We study the presence of copies of $l_p^n$’s uniformly in the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$. By using Dvoretzky’s theorem we deduce that if $X$ is an infinite-dimensional Banach space, then ${\Pi }_2(C[0,1],X)$ contains $\lambda \sqrt{2}$-uniformly copies of $l_{\infty }^n$’s and ${\Pi }_1(C[0,1],X)$ contains $\lambda$-uniformly copies of $l_2^n$’s for all $\lambda >1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$ are distinct, extending the well-known result that the spaces ${\Pi }_2(C[0,1],X)$ and $𝒩\left(C\right[0,1],X)$ are distinct.

The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles...

For $\varphi$ a normalized positive definite function on a locally compact abelian group $G$, let ${\pi }_{\varphi }$ be the unitary representation associated to $\varphi$ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,{\pi }_{\varphi })$ and reduced 1-cohomology ${\overline{H}}^1(G,{\pi }_{\varphi })$. For example, ${\overline{H}}^1(G,{\pi }_{\varphi })=0$ if and only if either $\text{Hom}(G,ℂ)=0$ or ${\mu }_{\varphi }\left(1_G\right)=0$, where $1_G$ is the trivial character of $G$ and ${\mu }_{\varphi }$ is the probability measure on the Pontryagin dual $\widehat{G}$ associated to $\varphi$ by Bochner’s Theorem. This streamlines an argument of Guichardet...

Let $F=(f_1,...,f_q)$ be a polynomial dominating map from ${ℂ}^n$ to ${ℂ}^q$. We study the quotient ${𝒯}^1\left(F\right)$ of polynomial 1-forms that are exact along the generic fibres of $F$, by 1-forms of type $\mathrm{d}R+\sum a_i\mathrm{d}{f}_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${𝒯}^1\left(F\right)$ is always a torsion $ℂ[t_1,...,t_q]$-module. Then we determine under which conditions on $F$ we have ${𝒯}^1\left(F\right)=0$. As an application, we study the behaviour of a class of algebraic $({ℂ}^p,+)$-actions on ${ℂ}^n$, and determine in particular when these actions are trivial.

We consider a biharmonic problem ${\Delta }^2{u}_{\omega }=f_{\omega }$ with Navier type boundary conditions $u_{\omega }=\Delta u_{\omega }=0$, on a family of truncated sectors ${\Omega }_{\omega }$ in ${ℝ}^2$ of radius $r$, $0<r<1$ and opening angle $\omega$, $\omega \in (2\pi /3,\pi ]$ when $\omega$ is close to $\pi$. The family of right-hand sides ${\left(f_{\omega }\right)}_{\omega \in (2\pi /3,\pi ]}$ is assumed to depend smoothly on $\omega$ in $L^2\left({\Omega }_{\omega }\right)$. The main result is that $u_{\omega }$ converges to $u_{\pi }$ when $\omega \to \pi$ with respect to the $H^2$-norm. We can also show that the $H^2$-topology is optimal for such a convergence result.

Let $k$ be a field of characteristic $p\&gt;0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

A nonincreasing sequence $\pi =(d_1,...,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. Given two graphs $G_1$ and $G_2$, A. Busch et al. (2014) introduced the potential-Ramsey number of $G_1$ and $G_2$, denoted by $r_{\mathrm{pot}}(G_1,G_2)$, as the smallest nonnegative integer $m$ such that for every $m$-term graphic sequence $\pi$, there is a realization $G$ of $\pi$ with $G_1\subseteq G$ or with $G_2\subseteq \overline{G}$, where $\overline{G}$ is the complement of $G$. For $t\ge 2$ and $0\le k\le \lfloor \frac{t}{2}\rfloor$, let $K_t^{-k}$ be the graph...

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if for any chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi (HK/K\cap L/K)$-number. We obtain some criteria for the $p$-supersolubility or $p$-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $\Pi$-property.

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

The Golomb space ${\mathbb{N}}_{\tau }$ is the set $\mathbb{N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb{N}}_{\tau }$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of ${\mathbb{N}}_{\tau }$, and each homeomorphism $h$ of ${\mathbb{N}}_{\tau }$ has the following properties: $h\left(1\right)=1$, $h\left(\Pi \right)=\Pi$, ${\Pi }_{h\left(x\right)}=h\left({\Pi }_x\right)$, and $h\left(x^{\mathbb{N}}\right)=h{\left(x\right)}^{\mathbb{N}}$ for all $x\in \mathbb{N}$. Here $x^{\mathbb{N}}:=\{x^n:n\in \mathbb{N}\}$ and ${\Pi }_x$ denotes the set of...

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

Let $(R,𝔪)$ be a commutative Noetherian local ring, $𝔞$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $𝔞M\ne M$ and $\mathrm{cd}(𝔞,M)-\mathrm{grade}(𝔞,M)\le 1$, where $\mathrm{cd}(𝔞,M)$ is the cohomological dimension of $M$ with respect to $𝔞$ and $\mathrm{grade}(𝔞,M)$ is the $M$-grade of $𝔞$. Let $D(-):={\mathrm{Hom}}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/𝔪)$ is the injective hull of the residue field $R/𝔪$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0⟶& H_{𝔞}^{n-2}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)⟶D\left(M\right)\\ \hfill ⟶& H_{𝔞}^{n-1}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^{n+1}\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)\\ \hfill ⟶& H_{𝔞}^n\left(D\left(H_{(x_1,...,x_{n-1})}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{(}^{n-1}M\right)\right))⟶...,\end{array}$$ where $n:=\mathrm{cd}(𝔞,M)$ is a non-negative integer, $x_1,...,x_{n-1}$ is a regular sequence in $𝔞$ on $M$ and, for an $R$-module $L$, $H_{𝔞}^i\left(L\right)$ is the $i$th local cohomology module...