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A left canonical factorization theorem for rational matrix functions relative to the unit circle is presented. The result is a time invariant version of a recent strict LU factorization theorem for certain semi-separable operators, due to Dewilde (2012) [6]. Explicit formulas for the factors are also given. The theorem is proved, first by using the state space method of Dewilde (2012) [6], and next by using an operator theory approach. In both cases the main part of the proof concerns rational matrix functions that are unitary on the unit circle. © 2012 Royal Dutch Mathematical Society (KWG).

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.

In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a , generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations. The recipe involves folding and contracting a -invariant and θ -twisted symmetric classical dynamical r -matrices along an involutive automorphism θ . In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r -matrices which are a -invariant and θ -twisted. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.

In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators { A ( t ) : t ∈ [ 0 , b ] } generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C 0 − semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

Let M m be a closed smooth manifold equipped with a smooth involution having fixed point set of the form F n ∪ F 3 , where F n and F 3 are submanifolds with dimensions n and 3, respectively, where 3 n m and with the normal bundles over F n and F 3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4 , which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M ( n − 3 ) + 6 , where M ( n ) is the Stong–Pergher number (see the definition of M ( n ) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M ( n − 3 ) + 5 , which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n .

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices., 17 pages; The paper was revised and some material is added to it. To appear in Indag. Math

In this paper, the Taylor sequence space t p θ of non-absolute type is introduced which consists of all sequences whose Taylor transforms of order θ , ( 0 θ 1 ) , are in the space l p . It is shown that the space t p θ is a normed space which includes the space l p where 1 ≤ p ≤ ∞ . Moreover, in this paper a general upper estimate is obtained for the norm of Hausdorff matrices and Norlund matrices as operators from l p into the Taylor sequence space t p θ . Finally, the results are extended to the matrix domain of an arbitrary invertible matrix E in the sequence space l p .

In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.

In the first part of the paper we introduce some geometric tools needed to describe slow–fast Hamiltonian systems on smooth manifolds. We start with a smooth bundle p : M → B where ( M , ω ) is a C ∞ -smooth presymplectic manifold with a closed constant rank 2-form ω and ( B , λ ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ω e = ω + e − 1 p ∗ λ on M for any positive small e , where p ∗ λ is the lift of the 2-form λ to M . Given a smooth Hamiltonian H on M one gets a slow–fast Hamiltonian system with respect to Ω e . We define a slow manifold S M for this system. Assuming S M is a smooth submanifold, we define a slow Hamiltonian flow on S M . The second part of the paper deals with singularities of the restriction of p to S M . We show that if dim M = 4 , dim B = 2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painleve-I, and if this singularity is a cusp, then the related equation is Painleve-II.

This paper is concerned with nonlinear general fractional differential equations. The existence and uniqueness of the solutions of above equations are obtained by using a new norm and Banach contraction principle. The results are the generalization of Theorem 1.1 of the paper (Jankowski, 2013).

Let C be the complex Levi-Civita field and let c 0 ( C ) or, simply, c 0 denote the space of all null sequences z = ( z n ) n ∈ N of elements of C . The natural inner product on c 0 induces the sup-norm of c 0 . In a previous paper Aguayo et al. (2013), we presented characterizations of normal projections, adjoint operators and compact operators on c 0 . In this paper, we work on some B ∗ -algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators. We finish the paper with a characterization of closed subspaces of the B ∗ -algebra of all adjoint and compact operators on c 0 which admit normal complements.

We study in this paper, a one parameter deformation of the q -Laguerre weight function. An investigation is made on the polynomials orthogonal with respect to such a weight. With the aid of the two compatibility conditions previously obtained in Chen and Ismail (1997) and the q -analog of a sum rule obtained in this paper, we derive expressions for the recurrence coefficients in terms of certain auxiliary quantities, and show that these quantities satisfy a pair of first order non linear difference equations. These difference equations are similar in form to the recognized asymmetric discrete Painleve systems such as α q -P-IV and α q -P-V.

