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.     

Non-crossing monotone paths and binary trees in edge-ordered complete geometric graphs

Acta Mathematica Hungarica - This paper is devoted to the studying of the weak Orlicz–Lorentz space $$\Lambda_X^{\varphi, \infty}(w)$$ , which can be regarded as an extension of weak Orlicz...     

Some regularity criteria of a weak solution to the 3D Navier–Stokes equations in a domain

Archiv der Mathematik - We give some regularity criterion (of a weak $$-L^{p}$$ Serrin type) of a weak solution to the 3D Navier–Stokes equations in a bounded domain $$\Omega \subset \mathbb...     

The completeness characterization of $$C(\mathcal {L})$$ C ( L ) , $$\mathcal {L}$$ L a locale

Positivity - The result of the title is: an archimedean $$\ell $$ -group with weak unit A is (isomorphic to) $$C(\mathcal {L})$$ for some (identifiable) locale $$\mathcal {L}$$ (or, $$\mathbb...     

Two Hypotheses on the Exponential Class in the Class Of O-subexponential Infinitely Divisible Distributions

Journal of Theoretical Probability - Two hypotheses on the class $${\mathcal {L}}(\gamma )$$ in the class $$\mathcal {OS}\cap \mathcal {ID}$$ are discussed. Two weak hypotheses on the class...     

Weakly multiplicative arithmetic functions and the normal growth of groups

Archiv der Mathematik - We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $$\log $$ -uniformly continuous normal...     

Some fixed point results for relation theoretic weak $$\varphi $$ φ -contractions in cone metric spaces equipped with a binary relation and application to the system of Volterra type equations

Positivity - In this paper, some fixed point results for relation theoretic weak $$\varphi $$ -contractions on cone metric spaces which is defined over a Banach algebra and equipped with a binary...     

A no expanding breather theorem for noncompact Ricci flows

Annals of Global Analysis and Geometry - In this note we show that, under certain curvature positivity conditions (the weak $${\text {PIC}}-2$$ condition or the nonnegative bisectional curvature...     

Representation of Synthesizable Differentiation-Invariant Subspaces of the Schwartz Space

Doklady Mathematics - We consider a differentiation-invariant subspace W in the Schwartz space $${{C}^{\infty }}(a;b)$$ which admits weak spectral synthesis. We obtain the conditions under which W...     

Polynomial versions of weak Dunford–Pettis properties in Banach lattices

Positivity - In this paper, we introduce polynomial versions of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property for Banach lattices. By using Fremlin...     

Asymptotics of Nahm sums at roots of unity

The Ramanujan Journal - We construct bases of the space of harmonic weak Maass forms of weight $$\kappa \in \frac{1}{2}{\mathbb {Z}}$$ . Using these bases, we obtain a Shintani lift from a positive...     

Weak Tensor Category and Related Generalized Hopf Algebras

There are at least two kinds of generalization of Hopf algebra, i.e. pre-Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, a） and used it to characterize almost bialgebra and pre-Hopf algebra in Comm. in Algebra, 32（2）： 397-441 （2004）. Our aim in this paper is to generalize tensor category L = （L, ×, I, a, l, r） by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll^-1 = rr^-1 = id into regular natural transformations lll= l, rrr = r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized （bialgebras） Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed.     

A large class of solutions for the instationary Navier–Stokes system

We investigate very weak solutions to the instationary Navier–Stokes system being contained in where is a bounded domain and . The chosen space of data is small enough to guarantee uniqueness of solutions and existence in case of small data or short time intervals. On the other hand, the data space is large enough that every vector field in is a very weak solution for appropriate data. The solutions and the data depend continuously on each other.      

Weak minimum aberration and maximum number of clear two-factor interactions in $$2_{IV}^{m - p} $$ designs

Both the clear effects and minimum aberration criteria are the important rules for the design selection. In this paper, it is proved that some designs have weak minimum aberration, by considering the number of clear two-factor interactions in the designs. And some conditions are provided, under which a design can have the maximum number of clear two-factor interactions and weak minimum aberration at the same time. Some weak minimum aberration designs are provided for illustrations and two nonisomorphic weak minimum aberration designs are constructed at the end of this paper.     

Limit Theorems for Non-Commutative Random Variables

We study the weak law of large numbers and the central limit theorem for non-commutative random variables. We first define the concepts of variance and expectation for probability measures on homogeneous spaces, and formulate the weak law of large numbers and the central limit theorem for probability measures on locally compact groups. Then, we consider the non-commutative case, where the homogeneous space is replaced by a C^*-algebra that is equipped with a locally compact group G of automorphisms. We define the concepts of variance and expectation in the non-commutative situation. Furthermore, we prove that the weak law of large numbers and the central limit theorem hold for non-commutative random variables on if they hold on the group G of automorphisms.     

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).     

On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \mathbb{R}^{N}

In this paper we establish a Serrin’s type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equations in It is proved that if the gradient of pressure belongs to L^{α, γ} with then the weak solution actually is regular and unique. Received: May 4, 2004     

The instationary Stokes equations in weighted Bessel-potential spaces

We investigate the solvability of the instationary Stokes equations with fully inhomogeneous data in , where is a Bessel-potential space with a Muckenhoupt weight w. Depending on the order of this Bessel-potential space we are dealing with strong solutions or with very weak solutions. Whereas in the context of lowest regularity one obtains solvability with respect to inhomogeneous data by dualization, this is more delicate in the case of higher regularity, where one has to introduce some additional time regularity. As a preparation, we introduce a generalization of the Stokes operator that is appropriate to the context of very weak solutions in weighted Bessel-potential spaces.      

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 $${\text {ext}}\,Y$$.     

Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

Journal of Nonlinear Science - We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $$L^2$$ weak...