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The advanced theory of the first and higher symmetric Frechet differentials and K-sub-differentials is constructed including the mean value theorem and the Taylor formula. We give simple sufficient conditions for symmetric K-subdifferentiability and consider some applications to Fourier series and variational functionals.

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

A symmetric top is considered, which is a particular case of a mechanical top that is usually described by the canonical Poisson structure on T*SE (3). This structure is invariant under the right action of the rotation group SO(3), but the Hamiltonian of the symmetric top is invariant only under the right action of the subgroup S 1, which corresponds to the rotation of the symmetric top around its axis of symmetry. This Poisson structure is obtained as the reduction T* SE (3) / S 1. A Hamiltonian and motion equations are proposed that describe a wide class of interaction models of the symmetric top with an axially symmetric external field.

The paper contains a description of symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\ell_{1}$-sum of the space $\mathbb{C}^{2}.$ We show that the specrum of such algebra does not coincide of point evaluation functionals and described characters of the algebra as functions of exponential type with plane zeros.

Rotation symmetric Boolean functions have been extensively studied in the recent years because of their applications in cryptography. In this study, a novel method to construct p-variable 1-resilient rotation symmetric functions over GF(p) is proposed based on a Latin square with maximum cycle structure, which is not required to solve any equation system. And a lower bound on the number of p-variable 1-resilient rotation symmetric functions is given. At last, an equivalent characterization of p-variable 1-resilient rotation symmetric functions over GF(p) is demonstrated, as a direct corollary, the number of p-variable 1-resilient rotation symmetric functions is represented by all the solutions of the equation system. Copyright © 2017 John Wiley & Sons, Ltd.

Via the chip-firing game, a class of Schur positive symmetric functions depending on four parameters is introduced for any labeled connected simple graph. Tableaux formulae are stated to expand such symmetric functions in terms of the complete symmetric functions. In the case of a simple path, the resulting symmetric functions reduce to the transformed Hall-Littlewood symmetric functions when the parameters are suitable specialized.

We propose effective constructions of dual bases for the noncommutative symmetric and quasi-symmetric functions. To this end, we use an effective variation of Schutzenberger's factorization adapted to the diagonal pairing between a graded space and its dual.

In this paper, we introduce the notion of symmetric bi-derivations of a B-algebra and investigate some related properties. We study the notion of symmetric bi-derivations of a 0-commutative B-algebra and state some related properties. © 2016 Korean Mathem Department of Mathematics, Faculty of Science, Yaşar University, Izmir, 35100, Turkey

The purpose of the paper is to continue the study of completeness for symmetric spaces. We study variants of completeness in symmetric spaces and present some properties regarding completeness in symmetric spaces. Finally, as an application of completeness we will present the fixed point result for multivalued maps.

We show that the image of any degree-three multilinear Jordan polynomial evaluated over the Jordan algebras of real and complex symmetric matrices forms a vector space.

The notion of left-right (resp. right-left) symmetric biderivation of KU-algebras is introduced and some related properties are investigated.

The object of the present paper is to study a necessary condition for the existence of weakly symmetric and weakly Ricci -symmetric LPSasakian manifolds admitting a quarter-symmetric metric connection.

An operator T on a complex Hilbert space H is called skew symmetric if T can be represented as a skew symmetric matrix relative to some orthonormal basis for H. In this paper, we study skew symmetric operators with eigenvalues. First, we provide an upper-triangular operator matrix representation for skew symmetric operators with nonzero eigenvalues. On the other hand, we give a description of certain skew symmetric triangular operators, which is based on the geometric relationship between eigenvectors.

Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less or equal than 3., 15 pages

We show that the classification of symmetric spaces can be achieved by K-theoretical methods. We focus on Hermitian symmetric spaces of non-compact type, and define K-theory for JB*-triples along the lines of C*-theory. K-groups have to be provided with further invariants in order to classify the spaces in question. Among these are cycles obtained from so called grids, intimately connected to root systems of an underlying Lie-algebra and thus reminiscent of the classical classification scheme.

