Symmetry Groups

In this work we explore two speculative approaches to the unification of fundamental physics: Quantum Diffusion, based on the hypothesis of a discrete spacetime at the Planck scale where matter-energy diffuses through nodes, and Hidden Dimensions, proposed in string and superstring theories, where additional non-observable dimensions sustain the mathematical coherence of the universe. We perform a comparative analysis of both conceptual frameworks, highlighting their advantages, limitations, and possible points of convergence. We propose that Quantum Diffusion and Hidden Dimensions could be interpreted as complementary descriptions of the same underlying physical reality.

The paper contains a complete theory of factors for ray representations acting in a Hilbert bundle, which is a generalization of the known Bargmann's theory. With the help of it we have reformulated the standard quantum theory such that the gauge freedom emerges naturally from the very nature of quantum laws. The theory is of primary importance in the investigations of covariance (in contradistinction to symmetry) of a quantum theory which possesses a nontrivial gauge freedom. In that case the group in question is not any symmetry group but it is a covariance group only -that case which has not been deeply investigated. It is shown in the paper that the factor of its representation depends on space and time when the system in question possesses a gauge freedom. In the nonrelativistic theories the factor depends on the time only. In the relativistic theory the Hilbert bundle is over the spacetime and in the nonrelativistic one it is over the time. We explain two applications of this generalization: in the theory of a quantum particle in the gravitational field in the nonrelativistic limit and in the quantum electrodynamics.

Symmetry breaking is crucial in many areas of physics, mathematics, biology, and engineering. We investigate the symmetry of regular convex polygons, non-convex regular polygons (stars), and symmetric Jordan curves/domains. We demonstrate that removing a single point from the boundary of regular convex and non-convex polygons and symmetrical Jordan curves reduces the symmetry group of the polygon to the trivial C 1 group when the point does not belong to the axis of symmetry of the polygon. The same is true for solid and open 2D regular convex polygons and symmetric Jordan curves. The only exception is a circle. Removing a single point from the boundary of a circle creates a curve characterized by the C 2 group. The symmetry of circles is reduced to the trivial C 1 group by removing a triad of non-symmetrical points. The same is true for a solid circle. The "effort" necessary to break the symmetry of a circle is maximal. A 3D generalization of the theorem is exemplified. Thus, the classification of symmetrical curves following the minimal number of points necessary to break their symmetry becomes possible. The demonstrated theorem shows that the symmetry group action on curves and domains becomes trivial when an asymmetric perturbation is introduced, when the curve is not a circle. An informational interpretation of the demonstrated theorem, which is related to the Landauer principle, is provided.

Both necessary and sufficient conditions for the existence of two complementary-dual extremum principles for geometrically exact finite strain (one-dimensional) beam models are investigated by means of two different approaches. One is based on the results published by Gao and Strang, and the other relies on the approach proposed by Noble and Sewell. While the former is limited to beam models restricted to moderate large deformations, the latter is valid for arbitrarily large deformations (and strains). The numerical implementation of the complementary-dual extremum principles can lead to simple true global upper bounds of the error of the approximate solutions.

The coupled motion, between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics, is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian Particle Path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two-and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two-and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani, Bridges & Turner [1] (J. Fluid Struct. 59 432-460) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.

numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed a puzzle in the decay rates of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed 'splitting'. We discuss far-field splitting in the full field and near-horizon splitting in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the splitting behavior. The far-field splitting is explained by competition between projected modes. The near-horizon splitting is due to excitation of lower multipole modes that back excite the multipole mode for which splitting is observed. In both cases splitting is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution.

The standard topological censorship theorems require asymptotic hypotheses which are too restrictive for several situations of interest. In this paper we prove a version of topological censorship under significantly weaker conditions, compatible, e.g., with solutions with Kaluza-Klein asymptotic behavior. In particular we prove simple connectedness of the quotient of the domain of outer communications by the group of symmetries for models which are asymptotically flat, or asymptotically anti-de Sitter, in a Kaluza-Klein sense. This allows one, e.g., to define the twist potentials needed for the reduction of the field equations in uniqueness theorems. Finally, the methods used to prove the above are used to show that weakly trapped compact surfaces cannot be seen from Scri.

