Symmetry and Approximation Methods

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 September 2022) | Viewed by 22918

Special Issue Editors

Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Interests: optimized perturbation theory; self-similar approximation theory; method of self-similar prediction; correlated iteration theory; theory of heterophase fluctuations
Special Issues, Collections and Topics in MDPI journals
Chair and Institute of General Mechanics, RWTH Aachen University, Eilfschornsteinstraße 18, D-52062 Aachen, Germany
Interests: asymptotology; nonlinear dynamics; composite materials; thin-walled structures
Special Issues, Collections and Topics in MDPI journals
Materialica + Research Group, Bathurst St. 3000, Apt. 606, Toronto, ON M6B 3B4, Canada
Interests: mathematical physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Contributions are welcome on any subject related to the Padè approximants with applications to realistic problems and on any subject related to renormalization group applications in physics, finance, and geophysics.

Tentative contents (need more work!):

  1. Pade and two-point Padè—interpolation, extrapolation, and bounds;
  2. Euler-transformed, modified Padè approximants and corrected Padè approximants;
  3. Self-similarity and renormalization group as the source of roots, factors, and superexponential approximants;
  4. Nonperturbative conditions for accelerating convergence and optimization with minimal difference and sensitivity conditions;
  5. Factor and root approximants, examples of factors and roots, and interpolation and critical point calculations;
  6. Direct methods for critical index calculation: factors, D-Log Padè, D-Log-Roots and combined Log-Padè approximants, and test examples;
  7. Critical index as a control parameter and calculation with roots;
  8. Accelerated convergence of factors and D-Log Padè approximants;
  9. Additive and D-Log additive approximants for interpolation;
  10. Phase transitions, effective viscosity of suspensions, conductivity, elasticity, and permeability of landslides, earthquakes, and market crashes.

Prof. Dr. Vyacheslav Yukalov
Prof. Dr. Igor Andrianov
Dr. Simon L. Gluzman
Guest Editors

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Published Papers (13 papers)

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Editorial

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5 pages, 182 KiB  
Editorial
Symmetry and Approximation Methods
by Igor V. Andrianov, Simon Gluzman and Vyacheslav I. Yukalov
Symmetry 2023, 15(1), 106; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15010106 - 30 Dec 2022
Viewed by 1062
Abstract
The overwhelming majority of mathematical problems, describing realistic systems and processes, contain two parts: first, the problem needs to be characterized by an effective mathematical model and, second, the appropriate solutions are to be found. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)

