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Analysis of mean-field models arising from self-attention dynamics in transformer architectures with layer normalization. [PDF]

Philos Trans A Math Phys Eng Sci
Burger M, Kabri S, Korolev Y, Roith T, Weigand L.   +4 more
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Almost Everywhere Convergent Fourier Series

Journal of Fourier Analysis and Applications, 2011
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Carro, M. J., Mastyło, M., Rodríguez-Piazza, L.   +2 more
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Almost everywhere convergence for noncommutative spaces

Banach Journal of Mathematical Analysis, 2022
The authors introduce various notions of almost uniform and almost everywhere convergence in Haagerup \(L^p\)-spaces over an arbitrary von Neumann algebra, using spectral projections of the operators in the sequence under consideration. These are first demonstrated in the case of a semifinite von Neumann algebra.
Christian Budde, Louis Labuschagne, Claud Steyn   +2 more
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Almost everywhere convergence of Fourier integrals

Archiv der Mathematik, 1995
In this paper, we prove that if \(f \in L^ p (\mathbb{R}^ n)\), for certain \(p\) and \(n\), satisfies \[ \lim_{\lambda \downarrow 1} \varlimsup_{R \to \infty} \int_{R < | \xi | \leq \lambda R} \bigl | \widehat f(\xi) \bigr | d \xi = 0, \] then, for almost all \(x \in \mathbb{R}^ n\), \(\int_{| \xi | \leq R} \widehat f (\xi) e^{ix \cdot \xi} d \xi ...
Chen, Chang-Pao, Lin, Chin-Cheng
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Almost Everywhere Convergence

2016
We have seen in Part II the importance of a.e. convergence in integration theory. The purpose of this last chapter of our book is to clarify its relationship to other convergence notions.
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Almost Everywhere Convergence of Convolution Measures

Canadian Mathematical Bulletin, 2012
AbstractLet (X, ℬ, m, τ) be a dynamical system with (X, ℬ, m) a probability space and τ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in L1(X) of a sequence of operators of weighted averages.
Karin Reinhold, Anna K. Savvopoulou, Christopher M. Wedrychowicz   +2 more
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Almost everywhere convergence of weighted averages

Mathematische Annalen, 1992
Given a sequence \((\mu_ n)\) of probability measures on \(Z\), and an invertible measure-preserving transformation \(\tau\) of a probability space \((X,\beta,m)\), the averages \(\mu_ nf(x)=\sum^{\infty}_{k=- \infty}\mu_ n(k)f(\tau^ kx)\) are bounded operators on \(L^ p(m)\), \(1\leq p\leq\infty\).
Bellow, Alexandra, Jones, Roger L., Rosenblatt, Joseph   +2 more
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Uniqueness of Almost Everywhere Convergent Vilenkin Series

Canadian Mathematical Bulletin, 2004
AbstractD. J. Grubb [3] has shown that uniqueness holds, under a mild growth condition, for Vilenkin series which converge almost everywhere to zero. We show that, under even less restrictive growth conditions, one can replace the limit function 0 by an arbitrary f ∈ Lq, when q > 1.
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Almost everywhere convergence of convolution powers

Ergodic Theory and Dynamical Systems, 1994
AbstractGiven an ergodic dynamical system (X,B,m, τ) and a probability measure μ on the integers, define for all f ∈ L1(X) The almost everywhere convergence of the convolution powers μnf(x) depends on the properties of μ. If μ has finite and then for all f ∈ Lp(X), 1< p < ∞, exists for a.e. x.
Bellow, Alexandra, Jones, Roger, Rosenblatt, Joseph   +2 more
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