Let $G$ be a graph and let $m_{ij}(G)$, $i,jge 1$, be the number of edges $uv$ of $G$ such that ${d_v(G), d_u(G)} = {i,j}$. The {em $M$-polynomial} of $G$ is introduced with $displaystyle{M(G;x,y) = sum_{ile j} m_{ij}(G)x^iy^j}$. It is shown that degree-based topological indices can be routinely computed from the polynomial, thus reducing the problem of their determination in each particular ca...

and Applied Analysis 3 holding for any t ∈ 0, T . Here ( a ij x e j k ) t : ∫ t 0 a ij x, t, τ · e j k x, τ dτ 2.4 are Volterra integral operators with kernels a ij x, t, τ ; a hk ij 0 x, t are instantaneous elastic coefficients out-of-integral terms and a ij x : a hk ij 0 x, t a ij x ; fi0 are components of a vector of external forces; φi x, t are components of a vector of boundary traction on...

For $m,n\in\mathbb{N}$ let $X=(X_{ij})_{i\leq m,j\leq n}$ be a random matrix, $A=(a_{ij})_{i\leq real deterministic and $X_A=(a_{ij}X_{ij})_{i\leq the corresponding structured matrix. We study expected operator norm of $X_A$ considered as between $\ell_p^n$ $\ell_q^m$ for $1\leq p,q \leq \infty$. prove optimal bounds up to logarithmic terms when underlying matrix $X$ has i.i.d. Gaussian entries...

A word w is obtained by an arbitrary n-pattern interpretation of a word x if there are n homomorphisms h1, h2, . . . , hn and a positive integer k such that w = hi1(x)hi2 (x) · · ·hik (x) with 1 ≤ ij ≤ n for all 1 ≤ j ≤ k. This arbitrary multiple pattern interpretation of words is naturally extended to languages. We investigate some closure properties of the families of languages obtained by ar...

Let A = (a ij) be a real n n matrix with non-negative entries which are weakly increasing down columns. If B = (b ij) is the nn matrix where b ij := a ij +z; then we conjecture that all of the roots of the permanent of B, as a polynomial in z; are real. Here we establish several special cases of the conjecture.

Inspired by the recent observation of $\chi_{c0}(3930)$, $X(4685)$ and $X(4630)$ LHCb Collaboration some exotic resonances such as $X(4350)$, $X(4500)$, etc. several experiment collaborations, $cs\bar{c}\bar{s}$ tetraquark systems with $IJ^{P}=00^+$, $01^+$ $02^+$ are systematically investigated in framework quark delocalization color screening model(QDCSM). Two structures, meson-meson diquark-...

(A) Let Q(A) be number of comparisons done on input array A: (A) R ij : event that rank i element is compared with rank j element, for 1 ≤ i < j ≤ n. (B) X ij is the indicator random variable for R ij. That is, X ij = 1 if rank i is compared with rank j element, otherwise 0. (B) Q(A) = ∑ 1≤i<j≤n X ij. (C) By linearity of expectation,

Histidine pK(a) values were measured in charge-reversal (K78E, K97E, K127E, and K97E/K127E) and charge-neutralization (E10A, E101A, and R35A) mutants of staphylococcal nuclease (SNase) by (1)H-NMR spectroscopy. Energies of interaction between pairs of charges (DeltaG(ij)) were obtained from the shifts in pK(a) values relative to wild-type values. The data describe the distance dependence and sa...

We consider the free non-commutative analogue , introduced by D. Voiculescu, of the concept of Fisher information for random variables. We determine the minimal possible value of (a; a), if a is a non-commutative random variable subject to the constraint that the distribution of a a is prescribed. More generally, we obtain the minimal possible value of (fa ij ; a ij g 1i;jd), if fa ij g 1i;jd i...