In this paper we present a variety of continuation (homotopy) theorems for general classes of maps in the literature.
In this paper we present a variety of continuation (homotopy) theorems for general classes of maps in the literature.
We reinterpret the Rhodes semilattices (R_n({mathfrak {G}})) of a group ({mathfrak {G}}) in terms of gain graphs and generalize them to all gain graphs, both as sets of partition-potential pairs and as sets of subgraphs, and for the latter, further to biased graphs. Based on this we propose four different natural lattices in which the Rhodes semilattices and its generalizations are order ideals.
We consider a map f from one abelian group into another that satisfies either an additive or quadratic functional equation on any given pair of elements of its domain. Particular emphasis is placed on the possibility that f itself is neither additive nor quadratic and a complete description of all those cases is obtained.
Let (gamma : S^nrightarrow mathbb {R}_+) be a convex integrand and (mathcal {W}_gamma ) be the Wulff shape of (gamma ). A d-apex point naturally arises in a non-smooth Wulff shape, in particular, as a vertex of a convex polytope. In this paper, we study the behavior of the convex integrand at a d-apex point of its Wulff shape. We prove that (gamma (P)) is locally maximum, and (mathbb {R}_+ Pcap partial mathcal {W}_gamma ) is a d-apex point of (mathcal {W}_gamma ) if and only if the graph of (gamma ) around the d-apex point is a piece of a sphere with center (frac{1}{2}gamma (P)P) and radius (frac{1}{2}gamma (P)). As an application of the proof of this result, we prove that for any spherical convex body C of constant width (tau >pi /2), there exists a sequence ({C_i}_{i=1}^infty ) of convex bodies of constant width (tau ), whose boundaries consist only of arcs of circles of radius (tau -frac{pi }{2}) and great circle arcs such that (lim _{irightarrow infty }C_i=C) with respect to the Hausdorff distance.
For a discrete group G, we consider certain ideals (mathcal {I}subset c_0(G)) of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C(^*)-algebra of G and the C(^*)-completion (textrm{C}^*_{mathcal {I}}(G)) in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that G is amenable.
A double Roman domination function (DRDF) on a graph (G=(V,E)) is a mapping (f :Vrightarrow {0,1,2,3}) satisfying the conditions: (i) each vertex with 0 assigned is adjacent to a vertex with 3 assigned or at least two vertices with 2 assigned and (ii) each vertex with 1 assigned is adjacent to at least one vertex with 2 or 3 assigned. The weight of a DRDF f is defined as the sum (sum _{vin V}f(v)). The minimum weight of a DRDF on a graph G is called the double Roman domination number (DRDN) of G. This study establishes the values on DRDN for several graph classes. The exact values of DRDN are proved for Kneser graphs (K_{n,k},nge k(k+2)), Johnson graphs (J_{n,2}), for a few classes of convex polytopes, and the flower snarks. Moreover, tight lower and upper bounds on SRDN are proved for some convex polytopes. For the generalized Petersen graphs (P_{n,3}, n not equiv 0,(mathrm {mod 4})), we make a further improvement on the best known upper bound from the literature.
The concept of Gromov hyperbolicity is a geometric concept that leads to a rich general theory. Johnson and Kneser graphs are interesting combinatorial graphs defined from systems of sets. In this work we compute the precise value of the hyperbolicity constant of every Johnson graph. Also, we obtain good bounds on the hyperbolicity constant of every Kneser graph, and in many cases, we even compute its precise value.
We introduce probabilistic Stirling numbers of the first kind (s_Y(n,k)) associated with a complex-valued random variable Y satisfying appropriate integrability conditions, thus completing the notion of probabilistic Stirling numbers of the second kind (S_Y(n,k)) previously considered by the first author. Combinatorial interpretations, recursion formulas, and connections between (s_Y(n,k)) and (S_Y(n,k)) are given. We show that such numbers describe a large subset of potential polynomials, on the one hand, and the moments of sums of i. i. d. random variables, on the other, establishing their precise asymptotic behavior without appealing to the central limit theorem. We explicitly compute these numbers when Y has a certain familiar distribution, providing at the same time their combinatorial meaning.
Let G be a connected graph of order n, where n is a positive integer. A spanning subgraph F of G is called a path-factor if every component of F is a path of order at least 2. A (P_{ge k})-factor means a path-factor in which every component admits order at least k ((kge 2)). The distance matrix ({mathcal {D}}(G)) of G is an (ntimes n) real symmetric matrix whose (i, j)-entry is the distance between the vertices (v_i) and (v_j). The distance signless Laplacian matrix ({mathcal {Q}}(G)) of G is defined by ({mathcal {Q}}(G)=Tr(G)+{mathcal {D}}(G)), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue (eta _1(G)) of ({mathcal {Q}}(G)) is called the distance signless Laplacian spectral radius of G. In this paper, we aim to present a distance signless Laplacian spectral radius condition to guarantee the existence of a (P_{ge 2})-factor in a graph and claim that the following statements are true: (i) G admits a (P_{ge 2})-factor for (nge 4) and (nne 7) if (eta _1(G)<theta (n)), where (theta (n)) is the largest root of the equation (x^{3}-(5n-3)x^{2}+(8n^{2}-23n+48)x-4n^{3}+22n^{2}-74n+80=0); (ii) G admits a (P_{ge 2})-factor for (n=7) if (eta _1(G)<frac{25+sqrt{161}}{2}).
We give an example of some iteration group in a ring of formal power series over a field of characteristic 0. It allows us to obtain an explicit formula for some one-parameter group of (truncated) formal power series under an additional condition. Consequently, we are able to show some non-commutative groups of solutions of the third Aczél-Jabotinsky differential equation in the ring of truncated formal power series.
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