Abstract A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation consists of pairs points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination s linear equalities inequalities in $d_1 d_2$ variables . We show that k , number edges $K_{k,k}$ -free H a...

for fr$acute{mathbf{text{e}}}$chet algebras $(a, (p_n))$ and $(b, (q_n))$, a linear map $t:arightarrow b$ is textit{almost multiplicative} with respect to $(p_n)$ and $(q_n)$, if there exists $varepsilongeq 0$ such that $q_n(tab - ta tb)leq varepsilon p_n(a) p_n(b),$ for all $n in mathbb{n}$, $a, b in a$, and it is called textit{weakly almost multiplicative} with respect to $(p_n)$ and $(q_n)$,...

In 1991 De Giorgi conjectured that, given $\lambda >0$, if $\mu_\varepsilon$ stands for the density of Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + \lambda] d\mu_\varepsilon$ should $\Gamma$-converge to $c\lambda \mathrm{Per}(E) k \mathcal{W}(\Sigma)$ some real constant $k$, where $\mathrm{Per}(E)$ is perimeter set $E$, $\Sigma=\partial E$,...

This paper is an attempt to prove the following result:Let $n>1$ be an integer and let $mathcal{R}$ be a $n!$-torsion-free ring with the identity element. Suppose that $d, delta, varepsilon$ are additive mappings satisfyingbegin{equation}d(x^n) = sum^{n}_{j=1}x^{n-j}d(x)x^{j-1}+sum^{n-1}_{j=1}sum^{j}_{i=1}x^{n-1-j}Big(delta(x)x^{j-i}varepsilon(x)+varepsilon(x)x^{j-i}delta(x)Big)x^{i-1}quadend{e...

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $(k - 1)$-colorable, $L$ a list-assignment for $G$, then \textit{$L$-critical} not $L$-colorable but induced is. In 2014, Kostochka Yancey proved lower bound on the average degree an $n$-vertex $k$-critical tending to $k \frac{2}{k 1}$ large $n$ that tight infinitely many values $n$, they asked how their may be i...

We show that for any finite set A and an arbitrary $$\varepsilon >0$$ there exists $$k=k(\varepsilon )$$ such the higher energy $$\textsf{E} _k(A)$$ is at most $$|A|^{k+\varepsilon }$$ unless has a very specific structure. As application we obtain subset of real numbers or prime field either contains additive Sidon-type size $$|A|^{1/2+c}$$ multiplicative .

Abstract Let $\mathrm{AP}_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\varepsilon>0$ we call a set $\mathrm{AP}_k(\varepsilon)=\{x_0,\ldots,x_{k-1}\}$ $\varepsilon$ -approximate progression if for some and d , $|x_i-(a+id)|<\varepsilon d$ holds all $i\in\{0,1\ldots,k-1\}$ . Complementing earlier results of Dumitrescu (2011, J. Comput. Geom. 2 (1) 16–29), in this paper ...

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. Th...