Weak and strong extensions of ordinary differential operators

Let A be a second order differential operator with positive leading term defined on an interval J of R. In this paper we study conditions for the equality D₀(A) = D₁(A) to hold. Here D₀(A) and D₁(A) are the domains of the minimal and maximal extensions of A respectively. Under the general assumption that $\text{A(1)}\text{and}\text{A}{}^1\text{(1)}$ are bounded above it is proven that under certain conditions D₀(A) = D₁(A) if functions which are constant near the boundaries of J are in $\text{D}{}_0\text{(A) ∩ D}{}_0\text{(A}{}^1\text{)}$ whenever they are in $\text{D}{}_1\text{(A) ∩ D}{}_1\text{(A}{}^1\text{)}$. In particular if A is formally selfadjoint and 1 ?D₁(A) then D₁(A) = D₀(A) if and only if 1 ?D₀(A). When the measure of J is infinite at both ends D₀(A) is always equal to D₁(A). This fact is used to show that the leading term of A as well as its terminal coefficient can be chosen arbitrarily (although not independently of one another) in such a way that the equality D₀ = D₁ holds.     

Applications of the Connes spectrum to C1-dynamical systems,II

We show that for a C¹-dynamical system (A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to $\text{G}\text{?}$ if and only if every nonzero closed ideal in G × _αA has a nonzero intersection with A. Denote by G_J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G_J are discrete, then GX_αA is simple if and only if A is G-simple and $\text{Γ(}\text{α}\text{) =}\text{G}\text{?}$. This result is applicable not only when G is discrete but also when G?$\text{R}$ or G?$\text{T}$ provided that A is not primitive. Specializing to single automorphisms (i.e., G=$\text{Z}$) we show that if (the transposed of) α acts freely on a dense set of points in $\text{A}\text{?}$, then Λ(α)=$\text{T}$. The converse is only proved when A is of type I.     

On two classes of matrices with positive diagonal solutions to the Lyapunov equation

We characterize real indecomposable quasi-Jacobi matrices of class $\text{D}$, i.e., those which satisfy the Lyapunov equation PA + A′P = ?Q with P diagonal and both P and Q positive definite. The subclass $\text{D}$₂ (of class $\text{D}$) when also Q is diagonal is also characterized in the case of general indecomposable real matrices.     

A rank characterization of the number of final classes of a nonnegative matrix

Let A be a nonnegative square matrix, and let D be a diagonal matrix whose iith element is $\text{(Ax)}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}$, where x is a (fixed) positive vector. It is shown that the number of final classes of A equals n?rank(A?D). We also show that null(A?D) = null(A?D)², and that this subspace is spanned by a set of nonnegative elements. Our proof uses a characterization of nonnegative matrices having a positive eigenvector corresponding to their spectral radius.     

Stopping times and an extension of stochastic integrals in the plane

Let (Ω,$\text{J}$,P;$\text{J}$_z) be a probability space with an increasing family of sub-σ-fields {$\text{J}$_z, z ∈ D}, where D = [0, ∞) × [0, ∞), satisfying the usual conditions. In this paper, the stochastic integral with respect to an $\text{J}$_z-adapted 2-parameter Brownian motion for integrand processes in the class $\text{C}$₂($\text{J}$_z) is extended, by means of truncations cations by {0, 1}-valued 2-parameter stopping times, to integrand processes that are $\text{J}$_z-adapted and continuous. The stochastic integral in the plane thus extended resembles a locally square integrable martingale in the 1-parameter setting. A definition of a parameter-space valued, i.e., D-valued, stopping time is also given and its characteristic process is related to a {0, 1}-valued 2-parameter stopping time.     

A decomposition of R into two homeomorphic rigid parts

Let X be separable, completely metrizable, and dense in itself. We show that if X admits a triple (D₁, D₂, h) of two countable dense subsets D₁ and D₂ and a homeomorphism h: X?D₁ → X?D₂, satisfying some special properties, then there is a rigid subspace A of X such that A is homeomorphic to X?A = h[A]; for $\text{X =}\text{R}$, such atriple is shown to exist.     

The nontriviality of certain Zl-extensions

Let k be a J-field, K the basic Z_l-extension of k, and A₀, A the l-class groups of k, K respectively. It is known that if A₀^1?J is nontrivial, then $\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>J</mtext><mtext>?</mtext>}}$ is infinite. It is shown that this result is still true if the classes represented by the primes lying over l are factored out from both groups. This is applied to $\text{k = Q((?m)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ and A₀ = (0) for information on the invariants λ and μ. There are such k for which λ ≥ 2.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.     

The additive and logical complexities of linear and bilinear arithmetic algorithms

Let C_A(±) be the additive complexity of a (bi)linear algorithm A for a given problem; D(A) and $\text{D}\text{(A)}$ are two acyclic diagraphs that represent A, each of them is obtained from another one by reversing directions of all edges; ir(D) and do(D) are two numbers that are introduced to measure the structural deficiencies of an acyclic digraph D. K and Q are the numbers of outputs and input-variables. $\text{do(}\text{D}\text{(A))}$, do(D(A)), and ir(D(A)) characterize the logical complexity of A. It is shown that C_A(±) + do(D(A)) + ir(D(A)) = ω(K + Q)log(K + Q) and $\text{C}{}_{\mathrm{A}}\text{(±) + do(}\text{D}\text{(A)) = ω(K + Q)}\text{log}\text{(K + Q)}$ in the cases of DFT, vector convolution, and matrix multiplication. Also lower bounds on C_A(±) + do(D(A)) and on C_A(±) are expressed in terms of algebraic quantities such as the ranks of matrices and of multidimensional tensors associated with the problems.     

