Notion Of Symmetry Research Articles

Dancing with math: Using Kleins quartic for music generation

Studying music from a mathematical perspective is often based on the notions of symmetry and topological space. A group G acting on some set S of musical objects in a meaningful way is seen as a symmetry group in music. This can be used in analysing motif development and chord progressions. In neo-Riemannian analysis, one has three principal chord transformations that generate a group GD_12, acting on the set S of 24 major and minor triads. Moreover, this theory is visualized through a simplicial complex whose underlying space is a topological torus. In this paper, we first introduce various symmetry groups and graphic presentations in music theory. We then propose a way of doing neo-Riemannian analysis on Kleins quartic, which is a genus 3 surface instead of a torus, realizing an idea of John Baez. Finally, we use our theory to perform a harmonic analysis of The Imperial March from Star Wars.

Anomaly enforced gaplessness for background flux anomalies and symmetry fractionalization

Anomalous symmetries are known to strongly constrain the possible IR behavior along any renormalization group (RG) flow. Recently, the extension of the notion of symmetry in QFT has provided new types of anomalies with a corresponding new class of constraints on RG flows. In this paper, we derive the constraints imposed on RG flows from anomalies that can only be activated in the presence of specific background fluxes even though they do not necessarily correspond to a symmetry. We show that such anomalies can only be matched by gapped theories that exhibit either spontaneous symmetry breaking or symmetry fractionalization. In addition, we exhibit previously unstudied examples of these flux background anomalies that arise in 4d QCD and 4d SUSY QCD.

Symmetries of differential delay systems with applications to observer design

AbstractIn this paper we construct a general observer theory for differential delay systems based on different types of symmetries (exact symmetry or asymptotic symmetry), ending up with a certain number of semi‐global and global observers, with bounded or unbounded system's solutions. We introduce the notions of symmetry for a differential delay system, being inspired by well‐known definitions of symmetry for an ordinary or partial differential system, and variational symmetry for the associated variational differential delay system. We illustrate observer design procedures in details, by proving that the existence of a (variational asymptotic) symmetry with system's detectability in the first approximation have a central role in the design of a state observer. The symmetry is a one‐parameter group of transformations which maps the system into itself (exact symmetry) or into a different system (asymptotic symmetry), approximating the original one with better and better accuracy as the parameter of the symmetry is larger. The types of symmetries we consider here show an important contractive action on the state and input spaces for which the system's solutions are mapped into arbitrarily small neighbourhoods of the origin in which the transformed system can be well‐approximated by its linearization. The parameter of the symmetry may be constant (semiglobal observers) or updated on‐line by a state norm estimator (global observers).

Spontaneous symmetry breaking as a law of nature

In a semi-review paper, it was discussed the notion of symmetry of polyatomic systems defined as invariance under transformations, and show that this important property of atomic matter is extremely vulnerable, and may undergo internal breakdown, subject to the presence of electronic degeneracy or pseudodegeneracy. First formulated by Landau, L. in 1934, later proved and published by Jahn and Teller, this Jahn-Teller effect (JTE) underwent tremendous developments with important applications in physics, chemistry, biology, and materials science. Less attention was paid to the roots of this phenomenon and its correct interpretation in the sense of its influence on observable properties. It is shown that electronic degeneracy and its extended form, called pseudodegeneracy, are actually the only source of spontaneous symmetry breaking (SSB) in nature, including all forms of matter, beginning with elementary particles, via nuclei, atoms, molecules, and solids. Theoretically, the vulnerability of the notion of symmetry is due to the fact that, following quantum mechanics, the separation of the motion of electrons and nuclei (and, similarly, the separation of motions of elementary particles) is approximate, and hence the classical notion of polyatomic space configuration is approximate too, with SSB as one of its main violation.

Routes to chaos and generation of cnoidal wave envelopes in exciton-polariton microcavities

We show the presence of intermittent chaos as a function of the variation of the chemical potential during the condensation of exciton-polaritons using the bifurcation diagram and the Lyapunov Exponent. We then show the possibility of controlling the chaos in the homogeneous solution by the variation of the chemical potential when the pump increases. The influence of the chemical potential is demonstrated in stationary state by studying stability using the Routh-Hurwitz criterion. We also prove the existence of cnoidal waves and through phase portraits, their stabilities in propagation are established using the notion of symmetry. For this, we use a dissipative Gross-Pitaevskii equation for the polariton field, coupled to a reservoir dynamics equation, commonly used to describe polariton condensates under non-resonant pumping.

