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An error is pointed out in the paper [J.R. Kang, Y.C. Kwun, J.Y. Park, Controllability of second order differential inclusion in Banach spaces, J. Math. Anal. Appl. 285 (2003) 537-550]. By an additional condition the considered system is controllable.

W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arandelovic [I.D. Arandelovic, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385] established somewhat more general version of Kirk's result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arandelovic's theorem is false. We also explain how to correct this result.

In this short paper, we study some trace inequalities of the products of the matrices and the power of matrices by the use of elementary calculations.

The field of linear algebra over the quaternions is a research area which is still in development. In this paper we continue our research on canonical forms for a matrix pair ( A , H ) , where the matrix A is H-unitary, H is invertible and with A as well as H quaternionic matrices. We seek an invertible matrix S such that the transformations from ( A , H ) to ( S − 1 A S , S ⁎ H S ) brings the matrix A in Jordan form and simultaneously brings H into a canonical form. Canonical forms for such pairs of matrices already exist in the literature, the goal of the present paper is to add one more canonical form which specifically keeps A in Jordan form, in contrast to the existing canonical forms.

Integral invariants obtained from Principal Component Analysis on a small kernel domain of a submanifold encode important geometric information classically defined in differential-geometric terms. We generalize to hypersurfaces in any dimension major results known for surfaces in space, which in turn yield a method to estimate the extrinsic and intrinsic curvature tensor of an embedded Riemannian submanifold of general codimension. In particular, integral invariants are defined by the volume, barycenter, and the EVD of the covariance matrix of the domain. We obtain the asymptotic expansion of such invariants for a spherical volume component delimited by a hypersurface and for the hypersurface patch created by ball intersections, showing that the eigenvalues and eigenvectors can be used as multi-scale estimators of the principal curvatures and principal directions. This approach may be interpreted as performing statistical analysis on the underlying point-set of a submanifold in order to obtain geometric descriptors at scale with potential applications to Manifold Learning and Geometry Processing of point clouds. We would like to thank Louis Scharf for very helpful discussions during the writing of this paper. This paper is based on research partially supported by the National Science Foundation under Grants No. DMS-1513633, and DMS-1322508.

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17] , the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17] , our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin–Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant r x defined by Kato (1994) [1] . The main theme of this paper is to construct the Young diagram Y ( X , D , x ) which is closely related to r x and to prove Kato's conjecture Kato (1994) [1] for an upper bound of r x for a good Artin–Schreier extension.

In an earlier paper (R. Bhatia, T. Jain, Higher order derivatives and perturbation bounds for determinants, Linear Algebra Appl. 431 (2009) 2102–2108) we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant. In this paper we continue that work, and find expressions for the derivatives of all orders for the antisymmetric tensor powers and for the coefficients of the characteristic polynomial. We then evaluate norms of these derivatives, and use them to obtain perturbation bounds.

Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G × g , which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula.

This paper is devoted to the investigation of Lie algebras of local infinitesimal CR automorphisms. Such algebras are naturally associated to germs of homogeneous CR manifolds. The authors introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair $(L,L_1)$(L,L1), consisting of a real Lie algebra $L$L and a subalgebra $L_1$L1 of the complexification $\bold C\otimes_{\bold R} L$C⊗RL, such that the factor space $L/L\cap L_1$L/L∩L1 is finite-dimensional. The authors investigate some formal properties of CR algebras and construct some "fibrations'' (i.e., $L$L-equivariant submersions) of such algebras. They introduce three new notions of nondegeneracy of CR algebras---strict, weak and ideal nondegeneracy. These three concepts are weaker than those used previously by some other authors. The authors intend to extend the application of the E. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds. One of the main ideas of this paper is a decomposition of arbitrary CR algebras into three "parts'': totally real, totally complex and weakly nondegenerate CR algebras (Theorems 5.3 and 5.4). There are some results about these three special classes of CR algebras. Some results about prolongations for transitive CR algebras are also obtained, in particular about maximality of parabolic CR algebras with respect to transitive prolongations.

The purpose of this paper is to characterize QM-abelian surfaces which has a model of GL 2 -type over Q . The “special” involutions on a corresponding indefinite quaternion algebra (which is defined in Section 2) play an essential role in this paper.