The Calkin theorem provides a one-to-one correspondence between all operator ideals A(H) over the separable infinite-dimensional Hilbert space H and all symmetric sequence ideals a(N) over the index set N≔{1,2,…}. The main idea of the present paper is to replace a(N) by the ideal z(N0) that consists of all sequences (αh) indexed by N0≔{0,1,2,…} for which (α0,α1,α1,…,αh,…,αh︷2hterms,…)∈a(N). This new kind of sequence ideals is characterized by two properties: (1) For (αh)∈z(N0) there is a non-increasing (βh)∈z(N0) such that ∣αh∣≤βh. (2) z(N0) is invariant under the operator S+:(α0,α1,α2,…)↦(0,α0,α1,…). Using this modification of the Calkin theorem, we simplify, unify, and complete earlier results of [4,5,7–9,13,14,19–21,25] The central theorem says that there are canonical isomorphisms between the linear spaces of all traces on A(H), all symmetric linear forms on a(N), and all 12S+-invariant linear forms on z(N0). In this way, the theory of linear forms on ideals of a non-commutative algebra that are invariant under the members of a non-commutative group is reduced to the theory of linear forms on ideals of a commutative algebra that are invariant under a single operator. It is hoped that the present approach deserves the rating “streamlined”. Our main objects are linear forms in the purely algebraic sense. Only at the end of this paper continuity comes into play, when the case of quasi-normed ideals is considered. We also sketch a classification of operator ideals according to the existence of various kinds of traces. Details will be discussed in a subsequent publication.

In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λ φ initiated in the papers of Foralewski et al. (2008) [16] , [17] (cf. also Foralewski (2011) [11] , [12] ). As we will show in Example 1.1 , Example 1.2 , Example 1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ 2 λ , then λ φ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λ φ , a strong condition of δ 2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λ φ , of their subspaces of order continuous elements ( λ φ ) a as well as of finite dimensional subspaces λ φ n of λ φ are presented.

In our paper [10] we discussed an error in the recent literature on abstract fractional differential equations and we proposed a different approach to treat these type of problems. By noting that the results by Hernandez et al. (2010) [10] are not applicable for problems with nonlocal conditions, in this paper we study the existence of mild solutions for a class of abstract fractional differential equations with nonlocal conditions. An application involving a partial differential equation with a fractional temporal derivative and nonlocal conditions is considered.

This paper deals with the semi-implicit scheme with respect to the t -variable combined with a finite element spatial approximation of evolutionary Hamilton–Jacobi–Bellman equations with nonlinear source terms. We establish a convergence and a quasi-optimal L ∞ -asymptotic behavior, involving a weakly coupled system of discrete parabolic quasi-variational inequalities (PQVIs), for the solution of which an iterative discrete scheme of monotone kind is introduced and analyzed. Furthermore, the simple numerical example shows that the estimates introduced in this paper are efficient.

Invertibility in Banach algebras generated by two idempotents can be checked with the help of a theorem by Roch, Silbermann, Gohberg, and Krupnik. This theorem cannot be used to study generalized invertibility. The present paper is devoted to group invertibility in two types of finite-dimensional algebras which are generated by two idempotents, algebras generated by two tightly coupled idempotents on the one hand and algebras of dimension at most four on the other. As a side product, the paper gives the classification of all at most four-dimensional algebras which are generated by two idempotents.

In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

This paper is mainly concerned with the structure of the centre of a vector lattice. A special attention is paid in the case when E = L p , p ≥ 1 . In this paper we give some characterizations of dense vector sublattices of the centre. Those characterizations will be applied in several directions. At the end of this work we compare various fullness and richness properties of the centre of a vector lattice.

