The Hochschild cohomology of a closed manifold

Let M be a closed orientable manifold of dimension d and be the usual cochain algebra on M with coefficients in a field k. The Hochschild cohomology of M, is a graded commutative and associative algebra. The augmentation map induces a morphism of algebras . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of , which is in general quite small. The algebra is expected to be isomorphic to the loop homology constructed by Chas and Sullivan. Thus our results would be translated in terms of string homology.     

Mild solutions for a class of fractional SPDEs and their sample paths

We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset ${D\subset\mathbb{R}^{d}}We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset and driven by an infinite-dimensional fractional noise. We prove the existence of such a solution, establish its relation with the variational solution introduced by Nualart and Vuillermot (J Funct Anal 232:390–454, 2006) and the H?lder continuity of its sample paths when we consider it as an L ²(D)-valued stochastic process. When h is an affine function, we also prove uniqueness. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness. M. Sanz-Solé was supported by the grant MTM 2006-01351 from the Dirección General de Investigación, Ministerio de Educación y Ciencia, Spain.     

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$ on ${M \times [0, + \infty)}Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation

\frac?u?t = Dfu +au log u + bu \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu      5. Hardy Spaces and bmo on Manifolds with Bounded Geometry      Michael Taylor 《Journal of Geometric Analysis》2009,19(1):137-190 We develop the theory of the “local” Hardy space and John-Nirenberg space when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include – duality, L p estimates on an appropriate variant of the sharp maximal function, and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L p -Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.       6. Attached primes of the top local cohomology modules with respect to an ideal      Mohammad T. Dibaei  Siamak Yassemi 《Archiv der Mathematik》2005,84(4):292-297 Let be a Noetherian local ring and let be an ideal of R. Let M be an R-module of dimension n. In this paper we study the attached primes of the top local cohomology module Received: 12 May 2004      7. Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems      Omar Lakkis  Charalambos Makridakis. 《Mathematics of Computation》2006,75(256):1627-1658 We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of and the higher order spaces, and , with optimal orders of convergence.       8. Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations      Quoc Hung Phan 《Journal of Evolution Equations》2013,13(2):411-442 We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.      9. Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics      Nicolas Burq  Colin Guillarmou  Andrew Hassell 《Geometric And Functional Analysis》2010,20(3):627-656 In [Do], Doi proved that the ${L^{2}_{t}H^{1/2}_{x}}In [Do], Doi proved that the L2tH1/2x{L^{2}_{t}H^{1/2}_{x}} local smoothing effect for Schr?dinger equations on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L 1 → L ∞ dispersive estimates still hold without loss for e itΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.      10. Perverse sheaves on real loop Grassmannians      David?NadlerEmail author 《Inventiones Mathematicae》2005,159(1):1-73 The aim of this paper is to identify a certain tensor category of perverse sheaves on the loop Grassmannian Gr of a real form G of a connected reductive complex algebraic group G with the category of finite-dimensional representations of a connected reductive complex algebraic subgroup of the dual group . The root system of is closely related to the restricted root system of G. The fact that is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of Beilinson-Bernstein-Deligne.      11. Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra      Bertram?KostantEmail author 《Inventiones Mathematicae》2004,158(1):181-226 If is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of , then the absolute value of the kth coefficient of the power of the Euler product may be given by the dimension of a subspace of defined by all abelian subalgebras of of dimension k. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Petersons 2rank theorem on the number of abelian ideals in a Borel subalgebra of , an element of type and my heat kernel formulation of Macdonalds -function theorem, a set Dalcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when ), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac–Moody Lie algebra.      12. The PENROSE Transform for Bundles Non-Trivial on the General Line      Jürgen Leiterer 《Mathematische Nachrichten》1983,112(1):35-67 Let L0 be a fixed projective line in CP 3 and let M ? C 4 be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP 3 with L ∩ L 0 ?? Ø. Let D ? M , D ′ ? CP 3/ L 0 be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.      13. Intersection Local Time for Two Independent Fractional Brownian Motions      David Nualart  Salvador Ortiz-Latorre 《Journal of Theoretical Probability》2007,20(4):759-767 Let B H and be two independent, d-dimensional fractional Brownian motions with Hurst parameter H∈(0,1). Assume d≥2. We prove that the intersection local time of B H and exists in L 2 if and only if Hd<2.       14. Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>      Piotr?GajdaEmail author 《Numerical Algorithms》2005,40(4):401-414 We are interested in numerical algorithms for weighted L1 approximation of functions defined on . We consider the space ℱr,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L1-norm. We approximate f∈ℱr,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L1-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.      15. A note on a paper by G. Mastroianni and G. Monegato      G. Criscuolo. 《Mathematics of Computation》2004,73(245):243-250 Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral where or or and is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.       16. Weak type (p,q)‐inequalities for the Haar system and differentially subordinated martingales      Adam Oseçkowski 《Mathematische Nachrichten》2012,285(7):794-807 For any $1\leq p,\,q<\infty$, we determine the optimal constant $C_{p,q}$ such that the following holds. If $(h_k)_{k\geq 0}$ is the Haar system on [0,1], then for any vectors ak from a separable Hilbert space $\mathcal{H}$ and $\varepsilon_k\in \{-1,1\}$, $k=0,\,1,\,2,\ldots,$ we have This is generalized to the sharp weak‐type inequality where X, Y stand for $\mathcal{H}$‐valued martingales such that Y is differentially subordinate to X.      17. Graph von Neumann Algebras      Ilwoo Cho 《Acta Appl Math》2007,95(2):95-134 In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × α where is the subset of consisting of all reduced words in {e, e –1} and M × α is a W *-subalgebra of as a new graph von Neumann algebra induced by a graph G e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.      18. Moments of Brownian Motions on Lie Groups      Michael Voit 《Monatshefte für Mathematik》2005,145(3):247-260 Let (Bt)t ≥ 0 be a Brownian motion on with the corresponding Gaussian convolution semigroup (μt)t ≥ 0 and generator L. We show that algebraic relations between L and the generators of the matrix semigroups lead to for t → s, k ≥ 1, and all coordinates i,j. These relations will form the basis for a martingale characterization of (Bt)t ≥ 0 in terms of generalized heat polynomials. This characterization generalizes a corresponding result for the Brownian motion on in terms of Hermite polynomials due to J. Wesolowski and may be regarded as a variant of the Lévy characterization without continuity assumptions.      19. On <Emphasis Type="Italic">N</Emphasis>-Summations,II      H.?R?hrlEmail author 《Applied Categorical Structures》2005,13(1):79-97 Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (DN,D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.      20. Symmetric groups and expander graphs      Martin Kassabov 《Inventiones Mathematicae》2007,170(2):327-354 We construct explicit generating sets S n and of the alternating and the symmetric groups, which turn the Cayley graphs and into a family of bounded degree expanders for all n.

