510 Mathematik
Refine
Year of publication
Document Type
- Preprint (373)
- Article (273)
- Doctoral Thesis (77)
- Postprint (45)
- Monograph/Edited Volume (13)
- Other (10)
- Master's Thesis (7)
- Conference Proceeding (6)
- Part of a Book (5)
- Review (3)
Language
- English (766)
- German (46)
- French (3)
- Multiple languages (1)
Keywords
- random point processes (18)
- statistical mechanics (18)
- stochastic analysis (18)
- index (14)
- boundary value problems (12)
- Fredholm property (10)
- regularization (10)
- cluster expansion (9)
- elliptic operators (9)
- Bayesian inference (8)
Institute
- Institut für Mathematik (750)
- Institut für Physik und Astronomie (15)
- Extern (14)
- Mathematisch-Naturwissenschaftliche Fakultät (14)
- Hasso-Plattner-Institut für Digital Engineering gGmbH (7)
- Institut für Biochemie und Biologie (6)
- Institut für Informatik und Computational Science (5)
- Department Psychologie (4)
- Hasso-Plattner-Institut für Digital Engineering GmbH (4)
- Department Grundschulpädagogik (3)
Peter Jones' theorem on the factorization of Ap weights is sharpened for weights with bounds near 1, allowing the factorization to be performed continuously near the limiting, unweighted case. When 1 < p < infinite and omega is an Ap weight with bound Ap(omega) = 1 + epsilon, it is shown that there exist Asub1 weights u, v such that both the formula omega = uv(1-p) and the estimates A1 (u), A1 (v) = 1 + Omikron (√epsilon) hold. The square root in these estimates is also proven to be the correct asymptotic power as epsilon -> 0.
On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.
Parabolic equations on manifolds with singularities require a new calculus of anisotropic pseudo-differential operators with operator-valued symbols. The paper develops this theory along the lines of sn abstract wedge calculus with strongly continuous groups of isomorphisms on the involved Banach spaces. The corresponding pseodo-diferential operators are continuous in anisotropic wedge Sobolev spaces, and they form an alegbra. There is then introduced the concept of anisotropic parameter-dependent ellipticity, based on an order reduction variant of the pseudo-differential calculus. The theory is appled to a class of parabolic differential operators, and it is proved the invertibility in Sobolev spaces with exponential weights at infinity in time direction.
Soit (A, H, F) un module de Fredholm p-sommable, où l'algèbre A = CT est engendrée par un groupe discret Gamma d'éléments unitaires de L(H) qui est de croissance polynomiale r. On construit alors un triplet spectral (A, H, D) sommabilité q pour tout q > p + r + 1 avec F = signD. Dans le cas où (A, H, F) est (p, infini)-sommable on obtient la (q, infini)-sommabilité de (A, H, D)pour tout q > p + r + 1.
We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C n with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology.
We consider a homogeneous pseudodifferential equation on a cylinder C = IR x X over a smooth compact closed manifold X whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators over X. When assuming the symbol to be independent on the variable t element IR, we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points since C can be thought of as a 'stretched' manifold with conical points at t = -infinite and t = infinite.
Linear and non-linear analogues of the Black-Scholes equation are derived when shocks can be described by a truncated Lévy process. A linear equation is derived under the perfect correlation assumption on returns for a derivative security and a stock, and its solutions for European put and call options are obtained. It is also shown that the solution violates the perfect correlation assumption unless a process is gaussian. Thus, for a family of truncated Lévy distributions, the perfect hedging is impossible even in the continuous time limit. A second linear analogue of the Black-Scholes equation is obtained by constructing a portfolio which eliminates fluctuations of the first order and assuming that the portfolio is risk-free; it is shown that this assumption fails unless a process is gaussian. It is shown that the di erence between solutions to the linear analogues of the Black-Scholes equations and solutions to the Black-Scholes equations are sizable. The equations and solutions can be written in a discretized approximate form which uses an observed probability distribution only. Non-linear analogues for the Black-Scholes equation are derived from the non-arbitrage condition, and approximate formulas for solutions of these equations are suggested. Assuming that a linear generalization of the Black-Scholes equation holds, we derive an explicit pricing formula for the perpetual American put option and produce numerical results which show that the difference between our result and the classical Merton's formula obtained for gaussian processes can be substantial. Our formula uses an observed distribution density, under very weak assumptions on the latter.
