Self-correcting approximate solution by the iterative method for linear and nonlinear stochastic differential equations

The iterative method of the first author makes possible a self-correcting scheme for the approximate solution obtained. This scheme accelerates the convergence and substantially decreases the number of terms necessary in computation for applications involving either linear or nonlinear stochastic systems. A “feedback” term compensates for the approximation of the system inverse operator by a partial sum. Further, errors are determined in calculating the mean solution 〈y〉 and the correlation $\text{R}{}^{\mathrm{y}}\text{(t}{}^1\text{,t}{}^2\text{) = 〈y(t}{}^1\text{)}\text{y}\text{1}\text{(t}{}^2\text{)〉}$ by using the approximations 〈φ_n〉 and 〈φⁿ(t¹)φⁿ(t²)〉 , where φ_n represents n terms of the solution by the iterative method for stochastic differential equations.     

Solving linear systems via Pfaffians

An explicit solution is given for the system of linear equations EφE^t=φ^′, where φ, φ^′ are alternating matrices of Pfaffian 1, which at most differ in their first row and column, and E is of the form . If v, v^′, w∈R^r+1 with 〈v,w〉=1=〈v^′,w〉 then a sequence of Cohn transforms with respect to (the fixed) w which takes (v,w) to (v^′,w) is prescribed.     

The linearity coefficient of metric projections onto one-dimensional Chebyshev subspaces of the space C

The paper deals with the operator of metric projection onto an arbitrary one-dimensional Chebyshev subspace 〈φ〉 of the space C[K] of real-valued functions defined and continuous on a Hausdorff compact set K. The linearity coefficient of the operator is calculated in terms of the parameters of the generating function φ. As a consequence, a new estimate of the Lipschitz constant of the operator is obtained.     

Complementary variational principles for linear equations

This paper presents a useful alternative to the classical complementary variational principles associated with the equation Aφ = f, where A is a bounded linear operator.     

Lower bound for the convergence rate of nonstationary Jacobi-like iteration

Stationary and nonstationary Jacobi-like iterative processes for solving systems of linear algebraic equations are examined. For a system whose coefficient matrix A is an H-matrix, it is shown that the convergence rate of any Jacobi-like process is at least as high as that of the point Jacobi method as applied to a system with 〈A〉 as the coefficient matrix, where 〈A〉 is a comparison matrix of A.     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

Singular continuous Floquet operator for systems with increasing gaps

Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e^−iH₀Te^{−iκT|φ〉〈φ|}, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H₀ is pure point, simple, bounded from below and the gaps between the eigenvalues (λ_n) grow like λ_n+1−λ_n∼Cn^d with d?2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H₀, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.     

Generalized slant Toeplitz operators on H2

For an integer k ≥ 2, k^th‐order slant Toeplitz operator U_φ [1] with symbol φ in L^∞(??), where ?? is the unit circle in the complex plane, is an operator whose representing matrixM = (α_ij ) is given by α_ij = 〈φ, z^ki–j〉, where 〈. , .〉 is the usual inner product in L²(??). The operator V_φ denotes the compression of U_φ to H²(??) (Hardy space). Algebraic and spectral properties of the operator V_φ are discussed. It is proved that spectral radius of V_φ equals the spectral radius of U_φ, if φ is analytic or co‐analytic, and if T_φ is invertible then the spectrum of V_φ contains a closed disc and the interior of the disc consists of eigenvalues of infinite multiplicities. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp bounds on the ranks of negativity of certain sums

If M is a complex vector space and 〈·, ·〉 a Hermitian sesquilinear form on M with a finite rank of negativity k (i.e., k is the maximal dimension of any linear subspace E of M satisfying 〈x, x〉 < 0 for each nonzero x in E), if n is a positive integer, and if a ₁, …, a _n are endomorphisms of M, then it is easy to see that the Hermitian sesquilinear form $ (x,y) \mapsto \sum\limits_{v = 1}^n {\left\langle {a_v x,a_v y} \right\rangle } $ on M has rank of negativity at most nk. It is also fairly easy to see that the bound nk cannot be improved in general. Less trivial is the fact that it cannot be improved by making the following assumption (a) the space M is the *-algebra A:= (C[[w ₁, w ₂]] of polynomials in two self-adjoint non-commuting indeterminates; there is a (necessarily Hermitian) linear form φ on A such that 〈x, y〉 = φ(y* x) (x, y ∈ A); and a _v is just left multiplication by some element of A (which we may denote by ‘a _v ’ at no great risk of confusion). Now suppose that, with M, 〈·, ·〉, k, n, and a ₁ , …, a _n as initially, the following two conditions are satisfied:

1. each a _v has a formal adjoint a* _v , being an endomorphism of M such that $ \left\langle {a_v x,y} \right\rangle = \left\langle {x,a_v^* y} \right\rangle (x,y \in M); $
2. the mappings a ₁, …, a _n , a*₁, …, a* _n commute pairwise.

Then the bound nk can be replaced by k (regardless of how large n may be). This result cannot be improved in general since it may happen that each a _v is a scalar multiple of the identical mapping of M into itself (not all a _v equal to 0), in which case the form (1) is a positive multiple of 〈·, ·〉 itself. There are ties with the subjects of ‘positive semidefinite submodules’ (‘positive semidefinite left ideals’) and ‘definitisation’.     