According to general relativity, the interaction of a matter field with gravitation requires the simultaneous introduction of a tetrad field, which is a field related to translations, and a spin connection, which is a field assuming values in the Lie algebra of the Lorentz group. These two fields, however, are not independent. By analyzing the constraint between them, it is concluded that the relevant local symmetry group behind general relativity is provided by the Lorentz group. Furthermore, it is shown that the minimal coupling prescription obtained from the Lorentz covariant derivative coincides exactly with the usual coupling prescription of general relativity. Instead of the tetrad, therefore, the spin connection is to be considered as the fundamental field representing gravitation.

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.

This paper gives a few new developments in mechanics, as well as some remarks of a historical nature. To keep the discussion focussed, most of the paper is confined to equations of "rigid body", or "hydrodynamic" type on Lie algebras or their duals. In particular, we will develop the variational structure of these equations and will relate it to the standard variational principle of Hamilton. Even this small area of mechanics is fascinating from the historical point of view. In fact, it is quite surprising how long it can sometimes take for fundamental results of the masters to be tied together and to filter into the main literature and to become "well-known". In particular, part of our story follows a few fragments of a thread through the works of Euler, Lagrange, Lie, Poincaré, Clebsch, Ehrenfest, Hamel, Arnold, and many others. Although Newton's discoveries were directly motivated by planetary motion, the realm of mechanics expanded well beyond particle mechanics with the work of Euler, Lagrange, and others to include fluid and solid mechanics. Today we see its methods permeating large areas of physical phenomena besides these, including electromagnetism, plasma physics, classical field theories, general relativity, and quantum mechanics. Part of what makes this unified point of view possible is the abstraction, often in a geometric way, of the underlying structures in mechanics. Two general points of view emerged early on concerning the basic structures in mechanics. One, which is commonly referred to as "Lagrangian mechanics" can be based in variational principles, and the other, "Hamiltonian mechanics", rests on symplectic and Poisson geometry. As we shall see shortly, the history of this development is actually quite complex. How rigid body mechanics, fluid mechanics and their generalizations fit into this story is quite interesting because of the way their equations fit into the schemes of Lagrange and Hamilton. For example, the way the equations are normally presented (in body representation for the rigid body, and in spatial representation for

Symmetries of input and latent vectors have provided valuable insights for disentanglement learning in VAEs. However, only a few works were proposed as an unsupervised method, and even these works require known factor information in the training data. We propose a novel method, Composite Factor-Aligned Symmetry Learning (CFASL), which is integrated into VAEs for learning symmetry-based disentanglement in unsupervised learning without any knowledge of the dataset factor information. CFASL incorporates three novel features for learning symmetry-based disentanglement: 1) Injecting inductive bias to align latent vector dimensions to factor-aligned symmetries within an explicit learnable symmetry codebook 2) Learning a composite symmetry to express unknown factors change between two random samples by learning factor-aligned symmetries within the codebook 3) Inducing a
group equivariant encoder and decoder in training VAEs with the two conditions. In addition, we propose an extended evaluation metric for multi-factor changes in comparison to disentanglement evaluation in VAEs. In quantitative and in-depth qualitative analysis, CFASL demonstrates a significant improvement of disentanglement in single-factor change, and multi-factor change conditions compared to state-of-the-art methods.

This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov-Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next papers. The importance of the work can be situated in fields such as cosmology, quantum gravity and -for the mathematicians -global Lorentzian geometry.

This paper addresses the research problem of the characterization of a class of tilings or tessellations called k-isogonal tilings in the Euclidean and hyperbolic plane. Tilings that satisfy the property of having transitivity classes or orbits of vertices under the action of their respective symmetry groups are called k-isogonal tilings. We contribute to answering the research problem by giving various constructions of k-isogonal tilings carried out with the aid of the dynamic geometry software GeoGebra.

