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...

A partition of a natural number n is any integer sequence λ = (λ1, λ2, . . . ) such that λ1 ≥ λ2 ≥ · · · ≥ 0 and λ1 + λ2 + · · · = n (notation: λ ⊢ n). In particular, λ1 = max{λi ∈ λ}. Every partition λ ⊢ n can be represented geometrically by a planar shape called the Young diagram, consisting of n unit cell arranged in consecutive columns, containing λ1, λ2, . . . cells, respectively. On the s...

Let A be an n×n complex Hermitian matrix and let λ(A)=(λ1,…,λn)∈Rn denote the eigenvalues of A, counting multiplicities arranged in non-increasing order. Motivated by problems arising theory low rank approximation, we study spectral spread denoted Spr+(A), given Spr+(A)=(λ1−λn,λ2−λn−1,…,λk−λn−k+1)∈Rk, where k=[n/2] (integer part). The is a vector-valued measure dispersion spectrum that allows o...

Hepatitis C virus (HCV) infection is a leading cause of chronic liver diseases and hepatocellular carcinoma (HCC) and Golgi protein 73 (GP73) is a serum biomarker for liver diseases and HCC. However, the mechanism underlying GP73 regulates HCV infection is largely unknown. Here, we revealed that GP73 acts as a novel negative regulator of host innate immunity to facilitate HCV infection. GP73 ex...

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) the diagonal degrees and RD(G) Harary matrix. Clearly, RDL(G) real symmetric matrix, we denote its eigenvalues λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). largest eigenvalue λ1(RDL(G)) RDL(G), denoted by λ(G), called spectral radius. In this paper, obtain several upper bounds for sum k in terms...

Eradication of hepatitis C virus (HCV) infection, both spontaneous and treatment-induced, is marked by the wildtype allele C of a single nucleotide polymorphism upstream of the IL28B gene, rs12979860. This favorable allele was recently described to be in linkage disequilibrium with the wildtype allele TT of a dinucleotide polymorphism, ss469415590, located within a new protein-coding gene. Whil...

Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≤ λ1(X). We show that if equality λ1(Γ\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to R1 ×N endowed wi...

Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≤ λ1(X). We show that if equality λ1(Γ\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to R1 ×N endowed wi...

A short proof to a recent theorem of Giambruno and Mishchenko is given in this note. 1 The theorem The following theorem was recently proved by Giambruno and Mishchenko. Theorem 1. [1, Theorem 1] For every 0 < α < 1, there exist β > 1 and n0 ∈ N, such that for every partition λ of n > n0 with max{λ1, λ1} < αn f > β. The proof of Giambruno and Mishchenko is rather complicated and applies a cleve...

Direct-acting antivirals (DAAs) for hepatitis C virus (HCV) have potent anti-HCV effects but may provoke resistance-associated variants (RAVs). In this study, we assessed the characteristics of these RAVs and explored efficacious anti-HCV reagents using recombinant HCV with NS5A from a genotype 1b strain. We replaced the NS5A of JFH1 with that of Con1 (JFH1/5ACon1) and introduced known NS5A inh...