We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
For elliptic operators on manifolds with boundary, we define spectral boundary value problems, which generalize the Atiyah-Patodi-Singer problem to the case of nonhomogeneous boundary conditions, operators of arbitrary order, and nonself-adjoint conormal symbols. The Fredholm property is proved and equivalence with certain elliptic equations on manifolds with conical singularities is established.
We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.
In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators.
We consider a G-invariant star-product algebra A on a symplectic manifold (M,ω) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden-Weinstein theorem we define a reduction of the algebra A with respect to the G-action. The reduced algebra turns out to be isomorphic to a canonical star-product algebra on the reduced phase space B. In other words, we show that the reduction commutes with the canonical G-invariant deformation quantization. A similar statement in the framework of geometric quantization is known as the Guillemin-Sternberg conjecture (by now completely proved).
For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
Let {Tsub(p) : q1 ≤ p ≤ q2} be a family of consistent Csub(0) semigroups on Lφ(Ω) with q1, q2 ∈ [1, ∞)and Ω ⊆ IRn open. We show that certain commutator conditions on Tφ and on the resolvent of its generator Aφ ensure the φ independence of the spectrum of Aφ for φ ∈ [q1, q2]. Applications include the case of Petrovskij correct systems with Hölder continuous coeffcients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coeffcients.
It is shown that the Hankel transformation Hsub(v) acts in a class of weighted Sobolev spaces. Especially, the isometric mapping property of Hsub(v) which holds on L²(IRsub(+),rdr) is extended to spaces of arbitrary Sobolev order. The novelty in the approach consists in using techniques developed by B.-W. Schulze and others to treat the half-line Rsub(+) as a manifold with a conical singularity at r = 0. This is achieved by pointing out a connection between the Hankel transformation and the Mellin transformation.The procedure proposed leads at the same time to a short proof of the Hankel inversion formula. An application to the existence and higher regularity of solutions, including their asymptotics, to the 1-1-dimensional edge-degenerated wave equation is given.
The paper is devoted to the construction of the exponential function of a matrix pseudo-differential operator which do not satisfy any of the known theorems (see, Sec.8 Ch.VIII and Sec.2 Ch.XI of [17]). The applications to the construction of the fundamental solution for the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity are given, too.
The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.
The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.
The aim of this book is to develop the Lefschetz fixed point theory for elliptic complexes of pseudodifferential operators on manifolds with edges. The general Lefschetz theory contains the index theory as a special case, while the case to be studied is much more easier than the index problem. The main topics are: - The calculus of pseudodifferential operators on manifolds with edges, especially symbol structures (inner as well as edge symbols). - The concept of ellipticity, parametrix constructions, elliptic regularity in Sobolev spaces. - Hodge theory for elliptic complexes of pseudodifferential operators on manifolds with edges. - Development of the algebraic constructions for these complexes, such as homotopy, tensor products, duality. - A generalization of the fixed point formula of Atiyah and Bott for the case of simple fixed points. - Development of the fixed point formula also in the case of non-simple fixed points, provided that the complex consists of diferential operarators only. - Investigation of geometric complexes (such as, for instance, the de Rham complex and the Dolbeault complex). Results in this direction are desirable because of both purely mathe matical reasons and applications in natural sciences.
A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .
We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time.
Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations whose solutions describe the orbits of closed p-branes under the effect of a "generalized physical force". For the corresponding evolution equation, which is a system of parabolic nonlinear partial differential equations associated to the elliptic PDE, we can establish existence of short time solutions.
In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).