Subcentrality of restrictions of boundary measures on state spaces of C1-algebras

Let F be a closed face of the weak¹ compact convex state space of a unital C¹-algebra A. The author has already shown that F is a Choquet simplex if and only if p_φ^Fπ_φ(A)″p_φ^F is abelian for any φ in F with associated cyclic representation ($\text{H}$_φ,π_φ,ξ_φ), where p_φ^F is the orthogonal projection of $\text{H}$_φ onto the subspace spanned by vectors η defining vector states a → 〈π_φ(a)η, η)〉 lying in F. It is shown here that if B is a C¹-subalgebra of A containing the unit and such that ξ_φ is cyclic in $\text{H}$_φ for π_φ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if p_φ^F(π_φ(A), π_φ(B)′)″p_φ^F is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (π_φ(A)′ ∩ π_φ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C¹-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central.     

Zero-dimensional proximities and zero-dimensional compactifications

We introduce zero-dimensional proximities and show that the poset 〈Z(X),?〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈Π(X),?〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈Π(X),?〉 is isomorphic to the poset 〈B(X),⊆〉 of Boolean bases of X, and derive Dwinger's theorem that 〈Z(X),?〉 is isomorphic to 〈B(X),⊆〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone-?ech compactification of X is a unique up to equivalence extremally disconnected compactification of X.     

Linear Optimization with Box Constraints in Banach Spaces

Let X be a partially ordered real Banach space, let a,b∈X with a≤b. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max 〈φ,x〉, s.t. a≤x≤b, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author’s doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions:

1. (i)
   When is a closed subspace of a BOP space a BOP space?
    
2. (ii)
   When is the range of a bounded linear map a BOP space?
    
3. (iii)
   Is the quotient space of a BOP space a BOP space?
    

     

Scattering pour l'équation de Schrödinger en présence d'un potentiel répulsif

We study scattering theory for linear Schrödinger equations, when the reference Hamiltonian is ?Δ?〈x〉^α, in $\text{R}{}^{\mathrm{n}}$, with 0<α?2. The notion of short range perturbative potential is much weaker than for the usual reference Hamiltonian ?Δ. We also consider the case where 〈x〉² is replaced by a general second order polynomial. To cite this article: J.-F. Bony et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Homogeneoys graphs

Let Γ be a finite graph with vertex set VΓ, and let U, V be arbitrary subsets of VΓ. Γ is homogeneoys (resp. ultrahomogeneous) if whenever the induced subgraphs 〈U〉, 〈V〉 are isomorphic, some isomorphism (resp. every isomorphism) of 〈U〉 onto 〈V〉 extends to an automorphism of Γ. We extend a theorem of Sheehan on ultrahomogeneous graphs to the homogeneous case, and complete his classification of ultrahomogenous graphs.     

Combinatorial structures in loops I. Elements of the decomposition theory

Difference sets have been extensively studied in groups, principally in Abelian groups. Here we extend the notion of a difference set to loops. This entails considering the class of 〈υ, k〉 systems and the special subclasses of 〈υ, k, λ〉 principal block partial designs (PBPDs) and 〈υ, k, λ〉 designs. By means of a certain permutation matrix decomposition of the incidence matrices of a system and its complement, we can isomorphically identify an abstract 〈υ, k〉 system with a corresponding system in a loop. Special properties of this decomposition correspond to special algebraic properties of the loop. Here we investigate the situation when some or all of the elements of the loop are right inversive. We identify certain classes of 〈υ, k, λ〉 designs, including skew-Hadamard designs and finite projective planes, with designs and difference sets in right inverse property loops and prove a universal existence theorem for 〈υ, k, λ〉 PBPDs and corresponding difference sets in such loops.     

The k-factor conjecture is true

A sequence 〈d_i〉, 1≤i≤n, is called graphical if there exists a graph whose i^th vertex has degree d_i for all i. It is shown that the sequences 〈d_i〉 and 〈d_i-k〉 are graphical only if there exists a graph G whose degree sequence is 〈d_i〉 and which has a regular subgraph with k lines at each vertex. Similar theorems have been obtained for digraphs. These theorems resolve comjectures given by A.R. Rao and S.B. Rao, and by B. Grünbaum.     

High frequency and numerical Eulerian methods for aeroacoustic problems

The aim of this work is the simulation of the acoustic propagation in a moving flow using the high-frequency approach. We linearize the Euler equations around a stationary state for which the resulting system of PDE cannot be in general reduced to a wave equation. We are however able to perform a high-frequency analysis of the acoustic perturbation, using the W.K.B. method, introducing a phase φ$\phi$ and an amplitude A  . The phase φ$\phi$ is solution of a Hamilton–Jacobi equation that we solve by a numerical Eulerian method using a monotone scheme [S.J. Osher, C.W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations, SIAM J. Numer. Anal, 28(4) (1991) 907–922] following Benamou et al. [A geometric optics method for high frequency electromagnetic fields computations near fold caustics Part I, J. Comput. Appl. Math. 156 (2003) 93–125]. Adopting the techniques of Lax and Rauch [Lectures on Geometric Optics, 〈$〈$http://www.lsa.umich.edu/rauch〉$〉$] for hyperbolic systems, we compute the leading order term of the amplitude A. Our results are still valid in the neighborhood of a fold caustic.     

Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.