Positive definite symmetric functions on finite-dimensional spaces II

1985 ◽  
Vol 3 (6) ◽  
pp. 325-329 ◽  
Author(s):  
Donald St.P Richards
Keyword(s):  
Symmetric Functions ◽  
Positive Definite ◽  
Finite Dimensional

Some sign properties of symmetric functions

1989 ◽  
Vol 105 (2) ◽  
pp. 193-196
Author(s):  
D. B. Hunter ◽  
I. G. Macdonald
Keyword(s):  
Symmetric Functions ◽  
Positive Definite

AbstractThis paper is concerned with the sign properties of the S-functions sλ for real arguments. We show first that sλ is indefinite if any part of the partition λ is odd. Thus it is only if all parts of λ are even that sλ can possibly be positive definite or semi-definite. In this case we show that sλ(x) is positive provided that at least l(λ) of the components of x are non-zero, where l(λ) is the number of parts of the partition λ.

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

Two inequalities for the complete symmetric functions

1978 ◽  
Vol 84 (1) ◽  
pp. 1-3 ◽  
Author(s):  
V. J. Baston
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Positive Definite ◽  
Even Order ◽  
Image Position ◽  
Complete Symmetric Function ◽  
Special Case

In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequalitywhere ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).

scholarly journals Skew compressions of positive definite operators and matrices

2020 ◽  
Vol 36 (36) ◽  
pp. 400-410
Author(s):  
Matteo Polettini ◽  
Albrecht Böttcher
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Singular Values ◽  
Positive Definite ◽  
Graph Theoretic ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Positive Definite Operators ◽  
Finite Dimensional Hilbert Space

The paper is devoted to results connecting the eigenvalues and singular values of operators composed by $P^\ast G P$ with those composed in the same way by $QG^{−1}Q^\ast$. Here $P +Q = I$ are skew complementary projections on a finite-dimensional Hilbert space and $G$ is a positive definite linear operator on this space. Also discussed are graph theoretic interpretations of one of the results.

The positive-definiteness of the complete symmetric functions of even order

1977 ◽  
Vol 82 (2) ◽  
pp. 255-258 ◽  
Author(s):  
D. B. Hunter
Keyword(s):  
Generating Function ◽  
Classical Theory ◽  
Symmetric Functions ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Homogeneous Product ◽  
Even Order ◽  
Image Position

An important role in the classical theory of symmetric functions of a real n-tuple x = (x1, x2, …, xn) is played by the complete symmetric functions or homogeneous product sums hr defined by the generating function(see Littlewood (5), p. 82). In an earlier paper (4) I conjectured that h2r is positive definite. The main object of the present paper is to prove this conjecture in a rather sharper form.

scholarly journals Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials

2006 ◽  
Vol 13 (01) ◽  
pp. 67-86
Author(s):  
Xiaoping Xu
Keyword(s):  
Lie Algebras ◽  
Theta Series ◽  
Positive Definite ◽  
Combinatorial Identities ◽  
Symmetric Polynomials ◽  
Simple Lie Algebras ◽  
Type D ◽  
Finite Dimensional ◽  
Hidden Symmetry ◽  
Root Lattices

Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E4(z) and the Ramanujan series Δ24(z). A natural question is what kind of polynomials in E4(z) and Δ24(z) could be the theta series of positive definite even unimodular lattices. In this paper, we find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of E4(z) and Δ24(z). Using these two identities, we prove that three families of weighted symmetric polynomials of two fixed families of polynomials of E4(z) and Δ24(z) are the theta series of certain positive definite even unimodular lattices, obtained by gluing finitely many copies of the root lattices of the finite-dimensional simple Lie algebras of type D2n. The results also show that the full permutation groups are the hidden symmetry of the theta series of certain unimodular lattices.

scholarly journals SOLVABILITY OF THE G2 INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (22) ◽  
pp. 3885-3903 ◽  
Author(s):  
MARCOS ROSENBAUM ◽  
ALEXANDER TURBINER ◽  
ANTONIO CAPELLA
Keyword(s):  
Integrable System ◽  
Configuration Space ◽  
Symmetric Functions ◽  
Differential Operators ◽  
Coupling Constants ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Exactly Solvable ◽  
Three Body

It is shown that the three-body trigonometric G2 integrable system is exactly solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly.

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