Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $m,n$ non-negative integers. $M$ $A$-module such that $\operatorname{Ext}^i\_A(A/\mathfrak{a},M)$ is finitely generated for all $i\leq m+n$. We define class $\mathcal{S}n(\mathfrak{a})$ modules we assume $H{\mathfrak{a}}^s(M)\in\mathcal{S}{n}(\mathfrak{a})$ $s\leq m$. show $H{\mathfrak{a}}^s(M)$ $\mathfrak{a}$-cofinit...

A 3-way \((v,k,t)\) trade \(T\) of volume \(m\) consists three pairwise disjoint collections \(T_{1}\), \(T_{2}\) and \(T_{3}\), each blocks size \(k\), such that for every \(t\)-subset \(v\)-set \(V\), the number containing this is same in \(T_{i}\) \(1\leq i\leq 3\). If any found(\(T\)) occurs at most once 3\), then called Steiner trade. We attempt to complete spectrum \(S_{3s}(v,k)\), set al...

We generalize the classical sharp bounds for largest eigenvalue of normalized Laplace operator, $\frac{N}{N-1}\leq \lambda_N\leq 2$, to case chemical hypergraphs.

Exploiting the recent work of Tao and Ziegler on a concatenation theorem factors, we find explicit characteristic factors for multiple averages along polynomials systems with commuting transformations, use them to study criteria joint ergodicity sequences form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $1\leq j\leq k$, where $T_{1},\dots,T_{d}$ are measure prese...

Using weaker versions of the cardinal function $$\psi_c(X)$$ , we derive a series new bounds for cardinality Hausdorff spaces and regular that do not involve nor its variants at all. For example, show if X is then $$|X|\leq 2^{c(X)^{\pi\chi(X)}}$$ 2^{c(X)\pi\chi(X)^{ot(X)}}$$ where $$ot(X)$$ introduced by Tkachenko, has property $$ot(X)\leq\min\{t(X),c(X)\}$$ . It follows from latter space with...

In this work, byemploying the Leggett-Williams fixed point theorem, we study theexistence of at least three positive solutions of boundary valueproblems for system of third-order ordinary differential equationswith $(p_1,p_2,ldots,p_n)$-Laplacianbegin{eqnarray*}left { begin{array}{ll} (phi_{p_i}(u_i''(t)))'  +  a_i(t) f_i(t,u_1(t), u_2(t), ldots, u_n(t)) =0 hspace{1cm} 0  leq t leq 1, alpha_i u...