This paper is a posthume continuation of the papers by Professor A.H. Boers about duplication of algebras, as drafted by Artibano Micali and Moussa Ouattara in collaboration with Nakelgbamba Boukary Pilabre. The aim of the paper is essentially to study the non commutative duplication of algebras. Conditions are studied in which the duplicate algebra is flexible, Lie admissible or n -associative. Connections between the dimensions of commutative and non commutative duplication of algebras are compared. In the list of references in this paper, several items have been given, which are directly connected to the subject of study. Moreover, for the interest of the reader, a (possible) complete list of publications of A.H. Boers is added to the references, i.e. known up to and including 1995 as well as a note explicative.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

A fundamental result of Springer says that a quadratic form over a field of characteristic ≠ 2 is isotropic if it is so after an odd degree field extension. In this paper we generalize Springer’s theorem as follows. Let R be an arbitrary semilocal ring, let S be a finite R -algebra of constant odd degree, which is etale or generated by one element, and let q be a nonsingular R -quadratic form whose base ring extension q S is isotropic. We show that then already q is isotropic.

This paper is a subsequent paper of Melnikov and Pagnon: Reducibility of the intersections of components of a Springer fiber, Indag. Mathem. 19 (4) (2008) 611–631. Here we consider the irreducible components of a Springer fibre (or orbital varieties) for the two-column case in GLn (ℂ). We describe the intersection of two irreducible components, and specially give the necessary and sufficient condition for this intersection to be of codimension one. Since an orbital variety in the two-column case is a finite union of the Borel orbits, we solve the initial question by determining orbits of codimension one in the closure of a given orbit. We show that they are parameterized by a specific set of involutions called descendants, already introduced by the first author in a previous work. Applying this result we show that the codimension one intersection of two components is irreducible and provide the combinatorial description in terms of Young tableaux of the pairs of such components.

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

For r = ( r 1 ,…, r d ) ∈ ℝ d the mapping τ r :ℤ d →ℤ d given by τ r ( a 1 ,…, a d ) = ( a 2 , …, a d ,−⌊ r 1 a 1 +…+ r d a d ⌋) where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ d there exists an integer k > 0 with τ r k ( a ) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β -expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β -expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral., Comment: 20 pages; added in v3: more references to earlier occurrences of the identity and its multivariable analogue, combinatorial proof of the identity and extension to noninteger m,n, proof of multivariable identity by splitting up Dirichlet's multivariable beta integral

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted., Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new results

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997) 307–321] the authors give a condition for a 1-form β to be the perturbation of a Riemannian manifold ( M , α) such that the manifold equipped with any (α, β) -metric is a Wagner manifold with respect to the Wagner connection induced by β. The condition shows that its covariant derivative with respect to the Levi-Civita connection must be of a special form ( ∇ β ) ( X , Y ) = ∥ β # ∥ 2 α ( X , Y ) − β ( X ) β ( Y ) In this paper we give a family of Riemannian manifolds admitting nontrivial solutions and it will be proved as a structure theorem that there are no further essentially different examples; in particular we consider the classical hyperbolic space together with a nontrivial solution. Examples of Wagnerian Finsler manifolds are given in the last section.

In the present paper we are investigating a certain point measure of a distribution function arising in a paper by Grabner et al. [Combinatorica 22 (2002) 245–267]. This distribution function is defined by means of the subtractive Euclidean algorithm and bears a striking resemblance to the singular?(x)-function of H. Minkowski. Beyond it, we will also consider a whole family of distribution functions arising in a natural way from the above ones. Nevertheless we will prove that all of the corresponding measures of the mentioned functions are mutually singular by using dynamical systems and the ergodic theorem.

A partial flag variety P K of a Kac–Moody group G has a natural stratification into projected Richardson varieties. When G is a connected reductive group, a Bruhat atlas for P K was constructed in He et al. (2013): P K is locally modelled with Schubert varieties in some Kac–Moody flag variety as stratified spaces. The existence of Bruhat atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac–Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff–Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the J -Schubert varieties for a Birkhoff–Bruhat atlas. The notion of the J -Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac–Moody group). The main result of this paper is the construction of a Birkhoff–Bruhat atlas for any partial flag variety P K of a Kac–Moody group. We also construct a combinatorial atlas for the index set Q K of the projected Richardson varieties in P K . As a consequence, we show that Q K has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams (2007) in the case where the group G is a connected reductive group.