1. Preliminaries. We consider a smooth pseudo-Riemannian manifold (M, 〈·, ·〉), with signature (r, s). For simplicity, we suppose the manifoldM is connected, the dimension ofM is n. First we need to consider the following question: We are given a smooth manifold M . Under which conditions does there even exist a smooth nondegenerate metric field of signature (r, s) on M? Lemma 1. The following statements are equivalent:

Symmetrization of a heat conduction model for a rigid medium W. DOMANSKI, T. F. JABLONSKI and W. KOSTNSKI (WARSZAWA) THE SYMME.IRIZATION o f the equations of a heat conduction model fo r a rigid medium in time and three space dimensions is petformed. The general symmctr izability conditio n is formulated in terms o f the entropy functio n. Examples o f particular moc.lcls (e.g. Dcbyc's model) are discussed.

An algorithmic method is presented for determining all domain configurations and their symmetries in domain average engineered structures. This method is applied to PZN-PT single crystals by determining the domain configurations of domain states which arise in a phase transition between G = m3m and F = 3m.

This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.

- - Key features:

-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.

-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.

-> Focuses on the clear presentation of the mathematical notions and calculational technique.


- - - Table of Contents:

- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.

- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.

- - Bibliography.
- - Index.

This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.

- - Key features:

-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.

-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.

-> Focuses on the clear presentation of the mathematical notions and calculational technique.


- - - Table of Contents:

- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.

- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.

- - Bibliography.
- - Index.

This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.

- - Key features:

-> Contains a modern, streamlined presentation of classical topics, which are normally taught separately.

-> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity.

-> Focuses on the clear presentation of the mathematical notions and calculational technique.


- - - Table of Contents:

- Chapter 1. Manifolds and Tensors.
- Chapter 2. Geometry and Integration on Manifolds.
- Chapter 3. Symmetries of Manifolds.
- Chapter 4. Newtonian Mechanics.
- Chapter 5. Lagrangian Methods and Symmetry.
- Chapter 6. Relativistic Mechanics.
- Chapter 7. Lie Groups.
- Chapter 8. Lie Algebras.
- Chapter 9. Representations.
- Chapter 10. Rotations and Euclidean Symmetry.
- Chapter 11. Boosts and Galilei Symmetry.
- Chapter 12. Lorentz Symmetry.
- Chapter 13. Poincare Symmetry.
- Chapter 14. Conformal Symmetry.
- Chapter 15. Lagrangians and Noether's Theorem.
- Chapter 16. Spacetime Symmetries of Fields.
- Chapter 17. Gauge Symmetry.
- Chapter 18. Connection and Geodesics.
- Chapter 19. Riemannian Curvature.
- Chapter 20. Symmetries of Riemannian Manifolds.
- Chapter 21. Einstein's Gravitation.
- Chapter 22. Lagrangian Formulation.
- Chapter 23. Conservation Laws and Further Symmetries.

- - Appendices
- A) Notation and Conventions.
- Physical Units and Dimensions.
- Mathematical Conventions.
- Abbreviations.
- B) Mathematical Tools.
- Tensor Algebra.
- Matrix Exponential.
- Pauli and Dirac Matrices.
- Dirac Delta Distribution.
- Poisson and Wave Equation.
- Variational Calculus.
- Volume Element and Hyperspheres.
- Hypersurface Elements.
- C) Weyl Rescaling Formulae.
- D) Spaces and Symmetry Groups.

- - Bibliography.
- - Index.