The class algebra and the double coset algebra are two classical commutative subalgebras of the group algebra of the symmetric group. The connexion coefficients of these two algebraic structures are important numbers with significant applications. From a combinatorial point of view, they give the number of factorizations of a given permutation into the ordered product of permutations with specific cyclic properties and count in some cases the number of hypermaps and constellations on (locally) orientable surfaces. They are also of notable interest in the study of Schur and zonal polynomials as well as in the theory of the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore as shown by Hanlon, Stanley, Stembridge (1992), the respective generating series of these coefficients in the basis of power sum symmetric functions are equal to the mathematical expectation of the trace of ( X U Y U ? ) n where X and Y are given symmetric (respectively hermitian) matrices, U is a random real (respectively complex) valued square matrix of standard normal distribution and n a non negative integer.This paper is devoted to the explicit evaluation of these series in terms of monomial symmetric functions. Morales and Vassilieva (2009, 2011) and Vassilieva (2013) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type a , b , 1 n - a - b ] .

The algebra $\mathcal{H}_{bs}(\ell_1)$ of symmetric analytic functions of bounded type is investigated. In particular, we study continuity of some homomorphisms of the algebra of symmetric polynomials on $\ell_p$ and composition operators of the algebra of symmetric analytic functions. The paper contains several open questions.

We present examples of fully symmetric function spaces without an equivalent Fatou norm and examples of symmetric function spaces, which do not admit an equivalent strongly symmetric norm.

In this paper we study complex symmetric weighted composition operators on the Hardy space. We provide some characterizations of ψ and φ when a weighted composition operator W ψ , φ is complex symmetric. We investigate which combinations of weights ψ and maps of the open unit disk φ give rise to complex symmetric weighted composition operators with a special conjugation. As some applications, we obtain several examples for nonnormal complex symmetric operators. In addition, we give spectral properties of complex symmetric weighted composition operators. We examine eigenvalues and eigenvectors of such operators and find some conditions for which a complex symmetric weighted composition operator is Hilbert–Schmidt. Finally, we consider cyclicity, hypercyclicity, and the single-valued extension property for complex symmetric weighted composition operators.

Positive definite matrices factor into A = L L T (Cholesky). Symmetric indefinite matrices need a symmetric middle factor in A = L P L T . Then A and P have the same inertia (eigenvalues of the same sign). We construct P through elimination, so the inertias agree for all leading minors of A and P. When restricting P to be a variant of a symmetric permutation in which diagonal 1's can be replaced by 0's or −1's, it is unique.

A collection of sets is symmetric-difference-free, respectively symmetric difference-closed, if the symmetric difference of any two sets in the collection lies outside, respectively inside, the collection. Recently Buck and Godbole (Size-maximal symmetric difference-free families of subsets of [n], Graphs Combin. (to appear), 2013) investigated such collections and showed, in particular, that the the largest symmetric difference-free collection of subsets of an n-set has cardinality 2 n-1. We use group theory to obtain shorter proofs of their results.

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.

In this paper, we establish a new resonance identity for symmetric closed characteristics on symmetric compact convex hypersurface Σ in R2n when there exist only finitely many geometrically distinct symmetric closed characteristics. As its applications, some interesting results about the stability and multiplicity of symmetric closed characteristics are obtained, and also we prove that if Σ is C∞-generic, it carries infinitely many symmetric closed characteristics.

The complete relationship between the kth anti-diagonals and the eigenvalues of an n × n ( k ⩽ n ) real symmetric matrix is obtained. When k is even, the relationship is weak majorization. The Hermitian case is also studied. Similar results are obtained for real and complex skew symmetric matrices.

