Given a graph H on vertex set {1, 2, · · · , n} and a function f : [0, 1] → R, define ‖f‖H := ∣∣∣∣∣ ∫ ∏ ij∈E(H) f(xi, xj)dμ |V (H)| ∣∣∣∣∣ 1/|E(H)| , where μ is the Lebesgue measure on [0, 1]. We say that H is norming if ‖·‖H is a semi-norm. A similar notion ‖·‖r(H) is defined by ‖f‖r(H) := ‖|f |‖H and H is said to be weakly norming if ‖·‖r(H) is a norm. Classical results show that weakly normin...

In this present paper, by utilizing ij−δ semi open sets, we introduce ij−∧δ, ij−∨δ, g− ij−∧δ, g− ij−∨δ sets and investigate some of their properties in bitopological spaces. Also, we present and study the notions of ij − T∨sδ space, ij−∧δ continuous, ij−∧δ irresolute, ij−∧δ open and ij−∧δ homeomorphism functions.

where <£fc(X)=2w=i dk(ri)e^ +residue term. Hopefully, one could expand the k integral and estimate the resulting infinite series. This does not seem feasible, however, unless k{z) satisfies a certain approximate functional equation (AFE); see [5, p. 147] and [6, p. 42]. This is one reason why only k=l, k=2 are known. However, Bellman [2] has shown that, if the e~ in k(z) are replaced b...

With the discovery of some hidden-charm pentaquark resonances by LHCb Collaboration, investigations states containing heavy quarks have aroused interest theorists. We study herein $qqq\bar{q}Q$ ($q = u$ or $d$, $Q=c$ $b$) system, in framework chiral quark model. In consequence, charmed and bottomed pentaquarks are considered to exist five-body dynamical calculations. charm sector, $\Sigma_c\pi(...

S s p s ij ij ijl ijl ij jl jl i M l j l D X D D f q β ∈ ≠ =   + − +       ∑ ∑ ∑ () () () () () () 1 1 () 1 2 4 3 2 4 3 s p s s p p iz iz iz iz ij iz zj iz ij iz zj iz ij iz zj s p S s s p p iz iz iz i z N l iz ij izl iz zj izl iz ij izl iz zj izl iz ij izl iz zj izl l w x w x w x w x w x w x T w x C w x C w x C w x C w x C w x C λ λ λ λ λ λ α () 2 1 0.5 s p s ij ij ijl ijl ij i M l j T...

Let $G$ be a strongly connected digraph with vertex set $\{v_1, v_2, \dots, v_n\}$. Denote by $D_{ij}$ the distance between vertices $v_i$ and $v_j$ in $G$. Two variant versions of matrix were proposed Yan Yeh (Adv. Appl. Math.), Bapat et al. (Linear Algebra Appl.) independently, one is $q$-distance matrix, other exponential matrix. Given nonzero indeterminate $q$, $\mathscr{D}_G=(\mathscr{D}_{...

G02BSF Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details. 1 Purpose G02BSF computes Kendall and/or Spearman non-parametric rank correlation coefficients for a set of data omitting cases with missing values from only those calculations involving the variables for whi...