Proper Subspace Research Articles

On secant updates for use in general constrained optimization

In this paper we present two new classes of successive quadratic programming (SQP) secant methods for the equality-constrained optimization problem. One class of methods uses the SQP augmented Lagrangian formulation, while the other class uses the SQP Lagrangian formulation. We demonstrate, under the standard assumptions, that in both cases the BFGS and DFP versions of the algorithm are locally q-superlinearly convergent. To our knowledge this is the first time that either local or q-superlinear convergence has been established for an SQP Lagrangian secant method which uses either the BFGS or DFP updating philosophy and assumes no more than the standard assumptions. Since the standard assumptions do not require positive definiteness of the Hessian of the Lagrangian at the solution, it is no surprise that our BFGS and DFP updates possess the hereditary positive definiteness property only on a proper subspace.

Translates of exponential box splines and their related spaces

Exponential box splines ( E B EB -splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space H \mathcal {H} spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space H {\mathbf {H}} of all entire functions spanned by the integer translates of an E B EB -spline. This investigation relies on a detailed description of H \mathcal {H} and its discrete analog S \mathcal {S} . The approach taken here is based on the observation that the structure of H \mathcal {H} is relatively simple when H \mathcal {H} is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex E B EB -splines. The first is when the Fourier transform of the E B EB -spline vanishes at some critical points, the second is when H \mathcal {H} cannot be embedded in S \mathcal {S} and the third is when H {\mathbf {H}} is a proper subspace of H \mathcal {H} . We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.

Symmetry diffeomorphism group of a manifold of nonpositive curvature

Let M ~ \tilde M denote a complete simply connected manifold of nonpositive sectional curvature. For each point p ∈ M ~ p \in \tilde M let s p {s_p} denote the diffeomorphism of M ~ \tilde M that fixes p p and reverses all geodesics through p p . The symmetry diffeomorphism group G ∗ {G^{\ast }} generated by all diffeomorphisms { s p : p ∈ M ~ } \{ {s_p}:\,p \in \tilde M\} extends naturally to group of homeomorphisms of the boundary sphere M ~ ( ∞ ) \tilde M(\infty ) . A subset X X of M ~ ( ∞ ) \tilde M(\infty ) is called involutive if it is invariant under G ∗ {G^{\ast }} . Theorem. Let X ⊆ M ~ ( ∞ ) X \subseteq \tilde M(\infty ) be a proper, closed involutive subset. For each point p ∈ M ~ p \in \tilde M let N ( p ) N(p) denote the linear span in T p M ~ {T_p}\tilde M of those vectors at p p that are tangent to a geodesic γ \gamma whose asymptotic equivalence class γ ( ∞ ) \gamma (\infty ) belongs to X X . If N ( p ) N(p) is a proper subspace of T p M ~ {T_p}\tilde M for some point p ∈ M ~ p \in \tilde M , then M ~ \tilde M splits as a Riemannian product M ~ 1 × M ~ 2 {\tilde M_1} \times {\tilde M_2} such that N N is the distribution of M ~ \tilde M induced by M ~ 1 {\tilde M_1} . This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if M ~ \tilde M is irreducible and M ~ ( ∞ ) \tilde M(\infty ) admits a proper, closed involutive subset X X , then M ~ \tilde M is isometric to a symmetric space of noncompact type and rank k ⩾ 2 k \geqslant 2 .

Optimal control of nonlinear hereditary systems

A function space Λ is introduced for the study of nonlinear hereditary differential equations. The properties of Λ include: it is a Banach space under the supremum norm, the continuous functions constitute a closed proper subspace, and the unit ball is sequentially compact in the weak- ∗ topology. Existence, uniqueness, and continuous dependence results are obtained for solutions of a broad class of initial value problems. An optimization problem is formulated for systems which are affine in the control, and solutions are approximated by means of a sequence of problems which are finite-dimensional in the control.