İÇ MİMARİ YÜZEY TASARIMINDA SİMETRİ ALGORİTMALARININ KULLANIMINA YÖNELİK BİR MODEL ÖNERİSİ

ÖZET
Doğada biçimlenme yöntemleri sayılamayacak kadar çeşitlidir. Bu çeşitlilikte verim, tasarruf ve yarar olguları ile doğrudan ilişkili olarak görev alan simetri, fiziksel çevrelerdeki tüm etkinliklerin temelindedir. İnsan yapısı çevrelerin inşasında uyum ve düzen sağlamak amacıyla simetriyi doğada oluşturan yollar taklit edilmiş; simetriye duyulan bu ilgi, tasarımı görsel, anlamsal ya da işlevsel düzeylerde ilgilendirebilecek çeşitli doğaya öykünme modelleri ortaya çıkarmıştır. Bu biyomimetik süreçler kimi zaman estetik, kimi zaman yapısal beklentilerle şekillenmiş; simetrinin kompaktlık getirileri ile yapısal ve üretimsel faydalar da sağlanmıştır.
Günümüzde tasarım süreçlerinde kullanılan kimi teknolojik araçlar, simetrinin doğadaki en az enerji ile en çok verim sağlayabilme amacını insan yapısı çevreler için yeniden düşünebilme imkanı sunmaktadır. Simetrinin doğadaki ihtişamlı örneklerinden kar tanelerinde su moleküllerinin sayısız çeşitlilikteki olasılıkların kısa yollarını takip etmesi ya da arıların en tasarruflu ve kompakt yöntemlerle bal peteklerini inşa etmesi gibi simetriye ilişkin değerler yapay dünyamızı şekillendiren tasarımlar içinde tasarruf potansiyeli taşıyabilir. Doğal ve yapay olan arasında bu bağlamda kurulan bir bağdan ve bu bilinçle doğadan edinilen bilgileri tasarım eylemiyle yeniden doğa ile ilişkili tutma düşüncesinden hareketle; mekânsal bütünlükleri algılamamızı sağlayan yüzeyler, simetri ışığında verimsel beklentilerle yeniden değerlendirilebilir.
Çalışma, simetrinin doğa, tasarımcı ve tasarım arasındaki döngü için olası yöntemsel katkılarını araştırma ve doğal modülerliğin malzeme, enerji ve zaman tasarrufu etkenlerini iç mimari yüzey tasarım süreci ile örtüştürme amaçları üzerine inşa
xxii
edilmiştir. Çalışmada, doğa bilimleri, sanat ve matematik gibi çeşitli alanların kesişimindeki simetri kavramına dair olan ve bu alanlardan derlenen nitel ve nicel bulgulara ve kavramın estetik ve pragmatik doğasını irdeleyen çeşitli dünya görüşlerine yer verilmiştir. Pek çoğu sanat ve bilim arasında geçişkenlik taşıyan söz konusu mevcut bulgular üzerine, araştırmaları iç mimarlık odağında değerlendirmeye olanak tanıyacak unsurlar derlenmiş, iç mimari yüzey tasarımları için sistematik bir yöntem tasarımı önerilmiştir. Geliştirilen algoritma tabanlı tasarım süreci için modül ve yüzey tasarımında verim beklentisini karşılamak adına, simetri grup kuramını temel alan tariflerin yer aldığı, dijital ve fiziksel ortamda üretim yollarını araştıran deneysel bir yöntem kurgulanmıştır. Doğa ile ilişkili bu kuramsal modelin tasarım sürecine adaptasyonu için biçim üretiminde sayısal değerler aracılığıyla denetlenebilir kalıtsal ilişkiler planlamaya olanak tanıyan üretken tasarım aygıtları kullanılmıştır. Üretken tasarım yöntemi, simetriye ve doğadaki modülerliğe dair analojik basitleştirmelere olanak tanımış ve simetrik etkinliklerin doğadakine benzer kalıtsal benzerlikler yoluyla değerlendirilebilmesini sağlamıştır.
Çalışma, simetrinin getirileri ile az sayıda işlemle daha fazla yöntem dizini oluşturmak, az enerji ile daha fazla sonuç elde edebilmek, az sayıda prototip ile daha fazla anlamlı bütün üretebilmek gibi verimsel kaygılara bağlıdır. Bu bağlamda simetrinin noktasal olasılıklarından yola çıkarak, doğrultusal ve düzlemsel düzenlilik belirtebilecek tüm etkinlik şablonları algoritma tabanlı tasarım yöntemiyle yeniden kurgulanmış, ikinci boyutta ifade edilebilecek bu şablonlar sonraki adımda mekânsal projeksiyonlara dönüştürülmüştür. İç mimari yüzey tasarım ölçeğinde kullanılabilir bu sayısal parametreler aracılığıyla 418 adet modüler yüzey tasarımı üretilmiş ve dijital ortamda tasnif edilmiştir. Doğanın fiziksel sınırlarını pragmatik anlamda temel alarak tasarlanan algoritmayla eşbiçimli modülerlik içeren farklı mekânsal dizgelerde de kullanılabilecek sistematik bir şablon üretilmiş; doğal ve yapay yüzeylerin zaman ve maliyet verimliliği çerçevesinde benzer roller üstlenebileceği sonucuna varılmıştır.