"Considerons une particule mobile se mouvant aleatoirement sur la droite (ou sur un segment de droite). Supposons qu'il existe une probabilite F(x,y;s,t) bien definie pour que la particule se trouvant a l'instant s dans la position x se trouve a l'instant t (> s) a gauche de y, probabilite independante du mouvement anterieur de la particule...." Mit diesen Worten beginnt eines der berühmtesten mathematischen Manuskripte des letzten Jahrhunderts. Es stammt vom Soldaten Wolfgang Döblin, Sohn des deutschen Schriftstellers Alfred Döblin, und trägt den Titel "Sur l'equation de Kolmogoroff". Seine Veröffentlichung verbindet sich mit einer unglaublichen Geschichte. Wolfgang Döblin, stationiert mit seiner Einheit in den Ardennen im Winter 1939/1940, arbeitete an diesem Manuskript. Er entschloss sich, es als versiegeltes Manuskript an die Academie des Sciences in Paris zu schicken. Aber er kehrte nie aus diesem Krieg zurück. Sein Manuskript blieb 60 Jahre unter Verschluss im Archiv, und wurde erst im Jahre 2000 geöffnet. Wie weit Döblin damit seiner Zeit voraus war, wurde erkannt, nachdem es von Bernard Bru und Marc Yor ausgewertet worden war. Im ersten Satz umschreibt W. Döblin gleichzeitig das Programm des Manuskripts: "Wir betrachten ein bewegliches Teilchen, das sich zufällig auf der Geraden (oder einem Teil davon) bewegt." Er widmet sich damit der Aufgabe, die Fundamente eines Gebiets zu legen, das wir heute als stochastische Analysis bezeichnen.
Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious of these have mild instabilities and drift problems. Consequently, stabilization techniques have been proposed A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice. Some of the simulation methods that have been proposed and used in computations are reviewed here, from a stability point of view. This involves concepts of differential-algebraic equation (DAE) and ordinary differential equation (ODE) invariants. An explanation of the difficulties that may be encountered using Baumgarte's method is given, and a discussion of why a further quest for better parameter values for this method will always remain frustrating is presented. It is then shown how Baumgarte's method can be improved. An efficient stabilization technique is proposed, which may employ explicit ODE solvers in case of nonstiff or highly oscillatory problems and which relates to coordinate projection methods. Examples of a two-link planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.
A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.
We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.
Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.
The overall program "arborescent numbers" is to similarly perform the constructions from the natural numbers (N) to the positive fractional numbers (Q+) to positive real numbers (R+) beginning with (specific) binary trees instead of natural numbers. N can be regarded as the associative binary trees. The binary trees B and the left-commutative binary trees P allow the hassle-free definition of arbitrary high arithmetic operations (hyper ... hyperpowers). To construct the division trees the algebraic structure "coppice" is introduced which is a group with an addition over which the multiplication is right-distributive. Q+ is the initial associative coppice. The present work accomplishes one step in the program "arborescent numbers". That is the construction of the arborescent equivalent(s) of the positive fractional numbers. These equivalents are the "division binary trees" and the "fractional trees". A representation with decidable word problem for each of them is given. The set of functions f:R1->R1 generated from identity by taking powers is isomorphic to P and can be embedded into a coppice by taking inverses.
The predictability problem
(2007)
We try to determine whether it is possible to approximate the subjective Cloze predictability measure with two types of objective measures, semantic and word n-gram measures, based on the statistical properties of text corpora. The semantic measures are constructed either by querying Internet search engines or by applying Latent Semantic Analysis, while the word n-gram measures solely depend on the results of Internet search engines. We also analyse the role of Cloze predictability in the SWIFT eye movement model, and evaluate whether other parameters might be able to take the place of predictability. Our results suggest that a computational model that generates predictability values not only needs to use measures that can determine the relatedness of a word to its context; the presence of measures that assert unrelatedness is just as important. In spite of the fact, however, that we only have similarity measures, we predict that SWIFT should perform just as well when we replace Cloze predictability with our measures.
This paper deals with the Mie scattering kernels for multi-spectral data. The kernels may be represented in form of power series. Furthermore, the singular-value spectrum and the degree of ill-posedness in dependence on the refractive index of the particles are numerically approximated. A special hybrid regularization technique allows us to determine via inversion the particle distributions of different types.
The ill-posed inversion of multiwavelength lidar data by a hybrid method of variable projection
(1999)
The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.
