scholarly journals Defining relations for quantum symmetric pair coideals of Kac–Moody type

2021 ◽  
Author(s):  
Hadewijch De Clercq
Keyword(s):  
Symmetric Pair ◽  
Defining Relations

scholarly journals Defining relations of quantum symmetric pair coideal subalgebras

2021 ◽  
Vol 9 ◽  
Author(s):  
Stefan Kolb ◽  
Milen Yakimov
Keyword(s):  
Chebyshev Polynomials ◽  
Hermite Polynomials ◽  
Symmetric Pair ◽  
Star Products ◽  
Graded Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
New Family

Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$ -graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

Symmetric-pair antennas for beam steering, direction finding or isotropic-reception gain

1991 ◽  
Vol 138 (4) ◽  
pp. 368 ◽  
Author(s):  
R. Benjamin ◽  
W. Titze ◽  
P.V. Brennan ◽  
H.D. Griffiths
Keyword(s):  
Beam Steering ◽  
Direction Finding ◽  
Symmetric Pair

Effects of amplitude and phase errors on performance of symmetric-pair arrays

1991 ◽  
Vol 27 (17) ◽  
pp. 1557
Author(s):  
W. Titze
Keyword(s):  
Symmetric Pair ◽  
Phase Errors ◽  
Amplitude And Phase Errors

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

LORENTZ COVARIANCE FOR THE QUANTUM ALGEBRA OF OBSERVABLES: NAMBU–GOTO STRINGS IN 3 + 1 DIMENSIONS

2002 ◽  
Vol 17 (17) ◽  
pp. 2331-2349 ◽  
Author(s):  
GERRIT HANDRICH
Keyword(s):  
Rest Frame ◽  
Poisson Algebra ◽  
Quantum Algebra ◽  
Substantial Part ◽  
Quantum Case ◽  
Generators And Relations ◽  
Defining Relations ◽  
Algebra Of Observables ◽  
The One ◽  
The Relationship

To postulate correspondence for the observables only is a promising approach to a fully satisfying quantization of the Nambu–Goto string. The relationship between the Poisson algebra of observables and the corresponding quantum algebra is established in the language of generators and relations. A very valuable tool is the transformation to the string's rest frame, since a substantial part of the relations are solved. It is the aim of this paper to clarify the relationship between the fully covariant and the rest frame description. Both in the classical and in the quantum case, an efficient method for recovering the covariant algebra from the one in the rest frame is described. Restrictions on the quantum defining relations are obtained, which are not taken into account when one postulates correspondence for the rest frame algebra. For the part of the algebra studied up to now in explicit computations, these further restrictions alone determine the quantum algebra uniquely — in full consistency with the further restrictions found in the rest frame.

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

Tits polygons

2022 ◽  
Vol 275 (1352) ◽  
Author(s):  
Bernhard Mühlherr ◽  
Richard Weiss ◽  
Holger Petersson
Keyword(s):  
Rational Points ◽  
Jordan Algebras ◽  
Parabolic Subgroups ◽  
Characteristic P ◽  
Arbitrary Group ◽  
Content Type ◽  
Defining Relations ◽  
Moufang Condition ◽  
Unique Vertex ◽  
Natural Way

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank  2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least  4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank  1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

A procedure for obtaining simplified defining relations for a subgroup

1982 ◽  
pp. 155-159
Author(s):  
D.G. Arrell ◽  
S. Manrai ◽  
M.F. Worboys
Keyword(s):  
Defining Relations

Solving some problems of geomechanics on the base of defining relations of post-limit deformation of rocks

2011 ◽  
pp. 189-194
Author(s):  
A Chanyshev ◽  
I Abdulin ◽  
O Belousova
Keyword(s):  
Defining Relations

scholarly journals Autonomous emergence of connectivity assemblies via spike triplet interactions

2019 ◽  
Author(s):  
Lisandro Montangie ◽  
Julijana Gjorgjieva
Keyword(s):  
Random Network ◽  
Transmission Function ◽  
Higher Order ◽  
External Input ◽  
Spike Timing ◽  
Symmetric Pair ◽  
Graph Theoretic ◽  
Stdp Rule ◽  
Triplet Model ◽  
External Stimuli

AbstractNon-random connectivity can emerge without structured external input driven by activity-dependent mechanisms of synaptic plasticity based on precise spiking patterns. Here we analyze the emergence of global structures in recurrent networks based on a triplet model of spike timing dependent plasticity (STDP) which depends on the interactions of three precisely-timed spikes and can describe plasticity experiments with varying spike frequency better than the classical pair-based STDP rule. We describe synaptic changes arising from emergent higher-order correlations, and investigate their influence on different connectivity motifs in the network. Our motif expansion framework reveals novel motif structures under the triplet STDP rule, which support the formation of bidirectional connections and loops in contrast to the classical pair-based STDP rule. Therefore, triplet STDP drives the spontaneous emergence of self-connected groups of neurons, or assemblies, proposed to represent functional units in neural circuits. Assembly formation has often been associated with plasticity driven by firing rates or external stimuli. We propose that assembly structure can emerge without the need for externally patterned inputs or assuming a symmetric pair-based STDP rule commonly assumed in previous studies. The emergence of non-random network structure under triplet STDP occurs through internally-generated higher-order correlations, which are ubiquitous in natural stimuli and neuronal spiking activity, and important for coding. We further demonstrate how neuromodulatory mechanisms that modulate the shape of triplet STDP or the synaptic transmission function differentially promote connectivity motifs underlying the emergence of assemblies, and quantify the differences using graph theoretic measures.

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