Cosine Similarity Measures of (m, n)-Rung Orthopair Fuzzy Sets and Their Applications in Plant Leaf Disease Classification

A fuzzy set is a powerful tool to handle uncertainty and ambiguity, and generally, the notions of symmetry and similarity are also exhibited in the fuzzy set theory. The class of (m, n)-rung orthopair fuzzy sets through two universes are more flexible and efficient than the q-rung orthopair fuzzy sets when discussing the symmetry and similarity between multiple objects. This research article comprehensively investigates ten similarity measures that employ cosine and cotangent functions for comparing (m, n)-rung orthopair fuzzy sets, which are a superclass of q-rung orthopair fuzzy sets. Moreover, the proposed weighted similarity measures are applied to real-world problems in building material analysis. A comparative analysis is conducted between the proposed measures and the existing cosine and cotangent measures of q-rung orthopair fuzzy sets, showing that the proposed measures are more efficient than existing ones. Additionally, a numerical example demonstrates the practical and scientific applications of these similarity measures in classifying plant leaf diseases. The sensitivity analysis shows that the existing measures cannot be applied to (m, n)-fuzzy data for distinct values of m and n. The results are supported by graphical interpretations, further illustrating the efficacy of the proposed measures.

Carrollian hydrodynamics from symmetries

In this work, we revisit Carrollian hydrodynamics, a type of non-Lorentzian hydrodynamics which has recently gained increasing attentions due to its underlying connection with dynamics of spacetime near null boundaries, and we aim at exploring symmetries associated with conservation laws of Carrollian fluids. With an elaborate construction of Carroll geometries, we generalize the Randers–Papapetrou metric by incorporating the fluid velocity field and the sub-leading components of the metric into our considerations and we argue that these two additional fields are compulsory phase space variables in the derivation of Carrollian hydrodynamics from symmetries. We then present a new notion of symmetry, called the near-Carrollian diffeomorphism, and demonstrate that this symmetry consistently yields a complete set of Carrollian hydrodynamic equations. Furthermore, due to the presence of the new phase space fields, our results thus generalize those already presented in the previous literatures. Lastly, the Noether charges associated with the near-Carrollian diffeomorphism and their time evolutions are also discussed.

CLAP: A New Algorithm for Promise CSPs

We propose a new algorithm for Promise Constraint Satisfaction Problems (PCSPs). It is a combination of the Constraint Basic LP relaxation and the Affine IP relaxation (CLAP). We give a characterization of the power of CLAP in terms of a minion homomorphism. Using this characterization, we identify a certain weak notion of symmetry which, if satisfied by infinitely many polymorphisms of PCSPs, guarantees tractability. We demonstrate that there are PCSPs solved by CLAP that are not solved by any of the existing algorithms for PCSPs; in particular, not by the algorithm of Brakensiek et al. [SIAM J. Comput., 49 (2020), pp. 1232--1248] and not by a reduction to tractable finite-domain CSPs.

Solution of Fredholm Integral Equation via Common Fixed Point Theorem on Bicomplex Valued B-Metric Space

The notion of symmetry is the main property of a metric function. The area of fixed point theory has a suitable structure for symmetry in mathematics. The goal of this paper is to find fixed point and common fixed point results in a bicomplex valued b-metric space for mixed type rational contractions with control functions. Some well-known literature findings were generalized in our main findings. We provide an example to strengthen and validate our main results. As an example, in the context of bicomplex-valued b-metric space, we develop fixed point and common fixed point results for the rational contraction mapping.

Recurrent connections facilitate symmetry perception in deep networks

Symmetry is omnipresent in nature and perceived by the visual system of many species, as it facilitates detecting ecologically important classes of objects in our environment. Yet, the neural underpinnings of symmetry perception remain elusive, as they require abstraction of long-range spatial dependencies between image regions and are acquired with limited experience. In this paper, we evaluate Deep Neural Network (DNN) architectures on the task of learning symmetry perception from examples. We demonstrate that feed-forward DNNs that excel at modelling human performance on object recognition tasks, are unable to acquire a general notion of symmetry. This is the case even when the feed-forward DNNs are architected to capture long-range spatial dependencies, such as through ‘dilated’ convolutions and the ‘transformers’ design. By contrast, we find that recurrent architectures are capable of learning a general notion of symmetry by breaking down the symmetry’s long-range spatial dependencies into a progression of local-range operations. These results suggest that recurrent connections likely play an important role in symmetry perception in artificial systems, and possibly, biological ones too.

Symmetry Protected Topological Order in Open Quantum Systems

We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channels that satisfy twisted versions of the strong symmetry condition.

Causation, Symmetry and Quantum Physics: Space-like Causality and Conserved Quantities

This work addresses the problem of causation in physics, as for quantum fields, the fundamental theory describing elementary particle physics. Three basic elements are used as methodological concepts: a) the notion of mass-point and the notion of a field (as quantum field) describing a physical system (both concepts are used in physics to characterize elementary particles); b) the notion of state of a mass-point and of a field; c) the notion of causality among states, which are defined by symmetries. Then time and space changes include a space-like causality. These elements are discussed by using the definition of measurement, associated with the notion of symmetry. The derived results give rise to a generalization of the concept of causality, since a mechanical system includes mass-points, fields, and systems composed of mass-points and fields, together. In addition, conserved quantities, founding some causality theories, are generalized due to the notion of symmetry associated with causality.

Symmetry and Landscape Representation

This work analyses the presence of graphic examples and geometric orientation models of certain landscape compositions or natural processes, arranged in artistic literature as a repertoire of practical cases. Among them, we specify those where the notion of symmetry had special im- portance for said transmission, understanding it under a historical and aesthetic meaning where it has been linked to the application of harmonic theories in art. In an etymological sense, we under- stand in this text the notion of symmetry as a correspondence between parts of a whole from the his- torical conception established by Vitruvius, according to which it is born from the proportion to be the cause of a final composition (compositio). Historical-artistic literature allows a graphic and conceptual itinerary to be established through images that end up forming an iconographic corpus of graphic examples that place the landscape image under precepts related to geometric rationali- zation or use these precepts to propose an intuitive alternative related to with direct observation of nature.

A Novel Multi-Population Artificial Bee Colony Algorithm for Energy-Efficient Hybrid Flow Shop Scheduling Problem

Considering green scheduling and sustainable manufacturing, the energy-efficient hybrid flow shop scheduling problem (EHFSP) with a variable speed constraint is investigated, and a novel multi-population artificial bee colony algorithm (MPABC) is developed to minimize makespan, total tardiness and total energy consumption (TEC), simultaneously. It is necessary for manufacturers to fully understand the notion of symmetry in balancing economic and environmental indicators. To improve the search efficiency, the population was randomly categorized into a number of subpopulations, then several groups were constructed based on the quality of subpopulations. A different search strategy was executed in each group to maintain the population diversity. The historical optimization data were also used to enhance the quality of solutions. Finally, extensive experiments were conducted. The results demonstrate that MPABC can achieve an outstanding performance on three metrics DIR, c and nd for the considered EHFSP.

Time-Reversal Symmetry in Healthy and Pathological Resting Brain Activity

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.

Non-invertible global symmetries and completeness of the spectrum

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

За природата на математиката и за математиката в природата – една приказка за симетрията с продължение за малки и големи

The paper describes activities for developing the notion of symmetries. The activities were performed by a group of children between the ages of 5 and 9 and a parent of two of them. Through play and active work the group explores symmetries in sports, architecture, biology, language, music and mathematics. The activities were carried out outdoors in nature. Ideas for complementary activities with computers are presented. As a result the children demonstrate the ability to recognize symmetries in areas and situations different from the ones set in the activities they performed. This proves the main thesis of the paper: the nature of mathematics can be captured successfully in an informal environment at any age.

Functorial Evolution of Quantum Fields

We present a compositional algebraic framework to describe the evolution of quantum fields in discretised spacetimes. We show how familiar notions from Relativity and quantum causality can be recovered in a purely order-theoretic way from the causal order of events in spacetime, with no direct mention of analysis or topology. We formulate theory-independent notions of fields over causal orders in a compositional, functorial way. We draw a strong connection to Algebraic Quantum Field Theory (AQFT), using a sheaf-theoretical approach in our definition of spaces of states over regions of spacetime. We introduce notions of symmetry and cellular automata, which we show to subsume existing definitions of Quantum Cellular Automata (QCA) from previous literature. Given the extreme flexibility of our constructions, we propose that our framework be used as the starting point for new developments in AQFT, QCA and more generally Quantum Field Theory.

Approximate Killing symmetries in non-perturbative quantum gravity

We study the notion of approximate Killing vector fields in several toy models of non-perturbative two-dimensional quantum gravity. Using the framework of discrete exterior calculus, we show how to formulate quantum observables related to such approximate Killing vector fields. Using these methods, we aim to investigate symmetry properties of the space–time geometry produced by the quantum gravitational model at hand. Since we expect quantum fluctuations to dominate at small scales, our goal is to construct a scale-dependent notion of symmetry that might be used to determine whether the emergent (semi-)classical geometry admits any approximate Killing symmetries. We have evaluated one particular choice of such an observable on three ensembles of discrete geometry. We find that the method is useful in the setting where fluctuations are small, but that more work is needed before these ideas can be applied in the deep quantum regime.

A Fuzzy Approach to Support Evaluation of Fuzzy Cross Efficiency

Cross-efficiency evaluation effectively distinguishes a set of decision-making units (DMUs) via self- and peer-evaluations. In constant returns to scale, this evaluation technique is usually applied for data envelopment analysis (DEA) models because negative efficiencies will not occur in this case. For situations of variable returns to scale, the negative cross-efficiencies may occur in this evaluation method. In the real world, the observations could be uncertain and difficult to measure precisely. The existing fuzzy cross-evaluation methods are restricted to production technologies with constant returns to scale. Generally, symmetry is a fundamental characteristic of binary relations used when modeling optimization problems. Additionally, the notion of symmetry appeared in many studies about uncertain theories employed in DEA problems, and this approach can be considered an engineering tool for supporting decision-making. This paper proposes a fuzzy cross-efficiency evaluation model with fuzzy observations under variable returns to scale. Since all possible weights of all DMUs are considered, a choice of weights is not required. Most importantly, negative cross-efficiencies are not produced. An example shows that this paper’s fuzzy cross-efficiency evaluation method has discriminative power in ranking the DMUs when observations are fuzzy numbers.