Theoretic Conditions Research Articles

Minimal flows with a closed proximal cell

It is shown that regional proximality is an equivalence relation in a minimal flow which contains a closed proximal cell. Also, a certain group theoretic condition is shown to be equivalent to a ‘local Bronstein’ condition.

VARIATIONS ON THOMPSON'S CHARACTER DEGREE THEOREM

If P is a Sylow- p -subgroup of a finite p -solvable group G , we prove that G^\prime \cap \bf{N}_G(P) \subseteq {P} if and only if p divides the degree of every irreducible non-linear p -Brauer character of G. More generally if π is a set of primes containing p and G is π-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p -Brauer character to be divisible by some prime in π. This can also be applied to degrees of ordinary characters.

The Split Property and the Symmetry Breaking¶of the Quantum Spin Chain

We consider the relationship between the symmetry breaking and the split property of pure states of quantum spin chains. We obtain a representation theoretic condition implying that the half-sided uniform mixing condition leads to symmetry breaking of translationally invariant pure states. This is a mathematical generalization of Dichotomy previously found by I. Affleck and E. Lieb and M. Aizenman and B. Nachtergaele for ground states of a special class of Hamiltonians.

Linear Material Flow Models

This paper presents static and dynamic linear models for material flow systems. Based on the a priori knowledge given e.g. by mass balances these models are highly structured in a natural way. In the static case this structure is exploited for the reconciliation of the flow measurements, the estimation of the parameters and for simulation. In the dynamic case graph theoretic conditions are provided for stability, reachability and identifiability of three types of stock dynamics.

A formal framework of multi-agent systems with requirement/service cooperative style

Adopting three kinds of speech acts: request, promise, and inform, this paper analyses the interaction among aents in a kind of multi-agent systems with requirements/services cooperation style (MASr-s). The paper gives the objective model the theoretic satisfaction conditions of three kinds of speech acts in MASr-s. The formal definition of MASr-s has been presented. To evaluate concrete implementation architecture and mechanism of the variant MASr-s, including client/server computing architecture and mechanism, a spectrum of MASr-s has been proposed, which capturesdirect request/passive service mechanism,direct request/active service mechanism,indirect request/active service mechanism, and peerto-peer request/service mechanism. The spectrum shows a thread to improve traditional client/server computing.

Diophantine approximation and deformation

It is well-known that while the analogue of Liouville's theorem on diophan- tine approximation holds in finite characteristic, the analogue of Roth's theorem fails quite badly. We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the diophantine approximation exponents of the power series, with more 'generic' curves (in the de- formation sense) giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree. R ´ ESUM´ E.— APPROXIMATION DIOPHANTIENNE ET D´ EFORMATION. — Alors que l'analogue

Diffusive energy scattering from weakly random surfaces

We derive transport theoretic boundary conditions for acoustic wave reflection at a weakly rough boundary in an inhomogeneous half space. We use the Wigner distribution to go from waves to energy transport in the high frequency limit. We generalize known results on the reflection of acoustic plane waves in a homogeneous medium. We analyze higher order corrections, which include a enhanced backscattering effect in the back direction.

Transport theory for acoustic waves with reflection and transmission at interfaces

Transport theoretic boundary conditions are derived for acoustic wave reflection and transmission at a rough interface with small random fluctuations. The Wigner distribution is used to go from waves to energy transport in the high frequency limit, and the Born expansion is used to calculate the effect of the random rough surface. The smoothing method is also used to remove the grazing angle singularity due to the Born approximation. The results are presented in a form that is convenient both for theoretical analysis and for numerical computations.

A character theoretic condition characterizing nilpotent groups

A character theoretic condition characterizing nilpotent groups

SOME REMARKS ON CCR AND WEYL OPERATORS

Using the symbol calculus developed by Berezin, Krée and Rączka, it is shown that the algebra generated by the canonical commutation relation (CCR) is splitted into creation part and annhilation part with holomorphic and anti-holomorphic symbols respectively. Further, in the Weyl representation of CCR, three Weyl operators will constitute an irreducible set of the full CCR-algebra if some number theoretic conditions are satisfied.

Stiefel—Whitney currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v1,…,vm, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ2-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle.

