Notion Of Symmetry Research Articles

Hidden symmetries in real and theoretical networks

Symmetries are ubiquitous in real networks and often characterize network features and functions. Here we present a generalization of network symmetry called latent symmetry, which is an extension of the standard notion of symmetry on networks, which can be directed, weighted or both. They are defined in terms of standard symmetries in a reduced version of the network. One unique aspect of latent symmetries is that each one is associated with a size, which provides a way of discussing symmetries at multiple scales in a network. We are able to demonstrate a number of examples of networks (graphs) which contain latent symmetry, including a number of real networks. In numerical experiments, we show that latent symmetries are found more frequently in graphs built using preferential attachment, a standard model of network growth, when compared to non-network like (Erdős–Rényi) graphs. Finally we prove that if vertices in a network are latently symmetric, then they must have the same eigenvector centrality, similar to vertices which are symmetric in the standard sense. This suggests that the latent symmetries present in real-networks may serve the same structural and functional purpose standard symmetries do in these networks. We conclude from these facts and observations that latent symmetries are present in real networks and provide useful information about the network potentially beyond standard symmetries as they can appear at multiple scales.

Extended Riemannian Geometry I: Local Double Field Theory

We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical data and constraints. In special cases, we recover general relativity with and without 1-, 2- and 3-form gauge potentials as well as DFT. We believe that our extended Riemannian geometry helps to clarify the role of various constructions in DFT. For example, it leads to a covariant form of the strong section condition. Furthermore, it should provide a useful step towards global and coordinate invariant descriptions of T- and U-duality invariant field theories.

Pairings and related symmetry notions

In this paper, we give a purely mathematical generalization of an information table. We call pairing on a given set $$\Omega $$ a triple $$\mathfrak {P}=(U, F, \Lambda )$$ , where U and $$\Lambda $$ are non-empty sets and $$F:U\times \Omega \rightarrow \Lambda $$ is a map. We provide several examples of pairings: graphs, digraphs, metric spaces, group actions and vector spaces endowed with a bilinear form. Moreover, we reinterpret the usual notion of indiscernibility (with respect to a fixed attribute subset of an information table) in terms of local symmetry on U and, then, we study a global version of symmetry, that we called indistinguishability. In particular, we interpret the latter relation as the symmetrization of a pre-order $$\le _{\mathfrak {P}}$$ , that describes the symmetry transmission between subsets of $$\Omega $$ . Hence, we introduce a global average of symmetry transmission and studied it for some basic digraph families. Finally, we prove that the partial order of any finite lattice can be described in terms of the above pre-order.

