Dynamical Reduction Theories as a Natural Basis for a Realistic Worldview

2020 ◽  
pp. 258-296
Author(s):  
Gian Carlo Ghirardi
Keyword(s):  
Natural Basis ◽  
Dynamical Reduction

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

scholarly journals A Natural Basis for Unsupervised Machine Learning on Scanning Diffraction Data

2018 ◽  
Vol 24 (S1) ◽  
pp. 490-491 ◽  
Author(s):  
Paul Cueva ◽  
Elliot Padget ◽  
David A. Muller
Keyword(s):  
Machine Learning ◽  
Diffraction Data ◽  
Natural Basis ◽  
Unsupervised Machine Learning

scholarly journals A Schur-Cohn theorem for matrix polynomials

1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Harry Dym ◽  
Nicholas Young
Keyword(s):  
Unit Circle ◽  
Matrix Polynomial ◽  
Hermitian Matrix ◽  
Krein Space ◽  
Gram Matrix ◽  
Natural Basis ◽  
Test Matrix ◽  
Matrix Polynomials ◽  
Rational Vector ◽  
Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

Sub-ppm/°C Bandgap References With Natural Basis Expansion for Curvature Cancellation

2021 ◽  
Vol 68 (9) ◽  
pp. 3551-3561
Author(s):  
Nanqi Liu ◽  
Randall L. Geiger ◽  
Degang Chen
Keyword(s):  
Natural Basis ◽  
Basis Expansion

scholarly journals Casselman’s Basis of Iwahori Vectors and the Bruhat Order

2011 ◽  
Vol 63 (6) ◽  
pp. 1238-1253 ◽  
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Weyl Group ◽  
Bruhat Order ◽  
Group Element ◽  
The Other ◽  
Intertwining Operators ◽  
Transition Matrices ◽  
Natural Basis ◽  
Spherical Representation ◽  
Langlands Parameters ◽  
Combinatorial Condition

AbstractW. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group W. On the other hand, there is a natural basis , and one seeks to find the transition matrices between the two bases. Thus, let and . Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then , where z are the Langlands parameters for the representation and α runs through the set S(u, v) of positive coroots (the dual root systemof G) such that with rα the reflection corresponding to α. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan–Lusztig polynomial Pw0v,w0u = 1 with w0 the long Weyl group element. There is a similar formula for conjecturally satisfied if Pu,v = 1. This leads to various combinatorial conjectures.

scholarly journals Quantum groups via cyclic quiver varieties I

2015 ◽  
Vol 152 (2) ◽  
pp. 299-326 ◽  
Author(s):  
Fan Qin
Keyword(s):  
Quantum Groups ◽  
Bilinear Form ◽  
Canonical Basis ◽  
Natural Basis ◽  
Grothendieck Ring ◽  
Quiver Varieties ◽  
Enveloping Algebra ◽  
Dual Canonical Basis ◽  
Quantized Enveloping Algebra ◽  
Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Robin Chhabra ◽  
M. Reza Emami ◽  
Yael Karshon
Keyword(s):  
Symmetry Group ◽  
Mechanical Systems ◽  
Hamiltonian Formalism ◽  
Geometrical Approach ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Relative Configuration ◽  
D’Alembert Equation ◽  
Dynamical Reduction ◽  
Configuration Manifold

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

scholarly journals New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

10.37236/6464 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Ba Nguyen
Keyword(s):  
Cluster Algebra ◽  
Theta Functions ◽  
Type A ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Combinatorial Formula ◽  
Natural Basis ◽  
Combinatorial Formulas ◽  
New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

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