Research

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11 pages, 323 KiB  
Article
Features of Fermion Dynamics Revealed by SU2 Symmetry
by Angelo Plastino, Gustavo Luis Ferri and Angel Ricardo Plastino
Symmetry 2022, 14(10), 2179; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14102179 - 17 Oct 2022
Cited by 1 | Viewed by 747
Abstract
We deal here with the notion of statistical order and apply it to a system of interacting fermions endowed with an SU2 × SU2 symmetry. The discussion takes place in a thermal quantum statistical scenario. Two distinct fermion–fermion interactions are at play. One [...] Read more.
We deal here with the notion of statistical order and apply it to a system of interacting fermions endowed with an SU2 × SU2 symmetry. The discussion takes place in a thermal quantum statistical scenario. Two distinct fermion–fermion interactions are at play. One of them is a well-known spin–flip interaction. The other is the pairing interaction responsible for nuclear superconductivity. We used novel statistical quantifiers that yield insights regarding changes in the statistical order produced when the values of the pertinent coupling constants vary. In particular, we show that judicious manipulation of the energy cost associated with statistical order variations with the fermion number is the key to understanding important details of the associated dynamics. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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21 pages, 867 KiB  
Article
New Progressive Iterative Approximation Techniques for Shepard-Type Curves
by Umberto Amato and Biancamaria Della Vecchia
Symmetry 2022, 14(2), 398; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020398 - 17 Feb 2022
Cited by 2 | Viewed by 1121
Abstract
Progressive iterative approximation (PIA) technique is an efficient and intuitive method for data fitting. In CAGD modeling, if the given data points are taken as initial control points, PIA process generates a series of shaping curves by adjusting the control points iteratively, while [...] Read more.
Progressive iterative approximation (PIA) technique is an efficient and intuitive method for data fitting. In CAGD modeling, if the given data points are taken as initial control points, PIA process generates a series of shaping curves by adjusting the control points iteratively, while the limit curve interpolates the data points. Such format was used successfully for Shepard-type curves. The aim of the paper is to construct simple variants of the PIA method for Shepard-type curves producing novel curves modeling data points, so the designer can choose among several pencils of shapes outlining original control polygon. Matrix formulations, convergence results, error estimates, algorithmic formulations, critical comparisons, and numerical tests are shown. An application to a progressive modeling format by truncated wavelet transform is also presented, improving in some sense analogous process by truncated Fourier transform. By playing on two shapes handles—the number of base wavelet transform functions and the iteration level of PIA algorithm—several new contours modeling the given control points are constructed. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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15 pages, 290 KiB  
Article
Methods of Retrieving Large-Variable Exponents
by Vyacheslav I. Yukalov and Simon Gluzman
Symmetry 2022, 14(2), 332; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020332 - 06 Feb 2022
Cited by 7 | Viewed by 2090
Abstract
Methods of determining, from small-variable asymptotic expansions, the characteristic exponents for variables tending to infinity are analyzed. The following methods are considered: diff-log Padé summation, self-similar factor approximation, self-similar diff-log summation, self-similar Borel summation, and self-similar Borel–Leroy summation. Several typical problems are treated. [...] Read more.
Methods of determining, from small-variable asymptotic expansions, the characteristic exponents for variables tending to infinity are analyzed. The following methods are considered: diff-log Padé summation, self-similar factor approximation, self-similar diff-log summation, self-similar Borel summation, and self-similar Borel–Leroy summation. Several typical problems are treated. The comparison of the results shows that all these methods provide close estimates for the large-variable exponents. The reliable estimates are obtained when different methods of summation are compatible with each other. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
23 pages, 1138 KiB  
Article
Online Streaming Features Selection via Markov Blanket
by Waqar Khan, Lingfu Kong, Brekhna Brekhna, Ling Wang and Huigui Yan
Symmetry 2022, 14(1), 149; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010149 - 13 Jan 2022
Cited by 2 | Viewed by 1570
Abstract
Streaming feature selection has always been an excellent method for selecting the relevant subset of features from high-dimensional data and overcoming learning complexity. However, little attention is paid to online feature selection through the Markov Blanket (MB). Several studies based on traditional MB [...] Read more.
Streaming feature selection has always been an excellent method for selecting the relevant subset of features from high-dimensional data and overcoming learning complexity. However, little attention is paid to online feature selection through the Markov Blanket (MB). Several studies based on traditional MB learning presented low prediction accuracy and used fewer datasets as the number of conditional independence tests is high and consumes more time. This paper presents a novel algorithm called Online Feature Selection Via Markov Blanket (OFSVMB) based on a statistical conditional independence test offering high accuracy and less computation time. It reduces the number of conditional independence tests and incorporates the online relevance and redundant analysis to check the relevancy between the upcoming feature and target variable T, discard the redundant features from Parents-Child (PC) and Spouses (SP) online, and find PC and SP simultaneously. The performance OFSVMB is compared with traditional MB learning algorithms including IAMB, STMB, HITON-MB, BAMB, and EEMB, and Streaming feature selection algorithms including OSFS, Alpha-investing, and SAOLA on 9 benchmark Bayesian Network (BN) datasets and 14 real-world datasets. For the performance evaluation, F1, precision, and recall measures are used with a significant level of 0.01 and 0.05 on benchmark BN and real-world datasets, including 12 classifiers keeping a significant level of 0.01. On benchmark BN datasets with 500 and 5000 sample sizes, OFSVMB achieved significant accuracy than IAMB, STMB, HITON-MB, BAMB, and EEMB in terms of F1, precision, recall, and running faster. It finds more accurate MB regardless of the size of the features set. In contrast, OFSVMB offers substantial improvements based on mean prediction accuracy regarding 12 classifiers with small and large sample sizes on real-world datasets than OSFS, Alpha-investing, and SAOLA but slower than OSFS, Alpha-investing, and SAOLA because these algorithms only find the PC set but not SP. Furthermore, the sensitivity analysis shows that OFSVMB is more accurate in selecting the optimal features. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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26 pages, 1651 KiB  
