scholarly journals Asymptotic Analysis of Regular Sequences

Algorithmica ◽  
2019 ◽  
Vol 82 (3) ◽  
pp. 429-508
Author(s):  
Clemens Heuberger ◽  
Daniel Krenn
Keyword(s):  
Asymptotic Analysis ◽  
Fourier Coefficients ◽  
Linear Representation ◽  
Regular Sequence ◽  
List Type ◽  
Absolute Value ◽  
Regular Sequences ◽  
Asymptotic Formulae ◽  
Periodic Fluctuations ◽  
Finite Sum

Abstract In this article, q-regular sequences in the sense of Allouche and Shallit are analysed asymptotically. It is shown that the summatory function of a regular sequence can asymptotically be decomposed as a finite sum of periodic fluctuations multiplied by a scaling factor. Each of these terms corresponds to an eigenvalue of the sum of matrices of a linear representation of the sequence; only the eigenvalues of absolute value larger than the joint spectral radius of the matrices contribute terms which grow faster than the error term. The paper has a particular focus on the Fourier coefficients of the periodic fluctuations: they are expressed as residues of the corresponding Dirichlet generating function. This makes it possible to compute them in an efficient way. The asymptotic analysis deals with Mellin–Perron summations and uses two arguments to overcome convergence issues, namely Hölder regularity of the fluctuations together with a pseudo-Tauberian argument. Apart from the very general result, three examples are discussed in more detail:sequences defined as the sum of outputs written by a transducer when reading a q-ary expansion of the input;the amount of esthetic numbers in the first N natural numbers; andthe number of odd entries in the rows of Pascal’s rhombus. For these examples, very precise asymptotic formulæ are presented. In the latter two examples, prior to this analysis only rough estimates were known.

On the Strong Summability by Triangular Means of the Derived Fourier Series and its Conjugate Series

1966 ◽  
Vol 9 (05) ◽  
pp. 647-654
Author(s):  
Narendra K. Govil
Keyword(s):  
Fourier Series ◽  
Triangular Matrix ◽  
Regular Sequence ◽  
Strong Summability ◽  
List Type ◽  
Conjugate Series

The triangular matrix (A) = (X ), where n = 0, 1, 2,…; k = 0, 1, 2, …; and λn, k = 0 for k > n is regular (in the sense of defining a regular sequence to sequence transform) if for every fixed k ; independently of n;

scholarly journals On Regular Sequences

1966 ◽  
Vol 27 (1) ◽  
pp. 355-356 ◽  
Author(s):  
J. Dieudonné
Keyword(s):  
Algebraic Geometry ◽  
Noetherian Ring ◽  
Regular Sequence ◽  
Local Rings ◽  
Regular Sequences

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES

2008 ◽  
Vol 07 (05) ◽  
pp. 575-591
Author(s):  
HAGEN KNAF
Keyword(s):  
Projective Dimension ◽  
Regular Sequence ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Regular Sequences ◽  
Local Algebras ◽  
Weak Dimension ◽  
Valuation Domains ◽  
Finitely Presented

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

scholarly journals Improved Bounds on Fourier Entropy and Min-entropy

2021 ◽  
Vol 13 (4) ◽  
pp. 1-40
Author(s):  
Srinivasan Arunachalam ◽  
Sourav Chakraborty ◽  
Michal Koucký ◽  
Nitin Saurabh ◽  
Ronald De Wolf
Keyword(s):  
Functional Analysis ◽  
Boolean Function ◽  
Shannon Entropy ◽  
Fourier Coefficients ◽  
Universal Constant ◽  
Spectral Norm ◽  
List Type ◽  
Boolean Cube ◽  
Total Influence

Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S) 2 . The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f ˆ2 ) ≤ C ⋅ Inf (f), where H (fˆ2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f In this article, we present three new contributions toward the FEI conjecture: (1) Our first contribution shows that H(f ˆ2 ) ≤ 2 ⋅ aUC ⊕ (f), where aUC ⊕ (f) is the average unambiguous parity-certificate complexity of f . This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞ fˆ2) ≤ C ⋅ Inf(f), where H ∞ fˆ2) is the min-entropy of the Fourier distribution. We show H ∞ (fˆ2) ≤ 2⋅C min ⊕ (f), where C min ⊕ (f) is the minimum parity-certificate complexity of f . We also show that for all ε≥0, we have H ∞ (fˆ2) ≤2 log⁡(∥f ˆ ∥1,ε/(1−ε)), where ∥f ˆ ∥1,ε is the approximate spectral norm of f . As a corollary, we verify the FMEI conjecture for the class of read- k DNFs (for constant  k ). (3) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2 ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

scholarly journals On a characterization of compactness and the Abel-Poisson summability of fourier coefficients in Banach spaces

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1061-1068
Author(s):  
Seda Öztürk
Keyword(s):  
Banach Space ◽  
Topological Group ◽  
Integral Representation ◽  
Unit Circle ◽  
Banach Spaces ◽  
Fourier Coefficients ◽  
Linear Representation ◽  
Complex Banach Space ◽  
Continuous Linear

In this paper, for an isometric strongly continuous linear representation denoted by ? of the topological group of the unit circle in complex Banach space, we study an integral representation for Abel-Poisson mean A?r (x) of the Fourier coefficients family of an element x, and it is proved that this family is Abel-Poisson summable to x. Finally, we give some tests which are related to characterizations of relatively compactness of a subset by means of Abel-Poisson operator A?r and ?.