The Road to Novelty

What characterises a new insight, what distinguishes ideas, when may we speak of inspiration, what induces the formation of new (general) concepts, and how do such concepts assume their contentual meaning? How does such novelty (hence the title, The Road to Novelty) come into the world and what constitutes its essence? How are the phenomena of meaning, insight and idea connected to information? And, as a limitation: What conditions must the nature of recognition and any approach to it meet so that our cognitive capacity is able to answer the preceding questions, i.e. so that it may recognise itself? This set of questions points towards the core topic of epistemology or, more generally, of cognitive science. The cycle The Road to Novelty offers a new and integrative scientific solution to these questions. The key to answering them is provided by a new category of information content, which we call CoIn. This new category derives from generalised symmetry considerations and complements the familiar Bits & Bytes. As our guiding idea, the CoIn provide the pivotal point of all reflections and results along The Road to Novelty. Using typical problems and providing their solutions, as well as on the basis of theoretical deliberations, we show that the phenomena of meaning, insight and ideas can be integratively traced back to this common informational root CoIn. The concurrence of insight, a new category of information content, idea and meaning thus also explains the interdisciplinary nature of The Road to Novelty. Knowing about this common informational root in turn enables a new foundation of epistemology, from which far-reaching conclusions may be drawn regarding questions that are central to a number of disciplines within cognitive science, including philosophy, linguistics and the logic of language, information theory together with artificial intelligence (AI), and up to the limits of formal approaches. Finally, the nature of the emergence of insights or ideas – how does such novelty come into the world? – also becomes addressable as what we call symmetry synthesis: The tree of knowledge is rooted in symmetry syntheses. As a consequence, both the meanings of general language concepts and scientific results such as (physical) laws or models manifest themselves in the form of generalised symmetries (CoIn). Bits & Bytes can likewise be interpreted in the symmetry context, namely as symmetry breaking. They serve the realm of decisions (symmetry breaking), which is complementary to recognition (to symmetry syntheses or to CoIn). As a result, both categories of information content are rooted in the notion of symmetry, yet they in a sense root at opposite ends of the concept. Together, symmetry syntheses and symmetry breaks as it were jointly provide the DNA building blocks for the informational perspective on the world. The fact that the two symmetry perspectives refer to incommensurable aspects of the world means that strong AI (creativity, cognitive faculty) cannot be realised solely by means of Turing machines. The reason is that Turing’s concept of a computer possesses only the competence of symmetry breaking. A proof of this is proposed in Section 7.2 of Part III. To facilitate the discipline-specific access to the material, as well as the analysis of the results, in Parts III, IV and V of The Road to Novelty, we relate the CoIn approach to the history of ideas of the discussions concerning recognition in philosophy and language theory, as well as in computer science / AI, and finally in logic / mathematics. In addition to the propositions already mentioned, the approach opens up many more in-depth insights into each of the sub-disciplines of cognitive science, extending to new aspects with respect to neuroscience and psychology. Last but not least, although economics is not commonly considered a part of cognitive science, this discipline assumes a special role in the argument. The model notions developed in economics enable the identification of a specific type of rationality as an essential characteristic of cognitive processes and the assignment of this rationality to an information-related economic principle.

Tensor-Centric Warfare II: Entropic Uncertainty Modeling

In the first paper of the tensor-centric warfare (TCW) series [1], we proposed a tensor model of combat generalizing earlier Lanchester-type systems with a particular emphasis on contemporary military thinking, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). In the present paper, we extend this initial tensor combat model with entropic Lie-derivative machinery in order to capture some aspects of this deep uncertainty, while, in the process, formalizing into our model military notion of symmetry and asymmetry in warfare as a commutator, also known as a Lie bracket. In doing so, we have sought to shift the question from the prediction of outcomes of combat, upon which previous combat models such as the Lanchester-type equations have been typically constructed, towards determining the uncertainty outcomes, using a rigorous analytical basis.

Magnonic topological insulators in antiferromagnets

Extending the notion of symmetry protected topological phases to insulating antiferromagnets (AFs) described in terms of opposite magnetic dipole moments associated with the magnetic $\mathrm{N}\stackrel{\ifmmode \acute{}\else \'{}\fi{}}{\mathrm{e}}\mathrm{el}$ order, we establish a bosonic counterpart of topological insulators in semiconductors. Making use of the Aharonov-Casher effect, induced by electric field gradients, we propose a magnonic analog of the quantum spin Hall effect (magnonic QSHE) for edge states that carry helical magnons. We show that such up and down magnons form the same Landau levels and perform cyclotron motion with the same frequency but propagate in opposite direction. The insulating AF becomes characterized by a topological ${\mathbb{Z}}_{2}$ number consisting of the Chern integer associated with each helical magnon edge state. Focusing on the topological Hall phase for magnons, we study bulk magnon effects such as magnonic spin, thermal, Nernst, and Ettinghausen effects, as well as the thermomagnetic properties of helical magnon transport both in topologically trivial and nontrivial bulk AFs and establish the magnonic Wiedemann-Franz law. We show that our predictions are within experimental reach with current device and measurement techniques.

