Symmetry

Review

40 pages, 953 KiB

Open AccessReview

Recent Advances in Cosmological Singularities

by Oem Trivedi

Symmetry 2024, 16(3), 298; https://doi.org/10.3390/sym16030298 - 3 Mar 2024

Cited by 12 | Viewed by 1956

The discovery of the Universe’s late-time acceleration and dark energy has led to a great deal of research into cosmological singularities, and in this brief review, we discuss all the prominent developments in this field for the best part of the last two [...] Read more.

The discovery of the Universe’s late-time acceleration and dark energy has led to a great deal of research into cosmological singularities, and in this brief review, we discuss all the prominent developments in this field for the best part of the last two decades. We discuss the fundamentals of spacetime singularities, after which we discuss in detail all the different forms of cosmological singularities that have been discovered in recent times. We then address methods and techniques to avoid or moderate these singularities in various theories and discuss how these singularities can also occur in non-conventional cosmologies. We then discuss a useful dynamical systems approach to deal with these singularities and finish up with some outlooks for the field. We hope that this work serves as a good resource to anyone who wants to update themselves with the developments in this very exciting area. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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19 pages, 643 KiB

Open AccessReview

The Problem of Moments: A Bunch of Classical Results with Some Novelties

by Pier Luigi Novi Inverardi, Aldo Tagliani and Jordan M. Stoyanov

Symmetry 2023, 15(9), 1743; https://doi.org/10.3390/sym15091743 - 11 Sep 2023

Cited by 3 | Viewed by 1116

Abstract

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger’s results a century ago and ending with the great progress [...] Read more.

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger’s results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition, we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) to exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices (these bounds are used for concluding indeterminacy); (c) to provide new arguments to confirm classical results; (d) to give new numerical illustrations involving commonly used probability distributions. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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27 pages, 12047 KiB

Open AccessFeature PaperReview

Symmetric Strange Attractors: A Review of Symmetry and Conditional Symmetry

by Chunbiao Li, Zhinan Li, Yicheng Jiang, Tengfei Lei and Xiong Wang

Symmetry 2023, 15(8), 1564; https://doi.org/10.3390/sym15081564 - 10 Aug 2023

Cited by 15 | Viewed by 3264

Abstract

A comprehensive review of symmetry and conditional symmetry is made from the core conception of symmetry and conditional symmetry. For a dynamical system, the structure of symmetry means its robustness against the polarity change of some of the system variables. Symmetric systems typically [...] Read more.

A comprehensive review of symmetry and conditional symmetry is made from the core conception of symmetry and conditional symmetry. For a dynamical system, the structure of symmetry means its robustness against the polarity change of some of the system variables. Symmetric systems typically show symmetrical dynamics, and even when the symmetry is broken, symmetric pairs of coexisting attractors are born, annotating the symmetry in another way. The polarity balance can be recovered through combinations of the polarity reversal of system variables, and furthermore, it can also be restored by the offset boosting of some of the system variables if the variables lead to the polarity reversal of their functions. In this case, conditional symmetry is constructed, giving a chance for a dynamical system outputting coexisting attractors. Symmetric strange attractors typically represent the flexible polarity reversal of some of the system variables, which brings more alternatives of chaotic signals and more convenience for chaos application. Symmetric and conditionally symmetric coexisting attractors can also be found in memristive systems and circuits. Therefore, symmetric chaotic systems and systems with conditional symmetry provide sufficient system options for chaos-based applications. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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26 pages, 5945 KiB

Open AccessReview

Chiral Organophosphorus Pharmaceuticals: Properties and Application

by Anastasy O. Kolodiazhna and Oleg I. Kolodiazhnyi

Symmetry 2023, 15(8), 1550; https://doi.org/10.3390/sym15081550 - 7 Aug 2023

Cited by 12 | Viewed by 2121

Abstract

This review considers the chiral phosphorus-containing drugs used to treat patients in the clinic, as well as the promising and experimental drugs that are in the process of being researched. Natural and synthetic representatives of phosphorus-containing drugs, such as tenofovir (hepatitis B and [...] Read more.