We study the oscillation problems for the second order half-linear differential equation [ p ( t ) Φ ( x ′ ) ] ′ + q ( t ) Φ ( x ) = 0 , where Φ ( u ) = | u | r − 1 u with r > 0 , 1 / p and q are locally integrable on R + ; p > 0 , q ⩾ 0 a.e. on R + , and ∫ 0 ∞ p − 1 / r ( t ) d t = ∞ . We establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When p ≡ 1 , our results are complete extensions of work by Huang [C. Huang, Oscillation and nonoscillation for second order linear differential equations, J. Math. Anal. Appl. 210 (1997) 712–723] and by Wong [J.S.W. Wong, Remarks on a paper of C. Huang, J. Math. Anal. Appl. 291 (2004) 180–188] on linear equations to the half-linear case for all r > 0 . These results provide corrections to the wrongly established results in [J. Jiang, Oscillation and nonoscillation for second order quasilinear differential equations, Math. Sci. Res. Hot-Line 4 (6) (2000) 39–47] on nonoscillation when 0 r 1 and on oscillation when r > 1 . The approach in this paper can also be used to fully extend Elbert's criteria on linear equations to half-linear equations which will cover and improve a partial extension by Yang [X. Yang, Oscillation/nonoscillation criteria for quasilinear differential equations, J. Math. Anal. Appl. 298 (2004) 363–373].

In this paper we apply the Garland–Lepowsky Theorem to compute the homology of the Lie algebra N=sl n (t C [t]) . By suitably taking Euler characteristics we derive a theorem of Stembridge which gives a combinatorial decomposition of a virtual character of SL n ( C ) . In his theorem, Stembridge indexes the Schur functions that appear in his decomposition by an integer vector of length n and a permutation in Sn. An additional feature of our approach is that we interpret the integer vector and permutation in terms of the affine Weyl group of the Kac–Moody Lie algebra of type A. A second result, given in the Garland–Lepowsky paper (extending an earlier result of Kostant), is an explicit formula for the eigenvalues of the Laplacian in the Koszul complex for computing the homology of N. We give an alternative quite different in nature description of those eigenvalues. The Garland–Lepowsky formula is algebraic, stated in terms of differences of squared norms. Our description is combinatorial, stated in terms of structural features of certain graphs. In order to make this combinatorial interpretation seemingly more natural, we link it directly with the paper of Kostant. We work in this part with the Lie algebra L=T⊕N where T is the span of x⊗1 with x taken from the strictly upper triangular matrices in sl n ( C ) .

The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S- and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets and strict RS-sets. Our results obtained in the present paper extend and improve all earlier known results in this direction.

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively.

In this paper, the authors first show that the classical Hardy space H 1 ( R n ) can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups e − t P and e − t P , respectively, where P is an elliptic operator with real constant coefficients of homogeneous order 2m ( m ⩾ 1 ). Moreover, the authors also prove that H 1 ( R n ) can be characterized by the Riesz transforms ∇ m P − 1 / 2 if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space H L 1 ( R n ) and give its characterizations by area integrals related to the semigroups e − t L and e − t L , respectively. Finally, authors give the ( H L 1 , L 1 ) boundedness of Riesz transforms, square functions and maximal functions associated with L.

Let R be a local ring essentially of finite type over a field k of characteristic p > 0 . In the paper [M. Furuya, H. Niitsuma, Regularity criterion of Noetherian local rings of prime characteristic, J. Algebra 247 (2002) 219–230], we constructed some regularity criterion for such a local ring R in terms of the higher differential algebra and the p n -basis. In this paper, we introduce the concept of a reduced index of a Noetherian ring and we show the sharpened result of the above criterion and further we also give a geometric regularity criterion in terms of the higher differential algebra and the p n -basis. The latter criterion yield the sharpened result of a part of Orbanz's theorem [U. Orbanz, Hohere Derivationen und Regularitat, J. Reine. Angew. Math. 262/263 (1973) 194–204, 4.2].

We consider a 2 m th order elliptic operator of divergence form in a domain Ω of R n , whose leading coefficients are uniformly continuous. In the paper [Y. Miyazaki, The L p theory of divergence form elliptic operators under the Dirichlet condition, J. Differential Equations 215 (2005) 320–356], we developed the L p theory including the construction of L p resolvents, assuming that the boundary of Ω is of class C m + 1 . The purpose of this paper is to show that the L p theory also holds when Ω is a C 1 domain, applying the inequalities of Hardy type for the Sobolev spaces.

This paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type; the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels. Furthermore, the paper shows that the normed rings on L 1 ( R d ) are constructed by using the obtained convolutions, and an arbitrary Hermite function and appropriate linear combination of those functions are the weight-function of four generalized convolutions associating F and F ˇ . The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.

In this paper, we prove that the Cauchy problem to a hyperbolic conservation laws with relaxation with singular initial data admits a unique global entropy solution in the sense of Definition 1.1. Compared with former results in this direction, the main ingredient of this paper lies in the fact that it contains a uniqueness result and we do not ask f (u) to satisfy any convex, monotonic conditions and the regularity assumption we imposed on f (u) is weaker. (C) 2003 Elsevier Inc. All rights reserved.

In a 1992 paper, Witten gave a formula for the intersection pairings of the moduli space of flat $G$-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities., 46 pages

In this paper we initiate the study of composition operators on the noncommutative Hardy space $H^2_{\bf ball}$. Several classical results about composition operators (boundedness, norm estimates, spectral properties, compactness, similarity) have free analogues in our noncommutative multivariable setting. The most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of $B(\cH)^n$, the operator algebras generated by the left creation operators on the full Fock space with $n$ generators, and the classical complex function theory in the unit ball of $\CC^n$., 39 pages

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number., Comment: 12 pages

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron–Frobenius operators, invariant measures and representations of the Cuntz–Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz–Krieger algebras O A for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of O A . We use these representations to describe the Perron–Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.

In this paper a numerical method to compute principal component geodesics for Kendall’s planar shape spaces–which are essentially complex projective spaces–is presented. Underlying is the notion of principal component analysis based on geodesics for non-Euclidean manifolds as proposed in an earlier paper by Huckemann and Ziezold [S. Huckemann, H. Ziezold, Principal component analysis for Riemannian manifolds with an application to triangular shape spaces, Adv. Appl. Prob. (SGSA) 38 (2) (2006) 299–319]. Currently, principal component analysis for shape spaces is done on the basis of a Euclidean approximation. In this paper, using well-studied datasets and numerical simulations, these approximation errors are discussed. Overall, the error distribution is rather dispersed. The numerical findings back the notion that the Euclidean approximation is good for highly concentrated data. For low concentration, however, the error can be strongly notable. This is in particular the case for a small number of landmarks. For highly concentrated data, stronger anisotropicity and a larger number of landmarks may also increase the error.

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock–Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for L d -almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L ∞ -setting (and, conversely, if existence and uniqueness of martingale solutions is known for L d -a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker–Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker–Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L 2 -setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L ∞ -setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions.

In this paper we study the monotonicity properties of some functions involving the Mills' ratio of the standard normal law. From these we deduce some new functional inequalities involving the Mills' ratio, and we show that the Mills' ratio is strictly completely monotonic. At the end of this paper we present some Turán-type inequalities for Mills' ratio.

For an Artinian $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n(\geq 2)$, we show that if $\Lambda$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\mod\Lambda$, then $\Lambda$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\mod\Lambda$ is at most $n-1$. As a result, for an Artinian Auslander algebra with global dimension 2, if $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of $\mod\Lambda$, then $\Lambda$ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra $\Lambda$ over an algebraically closed field $K$, we show that $\Lambda$ is a basic and connected $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n(\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\mod\Lambda$ if and only if $\Lambda$ is given by the quiver: $$\xymatrix{1 & \ar[l]_{\beta_{1}} 2 & \ar[l]_{\beta_{2}} 3 & \ar[l]_{\beta_{3}} ... & \ar[l]_{\beta_{n}} n+1} $$ modulo the ideal generated by $\{\beta_{i}\beta_{i+1}| 1\leq i\leq n-1 \}$. As a consequence, we get that a finite-dimensional algebra over an algebraically closed field $K$ is an $(n-1)$-Auslander algebra with global dimension $n(\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory if and only if it is a finite direct product of $K$ and $\Lambda$ as above. Moreover, we give some necessary condition for an Artinian Auslander algebra admitting a non-trivial maximal 1-orthogonal subcategory., Comment: 25 pages. This version is a combination of the orginal version of this paper with "From Auslander Algebras to Tilted Algebras" (arXiv:0903.0760). The latter paper has been withdrawn