Article
The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform
by Donnacha Daly and Didier Sornette
Symmetry 2021, 13(10), 1922; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101922 - 13 Oct 2021
Cited by 1 | Viewed by 1299
Abstract
This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. [...] Read more.
This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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11 pages, 377 KiB  
Article
A New Constructive Method for Solving the Schrödinger Equation
by Kazimierz Rajchel
Symmetry 2021, 13(10), 1879; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101879 - 05 Oct 2021
Cited by 1 | Viewed by 1698
Abstract
In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The method is based on solutions [...] Read more.
In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The method is based on solutions of the Ricatti equation in the form of a quadratic function with three parameters. The logarithmic derivative of the wave function transforms the Schrödinger equation to the Ricatti equation with arbitrary potential. The Ricatti equation is solved by exploiting the particular symmetry, where a family of discrete transformations preserves the original form of the equation. The method is applied to a one-dimensional Schrödinger equation with a bound states spectrum. By extending the results of the Ricatti equation to the Schrödinger equation the three-parametric solutions for wave functions and energy spectrum are obtained. This three-parametric family of exact solutions is defined on compact support, as well as on the whole real axis in the limiting case, and corresponds to a uniquely defined form of potential. Celebrated exactly solvable cases of special potentials like harmonic oscillator potential, Coulomb potential, infinite square well potential with corresponding energy spectrum and wave functions follow from the general form by appropriate selection of parameters values. The first two of these potentials with corresponding solutions, which are defined on the whole axis and half axis respectively, are achieved by taking the limit of general three-parametric solutions, where one of the parameters approaches a certain, well-defined value. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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17 pages, 308 KiB  
Article
Continued Roots, Power Transform and Critical Properties
by Simon Gluzman
Symmetry 2021, 13(8), 1525; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081525 - 19 Aug 2021
Cited by 5 | Viewed by 2576
Abstract
We consider the problem of calculation of the critical amplitudes at infinity by means of the self-similar continued root approximants. Region of applicability of the continued root approximants is extended from the determinate (convergent) problem with well-defined conditions studied before by Gluzman and [...] Read more.
We consider the problem of calculation of the critical amplitudes at infinity by means of the self-similar continued root approximants. Region of applicability of the continued root approximants is extended from the determinate (convergent) problem with well-defined conditions studied before by Gluzman and Yukalov (Phys. Lett. A 377 2012, 124), to the indeterminate (divergent) problem my means of power transformation. Most challenging indeterminate for the continued roots problems of calculating critical amplitudes, can be successfully attacked by performing proper power transformation to be found from the optimization imposed on the parameters of power transform. The self-similar continued roots were derived by systematically applying the algebraic self-similar renormalization to each and every level of interactions with their strength increasing, while the algebraic renormalization follows from the fundamental symmetry principle of functional self-similarity, realized constructively in the space of approximations. Our approach to the solution of the indeterminate problem is to replace it with the determinate problem, but with some unknown control parameter b in place of the known critical index β. From optimization conditions b is found in the way making the problem determinate and convergent. The index β is hidden under the carpet and replaced by b. The idea is applied to various, mostly quantum-mechanical problems. In particular, the method allows us to solve the problem of Bose-Einstein condensation temperature with good accuracy. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
18 pages, 2816 KiB  
Article
Revisiting the Predictability of the Haicheng and Tangshan Earthquakes
by Didier Sornette, Euan Mearns and Spencer Wheatley
Symmetry 2021, 13(7), 1206; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071206 - 05 Jul 2021
Cited by 4 | Viewed by 1878
Abstract
We analyze a set of precursory data measured before but compiled in retrospect of the MS7.5 Haicheng earthquake in February 1975 and the MS7.6–7.8 Tangshan earthquake in July 1976. We propose a robust and simple coarse-graining method that aggregates [...] Read more.
We analyze a set of precursory data measured before but compiled in retrospect of the MS7.5 Haicheng earthquake in February 1975 and the MS7.6–7.8 Tangshan earthquake in July 1976. We propose a robust and simple coarse-graining method that aggregates and counts how all the anomalies together (levelling, geomagnetism, soil resistivity, earth currents, gravity, earth stress, well water radon, well water level) develop as a function of time. We demonstrate strong evidence for the existence of an acceleration of the number of anomalies leading up to the major Haicheng and Tangshan earthquakes. In particular for the Tangshan earthquake, the frequency of occurrence of anomalies is found to be well described by the log-periodic power law singularity (LPPLS) model, previously proposed for the prediction of engineering failures and later adapted to the prediction of financial crashes. Using a mock real-time prediction experiment and simulation study, based on this methodology of monitoring accelerated rates of physical anomalies measured at the surface, we show the potential for an early warning system with a lead time of a few days. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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18 pages, 3129 KiB  
Article
Optimal Random Packing of Spheres and Extremal Effective Conductivity
by Vladimir Mityushev and Zhanat Zhunussova
Symmetry 2021, 13(6), 1063; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061063 - 13 Jun 2021
Cited by 4 | Viewed by 2065
Abstract
A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing [...] Read more.
A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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Review

