Space Of Symmetric Functions Research Articles

Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space $E$, and of the unit ball of the space $E({\cal M},\tau)

Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations

We study weighted Sobolev embeddings in radially symmetric function spaces and then investigate the existence of nontrivial radial solutions of inhomogeneous quasilinear elliptic equation with singular potentials and super-$(p, q)$-linear nonlinearity. The model equation is of the form $ -\Delta_p u+V(|x|)|u|^{q-2}u=Q(|x|)|u|^{s-2}u, x\in R^N,$ $ u(x) \rightarrow 0,$ as $ |x|\rightarrow\infty. $

Some common fixed point theorems for a pair of tangential mappings in symmetric spaces

A metrical common fixed point theorem for a pair of self mappings due to Sastry and Murthy (K.P.R. Sastry, I.S.R. Krishna Murthy, A common fixed points of two partially commuting tangential selfmaps on a metric space, J. Math. Anal. Appl. 250 (2000) 731734.) [8] is extended to symmetric spaces which in turn generalises a fixed point theorem due to Pant (R.P. Pant, Common fixed points of Lipschitz type mapping pairs, J. Math. Anal. Appl. 248 (1999) 280283.) [11] besides deriving some related results. Some illustrative examples to highlight the realised improvements are also furnished.

A commutative algebra on degenerate CP1 and Macdonald polynomials

We introduce a unital associative algebra A associated with degenerate CP1. We show that A is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby, we can regard A as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov–Pandharipande [e-print arXiv:math-ph/0411210].

THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

Integral [formula omitted]-type contractions and existence of continuous solutions for nonlinear integral inclusions

In this paper, we discuss a general common fixed point theorem of integral Φ -type for two pairs of weakly compatible mappings satisfying certain integral type implicit relations in symmetric spaces by using the notion of a pair of mappings satisfying property (E.A). Our main result improves and extends the recent results of Aliouche [A. Aliouche, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, J. Math. Anal. Appl. 322 (2006) 796–802], Pathak, Tiwari and Khan [H.K. Pathak, Rakesh Tiwari, M.S. Khan, A common fixed point theorem satisfying implicit relation, Appl. Math. E-Notes, 7 (2007) 222–228] and several other known results. Subsequently, a multi-valued version of Krasnoselski, fixed point theorem in a cone is used to discuss the existence of C [ 0 , T ] and L p [ 0 , T ] solutions to nonlinear integral inclusion y ( t ) ∈ ∫ 0 T k ( t , s ) [ a ( s ) g ( s , y ( s ) ) + f ( s , y ( s ) , y ′ ( s ) ) ] ds which improves the recent result of Agarwal, Meehan and O’Regan [R.P. Agarwal, M. Meehan, D. O’Regan, Positive L p and continuous solutions for Fredholm integral inclusions, in: Set Valued Mappings with Applications in Nonlinear Analysis, Gordon & Breach, Amsterdam, SIMAA 4 (2002) 1–9].

Brauer-Schur functions

A new class of functions is studied. We define the Brauer-Schur functions $B^{(p)}_{\lambda}$ for a prime number $p$, and investigate their properties. We construct a basis for the space of symmetric functions, which consists of products of $p$-Brauer-Schur functions and Schur functions. We will see that the transition matrix from the natural Schur function basis has some interesting numerical properties.

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E ( M ) be the associated noncommutative function space. Let ( ε k ) k ⩾ 1 be a Rademacher sequence, on some probability space Ω. For finite sequences ( x k ) k ⩾ 1 of E ( M ) , we consider the Rademacher averages ∑ k ε k ⊗ x k as elements of the noncommutative function space E ( L ∞ ( Ω ) ⊗ ¯ M ) and study estimates for their norms ‖ ∑ k ε k ⊗ x k ‖ E calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖ ∑ k ε k ⊗ x k ‖ E is equivalent to the infimum of ‖ ( ∑ y k ∗ y k ) 1 2 ‖ + ‖ ( ∑ z k z k ∗ ) 1 2 ‖ over all y k , z k in E ( M ) such that x k = y k + z k for any k ⩾ 1 . Dual estimates are given when E is 2-convex and has a nontrivial upper Boyd index. In this case, ‖ ∑ k ε k ⊗ x k ‖ E is equivalent to ‖ ( ∑ x k ∗ x k ) 1 2 ‖ + ‖ ( ∑ x k x k ∗ ) 1 2 ‖ . We also study Rademacher averages ∑ i , j ε i ⊗ ε j ⊗ x i j for doubly indexed families ( x i j ) i , j of E ( M ) .

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

AbstractWe introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a natural inclusion of the Hopf algebra of noncommutative symmetric functions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials and conclude two analogues of Chevalley’s theorem in the noncommutative setting.

Corrigenda to “On Non-Compact Operators in Weighted Ideal and Symmetric Function Spaces”

Abstract Corrections to [Oniani, Georgian Math. J. 13: 501–514, 2006] are listed.

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on \(H({\mathbb{C}}^{n})\) which can be obtained from the translation operator using polynomial automorphisms of \({\mathbb{C}}^{n}\) . In particular we show that if CS is a hypercyclic operator for an affine automorphism S on \(H({\mathbb{C}}^{n})\) , then \(S = \Theta \circ (I + b) \circ \Theta ^{-1} + a\) for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on l1.

