On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $$wt(X)\le t(X)$$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ under $$2^{\aleph _0}=2^{\aleph _1}$$. In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then $$|X|\le 2^{wt(X)\pi \chi (X)}$$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $$w(X)\le 2^{wt(X)}$$ for a homogeneous compactum. Third, we show that if X is a $$T_1$$ space, $$wt(X)\le \kappa $$, X is $$\kappa ^+$$-compact, and $$\psi (\overline{D},X)\le 2^\kappa $$ for any $$D\subseteq X$$ satisfying $$|D|\le 2^\kappa $$, then (a) $$d(X)\le 2^\kappa $$ and (b) X has at most $$2^\kappa $$-many $$G_\kappa $$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $$\kappa =L(X)wt(X)$$, and $$\lambda $$ is a caliber of X satisfying $$\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$$, then $$d(X)\le 2^{\kappa }$$. This extends of theorem of Arhangel$$'$$skiĭ (Topol Appl 104:13–26, 2000).     

Exponential density vs exponential domination

Acta Mathematica Hungarica - It is established that a space X with exponential $$\kappa$$ -domination has density not exceeding $$\kappa$$ if $$l(X) \leq\kappa$$ and $$t(X) \leq\kappa$$ . We also...     

The minimum principle for affine functions and isomorphisms of continuous affine function spaces

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if $$f:X\rightarrow {\mathbb {R}}$$ is an affine function with the point of continuity property such that $$f\le 0$$ on $${\text {ext}}\,X$$, then $$f\le 0$$ on X. As a corollary of this minimum principle, we obtain a generalization of a theorem by C.H. Chu and H.B. Cohen by proving the following result. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let $$T:\mathfrak {A}^c(X)\rightarrow \mathfrak {A}^c(Y)$$ be an isomorphism such that $$\left\| T\right\| \cdot \left\| T^{-1}\right\| <2$$. Then $${\text {ext}}\,X$$ is homeomorphic to $${\text {ext}}\,Y$$.     

On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$Z/nZ

A unit u in a commutative ring with unity R is called exceptional if $$u-1$$ is also a unit. We introduce the notion of a polynomial version of this (abbreviated as $$f\hbox {-exunits}$$) for any $$f(X) \in \mathbb {Z}[X]$$. We find the number of representations of a non-zero element of $$\mathbb {Z}/n\mathbb {Z}$$ as a sum of two f-exunits for an infinite family of polynomials f of each degree $$\ge 1$$. We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).     

On the irreducibility of the Severi variety of nodal curves in a smooth surface

Archiv der Mathematik - Let X be a smooth projective surface and $$L\in \mathrm {Pic}(X)$$ . We prove that if L is $$(2k-1)$$ -spanned, then the set $${\tilde{V}}_k(L)$$ of all nodal and...     

Semistar ascending chain conditions over power series rings

Ricerche di Matematica - Let D be an integral domain. We associate to a semistar operation $$\star $$ on D, a semistar operation $$*$$ on D[[X]]. We prove that if D satisfies the $$\star _f$$...     

Sparse graphs and an augmentation problem

Mathematical Programming - For two integers $$k&gt;0$$ and $$\ell $$ , a graph $$G=(V,E)$$ is called $$(k,\ell )$$ -tight if $$|E|=k|V|-\ell $$ and $$i_G(X)\le k|X|-\ell $$ for each...     

Weak finitely regular relations and applications

In this note, we introduce the concepts of weak finitely regular relations and weak hypercontinuous posets. It is proved that when a binary relation $$\rho :X/\rightarrow Y$$ satisfies a certain condition, $$\rho $$ is weak finitely regular if and only if $$(\varphi _{\rho }(X,Y),\subseteq )$$ is a weak hypercontinuous poset. For a poset P, we get that the relation $$\not \le $$ on P is weak finitely regular if and only if P is weak hypercontinuous.     

Coincidence Wecken Property for Nilmanifolds

Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.     

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Generalized Shift-Invariant Systems

A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf R})$ is said to be a Bessel system if the associated analysis operator $$ \txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X} $$ is well-defined and bounded. A Bessel system is a fundamental frame if $\txs{X}$ is injective and its range is closed. This paper considers the above two properties for a generalized shift-invariant system $X$. By definition, such a system has the form $$ X=\bigcup_{j\in J} Y_j, $$ where each $Y_j$ is a shift-invariant system (i.e., is comprised of lattice translates of some function(s)) and $J$ is a countable (or finite) index set. The definition is general enough to include wavelet systems, shift-invariant systems, Gabor systems, and many variations of wavelet systems such as quasi-affine ones and nonstationary ones. The main theme of this paper is the fiberization of $\txs{X}$, which allows one to study the frame and Bessel properties of $X$ via the spectral properties of a collection of finite-order Hermitian nonnegative matrices.     

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

非退化扩散过程的水平集和极集

设X(t)(t∈R )是一个d维非退化扩散过程．本文得到了比原有结果更一般的非退化扩散过程极性的充分条件,证明了对任意u∈Rd,紧集E(0, ∞),有若d=1,则对任意紧集F(?)R, 若d≥2,则对任意紧集E ∈(0, ∞), 其中B(Rd)为Rd上的Borel σ-代数,dim和Dim分别表示Hausdorff维数和Packing 维数．     

The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices

Exponential Dichotomy on the Real Line and Admissibility of Function Spaces

The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted and we prove that if with and the pair is admissible for an evolution family then is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs and with      

The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Geometriae Dedicata - Let $$\pi :\mathcal {X}\rightarrow M$$ be a holomorphic fibration with compact fibers and L a relatively ample line bundle over $$\mathcal {X}$$ . We obtain the asymptotic of...     

Un critère sur les petites boules dans le théorème limite central

We show that a Banach space valued random variableX such that t} \right\} = 0$$ " align="middle" border="0"> satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled:

The C-convexity of Banach spaces with unconditional bases

A Banach space is called C-convex if the space c₀ cannot be represented finitely in it. Necessary and sufficient conditions for the C-convexity of a space with an unconditional basis and of the product of a space Y with respect to the unconditional basis of a space X are obtained. These conditions are rendered concrete for two classes of spaces: The Orlich space of sequences is C-convex if and only if its normalizing function satisfies the δ₂-condition; the Lorentz space of sequences is C-convex if and only if its normalizing sequence satisfies the condition \(\mathop {\underline {\lim } }\limits_{n \to \infty } {{\sum\nolimits_{i = 1}^{2n} {c_i } } \mathord{\left/ {\vphantom {{\sum\nolimits_{i = 1}^{2n} {c_i } } {\sum\nolimits_{i = 1}^n {c_i > 1} }}} \right. \kern-0em} {\sum\nolimits_{i = 1}^n {c_i > 1} }}\) . We call a Banach space X a C-convex space if the following condition is fulfilled: $$\mathop {\sup }\limits_n \inf d\left( {X_n , l_\infty ^n } \right) = \infty $$ ,     

Quasi-locally finite polynomial endomorphisms

If F is a polynomial endomorphism of , let denote the field of rational functions such that . We will say that F is quasi-locally finite if there exists a nonzero such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each the sequence is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.