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Let G be a graph with vertex-set V = V ( G ) and edge-set E = E ( G ). A 1- factor of G (also called perfect matching ) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V . A 1- factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, | V ( G )| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r , 1 ≤ r ≤ | V | − 1. One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs. A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G . An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs. It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

In this paper, we define a new class of posets which are complemented and ideal-distributive, we call these posets strong Boolean. This definition is a generalization of Boolean lattices on posets, and is different from Boolean posets. We give a topology on the set of all prime Frink ideals in order to obtain the Stone’s topological representation for strong Boolean posets. A discussion of a duality between the categories of strong Boolean posets and BP-spaces is also presented.

In this note we find all the solutions to the equation x 2 + 2 α 3 β 19 γ = y n in nonnegative unknowns with n ≥ 3 and gcd( x , y ) = 1, and nonnegative solutions to x 2 + 2 α 3 β 13 γ = y n with n ≥ 3, gcd( x , y ) = 1, except when α = 0 and x . β . γ is odd.

In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number T p −1 is divisible by p if and only if p is of the form x 2 + 11 y 2 . We first use class field theory on the Galois closure of the number field corresponding to the polynomial x 3 − x 2 − x − 1 to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for T p −1 . We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.

Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper.

The class 𝒞ℛ of completely regular semigroups considered with the unary operation of inversion within maximal subgroups forms a variety. The B -relation on the lattice ℒ(𝒞ℛ) of subvarieties of 𝒞ℛ identifies two varieties if they contain the same bands. Its classes are intervals with the set Δ of upper ends of these intervals. Canonical varieties form part of Δ. Previously we determined the sublattice Ψ of ℒ(𝒞ℛ) generated by the variety 𝒞𝒮 of completely simple semigroups and six canonical varieties. The conjecture is that the sublattice of ℒ(𝒞ℛ) generated by 𝒞𝒮 and canonical varieties follows the pattern of the structure of Ψ.

Taking two and three shared set problems into background, the uniqueness problem of a meromorphic function together with its shift operator have been studied. Our results will improve a number of recent results in the literature. Some examples have been provided in the last section to show that certain conditions used in the paper, is the best possible.

We aim to present a new linear integral operator and a certain subclass of regular functions with bounded turning property in correlation with the extended generalised Mittag-Leffler functions. Several results concerning an application of differential subordination are also investigated.

Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the interpretation of solutions is given. A generalization of Mittag-Leffler stability with respect to non-instantaneous impulses is given and sufficient conditions are obtained. Lyapunov functions and the Razumikhin technique will be applied and appropriate derivatives among the studied fractional equations is defined and applied. Examples are given to illustrate our results.

In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2 and in particular existence and uniqueness results are established. Two examples are given to illustrate our results.

We adapt the construction from [HAUSEUX, L.—LE ROUX, F.: Polynomial entropy of Brouwer homeomorphisms , arXiv:1712.01502 (2017)] to obtain an easy method for computing the polynomial entropy for a continuous map of a compact metric space with finitely many non-wandering points. We compute the maximal cardinality of a singular set of Morse negative gradient systems and apply this method to compute the polynomial entropy for Morse gradient systems on surfaces.

In this paper we present general quantitative estimates in terms of the modulus of continuity and of a K -functional, in approximation by the generalized multivariate Stancu-Durrmeyer-Choquet-Šipoš operators Mn,Γ n,x(β,γ )$\begin{array}{} M_{n, \Gamma_{n, x}}^{(\beta, \gamma)} \end{array} $, with 0 ≤ β ≤ γ , written in terms of Choquet and Šipoš integrals with respect to a family of monotone and submodular set functions, Γ n , x , on the standard d -dimensional simplex. If d = 1 and the Choquet integrals are taken with respect to some concrete possibility measures, the estimate in terms of the modulus of continuity is detailed. Examples improving the estimates given by the classical operators also are presented.

Taghavi et al. in [TAGHAVI, A.—ROHI, H.—DARVISH, V.: Non-linear ∗-Jordan derivations on von Neumann algebras , Linear Multilinear Algebra 64 (2016), 426–439] proved that the map Φ: 𝓐 → 𝓐 which satisfies the following condition Φ (A⋄ B)=Φ (A)⋄ B+A⋄ Φ (B)$$\begin{array}{} \Phi(A\diamond B)=\Phi(A)\diamond B+A\diamond \Phi(B) \end{array} $$ where A ⋄ B = AB + BA * for every A , B ∈ 𝓐 is an additive ∗-derivation. In this short note, we prove that when A is a prime ∗-algebras and Φ: 𝓐 → 𝓐 satisfies the above condition, then Φ is ∗-additive. Moreover, if Φ( iI ) is self-adjoint then Φ is derivation.

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure λ, where λ is invariant under a continuous injective mapping φ : G → G . We characterize the disjoint hypercyclicity of finite weighted translations generated by φ acting on the weighted space L p ( G , ω ) (1 ≤ p < ∞).

Let M be a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator of M is transversally Killing with respect to the generalized Tanaka-Webster connection if and only if M is locally congruent to a type ( A ) or ( B ) real hypersurface. We also prove that shape operator of M commutes with Cho operator on holomorphic distribution if and only if M is locally congruent to a ruled real hypersurface.

In this paper, we present the joint study of two main branches of topology, relative properties and hyperspaces, with the aim of giving an answer to Problem 3 in [Arhangeľskii, A. V.: A generic theorem in the theory of cardinal invariants of topological spaces , Comment. Math. Univ. Carolin. 36 (1995), 305–325].

Let X be a 2 n -dimensional torus manifold with a locally standard T ≅ ( S 1 ) n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T -bundle p : E → B and let π : E ( X ) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E ( X ) as a H * ( B )-algebra and the topological K -ring of E ( X ) as a K * ( B )-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt . These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

For a Tychonoff space X and a family λ of subsets of X , we denote by C λ ( X ) the space of all real-valued continuous functions on X with the set-open topology. A Menger space is a topological space in which for every sequence of open covers 𝓤 1 , 𝓤 2 , … of the space there are finite sets 𝓕 1 ⊂ 𝓤 1 , 𝓕 2 ⊂ 𝓤 2 , … such that family 𝓕 1 ∪ 𝓕 2 ∪ … covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space C λ ( X ). Our main results state that C λ ( X ) is Menger if and only if C λ ( X ) is σ -compact; C p ( Y | X ) is projective Menger if and only if C p ( Y | X ) is σ -pseudocompact where Y is a dense subset of X .

In this paper, we study the limit distributions of upper and lower record values of a stationary Gaussian sequence under an equi-correlated set up. Moreover, the class of limit distribution functions (df’s) of the joint upper (and the lower) record values of a stationary Gaussian sequence is fully characterized. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-mid-range, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit df’s of these statistics are derived.