Klein-Weyl's program and the ontology of gauge and quantum systems

We distinguish two orientations in Weyl's analysis of the fundamental role played by the notion of symmetry in physics, namely an orientation inspired by Klein's Erlangen program and a phenomenological-transcendental orientation. By privileging the former to the detriment of the latter, we sketch a group(oid)-theoretical program—that we call the Klein-Weyl program—for the interpretation of both gauge theories and quantum mechanics in a single conceptual framework. This program is based on Weyl's notion of a “structure-endowed entity” equipped with a “group of automorphisms”. First, we analyze what Weyl calls the “problem of relativity” in the frameworks provided by special relativity, general relativity, and Yang-Mills theories. We argue that both general relativity and Yang-Mills theories can be understood in terms of a localization of Klein's Erlangen program: while the latter describes the group-theoretical automorphisms of a single structure (such as homogenous geometries), local gauge symmetries and the corresponding gauge fields (Ehresmann connections) can be naturally understood in terms of the groupoid-theoretical isomorphisms in a family of identical structures. Second, we argue that quantum mechanics can be understood in terms of a linearization of Klein's Erlangen program. This stance leads us to an interpretation of the fact that quantum numbers are “indices characterizing representations of groups” ((Weyl, 1931a), p.xxi) in terms of a correspondence between the ontological categories of identity and determinateness.

Continuous point symmetries in group field theories

We discuss the notion of symmetries in non-local field theories characterized by integro-differential equations of motion, from a geometric perspective. We then focus on group field theory (GFT) models of quantum gravity and provide a general analysis of their continuous point symmetry transformations, including the generalized conservation laws following from them.

Symmetries, coverings, and decomposition of systems and trajectory generation

We derive relations between the notions of symmetry, covering, and decomposition of systems and trajectory generation. We show that any decomposition of a system determines a finite-dimensional covering of that system and is determined by it. We present conditions on vector fields under which any covering is obtained by factorization along the Lie algebra of such fields. On the basis of these relations, we study whether a point-to-point steering problem can be transformed into a set of boundary value problems of lower dimension.

Variational Tricomplex, Global Symmetries and Conservation Laws of Gauge Systems

Using the concept of variational tricomplex endowed with a presymplectic structure, we formulate the general notion of symmetry. We show that each generalized symmetry of a gauge system gives rise to a sequence of conservation laws that are represented by on-shell closed forms of various degrees. This extends the usual Noether's correspondence between global symmetries and conservation laws to the case of lower-degree conservation laws and not necessarily variational equations of motion. Finally, we equip the space of conservation laws of a given degree with a Lie bracket and establish a homomorphism of the resulting Lie algebra to the Lie algebra of global symmetries.

Spacetime and observer space symmetries in the language of Cartan geometry

We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly introduced notion of symmetry agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian, and Weizenböck geometries, which are conventionally used as spacetime models. Further, we discuss the case of Cartan geometries which can be used to model observer space instead of spacetime. We show which vector fields on an observer space can be interpreted as symmetry generators of an underlying spacetime manifold, and may hence be called “spatio-temporal.” We finally apply this construction to Finsler spacetimes and show that symmetry generating vector fields on a Finsler spacetime are indeed in a one-to-one correspondence with spatio-temporal vector fields on its observer space.

Some refinement of the notion of symmetry for the Volterra integral equations and the construction of symmetrical methods to solve them

The theory of integral calculations is employed in most fields of the natural sciences for computing the volumes of rotating bodies, areas with different shapes, distances between objects, and other applications. They can be used to explore energy signals to study earthquakes, the distribution of telecommunications signals, and other uses. In particular, many problems formulated as mathematical models use integral equations with variable boundaries. The most interesting applications use integrals where the boundaries are symmetric functions. In the present study, to solve the Volterra integral equations with variable boundaries, we propose multi-step methods such as hybrids, and we construct some simple algorithms for their application where the order of accuracy is p⩽8(k=1).

The Role of Symmetry in Mathematics

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of mathematics and the relationship of mathematics to physics.

Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian k-symplectic formalisms

In this paper, we will present Lagrangian and Hamiltonian [Formula: see text]-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for [Formula: see text]-symplectic Hamiltonian systems and [Formula: see text]-symplectic Lagrangian systems.

