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 aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values $\alpha$, $\beta$ and $𝔭$. Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.

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.

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(mod2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product ${\prod }_{k\in R_m\left(p\right)}(1+tan(\pi ak/p))$, where $$R_m\left(p\right)=\{0<k<p:k\in \mathbb{Z}\text{is}\text{an}m\text{th}\text{power}\text{residue}\text{modulo}p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4\left(p\right)}\left(1+tan\pi \frac{ak}{p}\right)={(-1)}^y{(-2)}^{(p-1)/8}.$$

Let $\mathbb{N}$ be the set of positive integers and let $s\in \mathbb{N}$. We denote by $d^s$ the arithmetic function given by $d^s\left(n\right)={\left(d\left(n\right)\right)}^s$, where $d\left(n\right)$ is the number of positive divisors of $n$. Moreover, for every $\ell ,m\in \mathbb{N}$ we denote by ${\delta }^{s,\ell ,m}\left(n\right)$ the sequence $$\underset{m\text{-times}}{\underbrace{d^s(d^s(...d^s(d^s\left(n\right)+\ell )+\ell ...)+\ell )}}=\left\{\begin{array}{cc}{d}^s\left(n\right)\hfill & \text{for}m=1,\hfill \\ d^s(d^s\left(n\right)+\ell )\hfill & \text{for}m=2,\hfill \\ d^s(d^s(d^s\left(n\right)+\ell )+\ell )\hfill & \text{for}m=3,\hfill \\ ⋮\hfill \end{array}\right.$$ We present classical and nonclassical notes on the sequence ${\left({\delta }^{s,\ell ,m}\left(n\right)\right)}_{m\ge 1}$, where $\ell$, $n$, $s$ are understood as parameters.