Using Hankel operators and shift-invariant subspaces on Hilbert space, this paper develops the theory of the integrable operators associated with soft and hard edges of eigenvalue distributions of random matrices. Such Tracy–Widom operators are realized as controllability operators for linear systems, and are reproducing kernels for weighted Hardy spaces, known as Sonine spaces. Periodic solutions of Hill's equation give a new family of Tracy–Widom type operators. This paper identifies a pair of unitary groups that satisfy the von Neumann–Weyl anti-commutation relations and leave invariant the subspaces of L 2 that are the ranges of projections given by the Tracy–Widom operators for the soft edge of the Gaussian unitary ensemble and hard edge of the Jacobi ensemble.

In previous paper [Ann. Comb. 4 (2000) 347–373], we established and proved two new q -beta integrals and two multivariate basic hypergeometric series associated with the root system of A m . In this paper we give a detailed proof of the multivariate basic hypergeometric series obtained in [Ann. Comb. 4 (2000) 347–373].

This paper is a sequel to our previous paper [11] . We prove existence of nested index filtrations for tubular singular semiflow perturbations. This generalizes a corresponding result from [7] and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow. This result implies tubular homology index braid and connection matrix continuation principles.

Let (X,d,μ) be a metric measure space satisfying the upper doubling and the geometrically doubling conditions in the sense of T. Hytönen. In this paper, the authors prove that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to either of the boundedness of T from the atomic Hardy space H1(μ) to L1,∞(μ) or from H1(μ) to L1(μ). To this end, the authors first establish an interpolation result that a sublinear operator which is bounded from H1(μ) to L1,∞(μ) and from Lp0(μ) to Lp0,∞(μ) for some p0∈(1,∞) is also bounded on Lp(μ) for all p∈(1,p0). A main tool used in this paper is the Calderón–Zygmund decomposition in this setting established by B.T. Anh and X.T. Duong.

The (univariate) t-distribution and symmetric V.G. distribution are competing models [D.S. Madan, E. Seneta, The variance gamma (V.G.) model for share market returns, J. Business 63 (1990) 511–524; T.W. Epps, Pricing Derivative Securities, World Scientific, Singapore, 2000 (Section 9.4)] for the distribution of log-increments of the price of a financial asset. Both result from scale-mixing of the normal distribution. The analogous matrix variate distributions and their characteristic functions are derived in the sequel and are dual to each other in the sense of a simple Duality Theorem. This theorem can thus be used to yield the derivation of the characteristic function of the t-distribution and is the essence of the idea used by Dreier and Kotz [A note on the characteristic function of the t-distribution, Statist. Probab. Lett. 57 (2002) 221–224]. The present paper generalizes the univariate ideas in Section 6 of Seneta [Fitting the variance-gamma (VG) model to financial data, stochastic methods and their applications, Papers in Honour of Chris Heyde, Applied Probability Trust, Sheffield, J. Appl. Probab. (Special Volume) 41A (2004) 177–187] to the general matrix generalized inverse gaussian (MGIG) distribution.

In this paper the approximation properties of Gamma operators G n are studied to the locally bounded functions and the absolutely continuous functions, respectively. Firstly, in Section 2 of the paper a quantitative form of the central limit theorem in probability theory is used to derive an asymptotic formula on approximation of Gamma operators G n for sign function. And then, this asymptotic formula combining with a metric form Ω x ( f , λ ) is used to derive an asymptotic estimate on the rate of convergence of Gamma operators G n for the locally bounded functions. Next, in Section 3 of the paper the optimal estimate on the first order absolute moment of the Gamma operators G n ( | t − x | , x ) is obtained by direct computations. And then, this estimate and Bojanic–Khan–Cheng's method combining with analysis techniques are used to derive an asymptotically optimal estimate on the rate of convergence of Gamma operators G n for the absolutely continuous functions.

Estimates of Kolmogorov $n$-widths $d_n(B_p^r, L^q)$ and linear $n$-widths $\da_n(B_p^r, L^q)$, ($1\leq q\leq \infty$) of Sobolev's classes $B_p^r$, ($r>0$, $1\leq p\leq \infty$) on compact two-point homogeneous spaces (CTPHS) are established. For part of $(p, q)\in[1,\infty]\times[1,\infty]$, sharp orders of $d_n(B_p^r, L^q)$ or $\da_n (B_p^r, L^q) $ were obtained by Bordin, Kushpel, Levesley and Tozoni in a recent paper `` J. Funct. Anal. 202 (2) (2003), 307--326''. In this paper, we obtain the sharp orders of $d_n(B_p^r, L^q)$ and $\da_n (B_p^r, L^q)$ for all the remaining $ (p,q)$. Our proof is based on positive cubature formulas and Marcinkiewicz-Zygmund type inequalities on CTPHS.