Integrated depth for functional data: statistical properties and consistency

Several depths suitable for infinite-dimensional functional data that are available in the literature are of the form of an integral of a finite-dimensional depth function. These functionals are characterized by projecting functions into low-dimensional spaces, taking finite-dimensional depths of the projected quantities, and finally integrating these projected marginal depths over a preset collection of projections. In this paper, a general class of integrated depths for functions is considered. Several depths for functional data proposed in the literature during the last decades are members of this general class. A comprehensive study of its most important theoretical properties, including measurability and consistency, is given. It is shown that many, but not all, properties of the integrated depth are shared with the finite-dimensional depth that constitutes its building block. Some pending measurability issues connected with all integrated depth functionals are resolved, a broad new notion of symmetry for functional data is proposed, and difficulties with respect to consistency results are identified. A general universal consistency result for the sample depth version, and for the generalized median, for integrated depth for functions is derived.

The symmetric structure of Green–Naghdi type equations

The notion of symmetry classically defined for hyperbolic systems of conservation laws is extended to the case of evolution equations of conservative form for which the flux function can bean operator. We explain how such a symmetrization can work from a general point of view using an extension of the classical Godunov structure. We then apply it to the Green–Naghdi type equations which are a dispersive extension of the hyperbolic shallow-water equations. In fact, in the case of these equations, the general Godunov structure of the system is obtained from its Hamiltonian structure.

SYMMETRIES IN MODAL LOGICS

AbstractIn this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [5]. Hence, the results apply to a wide class of modal logics including, for example, hybrid logics. We present two graph constructions that enable the reduction of symmetry detection in modal formulas to the graph automorphism detection problem, and we evaluate the graph constructions on modal benchmarks.

Divergence and the Price of Uncertainty

Realized divergence gauges the distinct realized moments associated with time-varying uncertainty and is tradeable with divergence swaps engineered from delta-hedged option portfolios. Consistently with established notions of symmetry in arbitrage-free option markets, implied divergence systematically decomposes the price of uncertainty into the contributions of distinct implied moments. Empirically, implied market divergence and market divergence premia vary substantially, in the time series, in the cross-section and in dependence of the investment horizon, making the shortcomings of a model easily visible.

The role of symmetry in the interpretation of physical theories

The symmetries of a physical theory are often associated with two things: conservation laws (via e.g. Noether׳s and Schur׳s theorems) and representational redundancies (“gauge symmetry”). But how can a physical theory׳s symmetries give rise to interesting (in the sense of non-trivial) conservation laws, if symmetries are transformations that correspond to no genuine physical difference? In this paper, I argue for a disambiguation in the notion of symmetry. The central distinction is between what I call “analytic” and “synthetic“ symmetries, so called because of an analogy with analytic and synthetic propositions. “Analytic“ symmetries are the turning of idle wheels in a theory׳s formalism, and correspond to no physical change; “synthetic“ symmetries cover all the rest. I argue that analytic symmetries are distinguished because they act as fixed points or constraints in any interpretation of a theory, and as such are akin to Poincaré׳s conventions or Reichenbach׳s ‘axioms of co-ordination’, or ‘relativized constitutive a priori principles’.

Formula omitted]-symmetric Hamilton cycle decompositions of graphs

The existence of symmetric Hamilton cycle decompositions for complete graphs and cocktail party graphs has been defined and explored in recent work by Akiyama et al., Brualdi and Schroeder, and others. In these works, the notion of symmetry in cocktail party graphs K2m−F was integrally tied to the missing 1-factor. In this paper, we generalize the notion of symmetric decompositions in two ways. First, we require only that F is symmetric and show that if F is not the invariant 1-factor under the symmetry action, then K2m−F has a symmetric Hamilton cycle decomposition for every m≥2. Second, we consider other actions as symmetry, apply such definitions to appropriate complete graphs and complete multipartite graphs, and classify the existence of Hamilton cycle decompositions with such symmetry.