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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

, 2008

"... for c < 1 degenerate matter ..."

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... Let (X,T,µ) be a dynamical system, i.e. T: X → X is a map preserving a probability measure µ. The dynamics is seen as a sequence x0,x1,...,xn,... of points in X such that xn = T n (x0). Alternatively, we may only ‘see ’ the values F0,F1,...,Fn,... of some function F: X → R, where Fn = F(xn). We call ..."

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Let (X,T,µ) be a dynamical system, i.e. T: X → X is a map preserving a probability measure µ. The dynamics is seen as a sequence x0,x1,...,xn,... of points in X such that xn = T n (x0). Alternatively, we may only ‘see ’ the values F0,F1,...,Fn,... of some function F: X → R, where Fn = F(xn). We call F an observable. If the sequence {Xn}, or {Fn}, was independent (relative to the measure µ), then we could easily apply all major results of classical probability theory... But this is almost never the case in deterministic systems. So let us assume the simplest situation with dependence: the function F only takes finitely many values, say {1, 2,...,I}, and the sequence {Fn} is a Markov chain; i.e. Fn depends only on Fn−1 but not on the previous values Fn−m, m ≥ 2. This Markov chain has a stationary distribution P with components pi = µ(F0 = i) and its transition probability matrix Π has components πij = µ(F1 = j/F0 = i). If πij = pj for all i,j we would have an independent sequence. Suppose πij ≥ γpj for some γ> 0 (the components of the matrix Πn) converge to the stationary distribution exponentially fast in the following sense: and all i,j. Then the n-step transition probabilities π (n) ij (1) Var(Π (n) i,P) ≤ (1 − γ) n. = (π(n) i1,...,π(n) iI) is the ‘image ’ of the ith state at time n and P = (p1,...,pI) is the stationary vector; we denote by Var the distance in variation between probability vectors:

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

, 1992

"... Key words Noncommutative geometry, D-branes, deformation quantization ..."

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

, 2005

"... We present new solutions of the classical equations of motion of bosonic (matrix-) membranes. Those relating to minimal surfaces in spheres provide spinning membrane solutions in AdSp × S q, as well as in flat space-time. Non-trivial reductions of the BMN matrix model equations are also given. 1 1 ..."

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We present new solutions of the classical equations of motion of bosonic (matrix-) membranes. Those relating to minimal surfaces in spheres provide spinning membrane solutions in AdSp × S q, as well as in flat space-time. Non-trivial reductions of the BMN matrix model equations are also given. 1 1

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... Abstract. We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ R d, d ≥ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on Z(t) in domains of infinite volume. For domains of fi ..."

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Abstract. We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ R d, d ≥ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on Z(t) in domains of infinite volume. For domains of finite volume the bound on Z(t) decays exponentially as t tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of Z(t). To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means. 1. Introduction and

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... Abstract. Let G be a simply connected Chevalley group corresponding to an irreducible simply laced root system. Then the finite group G(Z/4Z) has a two fold central extension G ′ (Z/4Z) realized as a Steinberg group. In this paper we construct a natural correspondence between genuine representations ..."

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Abstract. Let G be a simply connected Chevalley group corresponding to an irreducible simply laced root system. Then the finite group G(Z/4Z) has a two fold central extension G ′ (Z/4Z) realized as a Steinberg group. In this paper we construct a natural correspondence between genuine representations of G ′ (Z/4Z) and representations of the Chevalley group G(Z/2Z). 1.

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... Abstract. In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □b-heat equation on quadric submanifolds M ⊂ C n × C m. As a consequence, we can also compute the heat kernel associated to the weighted ∂-equation in C n when the weight is ..."

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Abstract. In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □b-heat equation on quadric submanifolds M ⊂ C n × C m. As a consequence, we can also compute the heat kernel associated to the weighted ∂-equation in C n when the weight is given by exp(−φ(z,z) ·λ) where φ: C n ×C n → C m is a quadratic, sesquilinear form and λ ∈ R m. Our method involves the representation theory of the Lie group M and the group Fourier transform. 1.

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... Abstract. We obtain a number of Hardy type inequalities for domains involving both the distance to the boundary and the distance to the origin. In particular, this implies a Hardy-Sobolev inequality for the class of symmetric functions in a ball. Moreover, we prove that if the dimension d ≥ 3 then t ..."

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Abstract. We obtain a number of Hardy type inequalities for domains involving both the distance to the boundary and the distance to the origin. In particular, this implies a Hardy-Sobolev inequality for the class of symmetric functions in a ball. Moreover, we prove that if the dimension d ≥ 3 then the Hardy inequality involving the distance to the boundary is true with the constant 1/4 in a large family of domains which are not convex. At the end we give an example where we show that for any positive fixed constant there is an ellipsoid layer, such that Hardy’s inequality with the distance to the boundary fails. Key-words: Hardy inequalities; Sobolev inequalities MSC (2000): Primary: 35P15; Secondary: 81Q10

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... over supergroups OSp(2p|2q) ..."

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Supported by the *Austrian* *Federal* *Ministry* of Education, Science and Culture

"... Abstract. We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation in ..."

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Abstract. We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by constructing explicitly the focal set of the reflected line congruence (2-parameter family of oriented lines in R 3) with the aid of the natural complex structure on the space of all oriented affine lines. The purpose of this paper is to prove the following Theorem: Main Theorem: The focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. In contrast to the focal surface, the reflected wavefront is not translation invariant, in general.