This review considers the chiral phosphorus-containing drugs used to treat patients in the clinic, as well as the promising and experimental drugs that are in the process of being researched. Natural and synthetic representatives of phosphorus-containing drugs, such as tenofovir (hepatitis B and HIV treatment), fosfomycin (antibiotic), valinofos (antibiotic), phosphazinomycin A (antibiotic), (R)-phospholeucine, various antibacterial and antifungal agents, renin inhibitors, etc., have found practical applications as medicines and bioregulators and other medicines. The influence of the chirality of both carbon atoms and phosphorus atoms on the pharmacodynamics, pharmacokinetics, and toxicological properties of phosphorus drugs has been demonstrated. Therefore, the choice of enantiomers is critical since the wrong choice of a chiral drug can lead to undesirable consequences, carcinogenicity, and teratogenicity. New chiral technologies affecting drug development are discussed, such as the “chiral switch” of racemates already on the market, as well as phosphorus-containing prodrugs with a higher biological selectivity and low adverse effects. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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38 pages, 11844 KiB

Open AccessFeature PaperReview

Azulene, Reactivity, and Scientific Interest Inversely Proportional to Ring Size; Part 2: The Seven-Membered Ring

by Alexandru C. Razus

Symmetry 2023, 15(7), 1391; https://doi.org/10.3390/sym15071391 - 10 Jul 2023

Cited by 3 | Viewed by 5439

Abstract

The second part of the article Azulene, Reactivity, and Scientific Interest Inversely Proportional to Ring Size deals with the chemical behavior of the seven-atom azulenic ring. As the title states, the ability of this system to react is lesser compared to [...] Read more.

The second part of the article Azulene, Reactivity, and Scientific Interest Inversely Proportional to Ring Size deals with the chemical behavior of the seven-atom azulenic ring. As the title states, the ability of this system to react is lesser compared to that of the five-atom ring; despite this, a large number of syntheses contain it as a participant in the molecules of starting compounds. This review is focused on certain more frequent syntheses such as nucleophilic substitution of the seven-atom ring or its substituents, vicarious nucleophilic substitutions, substitutions of azulene metallic compounds, or reactions catalyzed by complexes of certain transition metals. The syntheses of tricyclic compounds, porphyrinogenic systems, or azulenocyanines containing an azulenyl moiety are also presented. The adopted presentation is mainly based on reaction schemes that include the reaction conditions, as well as the yields of the products formed. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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31 pages, 5754 KiB

Open AccessFeature PaperReview

Symmetry Perception and Psychedelic Experience

by Alexis D. J. Makin, Marco Roccato, Elena Karakashevska, John Tyson-Carr and Marco Bertamini

Symmetry 2023, 15(7), 1340; https://doi.org/10.3390/sym15071340 - 30 Jun 2023

Cited by 6 | Viewed by 4734

Abstract

This review of symmetry perception has six parts. Psychophysical studies have investigated symmetry perception for over 100 years (part 1). Neuroscientific studies on symmetry perception have accumulated in the last 20 years. Functional MRI and EEG experiments have conclusively shown that regular visual [...] Read more.

This review of symmetry perception has six parts. Psychophysical studies have investigated symmetry perception for over 100 years (part 1). Neuroscientific studies on symmetry perception have accumulated in the last 20 years. Functional MRI and EEG experiments have conclusively shown that regular visual arrangements, such as reflectional symmetry, Glass patterns, and the 17 wallpaper groups all activate the extrastriate visual cortex. This activation generates an event-related potential (ERP) called sustained posterior negativity (SPN). SPN amplitude scales with the degree of regularity in the display, and the SPN is generated whether participants attend to symmetry or not (part 2). It is likely that some forms of symmetry are detected automatically, unconsciously, and pre-attentively (part 3). It might be that the brain is hardwired to detect reflectional symmetry (part 4), and this could contribute to its aesthetic appeal (part 5). Visual symmetry and fractal geometry are prominent in hallucinations induced by the psychedelic drug N,N-dimethyltryptamine (DMT), and visual flicker (part 6). Integrating what we know about symmetry processing with features of induced hallucinations is a new frontier in neuroscience. We propose that the extrastriate cortex can generate aesthetically fascinating symmetrical representations spontaneously, in the absence of external symmetrical stimuli. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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41 pages, 496 KiB

Open AccessFeature PaperReview

Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras

by Sergey V. Bolokhov and Vladimir D. Ivashchuk

Symmetry 2023, 15(6), 1199; https://doi.org/10.3390/sym15061199 - 3 Jun 2023

Cited by 1 | Viewed by 1066

Abstract

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n [...] Read more.

