We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality \[\left|Df(x)\right|^n \leq K J_f(x) + \sigma^n(x) \left|f(x)\right|^n\] for almost every $x \mathbb{R}^n$. Here $K [1, \infty)$ is a constant and $\sigma \geq 0$ function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover class of $K$-qu...

Let (M, g) be a closed locally conformally flat Riemannian manifold of dimension $$n \ge 7$$ and positive Yamabe type. If $$h \in C^1(M)$$ $$\xi _0$$ is non-degenerate critical point the mass function we prove existence, for any $$ k 1$$ blowing-up solution $$u_\varepsilon $$\begin{aligned} \triangle _g u_\varepsilon +\big ( c_n S_g +\varepsilon h\big ) = ^{2^*-1} \end{aligned}$$ that blows up,...

For a given $\varepsilon > 0$, we say that graph $G$ is $\varepsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of color lists size $k$ on $V(G)$, preferred from list requested at set $R$ vertices, then least |R|$ these requests are satisfied by some $L$-coloring. We consider question flexible choosability in several classes with certain degeneracy conditions. charac...

This paper proves a tradeoff between the time it takes to search for elements in an implicit dictionary and the time it takes to update the value of elements in specified locations of the dictionary. It essentially shows that if the update time is constant, then the search time is $\Omega(n’)$ for some constant $\varepsilon >0$.

We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$ in whole space with global-in-time solutions bounded all Sobolev spaces; for simplicity, we restrict ourselves to model case $$K(x)=-|x|$$ . quantify mass concentration phenomenon, a genuinely nonlinear effect, radially symmetric of this small diffusivity $$\vare...

Let $$S= \{ p_1, \ldots, p_s\}$$ be a finite, non-empty set of distinct prime numbers and $$(U_{n})_{n \geq 0}$$ linear recurrence sequence integers order at least 2. For any positive integer k, $$w = (w_k, w_1)\in\mathbb{Z}^k, w_1, w_k\neq 0$$ we define $$(U_j^{(k, w)})_{j\geq 1}$$ an increasing composed the form $$|w_kU_{n_k} +\cdots + w_1U_{n_1}|$$ , $$ n_k>\cdots >n_1$$ . Under certain assu...

We prove that for every integer $k$, there exists $\varepsilon > 0$ such n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, exist disjoint sets $A,B \subseteq V(G)$ $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete $B$. This proves the analog of Erd\H{o}s-Hajnal conjecture class graphs $C_k$.