Weak dimensions and Gorenstein weak dimensions  of group rings

Yueming Xiang

Displaying similar documents to “Weak dimensions and Gorenstein weak dimensions of group rings”

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

Weak $n$ -injective and weak $n$ -fat modules

Umamaheswaran Arunachalam, Saravanan Raja, Selvaraj Chelliah, Joseph Kennedy Annadevasahaya Mani (2022)

Czechoslovak Mathematical Journal

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We introduce and study the concepts of weak $n$ -injective and weak $n$ -flat modules in terms of super finitely presented modules whose projective dimension is at most $n$ , which generalize the $n$ -FP-injective and $n$ -flat modules. We show that the class of all weak $n$ -injective $R$ -modules is injectively resolving, whereas that of weak $n$ -flat right $R$ -modules is projectively resolving and the class of weak $n$ -injective (or weak $n$ -flat) modules together with its left (or right) orthogonal class forms...

Relative weak derived functors

Panneerselvam Prabakaran (2020)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring, $n$ a fixed non-negative integer, $𝒲 ℐ$ the class of all left $R$ -modules with weak injective dimension at most $n$ , and $𝒲 ℱ$ the class of all right $R$ -modules with weak flat dimension at most $n$ . Using left (right) $𝒲 ℐ$ -resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that $- \otimes -$ is right balanced on $ℳ_{R} \times_{R} ℳ$ by $𝒲 ℱ \times 𝒲 ℐ$ , and investigate the global right $𝒲 ℐ$ -dimension of $_{R} ℳ$ by right derived functors of $\otimes$ .

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

Generalization of the weak amenability on various Banach algebras

Madjid Eshaghi Gordji, Ali Jabbari, Abasalt Bodaghi (2019)

Mathematica Bohemica

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The generalized notion of weak amenability, namely $(ϕ, ψ)$ -weak amenability, where $ϕ, ψ$ are continuous homomorphisms on a Banach algebra $𝒜$ , was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(ϕ, ψ)$ -weak amenability on the measure algebra $M (G)$ , the group algebra $L^{1} (G)$ and the Segal algebra $S^{1} (G)$ , where $G$ is a locally compact group, are studied. As a typical example, the $(ϕ, ψ)$ -weak amenability of a special semigroup algebra is shown as well.

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

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If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak...

The power set of ω Elementary submodels and weakenings of CH

István Juhász, Kenneth Kunen (2001)

Fundamenta Mathematicae

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We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C ₂^{s} (ω ₂)$ , but does not follow from $C ₂^{s} (ω ₂)$ , or even $C^{s} (ω ₂)$ .

Eventually semisimple weak $F I$ -extending modules

Figen Takıl Mutlu, Adnan Tercan, Ramazan Yaşar (2023)

Mathematica Bohemica

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In this article, we study modules with the weak $F I$ -extending property. We prove that if $M$ satisfies weak $F I$ -extending, pseudo duo, $C_{3}$ properties and $M / Soc M$ has finite uniform dimension then $M$ decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if $M$ satisfies the weak $F I$ -extending, pseudo duo, $C_{3}$ properties and ascending (or descending) chain condition on essential submodules then $M = M_{1} \oplus M_{2}$ for some semisimple submodule $M_{1}$ and Noetherian (or...

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

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We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$ ’s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

On weakened $(α, δ)$ -skew Armendariz rings

Alireza Majdabadi Farahani, Mohammad Maghasedi, Farideh Heydari, Hamidagha Tavallaee (2022)

Mathematica Bohemica

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In this note, for a ring endomorphism $α$ and an $α$ -derivation $δ$ of a ring $R$ , the notion of weakened $(α, δ)$ -skew Armendariz rings is introduced as a generalization of $α$ -rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(α, δ)$ -skew Armendariz ring if and only if $T_{n} (R)$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz if and only if $R [x] / (x^{n})$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz ring for any positive integer $n$ .

On weak sharp minima for a special class of nonsmooth functions

Marcin Studniarski (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a characterization of weak sharp local minimizers of order one for a function f: ℝⁿ → ℝ defined by $f (x) : = m a x f_{i} (x) | i = 1, . . ., p$ , where the functions $f_{i}$ are strictly differentiable. It is given in terms of the gradients of $f_{i}$ and the Mordukhovich normal cone to a given set on which f is constant. Then we apply this result to a smooth nonlinear programming problem with constraints.

Time regularity of generalized Navier-Stokes equation with $p (x, t)$ -power law

Cholmin Sin (2023)

Czechoslovak Mathematical Journal

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We show time regularity of weak solutions for unsteady motion equations of generalized Newtonian fluids described by $p (x, t)$ -power law for $p (x, t) \geq (3 n + 2) / (n + 2)$ , $n \geq 2,$ by using a higher integrability property and fractional difference method. Moreover, as its application we prove that every weak solution to the problem becomes a local in time strong solution and that it is unique.

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .

The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

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Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_{k} (X)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_{k} (X)$ is Ascoli iff $C_{k} (X)$ is a $k_{ℝ}$ -space iff X is locally compact. Moreover, $C_{k} (X)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability...

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

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It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...

Automorphism group of representation ring of the weak Hopf algebra $\tilde{H_{8}}$

Dong Su, Shilin Yang (2018)

Czechoslovak Mathematical Journal

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Let $H_{8}$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\tilde{H_{8}}$ based on $H_{8}$ , then we investigate the structure of the representation ring of $\tilde{H_{8}}$ . Finally, we prove that the automorphism group of $r (\tilde{H_{8}})$ is just isomorphic to $D_{6}$ , where $D_{6}$ is the dihedral group with order 12.

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

Zofia Adamowicz, Konrad Zdanowski (2011)

Fundamenta Mathematicae

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We prove that for i ≥ 1, the arithmetic $I Δ ₀ + Ω_{i}$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1 + ε) l o g^{i + 2}$ , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $l o g^{i + 3}$ .