This paper presents a connection between the defining basis presented by Beilinson-Lusztig-MacPherson [1] in their geometric setting for quantum GLn and the isomorphism classes of linear quiver representations. More precisely, the positive part of the basis in [1] identifies with the defining basis for the relevant Ringel-Hall algebra; hence, it is a PBW basis in the sense of quantum groups. This approach extends to q-Schur algebras, yielding a monomial basis property with respect to the Drinfeld-Jimbo type presentation for the positive (or negative) part of the q-Schur algebra. Finally, the paper establishes an explicit connection between the canonical basis for the positive part of quantum GLn and the Kazhdan-Lusztig basis for q-Schur algebras.

In this paper we make a contribution to the theory of necessary conditions for optimization problems. Tikhomirov has given in [T] a unified necessary condition for mixed (smooth-convex) problems. In this paper we develop the smooth case, one of the two building blocks. We offer a new, geometrical construction giving a necessary condition under minimal assumptions.

In this paper we determine explicit models for the unitary representations which occur discretely in the decomposition of canonical representations for hyperbolic spaces. This generalizes work of Gelfand, Graev and Vershik for the case SU(1,1). In a previous paper [1] we have introduced canonical representations π λ for hyperbolic spaces of Riemannian type, generalizing Berezin's definition for the case of Hermitian symmetric spaces. These unitary representations have a rich internal structure and prove that not only quasi-regular representations are important in harmonic analysis. They have an interesting decomposition into irreducible constituents: for large λ only unitary principal series representations occur, for small λ however a finite number of complementary series representations has to be added. A detailed description of the latter, as subrepresentations of π λ , was started in [1]. The present paper contains much more precise results. The classical notions of Fourier and Poisson transform play a dominant role and are treated in Section 4. In Section 5 the actual projection of a given function onto the complementary series is then given. Since the proof makes use of some results on maximal degenerate series representations of SL( n + 1), these representations are studied in Section 1 in some detail. As a by-product we obtain an alternative introduction of the canonical representations, see Sections 2 and 5. This paper has a considerable, but unevitable optical overlap with our joint paper [2] with V.F. Molchanov. The method described here has been introduced in a different setting by Molchanov, but the observation that it works also for hyperbolic spaces is new and surprising. To leave out parts of the proof would make this paper almost unreadable. We thank S.C. Hille and V.F. Molchanov for several discussions on these matters and for their encouragement to publish the results obtained in this paper.

In this paper, the mixed L p -surface area measures are defined and the mixed L p Minkowski inequality is obtained consequently. Furthermore, the mixed L p projection inequality for mixed projection bodies is established.

Let f ( x ) be a non-zero polynomial with complex coefficients, and M p = ∫ 0 1 f ( x ) p d x for p a positive integer. In a recent paper, Muger and Tuset showed that lim sup p → ∞ | M p | 1 ∕ p > 0 , and conjectured that this limit is equal to the maximum amongst the critical values of f together with the values | f ( 0 ) | and | f ( 1 ) | . We give an example that shows that this conjecture is false. It also may be natural to guess that lim sup p → ∞ | M p | 1 ∕ p is equal to the maximum of | f ( x ) | on [ 0 , 1 ] . However, we give a counterexample to this as well. We also provide a few more guesses as to the behavior of the quantity lim sup p → ∞ | M p | 1 ∕ p .