Hardy Spaces and bmo on Manifolds with Bounded Geometry

We develop the theory of the “local” Hardy space and John-Nirenberg space when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include – duality, L ^p estimates on an appropriate variant of the sharp maximal function, and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on L ^p -Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.      

Attached primes of the top local cohomology modules with respect to an ideal

Let be a Noetherian local ring and let be an ideal of R. Let M be an R-module of dimension n. In this paper we study the attached primes of the top local cohomology module Received: 12 May 2004     

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of and the higher order spaces, and , with optimal orders of convergence.

     

Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.     

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

In [Do], Doi proved that the ${L^{2}_{t}H^{1/2}_{x}}In [Do], Doi proved that the L²_tH^1/2_x{L^{2}_{t}H^{1/2}_{x}} local smoothing effect for Schr?dinger equations on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L ¹ → L ^∞ dispersive estimates still hold without loss for e ^itΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.     

Perverse sheaves on real loop Grassmannians

The aim of this paper is to identify a certain tensor category of perverse sheaves on the loop Grassmannian Gr of a real form G of a connected reductive complex algebraic group G with the category of finite-dimensional representations of a connected reductive complex algebraic subgroup of the dual group . The root system of is closely related to the restricted root system of G. The fact that is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of Beilinson-Bernstein-Deligne.     

Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra

If is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of , then the absolute value of the k^th coefficient of the power of the Euler product may be given by the dimension of a subspace of defined by all abelian subalgebras of of dimension k. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Petersons 2^rank theorem on the number of abelian ideals in a Borel subalgebra of , an element of type and my heat kernel formulation of Macdonalds -function theorem, a set D_alcove of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null m-core when ), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac–Moody Lie algebra.     

The PENROSE Transform for Bundles Non-Trivial on the General Line

Let L₀ be a fixed projective line in CP ³ and let M ? C ⁴ be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP ³ with L ∩ L ₀ ?? Ø. Let D ? M , D ′ ? CP ³/ L ₀ be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.     

Intersection Local Time for Two Independent Fractional Brownian Motions

Let B ^H and be two independent, d-dimensional fractional Brownian motions with Hurst parameter H∈(0,1). Assume d≥2. We prove that the intersection local time of B ^H and

Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on . We consider the space ℱ_r,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.     

A note on a paper by G. Mastroianni and G. Monegato

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral

where or or and is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

     

Weak type (p,q)‐inequalities for the Haar system and differentially subordinated martingales

For any $1\leq p,\,q<\infty$, we determine the optimal constant $C_{p,q}$ such that the following holds. If $(h_k)_{k\geq 0}$ is the Haar system on [0,1], then for any vectors a_k from a separable Hilbert space $\mathcal{H}$ and $\varepsilon_k\in \{-1,1\}$, $k=0,\,1,\,2,\ldots,$ we have This is generalized to the sharp weak‐type inequality where X, Y stand for $\mathcal{H}$‐valued martingales such that Y is differentially subordinate to X.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Moments of Brownian Motions on Lie Groups

Let (B_t)_{t ≥ 0} be a Brownian motion on with the corresponding Gaussian convolution semigroup (μ_t)_{t ≥ 0} and generator L. We show that algebraic relations between L and the generators of the matrix semigroups lead to for t → s, k ≥ 1, and all coordinates i,j. These relations will form the basis for a martingale characterization of (B_t)_{t ≥ 0} in terms of generalized heat polynomials. This characterization generalizes a corresponding result for the Brownian motion on in terms of Hermite polynomials due to J. Wesolowski and may be regarded as a variant of the Lévy characterization without continuity assumptions.     

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

Symmetric groups and expander graphs

We construct explicit generating sets S _n and of the alternating and the symmetric groups, which turn the Cayley graphs and into a family of bounded degree expanders for all n.