Subspaces,angles and pairs of orthogonal projections

We give a new proof for the Wedin theorem on the simultaneous unitary similarity transformation of two orthogonal projections and show that it is equivalent to Halmos' theorem on the unitary equivalence of projection pairs. As a consequence of these theorems, we derive several results on pairs of orthogonal projections, relative subspace positions and oblique projections as well.     

A reciprocity relation for WP-Bailey pairs

We derive a new general transformation for WP-Bailey pairs by considering a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.     

On pseudolinearity and generic pairs

We continue the study of the connection between the “geometric” properties of SU ‐rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU‐rank of the (complete) theory of generic pairs of models of an SU ‐rank 1 theory T can only take values 1 (if and only if T is trivial), 2 (if and only if T is linear) or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to linearity for ω ‐categorical SU ‐rank 1 structures, established in [7], from the conjecture that an ω ‐categorical supersimple theory has finite SU ‐rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some identities between basic hypergeometric series deriving from a new Bailey-type transformation

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.     

Condensation of fermion pairs in a domain

We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two-body potential which admits a bound state. The particles are confined to a domain with Dirichlet boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross–Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.     

Track formulas and derivations in simple jordan pairs

In finite dimensional Lie algebras, Jordan algebras, and other algebraic structures the study of derivations has been facilitated by having a nontrivial trace formula on hand (see for example [?]) . Tuere is no common pattern in proving these trace formulas, they all depend on the underlying structures. In this note we derive such a trace formula for finite dimensional central simple Jordan pairs. We use it to determine all derivations the Killing form and the dimensions of the derivation algebras of the Jordan pairs. Dur primary tool is a Trace Reduction Formula.     

An update on primitive ternary complementary pairs

In two recent papers we overhauled the theory of ternary complementary pairs, focusing on questions relating to the possible weights of pairs, and special pairs from which all others can be derived, which we call “primitive.”Of particular interest at this time is a new refinement of the concept of primitivity, which necessitates some revisions to our tables. In this article we report on the state of the art with respect to primitive pairs and elaborate on some conjectures in light of new data.30 new primitive pairs are given; the status of 12 previously “primitive” pairs is changed to “imprimitive.”     

Overpartition pairs and two classes of basic hypergeometric series

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.     

Families of Okamoto–Painlevé pairs and Painlevé equations

In [as reported by Saito et al. (J. Algebraic Geom. 11:311–362, 2002)], generalized Okamoto–Painlevé pairs are introduced as a generalization of Okamoto’s space of initial conditions of Painlevé equations (cf. [Okamoto (Jpn. J. Math. 5:1–79, 1979)]) and we established a way to derive differential equations from generalized rational Okamoto–Painlevé pairs through deformation theory of nonsingular pairs. In this article, we apply the method to concrete families of generalized rational Okamoto–Painlevé pairs with given affine coordinate systems and for all eight types of such Okamoto–Painlvé pairs we write down Painlevé equations in the coordinate systems explicitly. Moreover, except for a few cases, Hamitonians associated to these Painlevé equations are also given in all coordinate charts. Mathematics Subject Classification (2000) 34M55, 32G05, 14J26     

q-Inverting pairs of shape 1, 2, 1

Let V denote a vector space with finite positive dimension over an algebraically closed field 𝔽. Let K, K* be a q-inverting pair on V, an ordered pair of invertible linear transformations on V satisfying certain conditions. In this article we study the q-inverting pairs with shape 1, 2, 1. We define a pair of scalars called the parameter pair for K, K*. We give six bases for V and give the action of K, K ^?1, K* and (K*)^?1 on each of these bases, respectively. We classify the q-inverting pairs of shape 1, 2, 1 in terms of the parameter pairs. We conclude with some trace formulae involving the parameter pair.     

Classification of pairs consisting of a linear and a semilinear map

L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices [5, Chapter XII]. See also Dickson's paper [3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.     

Moment maps and Riemannian symmetric pairs

We study Hamiltonian actions of a compact Lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold. The fixed-point set of the involution on the manifold is a Lagrangian submanifold. We investigate its image under the moment map and conclude that the intersection with the Weyl chamber is an easily described subpolytope of the Kirwan polytope. Of special interest is the integral K?hler case, where much stronger results hold. In particular, we obtain convexity theorems for closures of orbits of the noncompact dual group (in the sense of the theory of symmetric pairs). In the abelian case these results were obtained earlier by Duistermaat. We derive explicit inequalities for polytopes associated with real flag varieties. Received: 8 February 1999 / in revised form: 25 October 1999 / Published online: 8 May 2000     

Fixed points on model solvmanifold pairs

In this paper we define model solvmanifold pairs and their diagonal type selfmaps in the tradition of Heath and Keppelmann. We derive an explicit formula for computing the relative Nielsen number N(F;X,A) on these spaces and selfmaps F:(X,A)→(X,A). We find that model solvmanifold pairs often exhibit interesting Schirmer theory, meaning N(F;X,A)>max{N(F),N(F|_A)}.     

Smoothness of isotopy for symplectic pairs

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.     

Applications of some transformations for several variable-coefficient nonlinear evolution equations from plasma physics,arterial mechanics,nonlinear optics and Bose–Einstein condensates

Variable-coefficient nonlinear evolution equations have occurred in such fields as plasma physics, arterial mechanics, nonlinear optics and Bose–Einstein condensates. This paper is devoted to giving some transformations to convert the original nonlinear evolution equations, e.g., the variable-coefficient nonlinear Schrödinger, generalized Gardner and variable-coefficient Sawada–Kotera equations to simpler ones or even constant-coefficient ones. Based on some constraints, we simplify the original equations and derive the associated chirp solitons, Lax pairs, and Bäcklund transformations from the original equations by means of the aforementioned transformations.     

Further explorations into ternary complementary pairs

In [R. Craigen, C. Koukouvinos, A theory of ternary complementary pairs, J. Combin. Theory Ser. A 96 (2001) 358-375], we proposed a systematic approach to the theory of ternary complementary pairs (TCPs) and showed how all pairs known then could be constructed using a single elementary product, the natural equivalence relations, and a handful of pairs which we called primitive. We also introduced more new primitive pairs than could be inferred previously, concluding with some conjectures reflecting the patterns that were beginning to arise in light of the new approach.In this paper we take what appears to be the natural next step, by investigating these patterns among those lengths and weights that are within easy computational distance from the last length considered therein, length 14. We give complete results up to length 21, and partial results up to length 28. (Ironically, although we proceed analytically by weight first then length, for computational reasons we are bound, in this empirical investigation, to proceed according to length first.)Thus we provide support for the previous conjectures, and shed enough new light to speculate further as to the likely ultimate shape of the theory. Since short term work on TCPs will require massive acquisition of data about small pairs, we also discuss affixes—a computational strategy that arose out of the investigations culminating in this article.     

Vortex pairs and dipoles

Point vortices have been extensively studied in vortex dynamics. The generalization to higher singularities, starting with vortex dipoles, is not so well understood.We obtain a family of equations of motion for inviscid vortex dipoles and discuss limitations of the concept. We then investigate viscous vortex dipoles, using two different formulations to obtain their propagation velocity. We also derive an integro-differential for the motion of a viscous vortex dipole parallel to a straight boundary.     

On pairs of free modules over a Dedekind domain

The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable. Work performed under the auspices of GNSAGA-INDAM     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

Matched pairs of Courant algebroids

We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We also consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.