Proper Subspace Research Articles

On existence of hyperinvariant subspaces for linear maps

Let $X$ be an infinite dimensional complex vector space. We show that a non-constant endomorphism of $X$ has a proper hyperinvariant subspace if and only if its spectrum is non-void. As an application we show that each non-constant continuous endomorphism of the locally convex space $(s)$ of all complex sequences has a proper closed hyperinvariant subspace.

A New Extension Theorem for Concave Operators

We present a new and interesting extension theorem for concave operators as follows. Let be a real linear space, and let be a real order complete PL space. Let the set be convex. Let be a real linear proper subspace of , with , where for some . Let be a concave operator such that whenever and . Then there exists a concave operator such that (i) is an extension of , that is, for all , and (ii) whenever .

Phase space localization of antisymmetric functions

Upper and lower bounds are written down for the minimum information entropy in phase space of an antisymmetric wave function in any number of dimensions. Similar bounds are given when the wave function is restricted to belong to any of the proper subspaces of the Fourier transform operator.

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n , with n > 2 ; (2) every triple of noncollinear points is contained in a unique member of Ω ; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n = 2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.

Linear spaces with small generated subspaces

The dimension of a linear space is the maximum positive integer d such that any d of its points generate a proper subspace. Given n, d, s, we consider linear spaces on n points such that any d points generate subspaces of size at most s. Certain design-theoretic constructions and applications are investigated. In particular, one consequence is the existence of proper n-edge-colourings of both K n + 1 (for n odd) and K n , n with a constant bound on the length of two-colored cycles.

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain Xn+1=AXn+Bn (mod p), where A is an integer matrix with nonzero eigenvalues, and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Qk invariant under A. If \(|\lambda_{i}|\not =1\) for all eigenvalues λi of A, then n=O((ln p)2) steps are sufficient and n=O(ln p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λi that are roots of positive integer numbers, |λ1|=1 and |λi|>1 for all \(i\not =1\) , then O(p2) steps are necessary and sufficient.

Estimates for Littlewood–Paley operators in [formula omitted

Let g ( f ) be the Littlewood–Paley g-function of f on R n . In this paper, the authors prove that if f ∈ BMO ( R n ) (the space of functions with bounded mean oscillation), then g ( f ) is either infinite everywhere or finite almost everywhere, and in the latter case, [ g ( f ) ] 2 is bounded from BMO ( R n ) into BLO ( R n ) (the space of functions with bounded lower oscillation), which is a proper subspace of BMO ( R n ) . Moreover, the authors also establish similar results for the Lusin-area function and the Littlewood–Paley g λ * -function.

Quantum chaos, delocalization, and entanglement in disordered Heisenberg models

We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin- and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.

Hyperbolic polynomials and multiparameter real-analytic perturbation theory

Let P(x,z)=zd+∑i=1dai(x)zd-i be a polynomial, where ai are real-analytic functions in an open subset U of Rn. If, for any x∈U, the polynomial z↦P(x,z) has only real roots, then we can write those roots as locally Lipschitz functions of x. Moreover, there exists a modification (a locally finite composition of blowups with smooth centers) σ:W→U such that the roots of the corresponding polynomial P~(w,z)=P(σ(w),z),w∈W, can be written locally as analytic functions of w. Let A(x),x∈U, be an analytic family of symmetric matrices, where U is open in Rn. Then there exists a modification σ:W→U such that the corresponding family A~(w)=A(σ(w)) can be locally diagonalized analytically (i.e., we can choose locally a basis of eigenvectors in an analytic way). This generalizes Rellich's well-known theorem (see [32]) from 1937 for 1-parameter families. Similarly, for an analytic family A(x),x∈U, of antisymmetric matrices, there exists a modification σ such that we can find locally a basis of proper subspaces in an analytic way

Unbounded operators commuting with restricted backward shifts

The closed densely defined operators on a proper invariant subspace of the backward shift that commute with the restricted backward shift are shown to be coanalytic Toeplitz operators induced by functions in the Nevanlinna class. The result can be interpreted as a kind of commutant lifting theorem for unbounded operators. It extends, and in a certain sense completes, earlier work of Daniel Suarez. Mathematics subject classification (2000): 47B35, 47A20, 30D50.

Quasi-exact solvability in a general polynomial setting

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrödinger operators beyond the Lie-algebraic class. Let be the space of nth degree polynomials in one variable. We first analyze exceptional polynomial subspaces , which are those proper subspaces of invariant under second-order differential operators which do not preserve . We characterize the only possible exceptional subspaces of codimension one and we describe the space of second-order differential operators that leave these subspaces invariant. We then use equivalence under changes of variable and gauge transformations to achieve a complete classification of these new, non-Lie algebraic Schrödinger operators. As an example, we discuss a finite gap elliptic potential which does not belong to the Treibich–Verdier class.

Adaptation of Aeroelastic Reduced-Order Models and Application to an F-16 Configuration

The proper orthogonal decomposition method has been shown to produce accurate reduced-order models for the aeroelastic analysis of complete aircraft configurations at fixed flight conditions. However, changes in the Mach number or angle of attack often necessitate the reconstruction of the reduced-order model to maintain accuracy, which destroys the sought-after computational efficiency. Straightforward approaches for reduced-order model adaptation-such as the global proper orthogonal decomposition method and the direct interpolation of the proper orthogonal decomposition basis vectors-that have been attempted in the past have been shown to lead to inaccurate proper orthogonal decomposition bases in the transonic flight regime. Alternatively, a new reduced-order model adaptation scheme is described in this paper and evaluated for changes in the freestream Mach number and angle of attack. This scheme interpolates the subspace angles between two proper orthogonal decomposition subspaces, then generates a new proper orthogonal decomposition basis through an orthogonal transformation based on the interpolated subspace angles. The resulting computational methodology is applied to a complete F-16 configuration in various airstreams. The predicted aeroelastic frequencies and damping coefficients are compared with counterparts obtained from full-order nonlinear aeroelastic simulations and flight test data. Good correlations are observed, including in the transonic regime. The obtained computational results reveal a significant potential of the adapted reduced-order model computational technology for accurate, near-real-time, aeroelastic predictions.

