Mathematical Symmetry Research Articles

Stochastic fractal and Noether's theorem.

We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability p or disappears with probability 1-p. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter α, which also determines the fragmentation rate. For a fractal dimension d_{f}, we find that the d_{f} th moment M_{d_{f}} is a conserved quantity, independent of p and α. While the scaling exponents do not depend on p, the self-similar distribution shows a weak p dependence. We use the idea of data collapse-a consequence of dynamical scaling symmetry-to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while M_{d_{f}} relates to a purely mathematical symmetry: Quantum-mechanical phase rotation in Euclidean time.

A story of mathematical symmetries

A story of mathematical symmetries

A novel mathematical model to measure individuals’ perception of the symmetry level of building facades

ABSTRACT Symmetry is one of the key parameters in aesthetic judgement, and development of computational aesthetic approaches. Various experimental, statistical, and machine learning-based methods have been developed to parametrically measuring the symmetry, yet mathematical symmetry measurement models were barely studied. The development of mathematical symmetry measurement models can contribute to the development of symmetry-related models, computational aesthetic approach, and architectural design review. Accordingly, this research aims 1) to develop a mathematical model (symmetry index), and 2) to investigate the relationship between symmetry and aesthetic judgment. For this purpose, two surveys were conducted amongst 337 participants, the symmetry and aesthetic of 13 building images were evaluated, and the survey results were compared to the developed symmetry index. According to the findings, a positive correlation between symmetry and aesthetic appreciation was found and the influence of demographic differences on the symmetry, and aesthetic perception was discussed. The developed model successfully predicts the symmetry of studied buildings’ images.

Improved Semiparametric Analysis of Polygenic Gene–Environment Interactions in Case–Control Studies

Standard logistic regression analysis of case–control data has low power to detect gene–environment interactions, but until recently it was the only method that could be used on complex polygenic data for which parametric distributional models are not feasible. Under the assumption of gene–environment independence in the underlying population, Stalder et al. (Biometrika, 104:801–812, 2017) developed a retrospective method that treats both genetic and environmental variables nonparametrically. However, the mathematical symmetry of genetic and environmental variables is overlooked. We propose an improvement to the method of Stalder et al. that increases the efficiency of the estimates with no additional assumptions and modest computational cost. This improvement is achieved by treating the genetic and environmental variables symmetrically to generate two sets of parameter estimates that are combined to generate a more efficient estimate. We employ a semiparametric framework to develop the asymptotic theory of the estimator, show its asymptotic efficiency gain, and evaluate its performance via simulation studies. The method is illustrated using data from a case–control study of breast cancer.

The Dao of Mathematical Creativity, Symmetry Theories, and Yin-Yang Philosophy With Their Classroom Applications

The Dao of Mathematical Creativity, Symmetry Theories, and Yin-Yang Philosophy With Their Classroom Applications

Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb

System performance and efficiency depends on the stability criteria. The lower limb prosthetic model design requires some prerequisites such as hardware design functionality and compatibility of the building block materials. Effective implementation of mathematical model simulation symmetry towards the achievement of hardware design is the focus of the present work. Different postures of lower limb have been considered in this paper to be analyzed for artificial system design of lower limb movement. The generated polynomial equations of the sitting and standing positions of the normal limb are represented with overall system transfer function. The behavioral analysis of the lower limb model shows the nonlinear nature. The Euler-Lagrange method is utilized to describe the nonlinearity in the field of forward dynamics of the artificial system. The stability factor through phase portrait analysis is checked with respect to nonlinear system characteristics of the lower limb. The asymptotic stability has been achieved utilizing the most applicable Lyapunov method for nonlinear systems. The stability checking of the proposed artificial lower extremity is the newer approach needed to take decisions on output implementation in the system design.

Embedded Element Pattern Loading Condition Transformations for Phased Array Modeling

Embedded element patterns (EEPs) depend on the loads attached to elements in an antenna array. When modeling antenna arrays, the loading condition used in numerically simulations may be different from the loads used in the operating environment. The ability to transform readily from one set to another is of great utility for both computed and measured data. We derive new expressions for the loading condition transformations that combine practical usefulness with a pleasing mathematical symmetry, compare array patterns synthesized from differently-loaded set of EEPs, and give numerical examples for arrays with different element port loading conditions.

