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We discuss the role of gravitational excitons/radions in different cosmological scenarios. Gravitational excitons are massive moduli fields which describe conformal excitations of the internal spaces and which, due to their Planck-scale suppressed coupling to matter fields, are WIMPs. It is demonstrated that, depending on the concrete scenario, observational cosmological data set strong restrictions on the allowed masses and initial oscillation amplitudes of these particles

We study the global singularity structure of solutions to 3-D semilinear wave equations with discontinuous initial data. More precisely, using Strichartz' inequality we show that the solutions stay conormal after nonlinear interaction if the Cauchy data are conormal along a circle. (C) 2003 Elsevier Inc. All rights reserved

We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points

The classical Lefschetz formula expresses the number of fixed points of a continuous map f: M -> M in terms of the transformation induced by f on the cohomology of M. In 1966, Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they obtained a holomorphic Lefschetz formula on compact complex manifolds without boundary. Brenner and Shubin (1981, 1991) extended the Atiyah-Bott theory to compact manifolds with boundary. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, therefore the Atiyah- Bott theory is not applicable. Bypassing difficulties related to the boundary behaviour of Dolbeault cohomology, Donnelly and Fefferman (1986) obtained a formula for the number of fixed points in terms of the Bergman metric. The aim of this paper is to obtain a Lefschetz formula on relatively compact strictly pseudoconvex subdomains of complex manifolds X with smooth boundary, that is, to find the total Lefschetz number for a holomorphic endomorphism f(*) of the Dolbeault complex and to express it in terms of local invariants of the fixed points of f.

Mixed elliptic problems for differential operators A in a domain Q with smooth boundary Y are studied in the form Au = f in Omega, T+/-u = g+/- on Y+/-, where Y+/- subset of Y are manifolds with a common boundary Z, such that Y- boolean OR Y+ = Y and Y- boolean AND Y+ = z, with boundary conditions T+/- on Y+/- (with smooth coefficients up to Z from the respective side) satisfying the Shapiro-Lopatinskij condition. We consider such problems in standard Sobolev spaces and characterise natural extra conditions on the interface Z with an analogue of Shapiro-Lopatinskij ellipticity for an associated transmission problem on the boundary; then the extended operator is Fredholm. The transmission operators on the boundary with respect to Z belong to a complete pseudo-differential calculus, a modification of the algebra of boundary value problems without the transmission property. We construct parametrices of elliptic elements in that calculus, and we obtain parametrices of the original mixed problems under additional conditions on the interface. We consider the Zaremba problem and other mixed problems for the Laplace operator, determine the number of extra conditions and calculate the index of associated Fredholm operators. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Given asymptotics types P, Q, pseudodifferential operators A is an element of L-cl(mu) (R+) are constructed in such a way that if u(t) possesses conormal asymptotics of type P as t --> +0, then Au(t) possesses conormal asymptotics of type Q as t --> +0. This is achieved by choosing the operators A in Schulze's cone algebra on the half-line R+, controlling their complete Mellin symbols {sigma(M)(u-j) (A); j is an element of N}, and prescribing the mapping properties of the residual Green operators. The constructions lead to a coordinate invariant calculus, including trace and potential operators at t = 0, in which a parametrix construction for the elliptic elements is possible. Boutet de Monvel's calculus for pseudodifferential boundary problems occurs as a special case when P = Q is the type resulting from Taylor expansion at t = 0.

The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f : M-->M in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschetz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah- Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschetz formula on a strictly convex domain in C-n, n>1

The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field Q(s) (t). It is an iterated twisted function field in two variables D (x, sigma) (y, tau) over a quaternion division algebra D which is defined over the number field Q(root3, root-7). The automorphisms sigma and tau are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field Q((s)) ((t)), given by an iterated twisted Laurent series ring D((x, sigma)) ((y, tau)) over the same quaternion division algebra D. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Multidimensional cosmological models : Cosmological and astrophysical implications and constraints
(2004)

