Characterizing face-number-related invariants of a given class simplicial complexes has been central topic in combinatorial topology. In this regard, one the well-known is g2. Let K be normal 3-pseudomanifold such that g2(K)≤g2(lk(v))+9 for some vertex v K. Suppose either only singularity or two singularities (at least) which an RP2-singularity. We prove obtained from boundary 4-simplices by se...

Abstract Background Crohn’s disease (CD) is a heterogeneous and complicated condition that often has delayed diagnoses poor outcomes. Disease location site-specific mechanisms have received increasing attention in recent studies. Suboptimal classification adds complexity to clinical management for CD the interpretation of its characteristics. This study aims clarify clinicopathological characte...

where Ni are known nonlinear functionals, g = (g1, · · · , gr) are unknown functions, and 2i iid ∼ N(0, σ) are random errors. Without loss of generality, we assume that r = 2. As in O’Sullivan (1990), we express design points x explicitly in the functional Ni: Ni(g1, g2) = η(g1, g2; xi), where η is a known nonlinear functional. In the following sections, η(g1, g2; x) is sometimes also represent...

Let G = G1 ∪ G2 be the sum of two simple graphs G1, G2 having a common edge or G = G1 ∪ e1 ∪ e2 ∪ G2 be the sum of two simple disjoint graphs G1, G2 connected by two edges e1 and e2 which form a cycle C4 inside G. We give a method of computing the determinant detA(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G1 and G2. To show the scope ...

Theorem 1.1 (Chudnovsky’s theorems). Let Λ = Zω+Zω′ ⊂ C be a lattice with invariants g2, g3, Weierstrass functions ℘, ζ and quasi-periods η, η′, all defined in the usual way (see [Sil94]). Each of the sets below contains at least two algebraically independent numbers : first (1) {g2, g3, η ω , π ω}, (2) {g2, g3, ω, η, ω′, η′}; if we assume g2, g3 ∈ Q̄ : (3) { η ω , π ω}, (4) {ω, ω′, η, η′}; and ...

The (k − 1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of an induced subgraph by deleting at most k − 1 vertices from G. This paper considers the fault diameter of the product graph G1 ∗G2 of two graphs G1 and G2 and proves that Dk1+k2(G1 ∗G2) Dk1(G1)+Dk2(G2)+ 1 if G1 is k1-connected and G2 is k2-connected. This generalizes some known results such as Banič and Žerovni...

Let G1 and G2 be graphs of order n with maximum degree ∆1 and ∆2, respectively. G1 and G2 are said to pack if there exist injective mappings of the vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer showed that if ∆1∆2 < n 2 , then G1 and G2 pack. We extend this result by showing that if ∆1∆2 ≤ n2 , then G1 and G2 do not pack if and only if one of G1...

Expanding on a recent definition by Bialostocki and Voxman, we define the rainbow ramsey number RR(G1, G2) of two graphs G1 and G2 to be the minimum integer N such that any edge-coloring of the complete graph KN with any number of colors must contain either a copy of G1 with every edge the same color or a copy of G2 with every edge a different color. This number is well-defined if G1 is a star ...