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17 pages, 671 KiB  
Review
Markov Moment Problems on Special Closed Subsets of Rn
by Octav Olteanu
Symmetry 2023, 15(1), 76; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15010076 - 27 Dec 2022
Cited by 1 | Viewed by 648
Abstract
First, this paper provides characterizing the existence and uniqueness of the linear operator solution T for large classes of full Markov moment problems on closed subsets F of Rn. One uses approximation by special nonnegative polynomials. The case when F is [...] Read more.
First, this paper provides characterizing the existence and uniqueness of the linear operator solution T for large classes of full Markov moment problems on closed subsets F of Rn. One uses approximation by special nonnegative polynomials. The case when F is compact is studied. Then the cases when F=Rn and F=R+n are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of Lν1(Rn), and respectively of Lν1(R+n), for a large class of measures ν. One solves the important difficulty created by the fact that on Rn, n2, there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
24 pages, 406 KiB  
Review
On Markov Moment Problem, Polynomial Approximation on Unbounded Subsets, and Mazur–Orlicz Theorem
by Octav Olteanu
Symmetry 2021, 13(10), 1967; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101967 - 18 Oct 2021
Cited by 1 | Viewed by 1147
Abstract
We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our [...] Read more.
We review earlier and recent results on the Markov moment problem and related polynomial approximation on unbounded subsets. Such results allow proving the existence and uniqueness of the solutions for some Markov moment problems. This is the first aim of the paper. Our solutions have a codomain space a commutative algebra of (linear) symmetric operators acting from the entire real or complex Hilbert space H to H; this algebra of operators is also an order complete Banach lattice. In particular, Hahn–Banach type theorems for the extension of linear operators having a codomain such a space can be applied. The truncated moment problem is briefly discussed by means of reference citations. This is the second purpose of the paper. In the end, a general extension theorem for linear operators with two constraints is recalled and applied to concrete spaces. Here polynomial approximation plays no role. This is the third aim of this work. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
19 pages, 926 KiB  
Review
Padé Approximants, Their Properties, and Applications to Hydrodynamic Problems
by Igor Andrianov and Anatoly Shatrov
Symmetry 2021, 13(10), 1869; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101869 - 04 Oct 2021
Cited by 11 | Viewed by 3056
Abstract
This paper is devoted to an overview of the basic properties of the Padé transformation and its generalizations. The merits and limitations of the described approaches are discussed. Particular attention is paid to the application of Padé approximants in the mechanics of liquids [...] Read more.
This paper is devoted to an overview of the basic properties of the Padé transformation and its generalizations. The merits and limitations of the described approaches are discussed. Particular attention is paid to the application of Padé approximants in the mechanics of liquids and gases. One of the disadvantages of asymptotic methods is that the standard ansatz in the form of a power series in some parameter usually does not reflect the symmetry of the original problem. The search for asymptotic ansatzes that adequately take into account this symmetry has become one of the most important problems of asymptotic analysis. The most developed technique from this point of view is the Padé approximation. Full article
(This article belongs to the Special Issue Symmetry and Approximation Methods)
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