scholarly journals Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2963
Author(s):  
Jocemar Q. Chagas ◽  
José A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Summation Formula ◽  
Summation Method ◽  
Divergent Series ◽  
Summation Methods ◽  
Classical Sense ◽  
Finite Sums

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

Hausdorff dimension of the set of almost convergent sequences

2022 ◽  
pp. 1-7
Author(s):  
Alexandr Usachev
Keyword(s):  
Hausdorff Dimension ◽  
Summation Method ◽  
Binary Representation ◽  
Hausdorff Dimensions ◽  
Summation Methods ◽  
Wide Range ◽  
Matrix Summation ◽  
Convergent Sequences

Abstract The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

Field theory: Perturbative expansion and summation methods

2019 ◽  
pp. 451-470
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Conformal Mapping ◽  
Critical Exponents ◽  
Perturbative Expansion ◽  
Divergent Series ◽  
Quantum Field ◽  
Summation Methods ◽  
Borel Transformation

Chapter 23 examines perturbative expansion and summation methods in field theory. In quantum field theory, all perturbative expansions are divergent series in the mathematical sense. This leads to a difficulty when the expansion parameter is not small. In the case of Borel summable series, using the knowledge of the large order behaviour, a number of summation techniques have been developed to derive convergent sequences from divergent series. Some methods apply directly on the series like Padé approximants or order–dependent mapping (the ODM method). Others involve first a Borel transformation, like the Padé–Borel method. The method of Borel transformation, suitably modified, followed by a conformal mapping, has been applied to renormalization group (RG) functions of the phi4 3 field theory and has led to precise values of critical exponents.

scholarly journals Transformation formulas for terminating Saalschützian hypergeometric series of unit argument

1995 ◽  
Vol 8 (2) ◽  
pp. 189-194
Author(s):  
Wolfgang Bühring
Keyword(s):  
Recurrence Relation ◽  
Hypergeometric Series ◽  
Summation Formula ◽  
Summation Formulas ◽  
Finite Sums ◽  
Transformation Formulas ◽  
Independent Parameters

Transformation formulas for terminating Saalschützian hypergeometric series of unit argument p+1Fp(1) are presented. They generalize the Saalschützian summation formula for 3F2(1). Formulas for p=3,4,5 are obtained explicitly, and a recurrence relation is proved by means of which the corresponding formulas can also be derived for larger p. The Gaussian summation formula can be derived from the Saalschützian formula by a limiting process, and the same is true for the corresponding generalized formulas. By comparison with generalized Gaussian summation formulas obtained earlier in a different way, two identities for finite sums involving terminating 3F2(1) series are found. They depend on four or six independent parameters, respectively.

scholarly journals Measurement Methods Affect the Observed Global Dimming and Brightening

2013 ◽  
Vol 26 (12) ◽  
pp. 4112-4120 ◽  
Author(s):  
Kaicun Wang ◽  
Robert E. Dickinson ◽  
Qian Ma ◽  
John A. Augustine ◽  
Martin Wild
Keyword(s):  
Important Implication ◽  
Summation Method ◽  
Measurement Methods ◽  
Term Trend ◽  
Summation Methods ◽  
Long Term Trends ◽  
Global Dimming ◽  
Long Term Trend ◽  
Parallel Measurements

Abstract Surface incident solar radiation G determines our climate and environment, and has been widely observed with a single pyranometer since the late 1950s. Such observations have suggested a widespread decrease between the 1950s and 1980s (global dimming), that is, at a rate of −3.5 W m−2 decade−1 (or −2% decade−1) from 1960 to 1990. Since the early 1990s, the diffuse and direct components of G have been measured independently, and a more accurate G has been calculated by summing these two measurements. Data from this summation method suggest that G increased at a rate of 6.6 W m−2 decade−1 (3.6% decade−1) from 1992 to 2002 (brightening) at selected sites. The brightening rates from these studies were also higher than those from a single pyranometer. In this paper, the authors used 17 years (1995–2011) of parallel measurements by the two methods from nearly 50 stations to test whether these two measurement methods of G provide similar long-term trends. The results show that although measurements of G by the two methods agree very well on a monthly time scale, the long-term trend from 1995 to 2011 determined by the single pyranometer is 2–4 W m−2 decade−1 less than that from the summation method. This difference of trends in the observed G is statistically significant. The dependence of trends of G on measurement methods uncovered here has an important implication for the widely reported global dimming and brightening based on datasets collected by different measurement methods; that is, the dimming might have been less if measured with current summation methods.

