scholarly journals Around the theory of the generalized weight system: Relations with singularity theory, the generalized Weyl group, and its invariant theory, etc.

1997 ◽  
pp. 101-143 ◽  
Author(s):  
Kyoji Saito
Keyword(s):  
Weyl Group ◽  
Invariant Theory ◽  
Singularity Theory ◽  
Weight System

Scale-Invariant Theory of Optical Properties of Fractal Clusters

1990 ◽  
Author(s):  
Vadim A. Markel ◽  
Leonid S. Muratov ◽  
Mark I. Stockman ◽  
Thomas F. George
Keyword(s):  
Optical Properties ◽  
Invariant Theory ◽  
Scale Invariant ◽  
Fractal Clusters

The Standard Theory

2017 ◽  
Author(s):  
John Iliopoulos
Keyword(s):  
Gauge Theory ◽  
Invariant Theory ◽  
Standard Theory ◽  
Strong Interactions ◽  
Gauge Invariant ◽  
Electromagnetic Interactions ◽  
Protons And Neutrons ◽  
Free Particles ◽  
Theoretical Predictions ◽  
Theory And Experiment

All ingredients of the previous chapters are combined in order to build a gauge invariant theory of the interactions among the elementary particles. We start with a unified model of the weak and the electromagnetic interactions. The gauge symmetry is spontaneously broken through the BEH mechanism and we identify the resulting BEH boson. Then we describe the theory known as quantum chromodynamics (QCD), a gauge theory of the strong interactions. We present the property of confinement which explains why the quarks and the gluons cannot be extracted out of the protons and neutrons to form free particles. The last section contains a comparison of the theoretical predictions based on this theory with the experimental results. The agreement between theory and experiment is spectacular.

Lie algebras graded by the weight system (Θ ,sl)

2021 ◽  
Vol 581 ◽  
pp. 1-44
Author(s):  
Alexander Baranov ◽  
Hogir M. Yaseen
Keyword(s):  
Lie Algebras ◽  
Weight System

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

scholarly journals Symmetry of Narayana Numbers and Rowvacuation of Root Posets

2021 ◽  
Vol 9 ◽  
Author(s):  
Colin Defant ◽  
Sam Hopkins
Keyword(s):  
Weyl Group ◽  
Catalan Number ◽  
Recent Past ◽  
Positive Roots ◽  
Narayana Numbers ◽  
The Subject

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

scholarly journals The Action of the Weyl Group on the $$E_8$$ Root System

2021 ◽  
Author(s):  
Rosa Winter ◽  
Ronald van Luijk
Keyword(s):  
Automorphism Group ◽  
Weyl Group ◽  
Root System ◽  
Sufficient Conditions ◽  
Maximal Clique ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Colored Graphs ◽  
Necessary And Sufficient ◽  
Root Polytope

AbstractLet $$\varGamma $$ Γ be the graph on the roots of the $$E_8$$ E 8 root system, where any two distinct vertices e and f are connected by an edge with color equal to the inner product of e and f. For any set c of colors, let $$\varGamma _c$$ Γ c be the subgraph of $$\varGamma $$ Γ consisting of all the 240 vertices, and all the edges whose color lies in c. We consider cliques, i.e., complete subgraphs, of $$\varGamma $$ Γ that are either monochromatic, or of size at most 3, or a maximal clique in $$\varGamma _c$$ Γ c for some color set c, or whose vertices are the vertices of a face of the $$E_8$$ E 8 root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $$\varGamma $$ Γ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism f from one such clique K to another, we give necessary and sufficient conditions for f to extend to an automorphism of $$\varGamma $$ Γ , in terms of the restrictions of f to certain special subgraphs of K of size at most 7.

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