scholarly journals Quasiconvex relaxation of isotropic functions in incompressible planar hyperelasticity

2019 ◽  
Vol 150 (5) ◽  
pp. 2620-2631 ◽  
Author(s):  
Robert J. Martin ◽  
Jendrik Voss ◽  
Patrizio Neff ◽  
Ionel-Dumitrel Ghiba
Keyword(s):  
Recent Result ◽  
Explicit Formula ◽  
Linear Group ◽  
Orthogonal Group ◽  
Special Linear Group ◽  
Energy Functions ◽  
Incompressible Nonlinear Elasticity ◽  
Rank One ◽  
Isotropic Functions ◽  
Quasiconvex Envelope

AbstractIn this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

Induced Representations and Invariants

1950 ◽  
Vol 2 ◽  
pp. 334-343 ◽  
Author(s):  
G. DE B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Explicit Formula ◽  
Linear Group ◽  
Schur Functions ◽  
The Other ◽  
Induced Representations ◽  
Direct Sum ◽  
Invariant Matrices ◽  
The One

1. Introduction. The problem of the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices is intimately associated with the representation theory of the full linear group on the one hand and with the representation theory of the symmetric group on the other. In a previous paper the author gave an explicit formula for this reduction in terms of characters of the symmetric group. Later J. A. Todd derived the same formula using Schur functions, i.e. characters of representations of the full linear group.

scholarly journals Integral cohomology and Chern classes of the special linear group over the ring of integers

2001 ◽  
Vol 131 (3) ◽  
pp. 445-457
Author(s):  
DOMINIQUE ARLETTAZ ◽  
CHRISTIAN AUSONI ◽  
MAMORU MIMURA ◽  
NOBUAKI YAGITA
Keyword(s):  
Linear Group ◽  
Upper Bound ◽  
Special Linear Group ◽  
Discrete Groups ◽  
Chern Classes ◽  
Additive Structure ◽  
Special Linear ◽  
Ring Of Integers ◽  
Integral Cohomology ◽  
Complete Calculation

This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(ℤ) over the ring of integers ℤ. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups.

scholarly journals The third homology of the special linear group of a field

2009 ◽  
Vol 213 (9) ◽  
pp. 1665-1680 ◽  
Author(s):  
Kevin Hutchinson ◽  
Liqun Tao
Keyword(s):  
Linear Group ◽  
Special Linear Group ◽  
Special Linear ◽  
The Third

On the existence of Engel pairs in certain linear groups

2016 ◽  
Vol 15 (04) ◽  
pp. 1650062
Author(s):  
S. G. Quek ◽  
K. B. Wong ◽  
P. C. Wong
Keyword(s):  
Linear Group ◽  
Field Extension ◽  
Special Linear Group ◽  
Linear Groups ◽  
Special Linear

Let [Formula: see text] be a group and [Formula: see text]. The 2-tuple [Formula: see text] is said to be an [Formula: see text]-Engel pair, [Formula: see text], if [Formula: see text], [Formula: see text] and [Formula: see text]. Let SL[Formula: see text] be the special linear group of degree [Formula: see text] over the field [Formula: see text]. In this paper, we show that given any field [Formula: see text], there is a field extension [Formula: see text] of [Formula: see text] with [Formula: see text] such that SL[Formula: see text] has an [Formula: see text]-Engel pair for some integer [Formula: see text]. We will also show that SL[Formula: see text] has a [Formula: see text]-Engel pair if [Formula: see text] is a field of characteristic [Formula: see text].

scholarly journals The special linear group for nonassociative rings

2020 ◽  
Vol 23 (2) ◽  
pp. 327-335
Author(s):  
Harry Petyt
Keyword(s):  
Linear Group ◽  
Special Linear Group ◽  
Large Classes ◽  
Special Linear ◽  
Lie Ring ◽  
Definition Of ◽  
Associative Rings

AbstractWe extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined.

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