High frequency asymptotic analysis of a string with rapidly oscillating density

2000 ◽  
Vol 11 (6) ◽  
pp. 595-622 ◽  
Author(s):  
C. CASTRO ◽  
E. ZUAZUA
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Asymptotic Analysis ◽  
Explicit Formula ◽  
High Frequency ◽  
Critical Size ◽  
The Other ◽  
Full Description ◽  
Eigenvalues And Eigenfunctions ◽  
Asymptotic Formulae

We consider the eigenvalue problem associated with the vibrations of a string with rapidly oscillating periodic density. In a previous paper we stated asymptotic formulae for the eigenvalues and eigenfunctions when the size of the microstructure ε is shorter than the wavelength of the eigenfunctions 1/√λε. On the other hand, it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions (ε ∼ 1/√λε) singular phenomena may occur. In this paper we study the behaviour of the eigenvalues and eigenfunctions when 1/√λε is larger than the critical size ε. We use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to ε. Our analysis provides all order correction formulae for the limit eigenvalues and eigenfunctions above the critical size. Each term of the asymptotic expansion requires one more derivative of the density. Thus, a full description requires the density to be C∞ smooth.

Singularity formation in three-dimensional motion of a vortex sheet

1995 ◽  
Vol 300 ◽  
pp. 339-366 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Integration ◽  
Finite Time ◽  
Fourier Coefficients ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Three Dimensions ◽  
Leading Order ◽  
Singularity Formation

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

FINITENESS PROPERTIES OF LOCAL COHOMOLOGY MODULES AND GENERALIZED REGULAR SEQUENCES

2010 ◽  
Vol 09 (02) ◽  
pp. 315-325
Author(s):  
KAMAL BAHMANPOUR ◽  
SEADAT OLLAH FARAMARZI ◽  
REZA NAGHIPOUR
Keyword(s):  
Nonnegative Integer ◽  
Local Cohomology ◽  
Regular Sequence ◽  
Associated Primes ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Local Cohomology Modules ◽  
Finiteness Properties ◽  
Cohomology Modules

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/𝔞M > 1, then the R-module ∪{N|N is a submodule of [Formula: see text] and dim N ≤ 1} is 𝔞-cominimax and for some x ∈ R is Rx + 𝔞-cofinite, where t ≔ gdepth (𝔞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of [Formula: see text] with dim N ≤ 1 the associated primes of [Formula: see text] are finite, whenever [Formula: see text] for all i < l. Finally, we show that if (R, 𝔪) is local, M is finitely generated, [Formula: see text] for all i < l, and [Formula: see text] then there exists a generalized regular sequence x1, …, xl ∈ 𝔞 on M such that [Formula: see text].

Distributions defined as limits I. Distributions as derivatives; continuity

1957 ◽  
Vol 53 (1) ◽  
pp. 76-92 ◽  
Author(s):  
J. R. Ravetz
Keyword(s):  
Regular Sequence ◽  
Continuous Functions ◽  
General Level ◽  
Natural Manner ◽  
Basic Theorem ◽  
Regular Sequences ◽  
Alternative Approaches ◽  
Definition Of ◽  
Generalized Limits ◽  
Products Of Distributions

The theory of distributions of L. Schwartz (3) provides a unified and rigorous foundation for special methods used in various branches of mathematics. Schwartz's treatment is on the most general level, and presupposes an understanding of modern abstract analysis. Several alternative approaches to distributions have been developed, all of them ‘elementary’ in one sense or another. We follow here the approach of Mikusiński (2) and Temple (4), in which distributions are defined as generalized limits of sequences of continuous functions. We find that, with this approach, it is possible to prove the basic theorem: every distribution is (locally) a derivative. The property of continuity of a distribution does not enter into the arguments establishing this result, but instead follows from it. Hence we are able to reduce the ‘regular sequence’ definition of a distribution to its simplest form. In a later paper we shall study convolution products of distributions, defined in the natural manner by regular sequences.

scholarly journals From symmetric product CFTs to AdS3

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Matthias R. Gaberdiel ◽  
Rajesh Gopakumar ◽  
Bob Knighton ◽  
Pronobesh Maity
Keyword(s):  
String Theory ◽  
Matrix Model ◽  
Spectral Curve ◽  
Underlying Mechanism ◽  
Covering Space ◽  
Symmetric Product ◽  
Branched Covers ◽  
World Sheet ◽  
Absolute Value ◽  
Finite Sum

Abstract Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator thus has the form of an integral over the entire world-sheet moduli space weighted with a Nambu-Goto-like action. Quite strikingly, at leading order this action can also be written as the absolute value of the Schwarzian of the covering map.Given the equivalence of the symmetric product CFT to tensionless string theory on AdS3, this provides an explicit realisation of the underlying mechanism of gauge-string duality originally proposed in [1] and further refined in [2].

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