Smallest Subspace Research Articles

Fredholmness of fringe operators over the bidisk

Let M be an invariant subspace of \(H^2\) over the bidisk. Associated with M, we have the fringe operator \(F^M_z\) on \(M\ominus w M\). For \(A\subset H^2\), let [A] denote the smallest invariant subspace containing A. Assume that \(F^M_z\) is Fredholm. If h is a bounded analytic function on \(\mathbb {D}^2\) satisfying \(h(0,0)\not =0\), then \(F^{[h M]}_z\) is Fredholm and \(\mathrm{ind}\,F^{[h M]}_z=\mathrm{ind}\,F^M_z\).

The Support Feature Machine: Classification with the Least Number of Features and Application to Neuroimaging Data

By minimizing the zero-norm of the separating hyperplane, the support feature machine (SFM) finds the smallest subspace (the least number of features) of a data set such that within this subspace, two classes are linearly separable without error. This way, the dimensionality of the data is more efficiently reduced than with support vector-based feature selection, which can be shown both theoretically and empirically. In this letter, we first provide a new formulation of the previously introduced concept of the SFM. With this new formulation, classification of unbalanced and nonseparable data is straightforward, which allows using the SFM for feature selection and classification in a large variety of different scenarios. To illustrate how the SFM can be used to identify both the smallest subset of discriminative features and the total number of informative features in biological data sets we apply repetitive feature selection based on the SFM to a functional magnetic resonance imaging data set. We suggest that these capabilities qualify the SFM as a universal method for feature selection, especially for high-dimensional small-sample-size data sets that often occur in biological and medical applications.

Extremal sizes of subspace partitions

A subspace partition Π of V = V(n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σq(n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρq(n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σq(n, t) and ρq(n, t) for all positive integers n and t. Furthermore, we prove that if n ≥ 2t, then the minimum size of a maximal partial t-spread in V(n + t -1, q) is σ q(n, t).

Jordan Derivations of Reflexive Algebras

Let \({\mathcal L}\) be a subspace lattice on a Banach space X and suppose that \({\vee\{L\in\mathcal L: L_- (0)\}=(0)}\) . Then each Jordan derivation from Alg\({\mathcal L}\) into B(X) is a derivation. This result can apply to completely distributive subspace lattice algebras, \({\mathcal J}\) -subspace lattice algebras and reflexive algebras with the non-trivial largest or smallest invariant subspace.

Algebraic cycles on the Jacobian of a curve with a linear system of given dimension

We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems that the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.

Iterative Stokes Solvers in the Harmonic Velte Subspace

We explore the prospects of utilizing the decomposition of the function space (H 1 0) n (where n=2,3) into three orthogonal subspaces (as introduced by Velte) for the iterative solution of the Stokes problem. It is shown that Uzawa and Arrow-Hurwitz iterations – after at most two initial steps – can proceed fully in the third, smallest subspace. For both methods, we also compute optimal iteration parameters. Here, for two-dimensional problems, the lower estimate of the inf-sup constant by Horgan and Payne proves useful and provides an inclusion of the spectrum of the Schur complement operator of the Stokes problem. We further consider the conjugate gradient method in the third Velte subspace and derive a corresponding convergence estimate. Computational results show the effectiveness of this approach for discretizations which admit a discrete Velte decomposition.

Monotonicity with respect to closed convex cones II

Consider the flow ψt for the system of differential equations open. Let K(t) be a closed convex expanding cone whenever and the smallest subspace containing K(t) for each t is the same for all t)k be a unit vector in K(0), and . If for some positive l leaves K(t) invariant for all t then for all . If in addition takes k into the relative interior of K(t) for all t then is in the relative interior of K(t) for all t > 0. These and related results are extensions of the Kamke—Möller theorem and Hirsch's theorem for strong monotone flows. Applications from chemical kinetics and epidemiology are given.

Noncommutative invariants of bialgebras

Suppose that H is a bialgebra over a field C and R = C〈V〉 is the tensor algebra of the C-space V endowed with the structure of an H-module algebra, so that V is a submodule of the H-module R, RH is the algebra of H-invariants, and W, the support of the algebra RH, is the smallest subspace of the C-space V such that\(R^H \subseteq C\left\langle W \right\rangle\). The main result of the paper is the theorem stating that if the algebra of H-invariants RH is finitely generated, then the support of RH is a finite-dimensional submodule of the H-module V, whose elements are H-semi-invariants of the same weight.

