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In Noether's original presentation of her celebrated theorem of 1918, allowances were made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to only point transformations. In recent decades, this diminution of the power of Noether's Theorem has been partly countered, in particular, in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether's Theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look at the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables.
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Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group.
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This reprint is dedicated to Professor Delfim F. M. Torres for his 50th birthday. Professor Delfim F. M. Torres is a recognized researcher, particularly in the calculus of variations, optimal control, and mathematical biology. In this reprint, several studies in his research fields are presented.
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The Highly Specialized Seminar on "Symmetries in Nuclear Structure", held in Erice, Italy, in March 2003, celebrated the career and the remarkable achievements of Francesco Iachello, on the occasion of his 60th birthday. Since the development of the interacting boson model in the early 1970's, the ideas of Iachello have provided a variety of frameworks for understanding collective behaviour in nuclear structure, founded on the concepts of dynamical symmetries and spectrum-generating algebras. The original ideas, which were developed for the description of atomic nuclei, have now been successful...
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The Max Born Symposia started in 1991 and have been held once or twice a year in different places in Lower Silesia. The scientific topics of the Symposia are closely related with front-line research subjects in theoretical physics. This volume deals with new concepts of symmetries in the theory of fundamental intersections as well as integrable dynamical systems.
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The reprint is devoted to the phenomena associated with exact or approximate axial symmetry in different areas of technical physics and mechanical engineering science. How can the symmetry of the problem be used most efficiently for its analysis? Why is the symmetry broken or why is it still approximately retained? These and other questions are discussed based on systems from different fields of engineering.
Symmetry --- History. --- Aesthetics --- Proportion
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Seeking Symmetry: Finding patterns in human health offers a guide through the overwhelming mass of data generated by contemporary science. Starved for the knowledge that would best help us stay healthy, we are simultaneously glutted with an overload of information about the human body. Amidst ubiquitous talk that patient-centered care and lifestyle changes are the keys to personal health, self-neglect and medical overtreatment nevertheless prevail.The body is rich with symmetries, many of them unknown to us who live in these bodies. Symmetry-seeking reveals certain patterns for understanding the information we have about the body, patterns whose roots lie in embryonic development and in evolution.The book's exploration will guide readers through the parts of their own bodies and introduce tangible, visible examples of symmetry, not only right and left but up and down, male and female, inside and out, as well as symmetries between humans and other species.It presents the symmetries of the body's internal structures that, despite their complexity, are nevertheless simple to understand when viewed with an eye for pattern.Through both words and images, this book will illustrate the most foundational of the principles, structures, and processes that decide how bodies function.A core purpose of the book is to present this knowledge through a lens that makes the information meaningful, by modeling the habit of symmetry-seeking.
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This monograph is a synthesis of the theory of the pairwise coupling of the angular momenta of arbitrarily many independent systems to the total angular momentum in which the universal role of doubly stochastic matrices and their quantum-mechanical probabilistic interpretation is a major theme. A uniform viewpoint is presented based on the structure of binary trees. This includes a systematic method for the evaluation of all 3n-j coefficients and their relationship to cubic graphs. A number of topical subjects that emerge naturally are also developed, such as the algebra of permutation matrice
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Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. ""Dynamical Symmetry"" introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in
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