Bad properties of the Bernstein numbers
Albrecht Pietsch (2008)
Studia Mathematica
Similarity:
We show that the classes associated with the Bernstein numbers bₙ fail to be operator ideals. Moreover, for 1/r = 1/p + 1/q.
Albrecht Pietsch (2008)
Studia Mathematica
Similarity:
We show that the classes associated with the Bernstein numbers bₙ fail to be operator ideals. Moreover, for 1/r = 1/p + 1/q.
Leokadia Bialas-Ciez (2012)
Annales Polonici Mathematici
Similarity:
The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function . Moreover, we show that one of these extremal-like functions is equal to if E is a nonpluripolar set with where , the supremum is taken over all polynomials P of N variables...
Alberto Farina, Berardino Sciunzi, Enrico Valdinoci (2008)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Similarity:
We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in and and of the Bernstein problem on the flatness of minimal area graphs in . A one-dimensional symmetry result in the half-space is also obtained as a byproduct...
Lin, Qun, Xu, Da, Zhang, Shuhua
Similarity:
In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem where A is an elliptic partial-differential operator and is positive, nonincreasing and log-convex on with . Error estimates are derived in the norm of , and some estimates for the first order time derivatives of the errors are also given.
Thomas J. Haines (2012)
Annales scientifiques de l'École Normale Supérieure
Similarity:
Let be an unramified group over a -adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with -level structure initiated by M. Rapoport and the author in [15].
Mirosław Baran, Agnieszka Kowalska (2014)
Annales Polonici Mathematici
Similarity:
It is known that for determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not determining. In this paper we give examples of sets which are not determining, but have the Bernstein and generalized Markov properties.
A. Kruse (1967)
Acta Arithmetica
Similarity:
Franklin D. Tall (2002)
Fundamenta Mathematicae
Similarity:
Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define to be X ∩ M with topology generated by . Suppose is homeomorphic to the irrationals; must ? We have partial results. We also answer a question of Gruenhage by showing that if is homeomorphic to the “Long Cantor Set”, then .
Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
Similarity:
Let be the uniform norm in the unit disk. We study the quantities where the infimum is taken over all polynomials of degree with and . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that . We find the exact values of and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.
Peter Holy, Philipp Lücke (2014)
Fundamenta Mathematicae
Similarity:
Given an uncountable cardinal κ with and regular, we show that there is a forcing that preserves cofinalities less than or equal to and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein...
Oleg Petrushov (2014)
Acta Arithmetica
Similarity:
Let . We prove that for each root of unity there is an a > 0 such that as r → 1-. For roots of unity e(l/q) with q ≤ 100 we prove that these omega-estimates are true with a = 1/2. From omega-estimates for (z) we obtain omega-estimates for some finite sums.
M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)
Journal of the European Mathematical Society
Similarity:
We present a novel approach for bounding the resolvent of for large energies. It is shown here that there exist a large integer and a large number so that relative to the usual weighted -norm, for all . This requires suitable decay and smoothness conditions on . The estimate (2) is trivial when , but difficult for large since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...
Wojciech Banaszczyk, Artur Lipnicki (2015)
Annales Polonici Mathematici
Similarity:
The paper deals with the approximation by polynomials with integer coefficients in , 1 ≤ p ≤ ∞. Let be the space of polynomials of degree ≤ n which are divisible by the polynomial , r ≥ 0, and let be the set of polynomials with integer coefficients. Let be the maximal distance of elements of from in . We give rather precise quantitative estimates of for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of for p ≠ 2. It follows that as n → ∞. The results...
Tie Zhu Zhang, Shu Hua Zhang (2015)
Applications of Mathematics
Similarity:
We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set , the gradient approximation possesses the superconvergence: , where denotes the average gradient on elements containing vertex . Furthermore, by using the interpolation post-processing...
Yuri Kifer (2014)
Annales de l'I.H.P. Probabilités et statistiques
Similarity:
We consider “nonconventional” averaging setup in the form , where , is either a stochastic process or a dynamical system with sufficiently fast mixing while , and , grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.
Nicolas Lerner (2006)
Bulletin de la Société Mathématique de France
Similarity:
For a principal type pseudodifferential operator, we prove that condition implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from for any (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition doesimply local solvability with a loss of 1...
Humio Ichimura (2013)
Acta Arithmetica
Similarity:
Let be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order (resp. order dividing q), and let ψₙ be an even character of conductor and order pⁿ. We put χ = δφψₙ, whose value is contained in . It is well known that the Bernoulli number is not zero, which is shown in an analytic way. In the extreme cases and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: for any...
Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)
Acta Arithmetica
Similarity:
We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in of degree d for which s consecutive coefficients are fixed. Our estimate asserts that , where . We also prove that , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of of degree d with s consecutive coefficients fixed as above. Finally, we show that , where ₂(d,0) denotes the average second moment for...
Tamás Erdélyi (2001)
Colloquium Mathematicae
Similarity:
Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials of the form , , by , (here 0/0 is interpreted as 1). We define the norms of the truncation operators by , . Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...
Yann Bugeaud (2015)
Acta Arithmetica
Similarity:
Let d be a positive integer and α a real algebraic number of degree d + 1. Set . It is well-known that , where ||·|| denotes the distance to the nearest integer. Furthermore, for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that for any integer n ≥ 1.