Gaussian distributions and phase space Weyl–Heisenberg frames

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a “phase space frame” associated with a Weyl–Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation.     

Integration over the Space of Functions and Poincaré Series Revisited

Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincar´e series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.     

Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points

In this paper we analyze a two-degree-of-freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the “roaming mechanism” whose reaction dynamics are of current interest in the chemistry community.     

Dynamical foundation and limitations of statistical reaction theory

We study the foundation and limitations of the statistical reaction theory. In particular, we focus our attention to the question of whether the rate constant can be defined for nonergodic systems. Based on the analysis of the Arnold web in the reactant well, we show that the survival probability exhibits two types of behavior: one where it depends on the residential time as the power-law decay and the other where it decays exponentially. The power-law decay casts a doubt on definability of the rate constant for nonergodic systems. We indicate that existence of the two types of behavior comes from sub-diffusive motions in remote regions from resonance overlap. Moreover, based on analysis of nonstationary features of trajectories, we can understand how the normally hyperbolic invariant manifold (NHIM) is connected with the Arnold web. We propose that the following two features play a key role in understanding the reactions where ergodicity is broken, i.e., whether the Arnold web is nonuniform and how the NHIM is connected with the Arnold web.     

Saddle surfaces in singular spaces

The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel' on compactness of saddle surfaces in a Euclidean space.

     

Quasi-random tournaments

We introduce a large class of tournament properties, all of which are shared by almost all random tournaments. These properties, which we term “quasi-random,” have the property that tournaments possessing any one of the properties must of necessity possess them all. In contrast to random tournaments, however, it is often very easy to verify that a particular family of tournaments satisfies one of the quasi-random properties, thereby giving explicit tournaments with “random-like” behavior. This paper continues an approach initiated in several earlier papers of the authors where analogous results for graphs (with R.M. Wilson) and hypergraphs are proved.     

Full nuclear cones associated to a normal cone. Application to Pareto efficiency

We introduce in this paper the notion of “full nuclear cone”, and we show that a nontrivial full nuclear cone can be associated to any normal cone in a locally convex space. We apply this notion to the study of Pareto efficiency.     

Extension of optimality conditions via supporting functions

In this study we introduce the notion of supporting functions and exploit some of their important properties. We make use of these functions to develop generalized optimality conditions of a mixed stationary-saddle type, where the supporting functions play either the role of the gradients/directional derivatives and/or the role of the original functions. These conditions may be applied to certain problems involving nondifferentiable and/or nonconvex functions. Classical, as well as new stationary and saddle optimality conditions follow from our approach in a natural fashion.     

Generic cuts in models of arithmetic

We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y. The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts (strength, regularity, saturation, coding properties) are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre. A new notion of “generic cut” is introduced and investigated and it is shown in the case of countable arithmetically saturated models M ? PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same “small interval” of the model are conjugate by an automorphism of the model.The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in model‐theoretic terms those properties for which there is an indicator Y. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bridging the Gap Between the Dense and the Discrete: The Number Line and the “Rubber Line” Bridging Analogy

In two experiments we explored the instructional value of a cross‐domain mapping between “number” and “line” in secondary school students' understanding of density. The first experiment investigated the hypothesis that density would be more accessible to students in a geometrical context (infinitely many points on a straight line segment) compared to a numerical context (infinitely many numbers in an interval). The participants were 229 seventh to eleventh graders. The results supported this hypothesis but also showed that students' conceptions of the line segment were far from that of a dense array of points. We then designed a text-based intervention that attempted to build the notion of density in a geometrical context, making explicit reference to the number-to-points correspondence and using the “rubber line” bridging analogy (the line as an imaginary unbreakable rubber band) to convey the no-successor principle. The participants were 149 eighth and tenth graders. The text intervention improved student performance in tasks regarding the infinity of numbers in an interval; the “rubber line” bridging analogy further improved performance successfully conveying the idea that these numbers can never be found one immediately next to the other.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.     

Noncommutative Generalizations of Theorems of Cohen and Kaplansky

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.     

Lagrange Duality in Set Optimization

Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. “Saddle sets” replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand, and the existence of a saddle set for the Lagrangian on the other hand. Applications to set-valued risk measures are indicated.     

Prototype Proofs in Type Theory

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ‐calculus and act as “proof‐schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type‐quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.     

The Reflection Structures of Generalized Co-Minkowski Spaces Leading to K-Loops

We extend the notion co- Minkowski plane to “unitary co- Minkowski space” (P, G, ～) and discuss the problem whether there are subsets Q of P which can be turned into a geometric K- loop.     

On Some Types of Rigid Sets in Banach Spaces

One of the properties characterizing Euclidean spaces says - roughly speaking- that their unit sphere has nice invariant properties. More precisely, a finite dimensional normed space has an Euclidean norm if and only if the group of isometries acts transitively on its unit sphere (the norm is “transitive”); such property of the sphere is also called “rigidity”. More recently, another notion of “rigidity” for compact sets, connected with “isometric sequences”, received some attention. Infinite rigid sets are diametral; moreover, under suitable assumptions on the space, they are also contained in the boundary of a sphere. These notions are connected with many problems, in different areas. Here we discuss and compare these two notions of rigid set, trying to indicate new relations among them and with some other properties of sets. Several examples complete the paper.     

Asymptotic gauges: Generalization of Colombeau type algebras

We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general “growth condition” formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restrictions on the coefficients can be solved.     

Large Cardinals and Ramifiability for Directed Sets

The notion of “ramifiability” (or “tree‐property”), usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.