We introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=sum_{i=1}^n |eta_i|, where eta_i=zeta_i-frac{2m}{n} and where zeta_1,zeta_2,ldots,zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if sum_{i=1}^n |eta_i|=sum_{i=1}^n |zeta_i|. Conditions ...

A family of non-Hermitian but ${\cal PT}-$symmetric $2J$ by toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\ldots$ and $t<J^2$ is studied, for which a real $2K$ tridiagonal-submatrix component $C(t)$ the Hamiltonian assumed coupled to its other two complex Hermitian $M$ components $A(t)$ $B(t)$. By construction, (i) all submatrices get decoupled at $t=t_M=M\,(2...

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...

In this paper, we show that if $M$ is a non-zero Artinian $R$-module and $\underline{x}:=x_1,\ldots,x_n$ an $M$-coregular sequence, then $x_1,\ldots,x_n$ $D(H_n^{\underline{x}}(M))$-coregular sequence. Moreover, $R$ complete with respect to $I$-adic topology $d=\mathrm{Ndim} M$, $\dim H^I_d(M) \le d$ $\mathrm{depth} H_I^d(M)\ge \min\{{2, d}\}$ whenever $H^I_d(M) \ne 0.$

Let $W(D_n)$ denote the walk matrix of Dynkin graph $D_n$. We prove that Smith normal form is $$\textup{diag}[\underbrace{1,1,\ldots,1}_{\frac{n}{2}-1},\underbrace{2,2,\ldots,2}_{\frac{n}{2}-1},0,0]$$ when $n\equiv 0\pmod{4}$. This gives an affirmative answer to a question in [W. Wang, C. S. Guo, On $D_n$, Linear Algebra Appl. 653 (2022) 193--206].

A data-driven procedure is developed to compute the optimal map between two conditional probabilities $$\rho (x|z_{1},\ldots ,z_{L})$$ and $$\mu (y|z_{1},\ldots , known only through samples depending on a set of covariates $$z_{l}$$ . The tested synthetic data from ACIC Data Analysis Challenge 2017 it applied non-uniform lightness transfer images. Exactly solvable examples simulations are perfo...

Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications remote sensing. Golay Shapiro independently devised a recursive construction that produces families of complementary pairs binary sequences. In the simplest case, Rudin–Shapiro sequences, general it what we call Golay–Rudin–Shapiro Calculations by Littlewood show sequences have mean square autocorrelati...

Let Γ ⊂ Q ¯ × $\Gamma \subset \overline{\mathbb {Q}}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. α 1 , … r ∈ $\alpha _1,\ldots ,\alpha _r\in {Q}}^\times$ numbers which are $\mathbb {Q}$ -linearly independent and let ε > 0 $\epsilon >0$ given real number. One the main results that we prove in this article is as follows: There exist only many tuples ( u q p ) Z +...