A Lattice-Theoretic Equivalent of the Invariant Subspace Problem

Every bounded linear operator on complex infinite-dimensional separable Hilbert space has a proper invariant subspace if and only if for every lattice automorphism $\phi$ of a certain abstract complete lattice $P$ (defined by N. Zierler) there exists an element $a \in P$ different from 0 and 1 such that ${\phi ^2}(a) \leq a$.

A geometric consequence of the classification of finite doubly transitive groups

We prove that the finite linear spaces containing a proper linear subspace and admitting an automorphism group which is transitive on the unordered pairs of intersecting lines are the projective and affine spaces of dimension ≥3, unless all lines have size 2.

A Note on Uniform Operators

An operator is uniform if its restriction to any infinite-dimensional invariant subspace is unitarily equivalent to itself. We show that a uniform operator having a proper infinite-dimensional invariant subspace resembles an analytic Toeplitz operator in the way that the weakly closed algebra generated by it and the identity operator is isomorphic to a subalgebra of the Calkin algebra; furthermore, this algebra contains no nonscalar operator which is quasi-similar to a normal operator.

Hyperreflexivity and a Dual Product Construction

We show that an example of a nonhyperreflexive CSL algebra recently constructed by Davidson and Power is a special case of a general and natural reflexive subspace construction. Completely different techniques of proof are needed because of absence of symmetry. It is proven that if $\mathcal {S}$ and $\mathcal {I}$ are reflexive proper linear subspaces of operators acting on a separable Hilbert space, then the hyperreflexivity constant of ${({\mathcal {S}_ \bot } \otimes {\mathcal {I}_ \bot })^ \bot }$ is at least as great as the product of the constants of $\mathcal {S}$ and $\mathcal {I}$.

A lattice-theoretic equivalent of the invariant subspace problem

Every bounded linear operator on complex infinite-dimensional separable Hilbert space has a proper invariant subspace if and only if for every lattice automorphism ϕ \phi of a certain abstract complete lattice P P (defined by N. Zierler) there exists an element a ∈ P a \in P different from 0 and 1 such that ϕ 2 ( a ) ≤ a {\phi ^2}(a) \leq a .

A note on uniform operators

An operator is uniform if its restriction to any infinite-dimensional invariant subspace is unitarily equivalent to itself. We show that a uniform operator having a proper infinite-dimensional invariant subspace resembles an analytic Toeplitz operator in the way that the weakly closed algebra generated by it and the identity operator is isomorphic to a subalgebra of the Calkin algebra; furthermore, this algebra contains no nonscalar operator which is quasi-similar to a normal operator.

Hyperreflexivity and a dual product construction

We show that an example of a nonhyperreflexive CSL algebra recently constructed by Davidson and Power is a special case of a general and natural reflexive subspace construction. Completely different techniques of proof are needed because of absence of symmetry. It is proven that if S \mathcal {S} and I \mathcal {I} are reflexive proper linear subspaces of operators acting on a separable Hilbert space, then the hyperreflexivity constant of ( S ⊥ ⊗ I ⊥ ) ⊥ {({\mathcal {S}_ \bot } \otimes {\mathcal {I}_ \bot })^ \bot } is at least as great as the product of the constants of S \mathcal {S} and I \mathcal {I} .

Series that converge on sets of null density

It is shown that a series of positive terms that converges on all sets of null density should be convergent. Using this result we construct examples of complete topological vector spaces that are proper subspaces of a Banach space, but whose dual spaces coincide with the dual space of the Banach space.

On the spectral mapping theorem for essential spectra

B. Gramsch and D. Lay have studied spectral mapping theorems for the essential spectra of an operator acting in a complex Banach space. Firstly they consider operators belonging to the Banach algebra of all bounded linear operators on the space, and later they derive the theorems for unbounded closed linear operators with non-empty resolvent from the aboye case ; but bounded closed linear operators with domain a proper subspace are not included. In this note we introduce a notion of extended essential spectra for any closed linear operator with non-empty resolvent, which covers the above cases. Then, in this more general context, we are able to prove the spectral mapping theorems by means of a more unified approach based on a factorization of the operators provided by the Dunford-Taylor calculus and well-known properties of products of operators present in Fredholm theory.

