and lower bounds for several small values of n. We also prove some variants of the crossing lemma and improve some constant factors.
Two hexagons in the space are said to intersect heavily if their intersection consists of at least one common vertex as well as an interior point. We show that the number of hexagons on n points in 3space without heavy intersections is o(n ^{2}), under the assumption that the hexagons are ‘fat’.
Let X be a smooth projective K3 surface over the complex numbers and let C be an ample curve on X. In this paper we will study the semistability of the LazarsfeldMukai bundle E_{C,A} associated to a line bundle A on C such that A is a pencil on C and computes the Clifford index of C. We give a necessary and sufficient condition for E_{C,A} to be semistable.
We prove criteria for a graph to be the Reeb graph of a function of a given class on a closed manifold: Morse–Bott, round, and in general smooth functions whose critical set consists of a finite number of submanifolds. The criteria are given in terms of whether the graph admits an orientation, which we call Sgood orientation, with certain conditions on the degree of sources and sinks, similar to the known notion of good orientation in the context of Morse functions. We also study when such a function is the height function associated with an immersion of the manifold. The condition for a graph to admit an Sgood orientation can be expressed in terms of the leaf blocks of the graph.
For each Montesinos knot K, we propose an efficient method to explicitly determine the irreducible SL(2, )character variety, and show that it can be decomposed as χ_{0}(K)⊔χ_{1}(K)⊔χ_{2}(K)⊔χ'(K), where χ_{0}(K) consists of tracefree characters χ_{1}(K) consists of characters of “unions” of representations of rational knots (or rational link, which appears at most once), χ_{2}(K) is an algebraic curve, and χ'(K) consists of finitely many points when K satisfies a generic condition.
We offer new properties of the special Gini mean S(a, b) = a^{a} ^{/(} ^{a} ^{+} ^{b} ^{)} ⋅ b^{b} ^{/(} ^{a} ^{+} ^{b} ^{)}, in connections with other special means of two arguments.
We treat a variation of graph domination which involves a partition (V _{1}, V _{2},..., V_{k} ) of the vertex set of a graph G and domination of each partition class V _{i} over distance d where all vertices and edges of G may be used in the domination process. Strict upper bounds and extremal graphs are presented; the results are collected in three handy tables. Further, we compare a high number of partition classes and the number of dominators needed.
Proctor and Scoppetta conjectured that

(1) there exists an infinite locally finite poset that satisfies their conditions VT and NTC but not SIS;

(2) there exists an infinite locally finite poset satisfying their conditions D3^{}C and D3MF but not both VT and FT; and

(3) there exists an infinite locally finite poset satisfying their conditions D3^{}C and D3MD but not NCC.
In this note, the conjecture of Proctor and Scoppetta, which is related to dcomplete posets, is proven.
In this paper we introduce the concept of the Hamilton triangle of a given triangle in an isotropic plane and investigate a number of important properties of this concept. We prove that the Hamilton triangle is homological with the observed triangle and with its contact and complementary triangles. We also consider some interesting statements about the relationships between the Hamilton triangle and some other significant elements of the triangle, like e.g. the Euler and the Feuerbach line, the Steiner ellipse and the tangential triangle.
We prove certain Menontype identities associated with the subsets of the set {1, 2,..., n} and related to the functions f, f_{k} , Ф and Ф _{k} , defined and investigated by Nathanson.
Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.