We reduce the problem of describing all $ℳf_m$-natural operators  transforming general affine connections on $m$-manifolds into general affine ones to the known description of all $GL\left({𝐑}^m\right)$-invariant maps ${𝐑}^{m*}\otimes {𝐑}^m\to {\otimes }^k{𝐑}^{m*}\otimes {\otimes }^k{𝐑}^m$ for $k=1,3$.

A ring extension $R\subseteq S$ is said to be strongly affine if each $R$-subalgebra of $S$ is a finite-type $R$-algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ is a Prüfer extension (i.e. $(R,S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown....

Let $X_i$, i∈ I, and $Y_j$, j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ${\prod }_{i\in I}{X}_i$ onto ${\prod }_{j\in J}{Y}_j$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_i$ onto $Y_{b\left(i\right)}$, i∈ I.

The aim of this paper is to extend the framework of the spectral method for proving property (T) to the class of reflexive Banach spaces and present a condition implying that every affine isometric action of a given group $G$ on a reflexive Banach space $X$ has a fixed point. This last property is a strong version of Kazhdan’s property (T) and is equivalent to the fact that $H^1(G,\pi )=0$ for every isometric representation $\pi$ of $G$ on $X$. The condition is expressed in terms of $p$-Poincaré constants and we...

Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $\left|detM\right|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set ${}_M$ of points in $A_M$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of ${}_M$ is positive. For this special class the full description of ${}_M$ is given as well. This paper continues our...

Let $\left[t\right]$ be the integral part of a real number $t$, and let $f$ be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum $S_f\left(x\right)={\sum }_{n\le x}f\left([x/n]\right)$, which improves the recent result of J. Stucky (2022).

Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element ${𝔽}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic...

Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $C_p\left(X\right)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p\left(X\right)$. We show that every exponentially $\kappa$-cofinal space $X$ has a ${\kappa }^{+}$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw\left(X\right)\le \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal...

We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\cdots ,u_m):U\to {ℝ}^m$ such that $U\cap X=\{u_1\cdots u_r=0\}$, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash...

Let $k$ be a field of characteristic $p\>0$. 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...

Let A be a d × d integral expanding matrix and let $S_j\left(x\right)=A^{-1}(x+d_j)$ for some $d_j\in {\mathbb{Z}}^d$, j = 1,...,m. The iterated function system (IFS) ${S_j}_{j=1}^m$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

Given $\{h_1,\cdots ,h_t\}$ a finite subset of ${ℝ}^d$, we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on ${ℝ}^d$ with the property that the forward differences ${\Delta }_{h_k}^{m_k}f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_1,\cdots ,m_t$.

Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

In $𝒟$-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+}\left(𝒪{\mathrm{e}}^g\right)$ of a $𝒟$-module twisted by the exponential of a polynomial $g$ by another polynomial $f$, where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can...

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$. Let $t\in {\mathbb{N}}_0$ be an integer and $M$ an $R$-module such that ${\mathrm{Ext}}_R^i(R/I,M)$ is minimax for all $i\le t+1$. We prove that if $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i<t$, then the $R$-modules $H_I^i\left(M\right)$ are $I$-cominimax for all $i<t$ and ${\mathrm{Ext}}_R^i(R/I,H_I^t\left(M\right))$ is minimax for $i=0,1$. Let $N$ be a finitely generated $R$-module. We prove that ${\mathrm{Ext}}_R^j(N,H_I^i\left(M\right))$ and ${\mathrm{Tor}}_j^R(N,H_I^i\left(M\right))$ are $I$-cominimax for all $i$ and $j$ whenever $M$ is minimax and $H_I^i\left(M\right)$ is ${\mathrm{FD}}_{\le 1}$ (or weakly Laskerian) for all $i$.

Let $I$ be an ideal of a commutative Noetherian ring $R$. It is shown that the $R$-modules $H_I^j\left(M\right)$ are $I$-cofinite for all finitely generated $R$-modules $M$ and all $j\in {\mathbb{N}}_0$ if and only if the $R$-modules ${\mathrm{Ext}}_R^i(N,H_I^j\left(M\right))$ and ${\mathrm{Tor}}_i^R(N,H_I^j\left(M\right))$ are $I$-cofinite for all finitely generated $R$-modules $M$, $N$ and all integers $i,j\in {\mathbb{N}}_0$.

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.

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...

We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty$ and any constant $\epsilon >0$, we can show that $$\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{[\alpha n+\beta ]\in {𝒬}_k}}1-\frac{x}{\zeta \left(k\right)}\ll x^{k/(2k-1)+\epsilon }+x^{1-1/(\tau +1)+\epsilon },$$ where ${𝒬}_k$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$ $\epsilon ,$ $k$ and $\beta .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $$\sum _{1\le h\le H}\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{n\in {𝒬}_k}}e\left(\vartheta hn\right).$$

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

Let $X$ denote either ${\mathrm{ℂ\mathbb{P}}}^m$ or ${ℂ}^m$. We study certain analytic properties of the space ${ℰ}^n(X,gp)$ of ordered geometrically generic $n$-point configurations in $X$. This space consists of all $q=(q_1,\cdots ,q_n)\in X^n$ such that no $m+1$ of the points $q_1,\cdots ,q_n$ belong to a hyperplane in $X$. In particular, we show that for a big enough $n$ any holomorphic map $f:{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ commuting with the natural action of the symmetric group $𝐒\left(n\right)$ in ${ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ is of the form $f\left(q\right)=\tau \left(q\right)q=(\tau \left(q\right)q_1,\cdots ,\tau \left(q\right)q_n)$, $q\in {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$, where $\tau :{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to \mathrm{𝐏𝐒𝐋}(m+1,ℂ)$ is an $𝐒\left(n\right)$-invariant holomorphic map. A similar result holds true for mappings of the configuration...

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${𝒟}_X^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${𝒟}_X^{\left(\infty \right)}$ on an ${𝒪}_X$-module $ℰ$ is equivalent to giving an infinite sequence of ${𝒪}_X$-modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it...