Search for Boolean Functions With Excellent Profiles in the Rotation Symmetric Class

For the first time Boolean functions on 9 variables having nonlinearity 241 are discovered, that remained as an open question in literature for almost three decades. Such functions are found by heuristic search in the space of rotation symmetric Boolean functions (RSBFs). This shows that there exist Boolean functions on n (odd) variables having nonlinearity >2n-1-2n-1/2 if and only if n>7. Using similar search technique, balanced Boolean functions on 9, 10, and 11 variables are attained having autocorrelation spectra with maximum absolute value <2lceiln/2rceil. On odd number of variables, earlier such functions were known for 15, 21 variables; there was no evidence of such functions at all on even number of variables. In certain cases, our functions can be affinely transformed to obtain first-order resiliency or first-order propagation characteristics. Moreover, 10 variable functions having first-order resiliency and nonlinearity 492 are presented that had been posed as an open question at Crypto 2000. The functions reported in this paper are discovered using a suitably modified steepest descent based iterative heuristic search in the RSBF class along with proper affine transformations. It seems elusive to get a construction technique to match such functions

A general fixed point theorem for converse commuting multivalued mappings in symmetric spaces

In this paper a general fixed point theorem for converse commuting multivalued mappings, which generalize Theorems 2.1 and 2.2 from [3], is proved. .

Common Fixed Points of Mappings and Set-Valued Mappings in Symmetric Spaces with Application to Probabilistic Spaces

The main purpose of this paper is to give some common fixed point theorems of mappings and set-valued mappings of a symmetric space with some applications to probabilistic spaces. In order to get these results, we define the concept of E-weak compatibility between set-valued and single-valued mappings of a symmetric space.

On Non-Compact Operators in Weighted Ideal and Symmetric Function Spaces

Abstract A resonance type theorem is proved, where conditions are given, which imply the non-compactness and a certain estimation of the measure of the non-compactness of operators in weighted ideal and symmetric function spaces. The application of the theorem to some concrete classes of operators is discussed.

A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type

In this paper, we proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property (E.A) introduced in [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188]. Our theorem generalizes Theorem 2.2 of [M. Aamri, D. El Moutawakil, Common fixed points under contractive conditions in symmetric spaces, Appl. Math. E-Notes 3 (2003) 156–162] and Theorem 2 of [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188].

Enumeration of m-tuples of permutations and a new class of power bases for the space of symmetric functions

In this paper, we continue a long line of research which shows that many generating function identities for various permutation statistics arise from well known symmetric function identities by applying certain ring homomorphisms on the ring of symmetric functions. This idea was first introduced in a 1993 paper of Brenti who used it to show that the generating functions of permutations of the symmetric group S n by descents and excedances could be derived in such a manner. In this paper, we define certain ( q , t )-analogues of Brenti's homomorphism that lead to generating functions for statistics on m-tuples of permutations. Our results generalize previous work of generating functions for permutations statistics due to Carlitz, Stanley, and Fedou and Rawlings. We also introduce some new bases of symmetric functions which are necessary to extend Brenti's results on generating functions for permutations by excedances to m-tuples of permutations. Finally, we study the image of our homomorphisms on analogues of the q-basis of symmetric functions studied by Ram and King and Wybourne to describe the irreducible characters of the Hecke algebras of type A.

Breakthroughs in the theory of Macdonald polynomials

In 1998, I. G. Macdonald (1) introduced a remarkable new basis for the space of symmetric functions. The elements of this basis are denoted , where λ is a partition and p, q are two free parameters. The 's, which are now called “Macdonald polynomials,” specialize to many of the well known bases for the symmetric functions, by suitable choices of the parameters q and t. In fact, we can obtain in this manner the Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the zonal symmetric functions, the zonal spherical functions, and the elementary and monomial symmetric functions.

Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach

We addressed, in a previous note [C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784], the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge–Ampère equation, namely: det D 2 ψ = f in Ω, ψ = g on ∂ Ω ( Ω ⊂ R 2 and f > 0 , here). The method discussed previously relies on an augmented Lagrangian algorithm operating in the space H 2 ( Ω ) and related functional spaces of symmetric tensor-valued functions. In the particular case where the above problem has no solution in H 2 ( Ω ) , while the data f and g verify { f , g } ∈ L 1 ( Ω ) × H 3 / 2 ( ∂ Ω ) , there is strong evidence that the augmented Lagrangian algorithm discussed in previously converges-in some sense-to a least squares solution belonging to V g = { φ | φ ∈ H 2 ( Ω ) , φ = g on ∂ Ω } . Our goal in this note is to discuss a least-squares based alternative solution method for the Monge–Ampère Dirichlet problem. This method relies on the minimization on the set V g × Q f (with Q f = { q | q = ( q i j ) 1 ⩽ i , j ⩽ 2 , q i j ∈ L 2 ( Ω ) , ∀ i , j , 1 ⩽ i , j ⩽ 2 , q = q t , det q = f } ) of a well-chosen least-squares functional. From a practical point of view we solve the above minimization problem via a relaxation type algorithm, operating alternatively in V g and Q f and very easy to combine to the mixed finite element approximations employed in the earlier work. Numerical experiments show that the above method has good convergence properties when the Monge–Ampère Dirichlet problem has solutions in V g ; they show also that, for cases where the above problem has no solution in V g , while neither V g nor Q f are empty, the new method reproduces the solutions obtained via the augmented Lagrangian approach, but faster. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Singular Symmetric Functionals

We present and study a new construction of singular symmetric functionals on Marcinkiewicz function spaces, which may be considered as a counterpart of an earlier construction by Dixmier of nonnormal traces on certain operator ideals. We also exhibit new examples of symmetric function spaces which are not subspaces of any Marcinkiewicz space and which admit nontrivial singular symmetric functionals. Bibliography: 9 titles.