We investigate four-dimensional effective theories which are obtained by dimensional reduction of multidimensional cosmological models with factorizable geometry and we consider the interaction between conformal excitations of the internal space (geometrical moduli excitations) and Abelian gauge fields. It is assumed that the internal space background can be stabilized by minima of an effective potential. The conformal excitations over such a background have the form of massive scalar fields (gravitational excitons) propagating in the external spacetime. We discuss cosmological and astrophysical implications of the interaction between gravexcitons and four-dimensional photons as well as constraints arising on multidimensional models of the type considered in our paper. In particular, we show that due to the experimental bounds on the variation of the fine-structure constant, gravexcitons should decay before nucleosynthesis starts. For a successful nucleosynthesis, the masses of the decaying gravexcitons should be mgreater than or similar to10(4) GeV. Furthermore, we discuss the possible contribution of gravexcitons to ultrahigh-energy cosmic rays. It is shown that, at energies Esimilar to10(20) eV, the decay length of gravexcitons with masses mgreater than or similar to10(4) GeV is very small, but that for mless than or similar to10(2) GeV it becomes much larger than the Greisen-Zatsepin-Kuzmin cutoff distance. Finally, we investigate the possibility for gravexciton-photon oscillations in strong magnetic fields of astrophysical objects. The corresponding estimates indicate that even the high-magnetic- field strengths B of magnetars (special types of pulsars with B>B(critical)similar to4.4x10(13) G) are not sufficient for an efficient and copious production of gravexcitons

If K is an algebraic function field of one variable over an algebraically closed field k and F is a finite extension of K, then any element a of K can be written as a norm of some b in F by Tsen's theorem. All zeros and poles of a lead to zeros and poles of b, but in general additional zeros and poles occur. The paper shows how this number of additional zeros and poles of b can be restricted in terms of the genus of K, respectively F. If k is the field of all complex numbers, then we use Abel's theorem concerning the existence of meromorphic functions on a compact Riemarm surface. From this, the general case of characteristic 0 can be derived by means of principles from model theory, since the theory of algebraically closed fields is model-complete. Some of these results also carry over to the case of characteristic p > 0 using standard arguments from valuation theory

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary

Let v be a valuation of terms of type tau, assigning to each term t of type tau a value v(t) greater than or equal to 0. Let k greater than or equal to 1 be a natural number. An identity s approximate to t of type tau is called k- normal if either s = t or both s and t have value greater than or equal to k, and otherwise is called non-k-normal. A variety V of type tau is said to be k-normal if all its identities are k-normal, and non-k-normal otherwise. In the latter case, there is a unique smallest k-normal variety N-k(A) (V) to contain V , called the k-normalization of V. Inthe case k = 1, for the usual depth valuation of terms, these notions coincide with the well-known concepts of normal identity, normal variety, and normalization of a variety. I. Chajda has characterized the normalization of a variety by means of choice algebras. In this paper we generalize his results to a characterization of the k-normalization of a variety, using k-choice algebras. We also introduce the concept of a k-inflation algebra, and for the case that v is the usual depth valuation of terms, we prove that a variety V is k-normal iff it is closed under the formation of k- inflations, and that the k-normalization of V consists precisely of all homomorphic images of k-inflations of algebras in V

Metastability in reversible diffusion processes : I. Sharp asymptotics for capacities and exit times
(2004)

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form -epsilonDelta+ delF(.) del on R-d or subsets of R-d, where F is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as epsilon down arrow 0, to the capacities of suitably constructed sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring - Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes

An intercomparison of aerosol backscatter lidar algorithms was performed in 2001 within the framework of the European Aerosol Research Lidar Network to Establish an Aerosol Climatology (EARLINET). The objective of this research was to test the correctness of the algorithms and the influence of the lidar ratio used by the various lidar teams involved in the EARLINET for calculation of backscatter-coefficient profiles from the lidar signals. The exercise consisted of processing synthetic lidar signals of various degrees of difficulty. One of these profiles contained height- dependent lidar ratios to test the vertical influence of those profiles on the various retrieval algorithms. Furthermore, a realistic incomplete overlap of laser beam and receiver field of view was introduced to remind the teams to take great care in the nearest range to the lidar. The intercomparison was performed in three stages with increasing knowledge on the input parameters. First, only the lidar signals were distributed; this is the most realistic stage. Afterward the lidar ratio profiles and the reference values at calibration height were provided. The unknown height- dependent lidar ratio had the largest influence on the retrieval, whereas the unknown reference value was of minor importance. These results show the necessity of making additional independent measurements, which can provide us with a suitable approximation of the lidar ratio. The final stage proves in general, that the data evaluation schemes of the different groups of lidar systems work well. (C) 2004 Optical Society of America

In this paper we present duality theory for compact groups in the case when the C*-algebra A, the fixed point algebra of the corresponding Hilbert C*-system (F, 9), has a nontrivial center Z superset of C1 and the relative commutant satisfies the minimality condition A' boolean AND F = Z, as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*- categories T-c < T, where T-c is a suitable DR-category and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of T-c and the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on (G) over cap, the dual object of G. The chain group, which is isomorphic to the character group of the center of g, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(g) encodes the possibility of defining a symmetry epsilon also for the larger category T of the previous inclusion

Consider the perturbed harmonic oscillator Ty=-y''+x(2)y+q(x)y in L-2(R), where the real potential q belongs to the Hilbert space H={q', xq is an element of L-2(R)}. The spectrum of T is an increasing sequence of simple eigenvalues lambda(n)(q)=1+2n+mu(n), ngreater than or equal to0, such that mu(n)-->0 as n-->infinity. Let psi(n)(x,q) be the corresponding eigenfunctions. Define the norming constants nu(n)(q)=lim(xup arrowinfinity)log |psi(n) (x,q)/psi(n) (-x,q)|. We show that {mu(n)}(0)(infinity) is an element of H {nu(n)}(0)(infinity) is an element of H-0 for some real Hilbert space and some subspace H-0 subset of H. Furthermore, the mapping Psi:q-- >Psi(q)=({lambda(n)(q)}(0)(infinity), {nu(n)(q)}(0)(infinity)) is a real analytic isomorphism between H and S x H-0, where S is the set of all strictly increasing sequences s={s(n)}(0)(infinity) such that s(n)=1+2n+h(n), {h(n)}(0)(infinity) is an element of H. The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -y"py, p is an element of L-2(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces H,H-0. We obtain their basic properties, using their representation as spaces of analytic functions in the disk

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus. (C) 2003 Elsevier Inc. All rights reserved

Pairwise proximity data, given as similarity or dissimilarity matrix, can violate metricity. This occurs either due to noise, fallible estimates, or due to intrinsic non-metric features such as they arise from human judgments. So far the problem of non-metric pairwise data has been tackled by essentially omitting the negative eigenvalues or shifting the spectrum of the associated (pseudo) covariance matrix for a subsequent embedding. However, little attention has been paid to the negative part of the spectrum itself. In particular no answer was given to whether the directions associated to the negative eigenvalues would at all code variance other than noise related. We show by a simple, exploratory analysis that the negative eigenvalues can code for relevant structure in the data, thus leading to the discovery of new features, which were lost by conventional data analysis techniques. The information hidden in the negative eigenvalue part of the spectrum is illustrated and discussed for three data sets, namely USPS handwritten digits, text-mining and data from cognitive psychology

A new efficient algorithm is presented for joint diagonalization of several matrices. The algorithm is based on the Frobenius-norm formulation of the joint diagonalization problem, and addresses diagonalization with a general, non- orthogonal transformation. The iterative scheme of the algorithm is based on a multiplicative update which ensures the invertibility of the diagonalizer. The algorithm's efficiency stems from the special approximation of the cost function resulting in a sparse, block-diagonal Hessian to be used in the computation of the quasi-Newton update step. Extensive numerical simulations illustrate the performance of the algorithm and provide a comparison to other leading diagonalization methods. The results of such comparison demonstrate that the proposed algorithm is a viable alternative to existing state-of-the-art joint diagonalization algorithms. The practical use of our algorithm is shown for blind source separation problems

For a general attractive Probabilistic Cellular Automata on SZd, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics exponentially fast in the uniform norm is equivalent to a condition (A). This condition means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on {;1, +1}Zd, with a naturally associated Gibbsian potential ;, we prove that a (spatial-) weak mixing condition (WM) for ; implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to ; holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.

The author considers the heat equation in dimension one with singular drift and inhomogeneous space-time white noise. In particular, the quadratic variation measure of the white noise is not required to be absolutely continuous w.r.t. the Lebesgue measure, neither in space nor in time. Under some assumptions the author gives statements on strong and weak existence as well as strong and weak uniqueness of continuous solutions.

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples.

Local asymptotic types
(2004)

Dynamical processes with particular reference to opposite phenomena as growth and decay, may be translated into the mathematical language of difference equations or recursive equations. Such equations can be treated in the discrete case(where the principal mathematical instrument is furnished by geometrical progressions) or in the continuous case (where the mathematical instrument is particularly represented by exponential functions and logarithms). The variety of problems and of resolution methods renders the proposed material apt to be used in a significant way in the high school.

Aspects of Boundary Problems in Analysis and Geometry : advances in partial differential equations
(2004)

Systems of elasticity theory
(2004)