Let Σ be a $C^2$ compact strictly convex hypersurface in R2n with $n\ge 2$. Suppose $PΣ=Σ$ with $P$ being a $2n\times 2n$ symplectic and orthogonal matrix and $P^r=I_{2n}$. We prove that there are at least two geometrically distinct $P$-cyclic symmetric closed characteristics $(\tau_j,x_j)$ on Σ in the sense that $x_j(t+\frac{\tau_j}{r})=Px_j(t)$ for all $t∈R$ with $j=1,2$. As a corollary we obtain the existence of two geometrically distinct central symmetric closed characteristics on any $C^2$ central symmetric compact convex hypersurface in R2n with $n\ge 2$.

We show that the spectrum of the algebra of bounded symmetric analytic functions on l p , 1 ≤ p + ∞ with the symmetric convolution operation is a commutative semigroup with the cancellation law for which we discuss the existence of inverses. For p = 1 , a representation of the spectrum in terms of entire functions of exponential type is obtained which allows us to determine the invertible elements.

A module of a finite group over a finite field with a symmetric non-degenerate bilinear form which is invariant by the group action is called a symmetric module. In this paper, a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained, and some applications to the self-dual permutation codes are shown.

The W-graph representations for symmetric groups Gn and their Hecke algebras have been studied by many authors, but the situation for a large n (for instance n ≥ 16) has not been completely solved. The aim of this note is to provide an algorithm of "inducing W-graphs" step by step from smaller n, to construct W-graphs for the left cells of the symmetric groups for bigger n.

Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of symmetric functions are studied. This is a detailed exposition of part of the results announced in arXiv:1009.2037. The work is motivated by a connection with a model of infinite-dimensional Markov dynamics., Comment: Latex, 52pp

A graph is called symmetric if its automorphism group acts transitively on its arc set. In this paper, we classify all connected cubic symmetric graphs of order 16p2 for each prime p.

In this paper we study properties of complex symmetric operators. In particular, we prove that every complex symmetric operator having property ( β ) or ( δ ) is decomposable. Moreover, we show that complex symmetric operator T has Dunfordʼs property ( C ) and it satisfies Weylʼs theorem if and only if its adjoint does.

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

We give explicit formulae of Whittaker vectors for Virasoro algebra in terms of the Jack symmetric polynomials. Our fundamental tools are the Feigin-Fuchs bosonization and the split expression of the Calogero-Sutherland model given by Awata-Matsuo-Odake-Shiraishi., Comment: revised, 22 pages

A complete list of indecomposable characters of the infinite symmetric semigroup is given. In comparison with a similar list for the infinite symmetric group, only one new parameter appears, which has a clear combinatorial meaning. The results rely on the representation theory of finite symmetric semigroups and the representation theory of the infinite symmetric group.

We completely describe the determinants of the sum of orbits of two real skew symmetric matrices, under similarity action of orthogonal group and the special orthogonal group respectively. We also study the Pfaffian case and the complex case.

We obtain formulas for certain weighted sums of values of the symmetric square and triple product L-functions. As a consequence, we get exact values at the right critical point for the symmetric square and symmetric cube L-functions attached to certain cuspforms. We also give applications to Fourier coefficients of modular forms.

By using as main tool the theory of prime submodules this article is devoted to describing the structure of the minimal prime ideals of the equidimensional symmetric algebra of a finitely generated module.

We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of the quaternion group.

In this contribution we will revisit the celebrated theorem that every square matrix is similar to a (complex) symmetric matrix and that every symmetric matrix is orthogonally similar to a given normal canonical form. Specifically we will re-examine the proof as well as the derivation of an explicit n-dimensional complex symmetric form. We will extend the formula to incorporate the various powers of the original normal form, a derivation not previously provided. Some examples of these complex symmetric forms in chemical and physical applications are indicated. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

We introduce a class of Riemannian symmetric spaces, called Jordan symmetric spaces, which correspond to real Jordan triple systems and may be infinite dimensional. This class includes the symmetric R-spaces as well as the Hermitian symmetric spaces.

An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric., 9 pages