Modeling a Thermochemical Reactor of a Solar Refrigerator by BaCl2-NH3 Sorption Using Artificial Neural Networks and Mathematical Symmetry Groups

The aim of this work is to present a model for heat transfer, desorbed refrigerant, and pressure of an intermittent solar cooling system’s thermochemical reactor based on backpropagation neural networks and mathematical symmetry groups. In order to achieve this, a reactor was designed and built based on the reaction of BaCl2-NH3. Experimental data from this reactor were collected, where barium chloride was used as a solid absorbent and ammonia as a refrigerant. The neural network was trained using the Levenberg–Marquardt algorithm. The correlation coefficient between experimental data and data simulated by the neural network was r = 0.9957. In the neural network’s sensitivity analysis, it was found that the inputs, reactor’s heating temperature and sorption time, influence neural network’s learning by 35% and 20%, respectively. It was also found that, by applying permutations to experimental data and using multibase mathematical symmetry groups, the neural network training algorithm converges faster.

The Deep Problems of Symmetry in Mathematical Physics and their Definitive Resolution

Symmetry stands for a central notion in Mathematics with extremely important applications in Theoretical Physics. Mathematician Emmy Noether in 1915 famously axiomatized this notion which resoundingly made its way in fundamental physics in its chapters of Lagrangian mechanics and Hamiltonian mechanics. However there is only one dimension to symmetry in mathematics and mathematical physics: an invariance that is preserved under a particular transformation. Contrary to this approach, there exist many other dimensions to the artifact of symmetry, which, if correctly prospected, could have helped accurately answer the long-running question in mathematical physics concerning the very origin of physical symmetries and the reason for the physical mechanism of spontaneous symmetry-breaking. The ignorance of origin of observable symmetries as well as the existence of other categories of symmetry is accompanied by a misunderstanding of the very states of symmetry, which is reflected in an inexact assessment of the ultra-structures subtending mathematical symmetry. In this writing, we aim to to expose the caveats of symmetry in mathematics, specifically in Geometry, as viewed from the overarching Quanto-Geometric perspective, and to show their resolution with the application of special Differential Geometry principles emanating from the Quanto-Geometric framework.

Structure-preserving strategy for conservative simulation of the relativistic nonlinear Landau-Fokker-Planck equation.

Mathematical symmetries of the Beliaev-Budker kernel are the most important structure of the relativistic Landau-Fokker-Planck equation. In most numerical simulations, however, one of the symmetries is not preserved in the discrete level resulting in a violation of the energy conservation. Recently, we proposed a charge-momentum-energy-conserving relativistic Vlasov-Maxwell scheme by preserving mathematical formulas in discrete form, and here we apply the concept to the relativistic Landau-Fokker-Planck equation. Through a numerical experiment of relativistic collisional relaxation, a mass-momentum-energy-conserving simulation has been demonstrated without any artificial constraints.

Revising the eclipse prediction scheme in the Antikythera mechanism

In 1901, an extraordinary ancient Greek artefact was discovered in a shipwreck just off the tiny island of Antikythera. It was later shown to be a complex astronomical calculating machine and is now known as the Antikythera Mechanism. In 2005, it was established that it predicted eclipses, using a 7th century BC Babylonian eclipse cycle of 223 lunar months, known as the Saros Cycle. Understanding the complex eclipse prediction scheme on the Antikythera Mechanism has resulted from a fascinating series of discoveries. The eclipse prediction scheme is implemented through descriptive glyphs, inscribed round a 223-month Saros Dial at the rear of the Mechanism: a glyph in a particular month indicates a predicted eclipse. A 2008 publication deciphered the meaning of the glyphs: they indicate whether the predicted eclipse is lunar or solar; the possible visibility of the eclipse; and its time of day. The glyphs also include an alphabetical Index Letter, referring to inscriptions round the Saros Dial, which describe eclipse characteristics. In that publication, the full eclipse prediction scheme was not understood but subsequent work in 2014 made substantial progress. The eclipse characteristics, such as the colour and magnitude of the eclipse, are listed together in the inscriptions, together with a group of Index Letters for the eclipses to which they apply. The deeply puzzling grouping and ordering of these Index Letter Groups was solved with a simple mathematical model, which both explained these groups and the distribution of the glyphs round the Saros Dial—revealing an eclipse prediction scheme of extraordinary sophistication and ambition. Later work in 2016 proposed a radical revision of this eclipse prediction scheme, though it did not challenge the mathematical basis of the scheme. The 2016 scheme implied a completely different picture of the whole of the back plate of the Antikythera Mechanism, destroying its essential mathematical symmetry. This revision is comprehensively refuted here, except for the identification of a new text character in one of the Index Letter Groups, which implies an interesting revision of the 2014 scheme but which preserves the elegant 2014 reconstruction of the back plate of the Antikythera Mechanism.