Critical exponents and equation of state from series summation

2021 ◽  
pp. 975-991
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Phase Transition ◽  
Superfluid Helium ◽  
Critical Exponents ◽  
Summation Method ◽  
Perturbative Expansion ◽  
Field Theories ◽  
Summation Methods ◽  
Universal Properties ◽  
Borel Transformation

Universal quantities near the phase transition of O(N) symmetric vector models, can be determined, in the framework of the (f2 )2 field theory, and the corresponding renormalization group (RG), in the form of perturbative series. The O(N) symmetric field theories describe, in particular for N = 0, the universal properties of the statistics of long polymers, for N = 1, the liquid–vapour transition, for N = 2, superfluid helium transition, and so on. Universal quantities have been calculated within two different schemes, the Wilson-Fisher ϵ = 4 − d expansion, and perturbative expansion at fixed dimensions 2 and 3 (as suggested by Parisi). In both cases, the series are divergent, and the expansion parameters are not small. In fixed dimensions smaller than 4, the series are proven to be Borel summable. For the ϵ expansion, there are reasons that the property is equally true, but a proof is lacking. With this assumption, in both cases, although the series are divergent, they define unique functions. Since the expansion parameters are not small, summation methods are then required to determine these functions. A specific summation method, based on a parametric Borel transformation and mapping, in which the knowledge of the large order behaviour has been incorporated, has been successfully applied to the series, and has led to a precise evaluation of critical exponents and other universal quantities.

A summation method for divergent series

1999 ◽  
Vol 54 (3) ◽  
pp. 626-627
Author(s):  
V V Belokurov ◽  
Yu P Solov'ev ◽  
E T Shavgulidze
Keyword(s):  
Summation Method ◽  
Divergent Series

scholarly journals Eulerian Gaussian beams for high-frequency wave propagation

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM61-SM76 ◽  
Author(s):  
Shingyu Leung ◽  
Jianliang Qian ◽  
Robert Burridge
Keyword(s):  
Gaussian Beam ◽  
High Frequency ◽  
Summation Formula ◽  
Summation Method ◽  
New Method ◽  
Gaussian Beams ◽  
Multiple Sources ◽  
Helmholtz Equations ◽  
Frequency Wave ◽  
Eulerian Formulation

We design an Eulerian Gaussian beam summation method for solving Helmholtz equations in the high-frequency regime. The traditional Gaussian beam summation method is based on Lagrangian ray tracing and local ray-centered coordinates. We propose a new Eulerian formulation of Gaussian beam theory which adopts global Cartesian coordinates, level sets, and Liouville equations, yielding uniformly distributed Eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources. The time harmonic wavefield can be constructed by summing up Gaussian beams with ingredients provided by the new Eulerian formulation. The conventional Gaussian beam summation method can be derived from the proposed method. There are three advantages of the new method: (1) We have uniform resolution of ray distribution. (2) We can obtain wavefields excited at different sources by varying only source locations in the summation formula. (3) We can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula. Numerical experiments indicate that the Gaussian beam summation method yields accurate asymptotic wavefields even at caustics. The new method may be used for seismic modeling and migration.

Mathematical Tools

2019 ◽  
pp. 657-666
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Finite Size ◽  
Summation Formula ◽  
Bernoulli Numbers ◽  
Quantum Spin ◽  
Mathematical Technique ◽  
Finite Sums ◽  
Mathematical Techniques ◽  
Linear Integral ◽  
Linear Integral Equations

Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the Bernoulli numbers are needed as a prerequisite for the Euler– Maclaurin summation formula, which allows to transform finite sums into integrals plus, in a systematic way, corrections taking into account the finite size of the system. Applying this mathematical technique to the Bethe ansatz equations results in linear integral equations of the Wiener–Hopf type for the solution of which an elaborate mathematical technique exists, the Wiener–Hopf technique.

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