We present a maximal $$L_{q}(L_{p})$$ -regularity theory with Muckenhoupt weights for the equation 0.1 $$\begin{aligned} \partial ^{\alpha }_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in {\mathbb {R}}^{d}. \end{aligned}$$ Here, $$\partial }_{t}$$ is Caputo fractional derivative of order $$\alpha \in (0,2)$$ and $$a^{ij}$$ are functions (t, x). Precisely, we show that \begin{ali...

Abstract In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + x ( h ) = D − 1 <m:mi ma...

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with r ≥ 2 vertices in each partite set contains a cycle with exactly r− 1 vertices from each partite set, with exception of the case that c = 4 and r = 2. Here we wi...

In this paper we recognize the topological type of the group D(R) of C diffeomorphisms of the real line, endowed with the Whitney C-topology (cf. [6]). Besides the whole group D(R) we shall also be interested in its subgroup D c (R) ⊂ D (R) consisting of diffeomorphisms f : R → R with compact support supp(f) = clR{x ∈ R : f(x) 6= x}. For n = 0 the topological group D(R) turns into the homeomorp...

We prove that the family of all invariant sets of iterated systems of contractions R → R is a nowhere dense Fσ type subset in the space of the non-empty compact subsets of R equipped with the Hausdorff metric. An iterated function system (IFS for short) is a finite collection (T1, . . . , Tn) of weak contractions of a metric space X. By a weak contraction we mean a mapping T : X → X such that d...

Let X be a smooth projective curve of genus g =2. For ah integers r,& with r > O let M(X;r,6) be the modulí seheme of rank r stable vector bundíes on X with degree 6. Here we prove that for ahí integers vi, d with vi > 3 and d > (5n — 8)g + 2n2 — 5» + 4 there is a» open dense suhset S2 of M<X; vi —1, (vi + 1)d+2g— 2) such that for every N Ef? there isa non-degenerate degree d embedding of X in ...

In this paper, we show that if $M$ is a non-zero Artinian $R$-module and $\underline{x}:=x_1,\ldots,x_n$ an $M$-coregular sequence, then $x_1,\ldots,x_n$ $D(H_n^{\underline{x}}(M))$-coregular sequence. Moreover, $R$ complete with respect to $I$-adic topology $d=\mathrm{Ndim} M$, $\dim H^I_d(M) \le d$ $\mathrm{depth} H_I^d(M)\ge \min\{{2, d}\}$ whenever $H^I_d(M) \ne 0.$

Considering singular Sturm--Liouville differential expressions of the type \[ \tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x), \quad x \in (0,b), \; \alpha \mathbb{R}, \] we employ some Sturm comparison-type results in spirit Kurss to derive criteria for $\tau_{\alpha}$ be limit point and circle case at $x=0$. More precisely, if $\alpha \mathbb{R}$ $0 0$ such that $0<x$ sufficiently small, \begi...

Let G be a graph. The vertex edge arboricity of G denoted by a G a1 G is the minimum number of subsets into which the vertex edge set of G can be partitioned so that each subset induces an acyclic subgraph. Let d be a graphical sequence and let R d be the class of realizations of d. We prove that if π ∈ {a, a1}, then there exist integers x π and y π such that d has a realization G with π G z if...

We demonstrate propagation rules of subsystem code constructions by extending, shortening and combining given subsystem codes. Given an [[n, k, r, d]]q subsystem code, we drive new subsystem codes with parameters [[n + 1, k, r,≥ d]]q, [[n − 1, k + 1, r,≥ d − 1]]q , [[n, k − 1, r + 1, d]]q . We present the short subsystem codes. The interested readers shall consult our companion papers for upper...