Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Unitary invariants of spectral measures with the CGS-property

A «CGS-property» for the spectral measures is introduced and the classical results of determining complete systems of unitary invariants for self-adjoint and bounded normal operators on separable Hilbert spaces are extended to the class of spectral measures with this property. As a consequence, the above mentioned results are extended to unbounded normal operators on separable Hilbert spaces. Moreover, three different kinds of multiplicity are defined and it is shown that for the measures with the «CGS-property» they all coincide. In the last section some analogues of the multiplicity functions defined by Stone [14] are related to the total multiplicity.     

Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂ ^m is a bounded domain. LetM ₀ ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M ₀ The invariants are given explicitly in the particular case ofk = 2.     

On the Consistency of the Quantum-Like Representation Algorithm for Hyperbolic Interference

Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]–[28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations. The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric, hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric interference was recently proved [29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning. For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H ^b|a and H ^a|b (in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary equivalent (in the sense of hyperbolic Hilbert space).     

On ɛ-representations

For certain classes of groups we show that a map to the group of unitary transformations of a Hilbert space which is “almost” a homomorphism is uniformly close to a unitary representation.     

Synaptic algebras

A synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective “synaptic”, borrowed from biology, is meant to suggest that such an algebra coherently “ties together” the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice.     

Yang-Baxter invariants for line configurations

We study line configurations in 3-space by means of “line diagrams”, projections into a plane with an indication of over and under crossing at the vertices. If we orient such a diagram, we can associate a “contracted tensor”T with it in the same spirit as is done in Knot Theory. We give conditions to makeT independent of the orientation, and invariant under isotopy. The Yang-Baxter equation is one such condition. Afterwards we restrict ourselves to Yang-Baxter invariants with a topological state model, and give some new invariants for line isotopy.     

Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group

We prove the pairwise disjointness of representations T _z,w of the infinite-dimensional unitary group. These representations are a natural generalization of the regular representation to the “big” group U(∞). They were introduced and studied by G. Olshanski and A. Borodin. The disjointness of these representations reduces to that of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.     

Finite-order invariants of ornaments

Anornament is a collection of oriented closed curves in a plane, no three of which intersect at the same point. We consider homotopy invariants of ornaments. Thefinite-order invariants of ornaments are a natural analog of the Vassiliev invariants of links. The calculation of them is based on the homological study of the corresponding space of singular objects. We perform the “local” part of these calculations and a part of the “global” one, which allows us to estimate the dimensions of the spaces of invariants of any order. We also construct explicity two large series of such invariants and establish some new algebraic structures in the space of invariants. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 35, Algebraicheskaya Geometriya-6, 1996.     

On the distribution of the sequence of fractional parts of a slowly increasing exponential function

In this paper for a slowly increasing exponential function we study the degree of deviation of the distribution of the sequence of its fractional parts from the “ideal” uniform distribution. An estimate is given which can appropriately be called the measure of irregularity of the distribution of a given sequence. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 148–152, January, 1999.     

Pseudoradial Spaces: Results and Problems

In this paper it is given a survey of principal results (old and new) concerning the class of pseudoradial spaces. In this class cardinal invariants and their inequalities are considered. The behaviour of pseudoradial spaces under the operations of taking topological products and subspaces are examined and a typical proof is given. A particular attention is dedicated to the so called “small cardinals” in connection with pseudoradiality. Pseudoradiality of 2^{ω 2} is also examined. It is proved that pseudoradiality can be ω¹ productive for spaces of weight at most ω². Finally, several open problems are presented. This work was supported by the National Group “Real Analysis, Measure Theory with Applications to Economy” of the Italian Ministery of Education, University and Research.     

On the reduction to a complete class in multiple decision problems (2)

Summary As a criterion for the reduction to a complete class of decision rule in case where actions, samples and states are finite in number, “regret-relief ratio” criterion and “incremental loss-gain ratio” criterion were introduced in 2-state of nature case [2]. In this paper, “generalized regret-relief ratio” criterion ink-state of nature case is introduced as an extension of “regret-relief ratio” criterion and its usefulness is shown with an example. The Institute of Statistical Mathematics     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Quasiperiodic spectra and orthogonality for iterated function system measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices. Research supported in part by a grant from the National Science Foundation DMS-0704191.     

First Colonization of a Hard-Edge in Random Matrix Theory

We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model, a phenomenon also known as the “birth of a cut” near a hard-edge. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials.     

Nonnegative unitary operators

Unitary operators in Hilbert space map an orthonormal basis onto another. In this paper we study those that map an orthonormal basis onto itself. We show that a sequence of cardinal numbers is a complete set of unitary invariants for such an operator. We obtain a characterization of these operators in terms of their spectral properties. We show how much simpler the structure is in finite-dimensional space, and also describe the structure of certain isometries in Hilbert space.

     

A KAM phenomenon for singular holomorphic vector fields

Let X be a germ of holomorphic vector field at the origin of Cⁿ and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.     

Stochastic Volterra equations driven by cylindrical Wiener process

In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of α-times resolvent families. Both authors are supported partially by project “Proyecto Anillo: Laboratorio de Analisis Estocastico; ANESTOC”.     

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated. This revised version was published online in August 2006 with corrections to the Cover Date.