In this paper we derive new properties complementary to an $2n\times 2n$ Brualdi-Li tournament matrix $B_{2n}$. We show that $B_{2n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

Assume that $\mathrm{\Omega }$, ${\mathrm{\Omega }}^{\prime }$ are planar domains and $f:\mathrm{\Omega }\stackrel{\text{onto}}{\to }{\mathrm{\Omega }}^{\prime }$ is a homeomorphism belonging to Sobolev space $W_{\text{loc}}^{1,1}(\mathrm{\Omega };{ℝ}^2)$ with finite distortion. We prove that if the distortion function $K_f$ of $f$ satisfies the condition , then the distortion function $K_{f^{-1}}$ of $f^{-1}$ belongs to $L_{\text{loc}}^1({\mathrm{\Omega }}^{\prime })$. We show that this result is sharp in sense that the conclusion fails if ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=1$. Moreover, we prove that if the distortion function $K_f$ satisfies the condition ${\mathrm{dist}}_{\text{EXP}}(K_f,L^{\mathrm{\infty }})=\lambda$ for some $\lambda >0$, then $K_{f^{-1}}$ belongs to $L_{\text{loc}}^p({\mathrm{\Omega }}^{\prime })$ for every $p\in (0,\frac{1}{2\lambda })$. As special case of this result we show that if...

We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either $F\left(x\right)=ux{u}^{-1}$ or $F\left(x\right)=u{x}^t{u}^{-1}$ for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\>0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \>0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

The function ${\sum }_{p\le x}1/p-loglog\left(x\right)-M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_1$ and $D_2$ such that $A^{-\mathrm{T}}=D_{1}A{D}_2$, where $A^{-\mathrm{T}}$ denotes the transpose of the inverse of $A$. Denote by $J=\mathrm{diag}(\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^{\mathrm{T}}JQ=J$. Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) ${\sum }_{i=1}^{n}X{₂}_i-d{\sum }_{i=1}^{n}Y{²}_i=-I$ with d ∈ N for nonsingular $X_i,Y_i\in M₂\left(Z\right)$, i=1,...,n.

The perturbed Laplacian matrix of a graph $G$ is defined as $L^D=D-A$, where $D$ is any diagonal matrix and $A$ is a weighted adjacency matrix of $G$. We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore,...

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $\Phi ={\sum }_{i+k=j+l}\frac{q_i{q}_k}{q_j{q}_l}$ satisfies $\Phi \ge N^2$, with equality if and only if $q=\left(q_i\right)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here...

We investigate the following quasilinear and singular problem, $$\text{t}o2.7cm\left\{\begin{array}{cc}-{\Delta }_{p}u=\frac{\lambda }{u^{\delta }}+u^q\hfill & \text{in}\Omega ;\hfill \\ {u|}_{\partial \Omega }=0,u\>0\hfill & \text{in}\Omega ,\hfill \end{array}\right.\text{t}o2.7cm\text{(P)}$$ where $\Omega$ is an open bounded domain with smooth boundary, $1\<p\<\infty$, $p-1\<q\le p^{*}-1$, $\lambda \>0$, and $0\<\delta \<1$. As usual, $p^{*}=\frac{Np}{N-p}$ if $1\<p\<N$, $p^{*}\in (p,\infty )$ is arbitrarily large if $p=N$, and $p^{*}=\infty$ if $p\>N$. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in $W_0^{1,p}\left(\Omega \right)$. While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

Our aim is to classify and compute zeros of the quadratic two sided matrix polynomials, i.e. quadratic polynomials whose matrix coefficients are located at both sides of the powers of the matrix variable. We suppose that there are no multiple terms of the same degree in the polynomial $𝐩$, i.e., the terms have the form ${𝐀}_j{𝐗}^j{𝐁}_j$, where all quantities $𝐗,{𝐀}_j,{𝐁}_j,j=0,1,...,N,$ are square matrices of the same size. Both for classification and computation, the essential tool is the description of the polynomial $𝐩$ by a matrix...

For a real square matrix $A$ and an integer $d\ge 0$, let $A_{\left(d\right)}$ denote the matrix formed from $A$ by rounding off all its coefficients to $d$ decimal places. The main problem handled in this paper is the following: assuming that $A_{\left(d\right)}$ has some property, under what additional condition(s) can we be sure that the original matrix $A$ possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists...