323 results match your criteria Algebras And Representation Theory[Journal]

Conserved quantities, optimal system and explicit solutions of a (1 + 1)-dimensional generalised coupled mKdV-type system.

J Adv Res 2021 03 26;29:159-166. Epub 2020 Oct 26.

International Institute for Symmetry Analysis and Mathematical Modelling & Focus Area for Pure and Applied Analytics, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa.

Introduction: The purpose of this paper is to study, a (1 + 1)-dimensional generalised coupled modified Korteweg-de Vries-type system from Lie group analysis point of view. This system is studied in the literature for the first time. The authors found this system to be interesting since it is non-decouplable and possesses higher generalised symmetries. Read More

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Cluster Algebras for Feynman Integrals.

Phys Rev Lett 2021 Mar;126(9):091603

DESY Theory Group, DESY Hamburg, Notkestrasse 85, 22607 Hamburg, Germany.

We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a C_{2} cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding C_{2} inside the A_{3} cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. Read More

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Duality Hierarchies and Differential Graded Lie Algebras.

Commun Math Phys 2021 18;382(1):277-315. Epub 2021 Feb 18.

Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany.

The gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by 'infinity-enhanced Leibniz algebras' defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. Read More

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February 2021

Integrable and Chaotic Systems Associated with Fractal Groups.

Entropy (Basel) 2021 Feb 18;23(2). Epub 2021 Feb 18.

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA.

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor's question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. Read More

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February 2021

Quantum Grothendieck rings as quantum cluster algebras.

Léa Bittmann

J Lond Math Soc 2021 Jan 27;103(1):161-197. Epub 2020 Jul 27.

Faculty of Mathematics Universität Wien Oskar-Morgenstern-Platz 1 Wien 1090 Austria.

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category of representations of the quantum loop algebra introduced by Hernandez-Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type , we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated quantum affine algebra. Read More

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January 2021

Fundamental solutions for semidiscrete evolution equations via Banach algebras.

Adv Differ Equ 2021 7;2021(1):35. Epub 2021 Jan 7.

Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain.

We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. Read More

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January 2021

Nonlinear generalized functions on manifolds.

Proc Math Phys Eng Sci 2020 Dec 16;476(2244):20200640. Epub 2020 Dec 16.

School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK.

In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. Read More

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December 2020

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants.

Entropy (Basel) 2020 Oct 31;22(11). Epub 2020 Oct 31.

Department of Physics, Mathematics and Computer Science, Cracov University of Technology, 31-155 Kraków, Poland.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. Read More

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October 2020

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation.

Entropy (Basel) 2020 Jun 9;22(6). Epub 2020 Jun 9.

Key Technology Domain PCC (Processing, Control & Cognition) Representative, Thales Land & Air Systems, Voie Pierre-Gilles de Gennes, F91470 Limours, France.

In 1969, Jean-Marie Souriau introduced a "Lie Groups Thermodynamics" in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau's model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g. Read More

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From the Jordan Product to Riemannian Geometries on Classical and Quantum States.

Entropy (Basel) 2020 Jun 8;22(6). Epub 2020 Jun 8.

Faculty for Mathematics, TU Dortmund University, 44221 Dortmund, Germany.

The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher-Rao metric tensor is recovered in the Abelian case, that the Fubini-Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures-Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. Read More

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Fact, Fiction, and Fitness.

Entropy (Basel) 2020 Apr 30;22(5). Epub 2020 Apr 30.

Department of Psychology and Center for Cognitive Science, Rutgers University, New Brunswick/Piscataway Campus, NJ 08854, USA.

A theory of consciousness, whatever else it may do, must address the structure of experience. Our perceptual experiences are richly structured. Simply seeing a red apple, swaying between green leaves on a stout tree, involves symmetries, geometries, orders, topologies, and algebras of events. Read More

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Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional -Algebras.

Entropy (Basel) 2020 Nov 23;22(11). Epub 2020 Nov 23.

Faculty of Mathematics, TU Dortmund University, 44221 Dortmund, Germany.

A geometrical formulation of estimation theory for finite-dimensional C∗-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer-Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented. Read More

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November 2020

Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems.

J Stat Phys 2020 27;178(2):319-378. Epub 2019 Nov 27.

Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria.

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional -algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein-Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates. Read More

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November 2019

Phylosymmetric Algebras: Mathematical Properties of a New Tool in Phylogenetics.

Bull Math Biol 2020 11 21;82(12):151. Epub 2020 Nov 21.

University of Tasmania, Churchill Avenue, Sandy Bay, TAS, 7005, Australia.

In phylogenetics, it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction in computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. Read More

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November 2020

Super Quantum Airy Structures.

Commun Math Phys 2020 13;380(1):449-522. Epub 2020 Oct 13.

Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland.

We introduce super quantum Airy structures, which provide a supersymmetric generalization of quantum Airy structures. We prove that to a given super quantum Airy structure one can assign a unique set of free energies, which satisfy a supersymmetric generalization of the topological recursion. We reveal and discuss various properties of these supersymmetric structures, in particular their gauge transformations, classical limit, peculiar role of fermionic variables, and graphical representation of recursion relations. Read More

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October 2020

The logic induced by effect algebras.

Soft comput 2020 26;24(19):14275-14286. Epub 2020 Jul 26.

Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic.

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras , we investigate a natural implication and prove that the implication reduct of is term equivalent to . Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. Read More

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A Unified Dynamic Programming Framework for the Analysis of Interacting Nucleic Acid Strands: Enhanced Models, Scalability, and Speed.

ACS Synth Biol 2020 10 10;9(10):2665-2678. Epub 2020 Sep 10.

Division of Biology & Biological Engineering, California Institute of Technology, Pasadena, California 91125, United States.

Dynamic programming algorithms within the NUPACK software suite enable analysis of nucleic acid sequences over complex and test tube ensembles containing arbitrary numbers of interacting strand species, serving the needs of researchers in molecular programming, nucleic acid nanotechnology, synthetic biology, and across the life sciences. Here, to enhance the underlying physical model, ensure scalability for large calculations, and achieve dramatic speedups when calculating diverse physical quantities over complex and test tube ensembles, we introduce a unified dynamic programming framework that combines three ingredients: (1) recursions that specify the dependencies between subproblems and incorporate the details of the structural ensemble and the free energy model, (2) evaluation algebras that define the mathematical form of each subproblem, (3) operation orders that specify the computational trajectory through the dependency graph of subproblems. The physical model is enhanced using new recursions that operate over the complex ensemble including coaxial and dangle stacking subensembles. Read More

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October 2020

Geometry of color perception. Part 2: perceived colors from real quantum states and Hering's rebit.

M Berthier

J Math Neurosci 2020 Sep 9;10(1):14. Epub 2020 Sep 9.

Laboratoire MIA, La Rochelle Université, Avenue Albert Einstein, BP 33060, 17031, La Rochelle, France.

Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space [Formula: see text] of perceived colors. Read More

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September 2020

Leibniz Gauge Theories and Infinity Structures.

Commun Math Phys 2020 6;377(3):2027-2077. Epub 2020 Jun 6.

Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany.

We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based on 'tensor hierarchies', which describe towers of -form gauge fields transforming under non-abelian gauge symmetries and which have been constructed up to low levels. Read More

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Algebraic Study of diatomic Molecules: homonuclear molecules H and N.

Sci Rep 2020 May 6;10(1):7663. Epub 2020 May 6.

Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz, 51664, Iran.

It is the aim of this study to discuss for two-body systems like homonuclear molecules in which eigenvalues and eigenfunctions are obtained by exact solutions of the solvable models based on SU(1, 1) Lie algebras. Exact solutions of the solvable Hamiltonian regarding the relative motion in a two-body system on Lie algebras were obtained. The U(1) ↔ O(2), U(3) ↔ O(4) and U(3) ↔ O(4) transitional Hamiltonians are employed to described for H and N molecules. Read More

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Classification of Rota-Baxter operators on semigroup algebras of order two and three.

Commun Algebra 2019 4;47(8):3094-3116. Epub 2019 Mar 4.

Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria.

In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions. Read More

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Pseudo-loop conditions.

Bull Lond Math Soc 2019 Oct 12;51(5):917-936. Epub 2019 Sep 12.

Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstrasse 8-10/104 1040 Wien Austria.

About a decade ago, it was realised that the satisfaction of a given (or ) of the form in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called , and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a in any arbitrary non-trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form , which we call . Read More

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October 2019

The boundary algebra of a GL -dimer.

Lukas Andritsch

Commun Algebra 2020 Feb 9;48(6):2555-2574. Epub 2020 Feb 9.

Institute for Mathematics and Scientific Computing, University of Graz, Graz, Austria.

We consider GL -dimers of triangulations of regular convex -gons, which give rise to a dimer model with boundary and a dimer algebra Λ . Let be the sum of the idempotents of all the boundary vertices, and the associated boundary algebra. In this article we show that given two different triangulations and of the -gon, the boundary algebras are isomorphic, i. Read More

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February 2020

A broad class of discrete-time hypercomplex-valued Hopfield neural networks.

Neural Netw 2020 Feb 18;122:54-67. Epub 2019 Oct 18.

Department of Applied Mathematics, University of Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas-SP, CEP 13083-859, Brazil. Electronic address:

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Read More

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February 2020

Non-locality, contextuality and valuation algebras: a general theory of disagreement.

Philos Trans A Math Phys Eng Sci 2019 Nov 16;377(2157):20190036. Epub 2019 Sep 16.

Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK.

We establish a strong link between two apparently unrelated topics: the study of conflicting information in the formal framework of valuation algebras, and the phenomena of non-locality and contextuality. In particular, we show that these peculiar features of quantum theory are mathematically equivalent to a general notion of between information sources. This result vastly generalizes previously observed connections between contextuality, relat- ional databases, constraint satisfaction problems and logical paradoxes, and gives further proof that contextual behaviour is not a phenomenon limited to quantum physics, but pervades various domains of mathematics and computer science. Read More

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November 2019

Integrated Information in Process-Algebraic Compositions.

Entropy (Basel) 2019 Aug 17;21(8). Epub 2019 Aug 17.

Institute of Information Science and Technologies, National Research Council (ISTI-CNR), 1, Via Moruzzi, 56124 Pisa, Italy.

Integrated Information Theory (IIT) is most typically applied to , a state transition model in which system parts cooperate by . By contrast, in , whose semantics can also be formulated in terms of (labeled) state transitions, system parts-"processes"-cooperate by with matching labels, according to interaction patterns expressed by suitable composition operators. Despite this substantial difference, questioning how much additional information is provided by the integration of the interacting partners above and beyond the sum of their independent contributions appears perfectly legitimate with both types of cooperation. Read More

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Rota-Baxter operators and post-Lie algebra structures on semisimple Lie algebras.

Commun Algebra 2019 11;47(5):2280-2296. Epub 2019 Jan 11.

Fakultät für Mathematik, Universität Wien, Wien, Austria.

Rota-Baxter operators of weight 1 on are in bijective correspondence to post-Lie algebra structures on pairs , where is complete. We use such Rota-Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras , where is semisimple. We show that for semisimple and , with or simple, the existence of a post-Lie algebra structure on such a pair implies that and are isomorphic, and hence both simple. Read More

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January 2019

On MV-Algebraic Versions of the Strong Law of Large Numbers.

Entropy (Basel) 2019 Jul 19;21(7). Epub 2019 Jul 19.

Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland.

Many-valued (MV; the many-valued logics considered by Łukasiewicz)-algebras are algebraic systems that generalize Boolean algebras. The MV-algebraic probability theory involves the notions of the state and observable, which abstract the probability measure and the random variable, both considered in the Kolmogorov probability theory. Within the MV-algebraic probability theory, many important theorems (such as various versions of the central limit theorem or the individual ergodic theorem) have been recently studied and proven. Read More

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Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms.

J Chem Phys 2019 Jul;151(1):014107

William R. Wiley Environmental Molecular Sciences Laboratory, Battelle, Pacific Northwest National Laboratory, K8-91, P.O. Box 999, Richland, Washington 99352, USA.

In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Read More

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Ideals and their complements in commutative semirings.

Soft comput 2019 31;23(14):5385-5392. Epub 2018 Aug 31.

1Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic.

We study conditions under which the lattice of ideals of a given a commutative semiring is complemented. At first we check when the annihilator of a given ideal of is a complement of . Further, we study complements of annihilator ideals. Read More

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