A study of nonperiodic sequences {w j } jϵ Z which are intertwinings of two periodic sequences with maximal overlap. Expressions in terms of Beatty sequences {⌋ci + d ⌋} iϵ Z are given in case (i) both periodic sequences have the same periodic cycle and in case(ii) the periodic sequences have coprime periods. The extreme sequences in case (i) occur in a paper of Morse and Hedlund [9] and in case (ii) in a paper by Coven and Hedlund [3].

The goal of this paper is to discuss a relation between microlocal solvability and an existence of microlocally complete set of eigenfunctions. Any differential operator with constant coefficients is locally solvable and this property can be derived from the existence of a complete set of eigenfunctions ei〈x,ξ〉. In this paper we prove that Hörmander's solvability condition {Rep, Imp} < 0 implies the existence of microlocally complete system of eigenfunctions and from this follows solvability. When a complete system of eigenfunctions is constructed, an operator with variable coefficients resembles very much the one with constant coefficients; for example one can introduce an analog of Fourier Transform and construct a parametrix exactly in the same way as it is done for operators with constant coefficients.

Let f be an analytic disc in C N attached to a maximal real submanifold M of C N . In a recent paper the author introduced partial indices κj, 1 ≤ j ≤ N, of M along the boundary of f and showed that if κj ≥ 0 for all j then the family of nearby analytic discs attached to M depends on κ1 + … + κN parameters. Y.-G. Oh sharpened this by proving the same when κj ≥ −1 for all j and showed that in terms of stability this is the best possible condition. In the present paper we explain why the latter condition is natural and give a simple proof of Oh's result in the orientable case.

This paper contains theorems of r -th order Frechet differentiability, with r ≥1, for the autonomous composition operator and for the inversion operator in Schauder spaces. The optimality of the differentiability theorems for the composition is indicated by means of an ‘inverse result’. A main point of this paper is that (higher order) ‘sharp’ differentiability theorems for the composition operator can be proved by approximating the operator by composition operators whose superposing function is a polynomial, an idea which may be employed in other function space settings.

The concept of characteristic manifold is very important in PDE, but it takes into account only the principal symbol. This paper is one of several papers devoted to the following problem: how can we construct the “characteristic manifold” that takes into account lower order terms and at the same time describes the intrinsic properties of the operator (like hypoellipticity or solvability) in the same way as the classical characteristic manifold does.

In recent years several papers have been published in quantum mechanical computation, kinetic theory, statistical applications and so forth, dealing with the so-called non-classical orthogonal polynomials. It happens that all these polynomials share a lot of properties, they belong to what we call semi-classical polynomials, i.e. orthogonal sequences whose derivative sequence is quasi-orthogonal of a certain order. This paper concentrates on describing the semi-classical linear functionals of class one. In the first place, we review the characterizations of semi-classical orthogonal polynomials, secondly, we produce the eight canonical forms of the functional equations satisfied by linear functionals of class one, thirdly, after rescaling the parameters (the rescaling suggests an obvious similarity with the classical situation) we establish the eight irreducible canonical functional equations (ICFE) of class one. As solutions to these ICFE we propose integral representations of the linear functionals of class one, and we point out the cases encountered in application fields. Finally, we give the non-linear systems satisfied by the coefficients of the three-term recurrence relation of semi-classical polynomials of class one.

In this paper a duality theory for morphism X→Y of rigid analytic spaces over some non-archimedean valued complete field K is developed. In order to keep the statements and the proofs reasonable we make the restrictions Y is affinoid and X→Y is smooth and proper. The successful approach of B. Chiarellotto [Ch] for Stein spaces is the basis for this paper. An essential step is to show a strong form of unicity for the residue maps Res. One uses further a suitable definition for cohomology with compact support; the work of R. Kiehl [Kl] on proper mappings; geometric points as introduced in [P]; the strong G-topology of [K1,K2] in the version of [BGR]. As one expects the paper is rather technical in nature and the results are not surprising. However for the investigation of proper rigid analytic manifolds the Serre-duality can not be missed. We thank B. Chiarellotto for his careful reading of the manuscript, which has led to improvements of the paper.