A partial converse to a theorem of Tamari

2010 ◽  
Vol 81 (2) ◽  
pp. 389-392
Author(s):  
John Donnelly
Keyword(s):  
Partial Converse

Extrapolation of Lp Data from a Modular Inequality

2002 ◽  
Vol 45 (1) ◽  
pp. 25-35
Author(s):  
Steven Bloom ◽  
Ron Kerman
Keyword(s):  
Orlicz Spaces ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Partial Converse

AbstractIf an operator T satisfies a modular inequality on a rearrangement invariant space Lρ(Ω, μ), and if p is strictly between the indices of the space, then the Lebesgue inequality holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form , and here, one can extrapolate to the (finite) indices i(Φ) and I(Φ) aswell.

scholarly journals SPACES OF SMALL METRIC COTYPE

2010 ◽  
Vol 02 (04) ◽  
pp. 581-597 ◽  
Author(s):  
E. VEOMETT ◽  
K. WILDRICK
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Ultrametric Space ◽  
Partial Converse ◽  
Definition Of ◽  
Hausdorff Limits

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

Approximable Algebras as Subalgebras of Section Rings

2020 ◽  
Author(s):  
Catriona Maclean
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Type Theorem ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Graded Ring ◽  
Partial Converse ◽  
Big Line Bundle

Abstract In [2], Huayi Chen introduced approximable graded algebras, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in [ 8]. Continuing the analysis started in [8], we show that while not every approximable graded algebra is a sub algebra of the section ring of a big line bundle, we can associate to any approximable graded algebra $\textbf{B}$ a projective variety $X(\textbf{B})$ and an infinite divisor $D(\textbf{B}) =\sum _{i=1}^\infty a_i D_i$ with $a_i\rightarrow 0$ such that $\textbf{B}$ is a subalgebra of $$\begin{equation*} R( D(\textbf{B}))=\oplus_n H^0(X(\textbf{B}), n D(\textbf{B})).\end{equation*}$$We also establish a partial converse to these results by showing that if an infinite divisor $D=\sum _i a_iD_i$ converges in the space of numerical classes, then any full-dimensional sub-graded algebra of $\oplus _mH^0(X, \lfloor mD \rfloor ))$ is approximable.

scholarly journals A partial converse to the Andreotti–Grauert theorem

2018 ◽  
Vol 155 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Xiaokui Yang
Keyword(s):  
Line Bundle ◽  
Ricci Curvature ◽  
Positive Eigenvalue ◽  
Projective Manifold ◽  
Hermitian Metric ◽  
Partial Converse

Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $\unicode[STIX]{x1D714}$ on $X$ such that its Ricci curvature $\text{Ric}(\unicode[STIX]{x1D714})$ has at least one positive eigenvalue everywhere.

scholarly journals Remark on the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights

2009 ◽  
Vol 7 (3) ◽  
pp. 301-311 ◽  
Author(s):  
Alexei Yu. Karlovich
Keyword(s):  
Integral Operator ◽  
Singular Integral Operator ◽  
Singular Integral ◽  
Sufficient Condition ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Partial Converse ◽  
Cauchy Singular Integral ◽  
Cauchy Singular Integral Operator

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.

scholarly journals Minimal complexes of cotorsion flat modules

2019 ◽  
Vol 124 (1) ◽  
pp. 15-33 ◽  
Author(s):  
Peder Thompson
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Homotopy Equivalence ◽  
Commutative Noetherian Ring ◽  
Partial Converse ◽  
Flat Modules ◽  
Cotorsion Pairs

Let $R$ be a commutative noetherian ring. We give criteria for a complex of cotorsion flat $R$-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the structure of cotorsion flat $R$-modules. More generally, we show that any complex built from covers in every degree (or envelopes in every degree) is minimal, as well as give a partial converse to this in the context of cotorsion pairs. As an application, we show that every $R$-module is isomorphic in the derived category over $R$ to a minimal semi-flat complex of cotorsion flat $R$-modules.

scholarly journals On some mean value theorems of the differential calculus

1971 ◽  
Vol 5 (2) ◽  
pp. 227-238 ◽  
Author(s):  
J.B. Diaz ◽  
R. Výborný
Keyword(s):  
Differential Calculus ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Value Theorems ◽  
Partial Converse ◽  
Generalized Mean ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Special Case

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

An Application of Ultraproducts to Lattice-Ordered Groups

1972 ◽  
Vol 24 (6) ◽  
pp. 1063-1064
Author(s):  
A. M. W. Glass
Keyword(s):  
Total Order ◽  
Lattice Ordered Group ◽  
Independent Subset ◽  
Ordered Group ◽  
Ordered Groups ◽  
Partial Converse ◽  
Lattice Ordered Groups ◽  
Total Orders

Using ultraproducts, N. R. Reilly proved that if G is a representable lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a total order which induces ≺ on J (see [5]). In [4], H. A. Hollister proved that a group G admits a total order if and only if it admits a representable order and, moreover, every latticeorder on a group is the intersection of right total orders. The purpose of this paper is to provide a partial converse, viz: if G is a lattice-ordered group and J is an independent subset totally ordered by ≺, then the order on G can be extended to a right total order which induces ≺ on J.

Games played on Boolean algebras

1983 ◽  
Vol 48 (3) ◽  
pp. 714-723 ◽  
Author(s):  
Matthew Foreman
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Closed Subset ◽  
Winning Strategy ◽  
Boolean Algebras ◽  
Algebra Structure ◽  
Partial Converse ◽  
Image Position ◽  
Special Case

In this paper we consider the special case of the Banach-Mazur game played on a topological space when the space also has an underlying Boolean Algebra structure. This case was first studied by Jech [2]. The version of the Banach-Mazur game we will play is the following game played on the Boolean algebra:Players I and II alternate moves playing a descending sequence of elements of a Boolean algebra ℬ.Player II wins the game iff Πi∈ωbi ≠ 0. Jech first considered these games and showed:Theorem (Jech [2]). ℬ is (ω1, ∞)-distributive iff player I does not have a winning strategy in the game played on ℬ.If ℬ has a dense ω-closed subset then it is easy to see that player II has a winning strategy in this game. This paper establishes a partial converse to this, namely it gives cardinality conditions on ℬ under which II having a winning strategy implies ω-closure.In the course of proving the converse, we consider games of length > ω and generalize Jech's theorem to these games. Finally we present an example due to C. Gray that stands in counterpoint to the theorems in this paper.In this section we give a few basis definitions and explain our notation. These definitions are all standard.

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