A note on matrix subalgebras containing a full Jordan block

Let A be a subalgebra of the full matrix algebra M_n(F), and suppose J∈A, where J is the Jordan block corresponding to xⁿ. Let $\text{S}$ be a set of generators of A. It is shown that the graph of $\text{S}$ determines whether A is the full matrix algebra M_n(F).     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

On the spectra of a family of positive matrices

It is shown that every positive matrix A can be embedded in an analytic family of positive matrices {A(ν) : ν∈$\text{R}$} in such a way that A(1)=A, $\text{A(0)≡}\text{A}\text{?}$ is symmetric, and A(-1)=A^T. A necessary and sufficient condition that A and Å have the same maximal eigenvalue and that their ergodic limits have the same diagonal elements is stated and proved.     

Un contre-exemple pour les formes invariantes sur un corps fini

Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements: (i) ∥ a ° b ∥ ≤ ∥ a ∥ ∥ b ∥, (ii) ∥ a² ∥ = ∥ a ∥², (iii) ∥ a² ∥ ≤ ∥ a² + b² ∥ for a, b?A. It is shown that A possesses a unique norm closed Jordan ideal J such that $\text{A}\text{J}$ has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every “irreducible” representation of A not annihilating J is onto the exceptional Jordan algebra M₃⁸.     

Extensions of the Ostrowski-Reich theorem for SOR iterations

The Ostrowski-Reich theorem states that for a system Ax =b of linear equations with A nonsingular, if A is hermitian and if the diagonal of A is positive, then the SOR method converges for each relaxation parameter in (0,2) if and only if A is positive definite. This is actually a special case of the Householder-John theorem, which states that for A=M?N with A,M nonsingular, if A is hermitian and $\text{M}{}^1\text{+N}$ is positive definite, then M^?1N is a convergent matrix if and only if A is positive definite. Our purposes here are to generalize the Householder-John theorem and to provide an insight into how and why the SOR method can converge. As a result the Ostrowski-Reich theorem is extended in two directions; one is when A is hermitian but the diagonal of A is not necessarily positive, so that A is not necessarily positive definite, and the other is when $\text{A+A}{}^1$ is positive definite but A is not necessarily hermitian. In the process, several other convergence results are obtained for general splittings of A. However, no claims are made concerning the case in which the convergence results obtained here can be applied to practical situations.     

Representation of functions of Markov processes as solutions of stochastic equations

Let X_t be a homogeneous Markov process generated by the weak infinitesimal operator A. Let $\text{H}$ be the class of functions f such that f, f²?D_A, the domain of A. The main result of this paper states that for ? ∈ $\text{H}$ can be represented by a stochastic integral and other terms. If the process is generated by a second order differential operator (with ‘poor’ coefficients possibly) on C₀²(R^d) then the process itself can be represented as the solution of an Itô stochastic differential equation.     

Algebraic integers and distributions on the N-torus

Let D(T_N) be the class of real-valued, infinitely differentiable, periodic functions, and let D′(T_N) be the class of real-valued distributions on T_N having D(T_N) as its test functions. Define A(T_N) ? D(T_N) as follows: S is in A(T_N) if there is a constant K such that $\text{¦ S^(m)¦ ? K}$ for all m, and, furthermore, lim_{min(|m1|,…,|mN|)←∞}|S^{^}(m)|=0. For $\text{0 < ξ <}\text{1}\text{2}$, let C(ξ) be the familiar Cantor set with constant ratio of dissection ξ, constructed on the interval [?π, π). The following result is established: A necessary and sufficient condition thatC(ξ₁) × … × C(ξ_N)be a set of uniqueness for the classA(T_N)is that eachξ_j^?1be an$\text{J}$number forj = 1,…, N.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Presentations for groups acting on simply-connected complexes

New expansions for global semigroup theory are developed. Many expansions have a left and a right version, each with specific (dual) properties; e.g., the Rhodes expansions ?^$\text{L}$, resp. ?^$\text{R}$, have unambiguous $\text{L}$-resp. $\text{R}$-order. In applications one sometimes needs expansions having both properties simultaneously; these can be constructed by alternately applying the left and the right expansion (possibly infinitely often) while keeping the same set of generators. Thus one obtains an expansion which is invariant under application of the old two expansions and thus has the properties of both (e.g., one obtains -+ with

, and so -+ has unambiguous $\text{L}$-and $\text{R}$-order). It is proved that, in the case of the Rhodes expansion, the new expansion is ‘close’ to the original semigroup; in particular (and this is the main result of the paper), ?⁺_A is finite (resp. finite $\text{J}$-above) if S is finite (resp. finite$\text{J}$-above).     

Sur l'η-expansion infinieOn infinite η-expansion

We give two characterizations of the ordering on Böhm trees induced by the D_∞ model, one of which formalizes a continuity property of infinite η-expansion: $\text{A}\text{?}{}_{\mathrm{\infty }}\text{B}$ if for any finite approximant A of $\text{A}$ there exists a finite approximant B of $\text{B}$ such that A is a sub-tree of B, modulo finitely many η-equalities and finitely many infinite η-expansions of variables. To cite this article: P.-L. Curien, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 77–82