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions

H_{s} (z)

comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters

p_{s} > 0

,

s = 1, \dots, n

. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks

n = 1, 2, 3, 4, 5

and exceptional algebra

E_{6}

. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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40 pages, 12642 KiB

Open AccessFeature PaperReview

Azulene, Reactivity, and Scientific Interest Inversely Proportional to Ring Size; Part 1: The Five-Membered Ring

by Alexandru C. Razus

Symmetry 2023, 15(2), 310; https://doi.org/10.3390/sym15020310 - 22 Jan 2023

Cited by 4 | Viewed by 2484

Abstract

The lack of azulene symmetry with respect to the axis perpendicular to a molecule creates an asymmetry of the electronic system, increasing the charge density of the five-atom ring and favoring its electrophilic substitutions. The increased reactivity of this ring has contributed to [...] Read more.

The lack of azulene symmetry with respect to the axis perpendicular to a molecule creates an asymmetry of the electronic system, increasing the charge density of the five-atom ring and favoring its electrophilic substitutions. The increased reactivity of this ring has contributed to ongoing interest about the syntheses in which it is involved. The aim of this review is to present briefly and mainly in the form of reaction schemes the behavior of this system. After a short chapter that includes the research until 1984, subsequent research is presented as generally accepted chapters and subchapters to describe the behavior of the azulene system: metal free catalyst reactions; reactions catalyzed by metals; various azulene five-ring substitutions. The author insists on reaction yields, and in some cases considers it useful to present the proposed reaction mechanisms. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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19 pages, 1050 KiB

Open AccessFeature PaperReview

Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons

by Irina Dymnikova and Anna Dobosz

Symmetry 2023, 15(2), 273; https://doi.org/10.3390/sym15020273 - 18 Jan 2023

Cited by 2 | Viewed by 1830

Abstract

We briefly overview the basic properties and generic behavior of circular equatorial particle orbits and light rings around regular rotating compact objects with dark energy interiors, which are described by regular metrics of the Kerr–Schild class and include rotating black holes and self-gravitating [...] Read more.

We briefly overview the basic properties and generic behavior of circular equatorial particle orbits and light rings around regular rotating compact objects with dark energy interiors, which are described by regular metrics of the Kerr–Schild class and include rotating black holes and self-gravitating spinning solitons replacing naked singularities. These objects have an internal de Sitter vacuum disk and can have two types of dark interiors, depending on the energy conditions. The first type reduces to the de Sitter disk, the second contains a closed de Sitter surface and an

S

surface with the de Sitter disk as the bridge and an anisotropic phantom fluid in the regions between the

S

surface and the disk. In regular geometry, the potentials decrease from

V (r) \to \infty

to their minima, which ensures the existence of the innermost stable photon and particle orbits that are essential for processes of energy extraction occurring within the ergoregions, which for the second type of interiors contain the phantom energy. The innermost orbits provide a diagnostic tool for investigation of dark interiors of de Sitter–Kerr objects. They include light rings which confine these objects and ensure the most informative observational signature for rotating black holes presented by their shadows. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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19 pages, 359 KiB

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Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems

by José F. Cariñena and José Fernández-Núñez

Symmetry 2022, 14(12), 2520; https://doi.org/10.3390/sym14122520 - 29 Nov 2022

Cited by 2 | Viewed by 1124

Abstract

The two-dimensional inverse problem for first-order systems is analysed and a method to construct an affine Lagrangian for such systems is developed. The determination of such Lagrangians is based on the theory of the Jacobi multiplier for the system of differential equations. We [...] Read more.

The two-dimensional inverse problem for first-order systems is analysed and a method to construct an affine Lagrangian for such systems is developed. The determination of such Lagrangians is based on the theory of the Jacobi multiplier for the system of differential equations. We illustrate our analysis with several examples of families of forces that are relevant in mechanics, on one side, and of some relevant biological systems, on the other. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

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