Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.     

Functions of Pairs of Unbounded Noncommuting Self-Adjoint Operators under Perturbation

Doklady Mathematics - For a pair $$(A,B)$$ of not necessarily bounded and not necessarily commuting self-adjoint operators and for a function f on the Euclidean space $${{\mathbb{R}}^{2}}$$ that...     

Multiplier and averaging operators in the Banach spaces $$\mathbf{ces(p), \, 1< p < \infty }$$ ces ( p ) , 1 < p < ∞

Positivity - The Banach sequence spaces ces(p) are generated in a specified way via the classical spaces $$ \ell _p, 1&lt; p &lt; \infty .$$ For each pair $$ 1&lt; p,q &lt; \infty...     

A family of WZ pairs and q-identities

The Ramanujan Journal - We observe that $$(F(n+k+1,k)+G(n+k,k), G(n+k,k))$$ is a WZ pair provided that (F(n,&nbsp;k),&nbsp;G(n,&nbsp;k)) is a WZ pair. This observation enables us to...     

On the occurrence of admissible representations in the real Howe correspondence in stable range

Let be an irreducible real reductive dual pair of type I in stable range, with G the smaller member. In this note, we prove that all irreducible genuine representations of occur in the Howe correspondence. The proof uses structural information about the groups forming a reductive dual pair and estimates of matrix coefficients.     

A conjecture on the lengths of filling pairs

Geometriae Dedicata - A pair $$(\alpha , \beta )$$ of simple closed geodesics on a closed and oriented hyperbolic surface $$M_g$$ of genus g is called a filling pair if the complementary components...     

Two-Sided Estimates of the $$K$$ -Functional for Spaces of Functions of Generalized Bounded Variation

Functional Analysis and Its Applications - A two-sided estimate is proposed for the $$K$$ -functional of the pair $$(C[0,1], BV(X))$$ , where $$BV(X)$$ is the space of functions of generalized...     

Integral formula for a generalized Sato-Levine invariant in magnetic hydrodynamics

Mathematical Notes - For a pair of divergence-free vector fields B and $$ \tilde B $$ respectively localized in two oriented tubes U and $$ \tilde U $$ in ?3, we propose a fourth-order...     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

On toric ideals arising from signed graphs

Journal of Algebraic Combinatorics - A signed graph is a pair $$(G,\tau )$$ of a graph G and its sign $$\tau $$ , where a sign $$\tau $$ is a function from $$\{ (e,v)\mid e\in E(G),v\in V(G), v\in...     

New partitionings of complete designs

Designs, Codes and Cryptography - A simple $$(2,3,v)$$ trade is a pair $$(T_0,T_1)$$ of disjoint sets of $$3$$ -subsets (blocks) of a $$v$$ -set such that any two elements meet the same number of...     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair As a consequence, we prove that a linear skew-product flow on is uniformly exponentially dichotomic if and only if the pair is uniformly admissible for .     

Minimality of invariant submanifolds in metric contact pair geometry

Annali di Matematica Pura ed Applicata (1923 -) - We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable structure tensor $$\phi $$ ....     

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

Theoretical and Mathematical Physics - The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously...     

On Global Optimality Conditions for Nonlinear Optimal Control Problems

Let a trajectory and control pair maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g( (T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x,u) from the above-mentioned level set. In particular, if the pair satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair on [0,T], and if there is no another pair (x,u) such that g(x(T))=g( (T)), then is a global maximizer.     

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set of K-orbits in the dual of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.     

A pair of equations in four prime squares and powers of 2

The Ramanujan Journal - In this paper, we obtain when $$k=98$$ , every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of four prime...