In this thesis we mainly generalize two theorems from Mackaay-Picken and Picken (2002, 2004). In the first paper, Mackaay and Picken show that there is a bijective correspondence between Deligne 2-classes $\xi \in \check{H}^2(M,\mathcal{D}^2)$ and holonomy maps from the second thin-homotopy group $\pi_2^2(M)$ to $U(1)$. In the second one, a generalization of this theorem to manifolds with boundaries is given: Picken shows that there is a bijection between Deligne 2-cocycles and a certain variant of 2-dimensional topological quantum field theories. In this thesis we show that these two theorems hold in every dimension. We consider first the holonomy case, and by using simplicial methods we can prove that the group of smooth Deligne $d$-classes is isomorphic to the group of smooth holonomy maps from the $d^{th}$ thin-homotopy group $\pi_d^d(M)$ to $U(1)$, if $M$ is $(d-1)$-connected. We contrast this with a result of Gajer (1999). Gajer showed that Deligne $d$-classes can be reconstructed by a different class of holonomy maps, which not only include holonomies along spheres, but also along general $d$-manifolds in $M$. This approach does not require the manifold $M$ to be $(d-1)$-connected. We show that in the case of flat Deligne $d$-classes, our result differs from Gajers, if $M$ is not $(d-1)$-connected, but only $(d-2)$-connected. Stiefel manifolds do have this property, and if one applies our theorem to these and compare the result with that of Gajers theorem, it is revealed that our theorem reconstructs too many Deligne classes. This means, that our reconstruction theorem cannot live without the extra assumption on the manifold $M$, that is our reconstruction needs less informations about the holonomy of $d$-manifolds in $M$ at the price of assuming $M$ to be $(d-1)$-connected. We continue to show, that also the second theorem can be generalized: By introducing the concept of Picken-type topological quantum field theory in arbitrary dimensions, we can show that every Deligne $d$-cocycle induces such a $d$-dimensional field theory with two special properties, namely thin-invariance and smoothness. We show that any $d$-dimensional topological quantum field theory with these two properties gives rise to a Deligne $d$-cocycle and verify that this construction is surjective and injective, that is both groups are isomorphic.
Since 1971, the Freudenthal Institute has developed an approach to mathematics education named Realistic Mathematics Education (RME). The philosophy of RME is based on Hans Freudenthal’s concept of ‘mathematics as a human activity’. Prof. Hans Freudenthal (1905-1990), a mathematician and educator, believes that ‘ready-made mathematics’ should not be taught in school. By contrast, he urges that students should be offered ‘realistic situations’ so that they can rediscover from informal to formal mathematics. Although mathematics education in Vietnam has some achievements, it still encounters several challenges. Recently, the reform of teaching methods has become an urgent task in Vietnam. It appears that Vietnamese mathematics education lacks necessary theoretical frameworks. At first sight, the philosophy of RME is suitable for the orientation of the teaching method reform in Vietnam. However, the potential of RME for mathematics education as well as the ability of applying RME to teaching mathematics is still questionable in Vietnam. The primary aim of this dissertation is to research into abilities of applying RME to teaching and learning mathematics in Vietnam and to answer the question “how could RME enrich Vietnamese mathematics education?”. This research will emphasize teaching geometry in Vietnamese middle school. More specifically, the dissertation will implement the following research tasks: • Analyzing the characteristics of Vietnamese mathematics education in the ‘reformed’ period (from the early 1980s to the early 2000s) and at present; • Implementing a survey of 152 middle school teachers’ ideas from several Vietnamese provinces and cities about Vietnamese mathematics education; • Analyzing RME, including Freudenthal’s viewpoints for RME and the characteristics of RME; • Discussing how to design RME-based lessons and how to apply these lessons to teaching and learning in Vietnam; • Experimenting RME-based lessons in a Vietnamese middle school; • Analyzing the feedback from the students’ worksheets and the teachers’ reports, including the potentials of RME-based lessons for Vietnamese middle school and the difficulties the teachers and their students encountered with RME-based lessons; • Discussing proposals for applying RME-based lessons to teaching and learning mathematics in Vietnam, including making suggestions for teachers who will apply these lessons to their teaching and designing courses for in-service teachers and teachers-in training. This research reveals that although teachers and students may encounter some obstacles while teaching and learning with RME-based lesson, RME could become a potential approach for mathematics education and could be effectively applied to teaching and learning mathematics in Vietnamese school.
Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions.
Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity
(2006)
We consider scattering in $\R^n$, $n\ge 2$, described by the Schr\"odinger operator $P(h)=-h^2\Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $h\to 0$ of the scattering amplitude $f(\omega_-,\omega_+;\lambda,h)$ $\omega_+\neq\omega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity.
We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.
We analyze the asymptotic behavior in the limit epsilon to zero for a wide class of difference operators H_epsilon = T_epsilon + V_epsilon with underlying multi-well potential. They act on the square summable functions on the lattice (epsilon Z)^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.
We consider a system of infinitely many hard balls in R<sup>d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.