Anahtar Kelimeler: İç Mimari Yüzey Tasarımı, Algoritma Destekli Tasarım, Üretken Tasarım, Döşeme Örüntüleri, Simetri Grupları, Doğa ve Modülerlik

A MODEL PROPOSAL FOR THE USAGE OF THE ALGORITHMS OF SYMMETRY ON THE INTERIOR SURFACE DESIGN
SUMMARY
Formation methods of nature are boundless. In this abundance, the symmetry, which is directly associated with cost and time efficiency, lies behind all activities in physical environments. Since the time immemorial, the symmetry, which has been counted as a factor of harmony and order in human made environments, as a visual, spiritual and functional tool, has been seen as a connection path between the designer and the nature. Biomimetic processes in which symmetry involves, have been shaped by aesthetic and structural expectations; providing compactness and productive contributions to the design processes beside its visual and semantic benefits.
The technological tools used in design practice today offer opportunities for designers to review the function of symmetry to provide more efficiency with less energy in order to build new methods for human-made environments. Correlatively with the water molecules following the shortest paths of numerous varieties of snowflakes or the honeycombs built by the bees with the most time and material efficient methods, the elements related with symmetry like compactness and modularity also may have saving potential for our artificial world. Based on the connection established within this framework between the natural and the artificial, the surfaces that enable us to perceive the spatial integrities may also be evaluated under the concepts of symmetry and efficiency with the aim of accommodating the design methods with the natural models in order to develop new point of view.
This study is based on the aims of detecting the possible methodological contributions of the symmetry and adopting the efficiency goals of the nature in favor of building a new design methodology for interior surface design. In the study, both qualitative and quantitative findings from distinct studies mainly from the fields of natural sciences, art, philosophy and mathematics emphasizing the relationship between symmetry, form and efficiency selected with the aim of building a systematical method for interior surface design. An experimental method, which includes the descriptions based on symmetry group theory, has been set up, to investigate production methods in both digital and physical environments in order to fulfil the efficiency criteria for the modularization of the surface with an algorithm aided design process. For the adaptation of this nature-related theoretical model to a surface design method, generative design tools that allow controlling genetic relationships through numerical values were used for form finding. The generative design method allowed geometrical simplifications of modularity and enabled the symmetrical activities of artificial surfaces to be evaluated through their numerical similarities which operates analogously with nature; thus, a total of 418 modular surface designs have been generated and classified in digital environment.
All the two-dimensional and regular activities of symmetry were tested in the context of interior architectural surfaces, depending on cost and time efficiency concerns such as creating more outputs with less number of operations, obtaining wide range of result with less energy, producing low cost physical integrities with less number of prototypes with the benefits of the symmetry. By means of the algorithm that designed, a systematic template which can be applied in different spatial systems containing isomorphic modularity was created in order to use the natural boundaries as a pragmatic tool and it has been concluded that natural and artificial surfaces can play similar roles in terms of cost and time efficiency.

Keywords: Interior Surface Design, Algorithms-Aided Design, Generative Design, Tiling, Symmetry Group, Nature and Modularity

F.Gramain [7] presente the situation in 1988 an attempt of the same kind: how to show by a method of transcendence, a result obtained in 1933 by A.O.Guelfond on entire functions taking integer values in all points of a geometric progression, like the previous multiplicative additive problem (see[6] theorem VIII). On the other hand, the same type of results for entire functions of several variables were obtained by P.Bundschuh1980 and J.P.Bézivin 1983 the first uses Newton interpolation series in several variables and the second linear récurentes suites. J.P.Bézivin[1] in 1984 studied a multivariate generalization of a result the Guelfond. Tanguy Rivoal and

o que?

Vetores no plano do ponto de vista geométrico (segmentos orientados) e algébrico (coordenadas); operações
com vetores (adição, subtração e multiplicação por escalar) e suas interpretações geométricas (translação
e homotetia); algumas aplicações na Física.

Por que?

