In [H4], Hamilton determined a sharp tensor Harnack inequality for complete solutions of the Ricci flow with non-negative curvature operator. This Harnack inequality is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [H3], [H5], [H6], and [H7]. Moreover, according to Hamilton, the discovery of a Harnack inequality i...

The purpose of this study is to identify the effective factors of optimal use of social media in the brand equity of startup companies. To this end, the researcher collected the required data using snowball sampling, protocol design, and exploratory interviews with sixteen university experts, the industry in this field, and senior managers of accelerator companies and startups. Maxqda 18 softwa...

A local cut point is by definition a point that disconnects its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop–Gromov inequality. As a corollary, we obtain an upper bound for the number of ends of such a space. We also obtain some obstruction conditions for the existence of...

In this paper, we extend two classical results about the density of subgraphs of hypercubes to subgraphs G of Cartesian products G1 × · · · × Gm of arbitrary connected graphs. Namely, we show that |E(G)| |V (G)| ≤ d2 max{dens(G1), . . . , dens(Gm)}e log |V (G)|, where dens(H) is the maximum ratio |E(H′)| |V (H′)| over all subgraphs H ′ of H. We introduce the notions of VC-dimension VC-dim(G) an...

Let G be a locally compact abelian group with dual group Ĝ. The Hausdorff–Young theorem states that if f ∈ Lp(G), where 1 ≤ p ≤ 2, then its Fourier transform Fp(f) belongs to Lq(Ĝ) (where 1 p + 1 q = 1) and ||Fp(f)||q ≤ ||f ||p. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G b...

1. Lenz and Stollman [3] pointed out that the “metric” ρ defined on page 266 of [4] does not satisfy the triangle inequality. There we used for two Delone sets Λ1 and Λ2, denoting by Br the ball of radius r centered at the origin: ρ(Λ1,Λ2) = min{2−1/2, ρ̃(Λ1,Λ2)}, where ρ̃(Λ1,Λ2) = inf{ > 0 : dH [Λ1 ∩B1/ , Λ2 ∩B1/ ] ≤ } . Here dH is the Hausdorff distance. There are several ways to fix this, desc...

Background: Spiritual is the most important dimension of existence in more than a third of the people and it has been at the center of their lives as strong force and so may have an influence on a person's quality of life. Infertile women are at risk of so much stress and anxiety. The purpose of this study was to investigate the relationship between spiritual health and quality of life in ...

The COVID-19 pandemic has changed the lives of people worldwide. Considering that no specific pharmaceutical treatment exists for this disease, the best way to deal with it is through prevention. However, many people in the community do not implement preventive behaviors. It is necessary to pay attention to the effective psychological and social factors of preventive behaviors. The present stud...