On Analyzing Diffusion Tensor Images by Identifying Manifold Structure Using Isomaps

This paper addresses the problem of statistical analysis of diffusion tensor magnetic resonance images (DT-MRI). DT-MRI cannot be analyzed by commonly used linear methods, due to the inherent nonlinearity of tensors, which are restricted to lie on a nonlinear submanifold of the space in which they are defined, namely R6. We estimate this submanifold using the Isomap manifold learning technique and perform tensor calculations using geodesic distances along this manifold. Multivariate statistics used in group analyses also use geodesic distances between tensors, thereby warranting that proper estimates of means and covariances are obtained via calculations restricted to the proper subspace of R6. Experimental results on data with known ground truth show that the proposed statistical analysis method properly captures statistical relationships among tensor image data, and it identifies group differences. Comparisons with standard statistical analyses that rely on Euclidean, rather than geodesic distances, are also discussed.

Subspaces of computable vector spaces

We show that the existence of a nontrivial proper subspace of a vector space of dimension greater than one (over an infinite field) is equivalent to WKL 0 over RCA 0 , and that the existence of a finite-dimensional nontrivial proper subspace of such a vector space is equivalent to ACA 0 over RCA 0 .

THE pp CONJECTURE FOR SPACES OF ORDERINGS OF RATIONAL CONICS

First counterexamples are given to a basic question raised in [10]. The paper considers the space of orderings (X,G) of the function field of a real irreducible conic [Formula: see text] over the field ℚ of rational numbers. It is shown that the pp conjecture fails to hold for such a space of orderings when [Formula: see text] has no rational points. In this case, it is shown that the pp conjecture "almost holds" in the sense that, if a pp formula holds on each finite subspace of (X,G), then it holds on each proper subspace of (X,G). For pp formulas which are product-free and 1-related, the pp conjecture is known to be true, at least if the stability index is finite [11]. The counterexamples constructed here are the simplest sort of pp formulas which are not product-free and 1-related.

Integrable Boehmians, Fourier transforms, and Poisson’s summation formula

Boehmians are classes of generalized functions whose construction is algebraic. The first construction appeared in a paper that was published in 1981 [6]. In [8], P. Mikusinski constructs a space of Boehmians, βL1(R), in which each element has a Fourier transform. Mikusinski shows that the Fourier transform of a Boehmian satisfies some basic properties, and he also proves an inversion theorem. However, the range of the Fourier transform is not investigated. Also, Mikusinski states that βL1(R) contains some elements which are not Schwartz distributions, but no examples are given. We will address these problems in this paper. In this note, we will construct a space of Boehmians β`(R). The space of integrable functions on the real line can be identified with a proper subspace of β`(R). Each element of β`(R) has a Fourier transform which is a continuous function and satisfies a growth condition at infinity. Conditions are given which ensure that a given function is the Fourier transform of an element of β`(R).

Enumerating Motzkin–Rabin geometries

AbstractThe main result of this paper is an enumeration of all Motzkin‐Rabin geometries on up to 18 points. A Motzkin‐Rabin geometry is a two‐colored linear space with no monochromatic line. We also study the embeddings of Motzkin‐Robin geometries into projective spaces over fields and division rings. We find no Motzkin‐Rabin geometries on up to 18 points embeddable in ℂ2 or 𝔽2(t)2. We find many examples of Motzkin‐Rabin geometries with no proper linear subspaces. We give an example of a proper linear space embeddable in ℙ(ℚ($\sqrt -7$)2). © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 179–194, 2007

Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In this paper we study the space $\mathcal {Z}_k(G/P, r[X_w])$ of effective $k$-cycles $X$ in $G/P$ with the homology class equal to an integral multiple of the homology class of Schubert variety $X_w$ of type $w$. When $X_w$ is a proper linear subspace $\mathbb {P}^k$ $(k<n)$ of a linear space $\mathbb {P}^n$ in $G/P \subset \mathbb {P}(V)$, we know that $\mathcal {Z}_k(\mathbb {P}^n, r[\mathbb {P}^k])$ is already complicated. We will show that for a smooth Schubert variety $X_w$ in a Hermitian symmetric space, any irreducible subvariety $X$ with the homology class $[X]=r[X_w]$, $r\in \mathbb {Z}$, is again a Schubert variety of type $w$, unless $X_w$ is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space $G/P$ is obtained by the action of the Lie group $G$.

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.

Asymptotic expression of the linear discrete best [formula omitted]-approximation

Let h p , 1 < p < ∞ , be the best ℓ p -approximation of the element h ∈ R n from a proper affine subspace K of R n , h ∉ K , and let h ∞ * denote the strict uniform approximation of h from K. We prove that there are a vector α ∈ R n ⧹ { 0 } and a real number a, 0 ⩽ a ⩽ 1 , such that h p = h ∞ * + a p p - 1 α + γ p , for all p > 1 , where γ p ∈ R n with ∥ γ p ∥ = o a p / p .