Moreau-Rockafellar equality for sublinear functionals

be the relative interior of the set A ~ X where lin A is the smallest subspace containing A. A functional p:X--~ ~* U1 {~ oo} is called sublinear if p(x + y) 5 p(x) + p(y) and p (lx) = %p (x) Vx, yEX and ~ 0. We denote by 5bl (X) the set of all sublinear functionals on X; Op ~ = {x* E X* Ix* (x) ~p (x) vxE X} is the subdifferential, epip : = {(~, x}E ~i x X 1 ~p(x)} is the supergraph and dom p ~ = {xE X [p (x) < + ~} is the effective domain of definition of p e Sbl (X). Cones K I and K 2 in the

On the minimal space of surjectivity question for linear transformations on vector spaces with applications to surjectivity of differential operators on locally convex spaces

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

A double commutant theorem for conjugate selfadjoint operators

Let A be a bounded linear transformation on the complex separable Hilbert space H. If there is a conjugation Q on H such that A = QA*Q, we say that A is conjugate selfadjoint. In this note we examine commutativity properties of conjugate selfadjoint operators which possess cyclic vectors. 1. Preliminaries. Let H be a complex Hilbert space with a countably infinite basis, and let (f, g) denote the inner product of two vectors in H. By H ff H we mean the Hilbert space of vectorsf E3 g having inner product (fl D gI,f2 E 92) = (fl,f2) + (g1, g2). A linear manifold is a subset which is closed under vector addition and under multiplication by complex numbers. A subspace is a linear manifold which is closed in the norm topology induced by the inner product. The smallest subspace containing the set U ??of fn) will be denoted by V{fn). If F is a subset of Hilbert space, clos F will denote the closure of F in the norm topology and F' = { gl( g, f) = 0, f E F). Whenever A is a continuous linear transformation on H, its graph F(A) = { fD Af If E H) is a subspace of H ff H. One can easily verify that F(A)= { A*(-f) Ef fIf E H), where A* is the adjoint of The germinal idea of representing a linear transformation through its associated graph subspace originated in the work of J. von Neumann [1]. Hereafter we shall refer to a continuous (or bounded) linear transformation as an operator. If A and B are operators on H, we define (A ff B)(f E g) = Af ff Bg. The set of all operators on H that commute with A is called the commutant of This algebra will be denoted by (A)'. The double commutant of A, designated {A}, is the algebra of all operators on H that commute with every member of (A }'. It is self evident that (A)' is abelian if and only if (A)' = (A A. A transformation Q on H is said to be a conjugation if Q2 = I and (Qf, Qg) = (g, f) for every f and g in H. Intuitively speaking, Q replaces each element of H by its conjugate with respect to the real subspace consisting of all fixed points of Q [3, p. 357]. If there is a conjugation Q such that A = QA*Q, we shall call A conjugate selfadjoint. The well-known Hankel operators [4] belong to this class. Received by the editors July 18, 1980 and, in revised form, January 12, 1981. 1980 Mathematics Subject Classification. Primary 47A05; Secondary 46J99.

On weak compactness in the space of Radon measures

Let X be a fixed compact space, C(X) the Banach lattice of real continuous functions on X, L(X) its dual, and M(X) its bidual. [L(X) is an (L) space and M(X) an (M) space, hence the notation.] We have two objects in the present paper. The first is to give a unified development of most of the characterizations of weakly compact sets in L(X)-hence also in -F(p), p EL(X)-using purely vector lattice tools (i.e., with no measure and integration theory). There has been a long succession of characterizations, beginning with the Dunford-Pettis theorem [S] and culminating in the remarkable theorem of Grothendieck [6]. We derive them all from a theorem of Nakano [12, section 281 on the dual EC of a vector lattice E, where EC consists of the linear functionals on E which are continuous with respect to order convergence in E. While L(X) [resp 9’(p)] is not the norm-dual of M(X) [resp Sm(p)], it is precisely the set of linear functionals on M(X) [resp J?“(p)] continuous with respect to order convergence, and it is this fact which we consider to be the basis of all the weak compactness properties. Our second object stems from the above theorem of Grothendieck, and from a theorem of DieudonnC [4], which he uses. These two theorems single out a very small subspace of M(X)-in some sense, the smallest subspace containing C(X). It is a proper subspace of the first Baire class Bar, and for want of a better name we denote it by Ba1j2. The above two theorems indicate that Bali2 bears study as a space in its own right, and Part II of the paper initiates such a study.

