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Displaying similar documents to “Optimal error estimates for finite elements on meshes containing bands of caps”

Optimality of the range for which equivalence between certain measures of smoothness holds

Z. Ditzian (2010)

Studia Mathematica

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Recently it was proved for 1 < p < ∞ that ω m ( f , t ) p , a modulus of smoothness on the unit sphere, and K ̃ ( f , t m ) p , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence ω m ( f , t ) p K ̃ ( f , t r ) p does not hold either for p = ∞ or for p = 1.

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula π = n = 0 ( ( 5 n + 3 ) n ! ( 2 n ) ! ) / ( 2 n - 1 ( 3 n + 2 ) ! ) with convergence as 13 . 5 - n , in much the same way as the Euler transformation gives π = n = 0 ( 2 n + 1 n ! n ! ) / ( 2 n + 1 ) ! with convergence as 2 - n . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

Convergence of greedy approximation I. General systems

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as T ε ( x ) : = j D ε ( x ) e * j ( x ) e j , where D ε ( x ) : = j : | e * j ( x ) | ε . We study a generalized version of T ε that we call the weak thresholding approximation. We modify the T ε ( x ) in the following way. For ε > 0, t ∈ (0,1) we set D t , ε ( x ) : = j : t ε | e * j ( x ) | < ε and consider...

Optimal Constants in Khintchine Type Inequalities for Fermions, Rademachers and q-Gaussian Operators

Artur Buchholz (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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For ( P k ) being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants C 2 n , n∈ ℕ, in the inequality k = 1 N A k P k 2 n [ C 2 n ] 1 / 2 n m a x ( k = 1 N A * k A k 1 / 2 L 2 n , ( k = 1 N A k A * k 1/2∥L2n , valid for all finite sequences of operators ( A k ) in the non-commutative L 2 n space related to a semifinite von Neumann algebra with trace. In particular, C 2 n = ( 2 n r - 1 ) ! ! for the Rademacher and Fermion sequences.

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let n = 1 a n be a convergent series of positive real numbers. L. Olivier proved that if the sequence ( a n ) is non-increasing, then lim n n a n = 0 . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n n a n = 0 ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence...

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D d and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by 𝒫 . We associate to Problem 𝒫 an optimal control problem, denoted by 𝒬 . Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex K ˜ , we provide results concerning the well-posedness...

Optimal estimators in learning theory

V. N. Temlyakov (2006)

Banach Center Publications

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This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs x i and outputs y i , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator f z on the base of given data z : = ( ( x , y ) , . . . , ( x m , y m ) ) that approximates well the regression function...

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

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Let A α f ( x ) = | B ( 0 , | x | ) | - α / n B ( 0 , | x | ) f ( t ) d t be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that A α is bounded from L p to L p α with p α = n p / ( α p - n p + n ) when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space S α , Y , which is strictly larger than X, and a ’target’ space T Y , which is strictly smaller than Y, under the assumption that A α is bounded from X into Y and the Hardy-Littlewood...

H 2 convergence of solutions of a biharmonic problem on a truncated convex sector near the angle π

Abdelkader Tami, Mounir Tlemcani (2021)

Applications of Mathematics

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We consider a biharmonic problem Δ 2 u ω = f ω with Navier type boundary conditions u ω = Δ u ω = 0 , on a family of truncated sectors Ω ω in 2 of radius r , 0 < r < 1 and opening angle ω , ω ( 2 π / 3 , π ] when ω is close to π . The family of right-hand sides ( f ω ) ω ( 2 π / 3 , π ] is assumed to depend smoothly on ω in L 2 ( Ω ω ) . The main result is that u ω converges to u π when ω π with respect to the H 2 -norm. We can also show that the H 2 -topology is optimal for such a convergence result.

