Generalized Gaussian bridges

A generalized bridge is a stochastic process that is conditioned on N$N$ linear functionals of its path. We consider two types of representations: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the process. Thus, the future knowledge of the path is needed. In the canonical representation the filtrations of the bridge and the underlying process coincide. The canonical representation is provided for prediction-invertible Gaussian processes. All martingales are trivially prediction-invertible. A typical non-semimartingale example of a prediction-invertible Gaussian process is the fractional Brownian motion. We apply the canonical bridges to insider trading.     

The canonical-boundary representation for automorphism groups of locally compact countable state Markov shifts

We introduce the canonical-boundary representation and study its range. This conjugacy invariant homomorphism captures information about the symmetry of the Markov shift near its (canonical) boundary and exhibits which actions on the boundary can be realized by automorphisms. The path-structure at infinity — a relation on the set of orbits of the canonical boundary — is a new conjugacy invariant, which is stronger than the canonical boundary and the periodic data at infinity. Moreover we determine its influence on the range of the canonical-boundary representation and the extendability of automorphisms from subsystems (ascending sequences of shifts os finite type (SFTs) and infinite subsets of periodic points) to the entire Markov shift.     

Representation theory and ADHM-construction on quaternion symmetric spaces

We determine all irreducible homogeneous bundles with anti-self-dual canonical connections on compact quaternion symmetric spaces. To deform the canonical connections, we give a relation between the representation theory and the theory of monads on the twistor space. The moduli spaces are described via the Bott-Borel-Weil Thereom. The Horrocks bundle is also generalized to higher-dimensional projective spaces.

     

Condition numbers and error bounds in convex programming

After a brief survey on condition numbers for linear systems of equalities, we analyse error bounds for convex functions and convex sets. The canonical representation of a convex set is defined. Other representations of a convex set by a convex function are compared with the canonical representation. Then, condition numbers are introduced for convex sets and their convex representations.     

On polynomial functions (mod m)

In this paper, we obtain a canonical representation for those polynomials (with integer coefficients) which vanish (mod m), a canonical representation for each polynomial function (mod m) and an expression for the number of polynomial functions (mod m). This number turns out to be (weakly) multiplicative in m.     

A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.      

A Fresh Perspective on Canonical Extensions for Bounded Lattices

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Plo??ica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphisms the Plo??ica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Plo??ica’s paper. This leads to a construction of canonical extension valid for all bounded lattices, which is shown to be functorial, with the property that the canonical extension functor decomposes as the composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors.     

Specific properties of canonical separated grammars

A class of canonical separated grammars capable of generating the same languages as general-type separated grammars is considered. The main properties and two criteria of canonical grammars are described. A way of unifying nonterminal symbols and proving the uniqueness of the canonical representation for separated grammars is proposed.     

Integration of Geodesic Flows on Homogeneous Spaces: The Case of a Wild Lie Group

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T ^* M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.     

A pair of linear canonical Hankel transformations and associated pseudo-differential operators

In this work, we have introduced a pair of linear canonical Hankel transformations and investigated some of its properties on Zemanian-type spaces. Moreover two versions of pseudo-differential operator associated with canonical Hankel transformations are defined and discussed its integral representation.     

Applying the Linblad equation to quantum dissipative systems

For quantum systems with linear dissipation, we obtain the representation of the Linblad equation in the canonical form via Hermitian operators. Based on this representation, we derive equations for the entropy density and for the statistical projection operator. We consider the quantum harmonic oscillator with linear dissipation as an example. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 288–294, August, 2006. An erratum to this article is available at .     

Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szegö recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [−1, 1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.     

A non-factorial algorithm for canonical numbering of a graph

A graph certificate or canonical form is a short unique (up to isomorphism) representation of the graph. Thus two graphs are isomorphic iff their certificates are identical. In this paper an O(cⁿ) graph isomorphism algorithm which also yields a certificate of the graph is presented. The certificate produced by this algorithm is a canonical numbering of the vertices of the graph.     

Extensions of Isometric Dual Representations of Semigroups

Let T be a dual representation of a suitable subsemigroup Sof a locally compact abelian group G by isometries on a dualBanach space X=(X_*)^*. It is shown that (X, T) can be extendedto a dual representation of G on a dual Banach space Y containingX, and that this extension can be done in a canonical way. Inthe case of a representation by *-monomorphisms of a von Neumannalgebra, the extension is a representation of G by *-automorphismsof a von Neumann algebra.     

Quantum Hamiltonian Systems on K-Orbits: Semiclassical Spectrum of the Asymmetric Top

We consider equations on Lie groups and classical and quantum Hamiltonian systems on coadjoint representation orbits. We show that the transition to canonical coordinates on orbits of the coadjoint representation allows constructing semiclassical solutions and the corresponding spectra of quantum equations such that all the symmetries of the original problem are preserved. Our method is used to find the semiclassical spectrum of the asymmetric quantum top.     

Left versus right canonical Wiener-Hopf factorization and realization

For an arbitrary rational matrix function, not necessarily analytic at infinity, the existence of a right canonical Wiener-Hopf factorization is characterized in terms of a left canonical Wiener-Hopf factorization. Formulas for the factors in a right factorization are given in terms of the formulas for the factors in a given left factorization. All formulas are based on a special representation of a rational matrix function involving a quintet of matrices.     

Combinatorial representation and convex dimension of convex geometries

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.Supported in part by NSF grant no. DMS-8501948.     

Irreducible function bases for simple fluids and liquid crystal films

We present a rigorous derivation of the canonical representation of a class of constitutive functions for liquid crystal films which has been widely used in various special forms in the fields of emulsion chemistry and cell-membrane biology. The representation yields the largest class of functions compatible with an appropriate definition of fluidity. The method used also furnishes established representation formulas in the classical theories of capillarity and three-dimensional compressible fluids.