The classical Eckmann–Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann–Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Sym n from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one ( n − 1 ) -arrow algebras of A is isomorphic to the category of algebras of Sym n ( A ) . Under some mild conditions, we present an explicit formula for Sym n ( A ) which involves taking the colimit over a remarkable categorical symmetric operad. We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.

This paper is a continuation of the paper [G. Parmeggiani, Pushing up point stabilizers, I, J. Algebra 319 (9) (2008) 3854–3884]. Under the same hypotheses, we determine those amalgams which involve SL n ( q ) , n ⩾ 3 , as factors.

In this paper, we study the L2 functions on U(2n)/O(2n) and Mp(n,R). We relate them using the oscillator representation. We first study some isometries between various L2 spaces using the compactification we defined in [H. He, An analytic compactification of the symplectic group, J. Differential Geom. 51 (1999) 375–399]. These isometries were first introduced by Betten and Ólafsson in [F. Betten, G. Ólafsson, Causal compactification and Hardy spaces for spaces of Hermitian type, Pacific J. Math. 200 (2) (2001) 273–312].11I was informed by Prof. Ólafsson of his work shortly after I finished this paper. We then give a description of the matrix coefficients of the oscillator representation ω in terms of algebraic functions on U(2n)/O(2n). The structure of L2(U(2n)/O(2n)) enables us to decompose the L2 space of odd functions on Mp(n,R) into a finite orthogonal direct sum, from which an orthogonal basis for L2(Mp(n,R)) is obtained. In addition, our decomposition preserves both left and right Mp(n,R)-action. Using this, we define the signature of tempered genuine representations of Mp(n,R). Our result implies that every genuine discrete series representation occurs as a subrepresentation in one and only one of (⊗pω)⊗(⊗2n+1−pω∗) for p with a fixed parity, generalizing some result in [M. Kashiwara, M. Vergne, On the Segal–Shale–Weil representations and harmonic polynomials, Invent. Math. 44 (1978) 1–47]. Consequently, we prove some results in the papers by Adams and Barbasch [J. Adams, D. Barbasch, Genuine representations of the metaplectic group, Compos. Math. 113 (1) (1998) 23–66] and by Móeglin [C. Móeglin, Correspondance de Howe pour les paires reductives duales: quelques calculs dans le cas archimédien, J. Funct. Anal. 85 (1) (1989) 1–85] without going through the details of the Langlands–Vogan parameter. In a weak sense, our paper also provides an analytic alternative to the Adams–Barbasch theorem on Howe duality [R. Howe, Transcending invariant theory, J. Amer. Math. Soc. 2 (1989) 535–552].

Denote by [ X , Y ] the additive commutator XY − YX of two square matrices X , Y over a field F . In a previous paper, the possible eigenvalues, ranks and numbers of nonconstant invariant polynomials of [⋯[[ A , X 1 ], X 2 ], …, X k ], when A is a fixed matrix and X 1 , …, X k vary, were studied. Moreover given any expression g ( X 1 , …, X k ), obtained from distinct noncommuting variables X 1 , …, X k by applying recursively the Lie product [· , ·] and without using the same variable twice, the possible eigenvalues, ranks and numbers of nonconstant invariant polynomials of g ( X 1 , …, X k ) when one of the variables X 1 , …, X k takes a fixed value in F n × n and the others vary, were studied. The purpose of the present paper is to show that analogous results can be obtained when additive commutators are replaced with multiplicative commutators or Jordan products.

In this paper, the authors study the boundedness of the singular integral operators associated to submanifolds on Triebel–Lizorkin spaces and Besov spaces. Moreover, the authors also establish an l q -valued inequality for the maximal operators associated to submanifolds in this paper.