The equation - Δ 𝑢 - λ 𝑢 | 𝑥 | 2 = | 𝑢 | 𝑝 + 𝑐 𝑓 ( 𝑥 ) : The optimal power

Boumediene Abdellaoui, Ireneo Peral (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We will consider the following problem - Δ u - λ u | x | 2 = | u | p + c f , u &gt; 0 in Ω , where Ω N is a domain such that 0 Ω , N 3 , c &gt; 0 and λ &gt; 0 . The main objective of this note is to study the precise threshold p + = p + ( λ ) for which there is novery weak supersolutionif p p + ( λ ) . The optimality of p + ( λ ) is also proved by showing the solvability of the Dirichlet problem when 1 p &lt; p + ( λ ) , for c &gt; 0 small enough and f 0 under some hypotheses that we will prescribe.

Weak convergence of mutually independent X B and X A under weak convergence of X X B - X A

W. Szczotka (2006)

Applicationes Mathematicae

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For each n ≥ 1, let v n , k , k 1 and u n , k , k 1 be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let X B ( t ) = ( 1 / c ) j = 1 [ n t ] ( v n , j - v ̅ ) , X A ( t ) = ( 1 / c ) j = 1 [ n t ] ( u n , j - u ̅ ̅ ) , t ≥ 0, and X = X B - X A . The main result gives conditions under which the weak convergence X X , where X is a Lévy process, implies X B X B and X A X A , where X B and X A are mutually independent Lévy processes and X = X B - X A .

Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese, Gian Paolo Leonardi (2013)

Journal of the European Mathematical Society

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We prove some results in the context of isoperimetric inequalities with quantitative terms. In the 2 -dimensional case, our main contribution is a method for determining the optimal coefficients c 1 , ... , c m in the inequality δ P ( E ) k = 1 m c k α ( E ) k + o ( α ( E ) m ) , valid for each Borel set E with positive and finite area, with δ P ( E ) and α ( E ) being, respectively, the 𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡 and the 𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 of E . In n dimensions, besides proving existence and regularity properties of minimizers for a wide class of 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠 including the lower semicontinuous extension of δ P ( E ) α ( E ) 2 , we...

Structure properties of D-R spaces

Hartmut von Trotha

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CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................

n-supercyclic and strongly n-supercyclic operators in finite dimensions

Romuald Ernst (2014)

Studia Mathematica

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We prove that on N , there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if N has an n-dimensional subspace whose orbit under T ( N ) is dense in N , then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator T ( N ) is strongly n-supercyclic if N has an n-dimensional subspace whose orbit under T is dense in ( N ) , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially...

Uniform L 1 error bounds for semi-discrete finite element solutions of evolutionary integral equations

Lin, Qun, Xu, Da, Zhang, Shuhua

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In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem u t + 0 t β ( t - s ) A u ( s ) d s = 0 , u ( 0 ) = v , t > 0 , where A is an elliptic partial-differential operator and β ( t ) is positive, nonincreasing and log-convex on ( 0 , ) with 0 β ( ) < β ( 0 + ) . Error estimates are derived in the norm of L t 1 ( 0 , ; L x 2 ) , and some estimates for the first order time derivatives of the errors are also given.

The topology of the space of ℋ𝒦 integrable functions in n

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

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It is known that there is no natural Banach norm on the space ℋ𝒦 of n -dimensional Henstock-Kurzweil integrable functions on [ a , b ] . We show that the ℋ𝒦 space is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . On each ℋ𝒦 ( X ) space, an F -norm · X is defined. A · X -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F -norm is also defined for a · X -continuous linear operator. Hence, many important results in functional analysis hold for the ℋ𝒦 ( X ) space. It is well-known that every...

Nonconventional limit theorems in averaging

Yuri Kifer (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider “nonconventional” averaging setup in the form d X ε ( t ) d t = ε B ( X ε ( t ) , 𝛯 ( q 1 ( t ) ) , 𝛯 ( q 2 ( t ) ) , ... , 𝛯 ( q ( t ) ) ) where 𝛯 ( t ) , t 0 is either a stochastic process or a dynamical system with sufficiently fast mixing while q j ( t ) = α j t , α 1 l t ; α 2 l t ; l t ; α k and q j , j = k + 1 , ... , grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.