Vetores são os objetos matemáticos adequados para se descrever, de modo apropriado, certas grandezas
(ditas vetoriais) tais como deslocamento, velocidade, aceleração e força. Mais ainda, muitas leis naturais
são expressas por equações que envolvem vetores e estudar tais equações permite entender os fenômenos
associados. Além disso, o material deste capítulo fornece um primeiro contato, mais simples, com uma
disciplina universitária denominada Álgebra Linear, que possui várias aplicações em Biologia, Economia,
Computação Gráfica, Engenharia, Física, Matemática, Química, Estatística, etc.

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler-Poincaré and Hamilton-Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton's principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler-Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange-Poincaré equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form. Contents David C. P. Ellis et al.

This paper looks at representing a group G as a group of transformations of an orientable compact bordered Klein surface. We construct visual representations (portraits) of three groups S4, Z2  S3 and a group L of order 32. These groups have real genus 3, 2 and 5 respectively. The first two groups are Mgroups; which means that they act on surfaces with maximal symmetry. They are also the only solvable M simple groups. 1. Mathematical Background The purpose of this paper is to construct visual representations (portraits) of finite groups on a compact orientable surface with one or more boundary components. A bordered Klein surface is a compact surface with one or more boundary components. The boundary components occur because an open disk is deleted from a surface. These will be referred to as holes. This should be distinguished from the tunnel that is found in a torus. The surface that results after filling in all of the holes has the same topological genus as the bordered Klein su...

Traditional clothing is a cultural heritage that should be preserved and expanded upon. Traditional fabrics that were once only used at traditional events are now catching the attention of the fashion industry. As a result, various innovations are developed to help traditional fabrics sell well in today's market. The concept of crystallographic groups is one of the innovations that can be used to develop motifs or patterns. Crystallographic groups are a mathematical classification of two-dimensional repeating patterns that are based on symmetry. Patterns like this are common in architecture and decorative arts, particularly in textiles, tiles, and wallpaper. In this study, the authors generated North Sumatran songket motifs based on 11 crystallographic group patterns using the MATLAB GUI program. The findings of this study can be used as a resource for art activists developing fabric motifs in the future.

Traditional clothing is a cultural heritage that should be preserved and expanded upon. Traditional fabrics that were once only used at traditional events are now catching the attention of the fashion industry. As a result, various innovations are developed to help traditional fabrics sell well in today's market. The concept of crystallographic groups is one of the innovations that can be used to develop motifs or patterns. Crystallographic groups are a mathematical classification of two-dimensional repeating patterns that are based on symmetry. Patterns like this are common in architecture and decorative arts, particularly in textiles, tiles, and wallpaper. In this study, the authors generated North Sumatran songket motifs based on 11 crystallographic group patterns using the MATLAB GUI program. The findings of this study can be used as a resource for art activists developing fabric motifs in the future.

The pattern of duality symmetries acting on the states of compactified superstring models reinforces an earlier suggestion that the Monster sporadic group is a hidden symmetry for superstring models. This in turn points to a supersymmetric theory of self-dual and anti-self-dual K3 manifolds joined by Dirac strings and evolving in a 13 dimensional spacetime as the fundamental theory. In addition to the usual graviton and dilaton this theory contains matter-like degrees of freedom resembling the massless states of the heterotic string, thus providing a completely geometric interpretation for ordinary matter.

In this paper we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) AdS setting.

In this paper, we discuss a method of arriving at colored three-dimensional uniform honeycombs. In particular, we present the construction of perfect and semi-perfect colorings of the truncated and bitruncated cubic honeycombs. If G is the symmetry group of an uncolored honeycomb, a coloring of the honeycomb is perfect if the group H consisting of elements that permute the colors of the given coloring is G. If H is such that [G : H] = 2, we say that the coloring of the honeycomb is semi-perfect.

In this study, a closed motion given in R 3 are considered in the dual space D 3. Thus this motion is generalized to the D 3. During this motion, some relations among the dual integral invariants of the closed ruled surfaces are given. In addition, separating real and dual parts of these dual integral invariants, some results and some theorems related to Holditch Theorem are given in the line space R 3 by Study's mapping.

