Abstract A generalization of the Riemannian Penrose inequality in n-dimensional space (3 ≤ n < 8) is done. We introduce a parameter α ($-\frac{1}{n-1}<\alpha \infty$) indicating strength gravitational field, and define refined attractive gravity probe surface (refined AGPS) with α. Then, we show area for AGPS, $A \le \omega _{n-1} \left[ (n+2(n-1)\alpha )Gm /(1+(n-1)\alpha ) \righ...

Let \(P(z)\) be a polynomial of degree \(n\), then concerning the estimate for maximum \(|P'(z)|\) on unit circle, it was proved by S. Bernstein that \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension this inequality. The polar derivative \(D_{\alpha}[P](z)\) \(P(z)\), with respect to point \(\alpha \in \mathbb{C}\), generalizes ordinary in sense \(\lim...

The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1 > 0$ between first neighbors and $J_2 = \alpha J_1$ second neighbors. supports both gapless quantum phases nondegenerate ground states gapped $\Delta(\alpha) doubly degenerate states. Exact thermodynamics limited to $\alpha 0$, the linear Heisenberg antiferromagnet (HAF). diagonalization of small systems at f...

For every non-integral $\alpha \gt 1$, the sequence of integer parts $n^{\alpha }$ $(n=1,2,\ldots )$ is called Piatetski-Shapiro with exponent $ and let $\mathrm {PS}(\alpha denote set all terms this sequence.

Let $f \in L^{2}(\mathbb{R}^n)$ and suppose we are interested in computing its average at a fixed scale. This is easy: pick the density $u_{}$ of probability distribution with mean 0 some moment desired scale compute convolution $u_{} * f$. Is there particularly natural choice for $u$? question studied space theory Gaussian popular answer. We were whether canonical $u$ can arise from new axiom:...

Let $$p>5$$ be a fixed prime and assume that $$\alpha _1,\alpha _2,\alpha _3$$ are coprime to p. We study the asymptotic behavior of small solutions congruences form _1x_1^2+\alpha _2x_2^2+\alpha _3x_3^2\equiv 0\bmod {q}$$ with $$q=p^n$$ , where $$\max \{|x_1|,|x_2|,|x_3|\}\le N$$ $$(x_1x_2x_3,p)=1$$ . (In fact, we consider smoothed version this problem.) If $$n\rightarrow \infty $$ establish a...

begin{abstract} the relations between the spectrum of the matrix $q+r$ and the spectra of the matrices $(gamma + delta)q+(alpha + beta)r-qr-rq$, $qr-rq$, $alpha beta r-qrq$, $alpha rqr-(qr)^{2}$, and $beta r-qr$ have been given on condition that the matrix $q+r$ is diagonalizable, where $q$, $r$ are ${alpha, beta}$-quadratic matrix and ${gamma, delta}$-quadratic matrix, respectively, of order $...

We construct a family of linear maximally recoverable codes with locality $r$ and dimension $r+1.$ For length $n$ $r\approx n^\alpha, 0\le\alpha\le 1$ the code alphabet is order $n^{1+3\alpha},$ which improves upon previously known constructions codes.