Minimal projections in Lp-spaces

Let X be a normed linear space and let W be a proper subspace of X. A projection is a surjective linear map P: X → W such that P is idempotent. It is immediately clear that P has norm at least unity. Thus the problem of calculating the numberhas some interest. The number λ(W, X) is often called the relative projectiion constant of W in X. If the infimum is attained, any attaining projection is called a minimal projection. The problems of calculating λ(W, X) for a fixed X and W or finding a minimal projection turn out to be very dificult. For example, if X = C [0, 1] with the usual supremem norm and W is the subspace of polynominals of degree at most two then λ(W, X) remains unknown as does any example of a minimal projection. One of the few places where the problem shows much tractability is the case

The exterior dirichlet problem for the Laplace equation

AbstractUsing a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex‐valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.

Vanishing Theorems on Complete Kähler Manifolds

Let X be a complex manifold of dimension n and let E be a holomorphic vector bundle over X. We shall here try to continue the study on the vanishment of the sheaf cohomology groups H(X3 0(E)) which has been performed by Kodaira [10], [11], GrauertRiemenschneider [5], Andreotti-Vesentini [1], [2], Nakano [14], [15], Kazama [9], and others. The purpose of the present paper is to study the cohomology groups on complete Kahler manifolds. Although the spirit is the same as in [1] and [14], we restrict ourselves to L-cohomology groups' and aim at finding a proper subspace of L-forms for which 8-equation is solvable. We shall prove the following theorem.

A note on hyponormal weighted shifts

If T T is a hyponormal weighted shift (unilateral or bilateral) and if p p is a polynomial, p ( T ) p(T) may not be hyponormal.

Aggregation of Models I-lith Restricted Domains: An Application of the Pseudoinverses

The paper is concerned with the aggregation problem for models with restricted domains, i. e. when the input variables are restricted to lie in some proper subset of their domain. First, nесеssагу and sufficient conditions for aggregation of models with restricted domains are discussed. Both the input-output models and the models defined by implicit relation are considered. We obtain conditions for aggregation that are milder than the conditions that would be required if the domains were unrestricted. Next, the order reduction problem for time-invariant dynamic linear systems is considered. The control variables and state variables are restricted to some proper subspaces of their respective domains.The pseudoinverses are used to obtain conditions that must be satisfied by the aggregation matrices. Moreover the closed form solutions for a class of aggregated models are presented.

On Completion of Spaces of Weakly continuous Functions

Let E and F be two Banach spaces and let A be a nonempty subset of E . A mapping f:A→F is said to be weakly continuous if it is continuous when A has the relative weak topology and F has the topology of its norm. Let A={E} , B= {A⊂E:A is bounded} and C= {A⊂E:A is weakly compact}. Then C w (E;F) , C wb (E;F) and C wk (E;F) are the spaces of all mappings f:E→F whose restrictions to subsets A⊂E belonging to A , B and C , respectively, are weakly continuous. Clearly, C w (E;F)⊂C wb (E;F)⊂C wk (E;F) , and they are all endowed with the topology of uniform convergence on weakly compact subsets of E . The authors show that C wk (E;F) is the completion of C w (E;F) . They also show that, when E has no subspace isomorphic to l 1 , then C wb (E;F)=C wk (E;F) . When E has the Dunford-Pettis property and contains a subspace isomorphic to l 1 , the authors prove that C wb (E;F) is a proper subspace of C wk (E;F) . The same conclusion holds when E is a Banach space that contains a subspace isomorphic to l ∞ .

A Space which will not go where its Proper Subspaces Want it to go

A Space which will not go where its Proper Subspaces Want it to go