Supercharacter theory is developed by P. Diaconis and I. M. Isaacs as a natural generalization of the classical ordinary character theory. Some classical sums of number theory appear as supercharacters which are obtained by the action of certain subgroups of GL d (Zn) on Z d n. In this paper we take Z d p , p prime, and by the action of certain subgroups of GL d (Zp) we find supercharacter table of Z d p .

Supercharacter theory is developed by P. Diaconis and I. M. Isaacs as a natural generalization of the classical ordinary character theory. Some classical sums of number theory appear as supercharacters which are obtained by the action of certain subgroups of GL d (Zn) on Z d n. In this paper we take Z d p , p prime, and by the action of certain subgroups of GL d (Zp) we find supercharacter table of Z d p .

‎Supercharacter theory is developed by P‎. ‎Diaconis and I‎. ‎M‎. ‎Isaacs as a natural generalization of the classical ordinary character theory‎. ‎Some classical sums of number theory appear as supercharacters which are obtained by the action of certain subgroups of GL_d(Z_n) on Z_n^d‎. ‎In this paper we take Z_p^d‎, ‎p prime‎, ‎and by the action of certain subgroups of GL_d(Z_p) we find supercharacter table of Z_p^d‎.

The classification of the dihedral folding tessellations of the sphere and the plane whose prototiles are a kite and an equilateral triangle were obtained in [1]. Recently, this classification was extended to isosceles triangles so that the classification of spherical folding tesselations by kites and isosceles triangles in three cases of adjacency was presented in [2, 3, 4]. In this paper we finalize this classification presenting all the dihedral folding tessellations of the sphere by kites and isosceles triangles in the remaining three cases of adjacency, that consists of five sporadic isolated tilings. A list containing these tilings including its combinatorial structure is presented in Table 1.

We prove that there is a unique folding tessellation of the sphere and an infinite family of folding tessellations of the plane with prototiles a kite and an equilateral triangle. Each tiling of this family is obtained by successive gluing of two patterns composed of triangles and kites, respectively. The combinatorial structure and the symmetry group is achieved.

The study of the dihedral f-tilings of the sphere S 2 whose prototiles are a scalene triangle and an isosceles trapezoid was initiated in a previous work. In this paper we continue this classification presenting the study of all dihedral spherical f-tilings by scalene triangles and isosceles trapezoids in some cases of adjacency.

The study of dihedral f-tilings of the sphere S 2 by spherical triangles and equiangular spherical quadrangles (which includes the case of 4-sided regular polygons) was presented in [3]. Also, in [6], the study of dihedral f-tilings of S 2 whose prototiles are an equilateral triangle (a 3-sided regular polygon) and an isosceles triangle was described (we believe that the analysis considering scalene triangles as the prototiles will lead to a wide family of f-tilings). In this paper we extend these results, presenting the study of dihedral f-tilings by spherical triangles and r-sided regular polygons, for any r ≥ 5. The combinatorial structure, including the symmetry group of each tiling, is given in Table 1.

In this paper we present some spherical f-tilings by two (distinct) right triangles. We classify the group of symmetries of the presented tilings and the transitivity classes of isohedrality are also determined. The combinatorial structure is given in Table 1.

Wachspress quadrilateral patches have been recently studied from the point of view of applications to surface modelling in CAGD [1], [3], [4]. Some more applications for defining barycentric coordinates for arbitrary polygons have also been presented in [5] [9]. The purpose of the present paper is to introduce non‐negative Wachspress rational basis functions for surface modelling on pentagonal partitions. Interpolation formula for function values and directional derivatives at the vertices of pentagon has been presented. Conditions for C1– continuity of the composite surface have also been studied in the paper. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

It is well known that the Einstein equation on a Riemannian flag manifold (G/K, g) reduces to an algebraic system if g is a G-invariant metric. In this paper we obtain explicitly new invariant Einstein metrics on generalized flag manifolds of Sp(n) and SO(2n); and we compute the Einstein system for generalized flag manifolds of type Sp(n). We also consider the isometric problem for these Einstein metrics.

In this paper, we review and extend some results in the literature pertaining to spacetime topology while naturally utilizing properties of the codimension 2 null cut locus. Our results fall into two classes, depending on whether or not one assumes the presence of horizons. Included among the spacetimes we consider are those that apply to the asymptotically (locally) anti-de sitter (AdS) setting.