Finite Dimensional Range Research Articles

Normal subgroups in the group of column-finite infinite matrices

Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

Stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel

In this article, we study the local boundary stabilization of the non-homogeneous Navier–Stokes equations in a 2d channel around Poiseuille flow which is a stationary solution for the system under consideration. The feedback control operator we construct has finite dimensional range. The homogeneous Navier–Stokes equations are of parabolic nature and the stabilization result for such system is well studied in the literature. In the present article we prove a stabilization result for non-homogeneous Navier–Stokes equations which involves coupled parabolic and hyperbolic dynamics by using only one boundary control for the parabolic part.

Duality in non-linear programming

In this paper we consider duality and converse duality for a programming problem involving convex objective and constraint functions with finite dimensional range. We do not assume any constraint qualification. The dual is presented by reducing the problem to a standard Lagrange multiplier problem.

Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

The notion of asymptotic almost periodicity was first introduced by Fréchet in 1941 in the case of finite dimensional range spaces. Later, its extension to the case of Banach range spaces and locally convex range spaces has been considered by several authors. In this paper, we have generalized the concept of asymptotic almost periodicity to the case where the range space is a general topological vector space, not necessarily locally convex. Our results thus widen the scope of applications of asymptotic almost periodicity.

Compact homomorphisms of regular banach algebras

Let A be a complex commutative Banach algebra and let MA be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points ϕ1, ϕ2 ∈ MA, there is an element a ∈ A for which , and ‖a‖ ⩽ C, where is the Gelfand transform of a ∈ A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Representation and Approximation of Positivity Preservers

We consider a closed set S⊆ℝn and a linear operator $$\Phi \colon \mathbb{R}[X_1,\ldots,X_n]\rightarrow \mathbb{R}[X_1,\ldots,X_n]$$ that preserves nonnegative polynomials, in the following sense: if f≥0 on S, then Φ(f)≥0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which concerns linear functionals on ℝ[X1,…,Xn]. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range.

On [formula omitted]-stable mappings with values in infinite-dimensional Banach spaces

The aim of the paper is to collect some results concerning ℓ -stability of mappings with values in possibly infinite-dimensional Banach spaces. We show that any ℓ -stable function from finite-dimensional space to an arbitrary Banach space is Lipschitz near the reference point. Further, we show that any ℓ -stable function from finite-dimensional space to a Banach space having Radon–Nikodým property is strictly differentiable at the reference point. As an application we present the second-order sufficient condition for the unconstrained optimization problem previously obtained for finite-dimensional range space.

Stable-Range Approach to Short Wave and Khokhlov-Zabolotskaya Equations

Short wave equations were introduced in connection with the nonlinear reflection of weak shock waves. They also relate to the modulation of a gas-fluid mixture. Khokhlov-Zabolotskaya equation is used to describe the propagation of a diffraction sound beam in a nonlinear medium. We give a new algebraic method of solving these equations by using certain finite-dimensional stable range of the nonlinear terms and obtain large families of new explicit exact solutions parameterized by several functions for them. These parameter functions enable one to find the solutions of some related practical models and boundary value problems.

Stable-range approach to the equation of nonstationary transonic gas flows

Using a certain finite-dimensional stable range of the nonlinear terms, we obtain large families of exact solutions parameterized by functions for the equation of nonstationary transonic gas flows discovered by Lin, Reissner and Tsien and its three-dimensional generalization.

Sections of convex bodies and splitting problem for selections

Let F 1 : X → Y 1 and F 2 : X → Y 2 be any convex-valued lower semicontinuous mappings and let L : Y 1 ⊕ Y 2 → Y be any linear surjection. The splitting problem is the problem of representation of any continuous selection f of the composite mapping L ( F 1 ; F 2 ) in the form f = L ( f 1 ; f 2 ) , where f 1 and f 2 are some continuous selections of F 1 and F 2 , respectively. We prove that the splitting problem always admits an approximate solution with f i being an ε-selection (Theorem 2.1). We also propose a special case of finding exact splittings, whose occurrence is stable with respect to continuous variations of the data (Theorem 3.1) and we show that, in general, exact splittings do not exist even for the finite-dimensional range.

Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism

We study the existence of solutions to the orthogonal dynamics equation, which arises in the Mori-Zwanzig formalism in irreversible statistical mechanics. This equation generates the random noise associated with a reduction in the number of variables. IfL is the Liouvillian, or Lie derivative associated with a Hamiltonian system, andP an orthogonal projection onto a closed subspace ofL 2, then the orthogonal dynamics is generated by the operator (I −P)L. We prove the existence of classical solutions for the case whereP has finite-dimensional range. In the general case, we prove the existence of weak solutions.

LIMITS OF HOMOMORPHISMS WITH FINITE-DIMENSIONAL RANGE

Let X be a compact metric space and A be a unital simple C*-algebra with TR (A)=0. Suppose that ϕ : C(X) → A is a unital monomorphism. We study the problem when ϕ can be approximated by homomorphisms with finite-dimensional range. We give a K-theoretical necessary and sufficient condition for ϕ being approximated by homomorphisms with finite-dimensional range.

Geometry of foliations on the Wiener space and stochastic calculus of variations

Stochastic Calculus of variations deals with maps defined on the Wiener space, with finite dimensional range; within this context appears the notion of non-degenerate map, which corresponds roughly speaking to some kind of infinite dimensional ellipticity; a non-degenerate map has a smooth law; by conditioning, it generates a continuous desintegration of the Wiener measure. Infinite dimensional Stochastic Analysis and particularly SPDE theory raise the natural question of what can be done for maps with an infinite dimensional range; our approach to this problem emphasizes an intrinsic geometric aspect, replacing range by generated σ-field and its associated foliation of the Wiener space. To cite this article: H. Airault et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

When almost multiplicative morphisms are close to homomorphisms

ABsrRACT. It is shown that approximately multiplicative contractive positive morphisms from C(X) (with dim X < 2) into a simple C*-algebra A of real rank zero and of stable rank one are close to homomorphisms, provided that certain K-theoretical obstacles vanish. As a corollary we show that a homomorphism h: C(X) -+ A is approximated by homomorphisms with finite dimensional range, if h gives no K-theoretical obstacle.

Fragility of Subhomogeneous C *-Algebras with One-Dimensional Spectrum

We prove that a large and natural class of subhomogeneous C*-algebras with one-dimensional spectrum have the fragility property, that is, every morphism out of such a C*-algebra and into a C*-algebra of real rank zero can be approximated by a morphism with finite-dimensional range, provided that a K-theoretical obstruction vanishes. 1991 Mathematics Subject Classification 46L05.

Homomorphisms From C(X) Into C*-Algebras

AbstractLet A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

On the Limits of (Linear Combinations of) Iterates of Linear Operators

We give necessary and sufficient conditions such that iterates or certain linear combinations of iterates of linear operators of finite dimensional range, respectively, converge. In case of convergence, we give an expression for the limit as well as estimates for the rate of convergence. Our results are then applied to Schoenberg type and Sablonnière operators as well as tensor product Schoenberg type operators, Bernstein and Bernstein–Durrmeyer operators over triangles.

A note concerning the ideal of nuclear operators

AbstractLet be a Banach algebra of bounded linear operators such that contains every operator with finite dimensional range. Then contains every nuclear operator.

Second-order conditions in extremal problems with finite-dimensional range. 2-normal maps

A minimization problem with constraints that includes problems for which the constraints are of equality and inequality type is considered. First- and second-order necessary conditions in the Lagrangian form are obtained for this problem. The main difference between these conditions and most of the previously known ones is the fact that they also remain meaningful for abnormal problems, in both the finite-dimensional and infinite-dimensional cases. The notion of 2-normal map is introduced. It is proved that if the map that defines a constraint is 2-normal, then the necessary conditions obtained turn into second-order sufficient conditions after an arbitrarily small perturbation of the problem by terms of second order of smallness. It is also proved that in the space of smooth maps, the set of 2-normal maps is everywhere dense in the Whitney topology.

Locally Compact Groups which have the Weakly Compact Homomorphism Property

A locally compact group G is WCHP if every weakly compact homomorphism from ${L^1}(G)$ into a Banach algebra has finite-dimensional range, and is ${\text {WCHP}^ + }$ if every extension of G by an abelian group is WCHP. We verify the ${\text {WCHP}^ + }$ property for certain locally compact groups, including all Moore groups and all connected groups.