In this paper we study the properties of coefficients appearing in the series expansions for zeros of generalized Rogers–Ramanujan series. Our primary purpose is to address several conjectures made by M.E.H. Ismail and C. Zhang. We prove that the coefficients in the series expansion of each zero approach rational multiples of π and π2 as q→1−. We also show that certain polynomials arising in connection with the zeros of Rogers–Ramanujan series generalize the coefficients appearing in the Taylor expansion of the tangent function. These polynomials provide an enumeration for alternating permutations different from that given by the classical q-tangent numbers. We conclude the paper with a method for inverting an elliptic integral associated with the zeros of generalized Rogers–Ramanujan series. Our calculations provide an efficient algorithm for the computation of series expansions for zeros of generalized Rogers–Ramanujan series.

In a recent paper, Granville and Soundararajan (2007) [5] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be well-distributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of Granville and Soundararajan (2007) [5] and prove that a similar phenomenon holds in F q [ t ] .

In our previous papers (Adv. in Math. 138 (1) (1998) 182; Potential Anal. 12 (2000) 419), we have obtained a decomposition of |f|, where f is a function defined on Rn, that is analogous to the one proved by H. Tanaka for martingales (the so-called “Tanaka formula”). More precisely, the decomposition has the form |f|=f̃+D∗0(f), where D∗0(f) is (a variant of) the density of the area integral associated with f. This functional (introduced by R.F. Gundy in his 1983 paper (The density of area integral, Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth, Belmont, CA, 1983, pp. 138–149.)) can be viewed as the counterpart of the local time in Euclidean harmonic analysis. In this paper, we are interested in boundedness and continuity properties of the mapping f↦f̃ (which we call the Lévy transform in analysis) on some classical function or distribution spaces. As was shown in [4,5], the above (non-linear) decomposition is bounded in Lp for every p∈[1,+∞[, i.e. one has ||f̃||p⩽Cp||f||p, where Cp is a constant depending only on p. Nevertheless our methods (roughly speaking, the Calderón–Zygmund theory in [4], stochastic calculus and martingale inequalities in [5]) both gave constants Cp whose order of magnitude near 1 is O(1/(p−1)). The aim of this paper is two-fold: first, we improve the preceding result and we answer a natural question, by proving that the best constants Cp are bounded near 1. Second, we prove that the Lévy transform f↦f̃ is continuous on the Hardy spaces Hp with p>n/(n+1).

This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues of quaternionic matrices.

This is the second paper of a series (started by Braverman and Finkelberg, 2010 [2] ) which describes a conjectural analog of the affine Grassmannian for affine Kac–Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian. In the case when G = SL ( n ) our conjectures can be derived from Nakajima (2009) [12] .

In a recent paper [7], Gowda et al. extended Ostrowski–Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that In ( a ) = In ( x ) whenever a ∘ x > 0 by the min–max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined by In ( x ) : = ( π ( x ) , ν ( x ) , δ ( x ) ) , with π ( x ) , ν ( x ) , and δ ( x ) denoting, respectively, the number of positive, negative, and zero eigenvalues, counting multiplicities. In this paper, we present a Peirce decomposition version of Wimmer’s result [13] and show that it is equivalent to the above result. In addition, we extend Higham and Cheng’s result ([8], Lemma 4.2) to the setting of Euclidean Jordan algebras.

In this paper, we consider a class of operator equilibrium problems (OEP for short) with operator solutions and derive a Minty type lemma for this class of problems. Further, using this lemma and KKM theorem, we establish some existence theorems for OEP. The theorems presented in this paper generalize, improve and unify many known results.

The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott-Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied. (C) 2008 Elsevier Inc. All rights reserved.

This paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras.

Let M be a smooth and compact moduli space of stable coherent sheaves on a projective surface S with an effective (or trivial) anti-canonical line bundle. We find generators for the cohomology ring of M, with integral coefficients. When S is simply connected and a universal sheaf E exists over SxM, then its class [E] admits a Kunneth decomposition as a class in the tensor product of the topological K-rings K(S) and K(M). The generators are the Chern classes of the Kunneth factors of [E] in K(M). The general case is similar, Comment: v3: Latex, 27 pages. Final version, to appear in Advances in Math. The proof of Lemma 21 is corrected and several other minor changes have been made. v2: Latex, 26 pages. The paper was split. The new version is a rewrite of the first three sections of version 1. The omitted results, about the monodromy of Hilbert schemes of point on a K3 surface, constitute part of the new paper arXiv:math.AG/0601304. v1: Latex, 53 pages