Graph theoretic and boolean conditions for structurally fixed modes of augmented system forms

The structurally fixed modes are studied for systems with state and output feedback and outputs which generally influence the system and measure states and inputs. Conditions for structurally fixed modes are produced and are compared with the subsystem unreachability. A condition on the graph of the augmented system and a condition based on the decomposition of the reduced x-dimensional system are thus established for this form of LTI augmented systems.

Classification of hereditary matrices

A classical approach used to obtain basic facts in the theory of square matrices involves an analysis of the relationship between polynomials p in one variable and square matrices T such that p( T) = 0. We consider matrices and operators which satisfy a different type of polynomial constraint. Let H be a complex Hilbert space, T be a bounded linear transformation of H , T∗ be the adjoint of T, and C[ x, y] be the algebra of polynomials in x and y with complex coefficients. For a polynomial p ∈ C[ x, y] in two variables with complex coefficients, define p(T) := Σ m, n ⩾ 0 p ^(m, n)T∗ nT m , where p ^( m, n) is the coefficient of y n x m in the expansion of p in a power series about the point (0, 0). T is called a root of p if and only if p( T) = 0. Note that if p ∈ C[ x, y] is a polynomial in the single variable x, then the definition of p( T) given here agrees with the classical definition. In this paper, we study the relationships which p( T) = 0 forces between p and T when T is an algebraic operator (i.e., there exists n ⩾ 1 and complex numbers a 0, …, a n − 1 such that 0 = a 0 + a 1 T + … + a n − 1 T n − 1 + T n ). The classification starts with the following observation: Suppose p ∈ C[ x, y] and an algebraic operator T ∈ L( H) satisfy p( T) = 0. Then certain subspaces of H which are invariant for T must be orthogonal or certain coefficients of p must vanish. This leads to the notions of a graph attached to each p ∈ C[ x, y] and a graph attached to each square matrix T. For diagonalizable T, a necessary and sufficient graph theoretic condition for solving p( T) = 0 is given. For nondiagonalizable T, this condition is necessary, but not sufficient. The use of these graphs does, however, reduce the problem to the problem of solving the equation p( T) = 0 for T with exactly one or two eigenvalues. For T with one eigenvalue, we give a necessary and sufficient condition for solving p( T) = 0. This leaves the case of solving p( T) = 0 when T has exactly two eigenvalues. This problem mixes algebra involving polynomials with matrix theory. We show that it is equivalent to the purely algebraic problem of determining if equations of the form Σ ( i, j) ∈ E C i, j X i + r, j + s = 0 have solutions of finite support with certain nonvanishing properties. We call these equations bi-Hankel equations subordinate to a given subset E of the lattice of integer pairs {( i, j) : 0 ⩽ i ⩽ m − 1, 0 ⩽ j ⩽ n − 1}. It turns out that there is an algorithm (which uses Gröbner bases) for determining if the type of solution we seek exists and for computing it.

The Coherence Argument Against Conditionalization

I re-examine Coherence Arguments (Dutch Book Arguments, No Arbitrage Arguments) for diachronic constraints on Bayesian reasoning. I suggest to replace the usual game–theoretic coherence condition with a new decision–theoretic condition ('Diachronic Sure Thing Principle'). The new condition meets a large part of the standard objections against the Coherence Argument and frees it, in particular, from a commitment to additive utilities. It also facilitates the proof of the Converse Dutch Book Theorem. I first apply the improved Coherence Argument to van Fraassen's (1984) Reflection principle. I then point out the failure of a Coherence Argument that is intended to support Conditionalization as a naive, universal, update rule. I also point out that Reflection is incompatible with the universal use of Conditionalization thus interpreted. The Coherence Argument therefore defeats the naive view on Bayesian learning that it was originally designed to justify.

On symmetric attractors in reversible dynamical systems

Let Γ ⊂ O( n) be a finite group acting orthogonally on R n . We say that Γ is a reversing symmetry group of a homeomorphism, diffeomorphism or flow f t: R n ↦ R n ( t ∈ Z or t ∈ R ) if Γ has an index two subgroup Γ whose elements commute with f t and for all elements ϱ ∈ Γ - Γ and all t, f t ○ ϱ( x) = ϱ ○ f − t ( x). We give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors (Lyapunov stable ω-limit sets). These conditions arise due to topological obstructions. In dimensions 1 and 2 we present a complete description of possible symmetry groups of asymptotically stable attractors for homeomorphisms and diffeomorphisms (these attractors cannot possess reversing symmetries). We also have a fairly complete description in the context of subgroups which contain reversing symmetries. For all dimensions n we present complete results on the possible symmetry groups of connected asymptotically stable attractors. In addition, we summarize our results in the context of reversible equivariant flows, which are complete in dimensions 1 and 2 as well as for subgroups of Γ ⊂ O(n) in any dimension n. A survey of the corresponding results in the equivariant context is also given.