Energy eigenvalues of the three-dimensional quantum harmonic oscillator from SU(3) cubic Casimir operator

The three-dimensional isotropic quantum harmonic oscillator (QHO) plays important roles in physics. Despite its importance, much of its rich mathematical structure and symmetry are often not sufficiently stressed, particularly at the advanced undergraduate or beginning graduate levels. In this paper, the well-known yet less apparent SU(3) symmetry of the three-dimensional isotropic QHO is used to yield its energy eigenvalues. In particular, the relatively less explored cubic Casimir operator of the SU(3) Lie group is exploited to obtain these well-established energy eigenvalues. The relationship between the degeneracy of each energy level of the three-dimensional QHO and the dimensions of irreducible representations of the SU(3) group is revisited and elaborated.

The Idea of Principles in Early Modern Thought: Interdisciplinary Perspectives ed. by Peter R. Anstey

Reviewed by: The Idea of Principles in Early Modern Thought: Interdisciplinary Perspectives ed. by Peter R. Anstey Daniel Schneider Peter R. Anstey, editor. The Idea of Principles in Early Modern Thought: Interdisciplinary Perspectives. New York: Routledge, 2017. Pp. xvi + 303. Cloth, $140.00. This book is a collection of essays that relate in some way to the notion of a principle as it appears in early modern thought. Essays by James Franklin, J. C. Campbell, Alberto Vanzo, Anstey, and William R. Newman provide a survey of the usage of principles within particular subjects: the principles of early modern mathematics, equity law, corpuscularism, and chemistry or alchemy, respectively. Other essays, by Kristen Walsh and Michael LeBuffe, clarify a particular early modern thinker's understanding and usage of the term 'principle.' Walsh's essay concerns Newton's usage and understanding of the term, while LeBuffe's concerns Spinoza's. Other essays, by Daniel Garber and Kiyoshi Shimokawa, clarify how an early modern thinker, or a pair of thinkers, understand and use a particular principle or set of principles to do philosophical work. Garber's essay considers Leibniz's principle of the equality of cause and effect, while Shimokawa's considers the principles that Grotius and Hume use in their accounts of the origin of justice and its relation to utility. Sophie Roux's essay (a wonderful example of the high level of scholarship found in this collection) examines disputes concerning, and contemporaneous to, the two comets of 1664, and shows the role principles played (and failed to play) in demarcating the positions within these disputes. Scholars looking for a sustained investigation into the idea of principles in the early modern period will be disappointed. This is not on account of any failure on the part of these authors, but rather on account of there being no such thing. As this collection makes clear, the early modern period reveals no single idea of principles. Indeed, the only attempt at offering a broad discussion of various early modern ideas of principles is in the excellent introduction by Anstey. Still, this collection succeeds in satisfying its description within this same introduction, namely, as a "preliminary sounding of the subject" (1) of the nature and role of principles in the early modern period. Of the essays offering a "sounding" of principles as they appear within a particular subject, James Franklin's insightful and fascinating "Early Modern Mathematical Principles and Symmetry Arguments" stands out as particularly useful. Franklin's survey of mathematical principles effectively illustrates a dominant early modern conception of mathematics as a science of quantity that yields deductive knowledge of the real world. Likewise, among the essays offering an analysis of particular thinkers' use of the term 'principle,' Kristen Walsh's essay "Principles in Newton's Natural Philosophy" stands out in its significance. Walsh argues persuasively that, in the Principia, Newton used the term 'principle' to signify the functional role of a proposition within his theories. Walsh argues that if, in the Principia, Newton describes a proposition as a principle, he does so because the proposition satisfies two conditions: one, it has been deduced from the phenomena, and two, it serves as a premise in further deductions. Walsh shows how these two simple constraints systematize Newton's otherwise unsystematic usage of 'principle' in the Principia. Walsh's essay is well argued, well researched, and important. Newton scholars will undoubtedly find it of interest. The essays in this collection are all solid works of scholarship. As a collection however, they do not fit together in a very useful way. On the one hand, it will be the rare scholar who can engage deeply with the material covered in each and every essay. On the other hand, it will be the rare researcher of early modern thought who does not find at least some essays in the collection to be of use. The collection would have benefited from a tightening of its theme or perhaps even a tightening of the approaches with which the essays address the theme. As it stands, this "preliminary sounding" of the study of early modern principles lacks a cohesive focus. But the quality of the essays shows that further volumes investigating principles...

Symmetrical group theory for mathematical complexity reduction of digital holograms

This work presents the use of mathematical group theory through an algorithm to reduce the multiplicative computational complexity in the process of creating digital holograms. An object is considered as a set of point sources using mathematical symmetry properties of both the core in the Fresnel integral and the image, where the image is modeled using group theory. This algorithm has multiplicative complexity equal to zero and an additive complexity (k − 1) × N for the case of sparse matrices and binary images, where k is the number of pixels other than zero and N is the total points in the image.