An axiom system for the modular logic

G. Birkhoff and J. v. Neumann in their paper [4] concerning logic of the quantum mechanics come to the conclusion that not all laws of the Boolean algebra of logic apply to the statements of the quantum mechanics. Linear subspaces play an important part in the quantum mechanics. This fact suggested the authors the thought to con? sider the lattice of all linear subspaces of some linear space as a model of the sentential calculus of the quantum mechanics, analogously to the lattice of all subsets of a given set which can be considered as a model of the Boolean logic. Birkhof fand v. Neumann de? fine the modular logic as a free, modular and orthocomplementary lattice 3J? = freely generated by a set P. Elements belonging to P are called sentential variables, elements of M are called sentences, "v" and "a" are binary operations called alternative and conjunction respectively. " ' " is an unary operation called negation. One of the simplest models of the modular logic is the lattice of all linear subspaces of an ^-dimensional projective space. These subspaces are considered as sentences, the meet and linear union of two linear subspaces are considered as the conjunction and alternative of sentences. The orthogonal complementation of given subspace to the ^-dimensional space is referred to as negation. By a meet of two linear subspaces we understand their common part and by their linear union ? the smallest subspace containing both of them. In the modular logic the distributive law is abandoned. It is replaced by a weaker law called modular identity: (a a b) v (a a c) = a a (b v (a a c)). The modular logic must necessarily be an orthocomplementary lattice as in modular non-distributive lattices the complementarity relation is not a one-to-one relation, whereas the ortho complementarity is a relation of this kind [5]. It follows from the definition of the modular logic that there exist two fixed pro? positions denoted by 0 and 1. 0 is the false proposition, 1 is the true proposition. No implication operation is defined in the Birkhoff? v. Neumann modular lo? gic, however, the authors define an implication relation of partial order type. In the Boolean algebra "a implies b" if and only if "a v b = b". The implication is in the Boolean algebra uniquely defined as the operation a! ^ b (see [5]). This operation has in the modular logic few properties of the ordinary implication, e.g. a! v b = 1 does not imply the implication relation a v b = b.

Modules of Finite Rank Over Prufer Rings

In this paper the results of [4] are rederived and completed (in particular the question raised on p. 338 is completely settled) by a systematic use of extension theory [2, 3]. Instead of handling Dedekind rings and valuation rings separately, the treatment throughout is in terms of Prilfer rings; the modified descending chain condition in the one case and the hypothesis of almost maximality in the other are reformulated as linear precompactness (which serves also to exhibit their relationship to each other). The technique for obtaining the decomposition theorems is by and large the same: search for a pure submodule and proof that it is a direct summand. However, the concept of purity undergoes a closer analysis which permits one to disengage the relevant hypotheses. The rather successful definition of rank might also be mentioned. We shall be dealing exclusively with modules over commutative integral domains R for which the unit is the identity endomorphism. The most general rings we shall consider are the Prufer rings in which every finitely generated ideal is invertible: among these are the finitely principal domains (every two elements have a greatest common divisor) and,' more particularly, the valuation rings (in which this divisor is equal to one of the two elements). A module M is torsion-free if for every non-zero x E M, ax = 0 implies a = 0. A torsion-free module can be embedded in a vector space over Q, the quotient field of R; the dimension of the smallest subspace containing M (this is independent of the embedding) is called the rank of M. The rank of an arbitrary module M is defined as the smallest of the ranks of the torsion-free modules of which M is a quotient module. A module M is divisible if for every x E M and a E R (a $ 0) there exists y E M such that ay = x. This concept is in a way dual to that of torsion-free; in a divisible module the non-zero scalar multiplications are onto endomorphisms while in a torsion-free module they are one-to-one. The field Q (and more generally any vector space), as well as its quotient modules, are divisible; thus the rank of a module could also be defined as the smallest rank of a divisible module containing it.