Capacitary asymptotic expansion of the groundstate to second order

Let H be the non-negative selfadjoint operator associated to a regular irreducible Dirichlet form on Its spectrum will be assumed discrete and λ will denote its lowest eigenvalue. A family of compact subsets of X whose capacity vanishes in the limit is given. Additional Dirichlet boundary conditions on Kεgive rise to a perturbed eigenvalue λε. Potential theoretic conditions involving the reduction of the groundstate eigenfunction are found which ensure first and second order expansions of λε, in terms of a weighted capacity. These results are then applied to the situation when H is a fractional power of the Laplace-Beltrami operator on a compact Riemannian manifold, and Kε is the e-neighbourhood of some closed submanifold. Conditions in terms of the codimension are given for the existence of first and second order expansions. We show finally that the above mentioned potential theoretic conditions are optimal in a weak sense.

Critical Graphs for the Positive Definite Completion Problem

Among various matrix completion problems that have been considered in recent years, the positive definite completion problem seems to have received the most attention. Indeed, in addition to being a problem of great interest, it is related to various applications as well as other completion problems. It may also be viewed as a fundamental problem in Euclidean geometry. A partial positive definite matrix A is "critical" if A has no positive definite completion, though every proper principal submatrix does. The graph G is called critical for the positive definite completion problem if there is a critical partial positive definite matrix A, the graph of whose specified entries is G. Complete analytical understanding of the general positive definite completion problem reduces to understanding the problem for critical graphs. Thus, it is important to try to characterize such graphs. The first, crucial step toward that understanding is taken here. A novel and restrictive topological graph theoretic condition necessary for criticality is identified. The condition, which may also be of interest on pure graph theoretic grounds, is also shown to be sufficient for criticality of graphs on fewer than 7 vertices, and the authors suspect it to be sufficient in general. In any event, the condition, which may be efficiently verified, dramatically narrows the class of graphs for which completability conditions on the specified data are needed. The concept of criticality and the graph theoretic condition extend to other completion problems, such as that for Euclidean distance matrices.

Preservation Theorems Without Continuum Hypothesis

Many results concerning the equivalence between a syntactic form of formulas and a model theoretic conditions are proven directly without using any form of a continuum hypothesis. In particular, it is demonstrated that any reduced product sentence is equivalent to a Horn sentence. Moreover, in any first order language without equality one now has that a reduced product sentence is equivalent to a Horn sentence and any sentence is equivalent to a Boolean combination of Horn sentences.

Deadbeat Controllability for Structured Linear Systems

In this note we study linear structured discrete time systems. We assume that we know which entries in the system matrices are nonzero. We refer to these entries as free parameters. We say that the system is generically deadbeat controllable if it is deadbeat controllable for almost all possible free parameters values. Here deadbeat controllability means that every initial state can be steered to the zero state in a finite number of step by means of an appropriate sequence of controls and for almost all means for all except for those in some proper algebraic variety. In this note we present a graph theoretic condition for a structured system to be generically deadbeat controllable.

Essential norms of composition operators and Aleksandrov measures

culated in terms of the Aleksandrov measures of the inducing holomorphic map. The argument provides a purely functiontheoretic proof of the equivalence of Sarason’s compactness condition for composition operators on L 1 and Shapiro’s compactness condition for composition operators on Hardy spaces. An application is given relating the essential norm to angular derivatives. x1. If is a holomorphic map of the unit disk D into itself, it is a consequence of Littlewood’s subordination principle [5] that composition with induces a bounded operator C on each Hardy space H p . A recurring theme in the study of composition operators has been the search for function theoretic conditions on which guarantee the compactness of C on H p .I t was shown by Shapiro and Taylor [11] that if C is compact on H p for some 0 <p< 1, then C is compact on H p for all 0 <p< 1, and so it is enough to study compactness on H 2 . In this context Shapiro [9] gave an expression for the essential norm of C on H 2 in terms of the Nevanlinna counting function of , thus providing a complete function theoretic characterization of compact composition operators on H 2 .