Application of Mathematical Symmetrical Group Theory in the Creation Process of Digital Holograms

This work presents an algorithm to reduce the multiplicative computational complexity in the creation of digital holograms, where an object is considered as a set of point sources using mathematical symmetry properties of both the core in the Fresnel integral and the image. The image is modeled using group theory. This algorithm has multiplicative complexity equal to zero and an additive complexity (k-1)N2 for the case of sparse matrices or binary images, where k is the number of pixels other than zero and N2 is the total of points in the image.

RETRACTED ARTICLE: On the unification of nuclear-structure theory: A response to Bortignon and Broglia

Nuclear-structure theory is unusual among the diverse fields of quantum physics. Although it provides a coherent description of all known isotopes on the basis of a quantum-mechanical understanding of nucleon states, nevertheless, in the absence of a fundamental theory of the nuclear force acting between nucleons, the prediction of all ground-state and excited-state nuclear binding energies is inherently semi-empirical. I suggest that progress can be made by returning to the foundational work of Eugene Wigner from 1937, where the mathematical symmetries of nucleon states were first defined. Those symmetries were later successfully exploited in the development of the independent-particle model ( IPM ∼ shell model , but the geometrical implications noted by Wigner were neglected. Here I review how the quantum-mechanical, but remarkably easy-to-understand geometrical interpretation of the IPM provides constraints on the parametrization of the nuclear force. The proposed “geometrical IPM” indicates a way forward toward the unification of nuclear-structure theory that Bortignon and Broglia have called for.

Non-relativistic time, existence and adaptation

By using non-dimensional Navier–Stokes equations in systems where partial modelling complications can be avoided, it is shown that geometrically similar systems can exhibit identical behaviour but according to different time scales where the latter are also within our own control. These models indicate that the physical nature of time is not related to a fundamental constant, but the passage of time can have a different but still constant value in a particular system of our own choosing. This observation also adds to our knowledge relating to the dichotomy of interpreting time either as a flowing parameter or, alternatively, just accepting it as a part of space-time. The observation adds evidence in favour of both viewpoints. The conclusions also seem relevant to biological species, chemical processes in general and other branches of physics. Using earlier work of Prigogine and others, it also appears that instantaneous time has important properties dividing past and future time into two segments which differ fundamentally in physical characteristics. This seems to explain the ‘raison d’tre’ of the second law of thermodynamics and the necessarily asymmetrical nature of time in our own world where mathematical symmetry of physical laws is the norm. This entropic time barrier also seems to explain the physically intangible nature of past events and their possible natural importance as former lower entropy forms of existence. It is concluded that these physical features of time are necessary for the existence of any form of life and Darwinian evolution, in particular. The conclusions may also help to throw further light on other suggested theories and observations in physics and the role of quantum decoherence in life and its adaptive ability.

Modeling of Swine Diaphragmatic Tissue Under Uniaxial Loading1

Modeling of Swine Diaphragmatic Tissue Under Uniaxial Loading1

Smile Enhancement Through Periodontal Surgery

Illustrations of the periodontal plastic surgery cases are presented. Results: Standard measures, golden proportions, and symmetry are concepts that mostly applied for guidance of image designing. Nevertheless, aesthetic attainment is possible with the presence of different measures and proportions and symmetry or asymmetry. Specific mathematical measures and symmetry should not be considered as beauty criteria and the violation of computational formulas and asymmetry should not be introduced as inconsistency. Symmetric smile, which follows proved measures, is not necessarily aesthetic. However, measures, proportions, and symmetry are all aspects of beautiful art work but rarely suffice for attaining beauty phenomenon. Conclusions: In modern society, we insist on finding a form that follows determined rules. Medical and periodontics sciences can have rules in repair and reconstruction of the beauty and health by anatomic symmetric smile principles. Nonetheless, physicians may not be able to achieve beauty despite of applying most complete world standards. Countless individual differences in facial features lead to achieving countless outcomes in aesthetic proceedings. Eventually, individual’s kindness and personality can create a wonderful attraction while smiling that is unjustifiable.

Equivalence relations and symmetries for laboratory, LIDAR, and planetary Müeller matrix scattering geometries.

Symmetry relationships for optical observations of matter generally fall into several common scattering geometries. The "planetary" configuration is preferred by observers of extraterrestrial planets, "laboratory" observations are performed in the biomedical research field, and the LIDAR configuration is preferred by those using lasers to probe optical properties of horizontal surfaces with mirror or axial symmetry. This paper begins with the Stokes matrix formalism and uses symmetries of Müller matrix scattering to establish links among the mathematical symmetries of each geometric configuration. We finish the paper by identifying and correcting an influential misapplication of rotational scattering matrices in the literature. The corrected equation should find wide application in models of the LIDAR scattering process.