Algebraic Subspace Research Articles

Parafermionic Bases of Standard Modules for Twisted Affine Lie Algebras of Type $A_{2l-1}^{(2)}$, $D_{l+1}^{(2)}$, $E_{6}^{(2)}$ and $D_{4}^{(3)}$

Using the bases of principal subspaces for twisted affine Lie algebras except \(A_{2l}^{(2)}\) by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight kΛ0 and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in Hatayama et al. (2001).

Canonical projection tilings defined by patterns

We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of {\em coincidence} and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

Algebraic List-Decoding in Projective Space: Decoding With Multiplicities and Rank-Metric Codes

The problem of list decoding algebraic subspace codes and rank-metric codes is considered. We develop two separate methods, via two different approaches, for list decoding subspace codes and rank-metric codes. These methods provide, for certain code parameters, improved tradeoffs between rate and error-correction capability than that of the Koetter-Kschischang codes, in the domain of subspace codes, and than that of the Gabidulin codes, in the domain of rank-metric codes, and several other extensions thereof. In the first approach, we introduce the notion of root multiplicities for a certain sub-ring of the ring of linearized polynomials. In the list-decoding algorithm, multiple roots are enforced for the interpolation polynomial in order to achieve an improved error-correction radius for an extended family of Koetter-Kschischang subspace codes. The normalized error-correction radius for this approach is τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> =2(L+1)/(r+1)-1-L(L+1)(L+n)/r(r+1)R, where L is the maximum list size, n is the subspace code dimension, R is the rate of the code, and r is the multiplicity parameter. In the second approach, we construct a folded version of Koetter-Kschischang codes. A linear-algebraic list-decoding algorithm is proposed for these codes that achieves the error-correction radius τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> =s(1-sR), where s is the folding parameter. As opposed to the first approach, the size of output list in the second approach depends on the underlying field size and is at most q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m(s-1)</sup> , where q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is the size of the field that message symbols are chosen from. It is also shown that the output list size is 1, with high probability, in a probabilistic setting. We utilize the techniques of the second approach in the domain of rank-metric codes to construct folded Gabidulin codes to enable a linear-algebraic list-decoding algorithm for such codes. Our algorithm makes it possible to recover.

Weak colored local rules for planar tilings

A linear subspace $E$ of $\mathbb{R}^{n}$ has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of $E$. The local rules are weak if the digitizations can slightly wander around $E$. We prove that a linear subspace has weak colored local rules if and only if it is computable. This goes beyond previous results, all based on algebraic subspaces. We prove an analogous characterization for sets of linear subspaces, including the set of all the linear subspaces of $\mathbb{R}^{n}$.

Algebraic Clustering of Affine Subspaces.

Subspace clustering is an important problem in machine learning with many applications in computer vision and pattern recognition. Prior work has studied this problem using algebraic, iterative, statistical, low-rank and sparse representation techniques. While these methods have been applied to both linear and affine subspaces, theoretical results have only been established in the case of linear subspaces. For example, algebraic subspace clustering (ASC) is guaranteed to provide the correct clustering when the data points are in general position and the union of subspaces is transversal. In this paper we study in a rigorous fashion the properties of ASC in the case of affine subspaces. Using notions from algebraic geometry, we prove that the homogenization trick , which embeds points in a union of affine subspaces into points in a union of linear subspaces, preserves the general position of the points and the transversality of the union of subspaces in the embedded space, thus establishing the correctness of ASC for affine subspaces.

Filtrated Algebraic Subspace Clustering

Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. Existing algebraic subspace clustering methods are based on fitting the data with an algebraic variety and decomposing this variety into its constituent subspaces. Such methods are well suited to the case of a known number of subspaces of known and equal dimensions, where a single polynomial vanishing in the variety is sufficient to identify the subspaces. While subspaces of unknown and arbitrary dimensions can be handled using multiple vanishing polynomials, current approaches are not robust to corrupted data due to the difficulty of estimating the number of polynomials. As a consequence, the current practice is to use a single polynomial to fit the data with a union of hyperplanes containing the union of subspaces, an approach that works well only when the dimensions of the subspaces are high enough. In this paper, we propose a new algebraic subspace clustering algorithm, which can identify the subspace ...

Finite dimensional invariant subspaces for algebras of linear operators and amenable Banach algebras

We study a finite dimensional invariant subspace property similar to Fan's Theorem on semigroups for arbitrary Banach algebras A in terms of amenability of X(A,ϕ), the closed subalgebra of A generated by the set of all maximal elements in A with respect to a character ϕ. As a consequence, we offer some applications to the measure algebra M(G) and the generalized Fourier algebra Ap(G) of a locally compact group G.

On almost-invariant subspaces and approximate commutation

A closed subspace Y of a Banach space X is almost-invariant for a collection S of bounded linear operators on X if for each T∈S there exists a finite-dimensional subspace FT of X such that TY⊆Y+FT. In this paper, we study the existence of almost-invariant subspaces for algebras of operators. We show, in particular, that if a closed algebra of operators on a Hilbert space has a non-trivial almost-invariant subspace then it has a non-trivial invariant subspace. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if T is an operator on a separable Hilbert space and if TP−PT has finite rank for all projections P in a given maximal abelian self-adjoint algebra M then T=M+F where M∈M and F is of finite rank.

Remarks on a Theorem of Amitsur

Amitsur proved the following theorem for an uncountable field k: Let A be a k-algebra, then any finite-dimensional nil subspace of A has bounded index, and any finite dimensional algebraic subspace of A has bounded degree. Here we give a new proof of Amitsur's theorem, a proof which is more elementary and direct, than Amitsur's original proof.

Weighted approximation in mean of classes of analytic functions by algebraic polynomials and finite-dimensional subspaces

We establish estimates for classic approximation quantities for sets from functional spaces (classes of functions analytic in Jordan domains), namely, for the best polynomial approximations and Kolmogorov widths.

Constrained least-squares solutions of linear inclusions and singular control problems in Hilbert spaces

LetH1 andH2 be Hilbert spaces and letN be an algebraic subspace ofH1. The least-squares problem for a linear relationLźH1źH2 restricted to an algebraic cosetS:=g+N, g ź H1, is considered. Various characterizations of a minimizer are derived in the form of inclusion relations that incorporate the constraints, and conditions under which a minimizer exists are developed. In particular, it is shown that generalized forms of the "normal equations" for constrained least-squares problems become "normal inclusions" that involve a multivalued adjoint. The setting includes large classes of constrained least-squares minimization problems and optimal control problems subject to generalized boundary conditions. An application of the abstract theory is given to a singular control problem involving ordinary differential equations with generalized boundary conditions, where the control may generate multiresponses. Characterizations of an optimal solution are developed in the form of inclusions that involve either an integrodifferential equation or a differential equation, and adjoint subspaces and/or solutions of certain linear equations.

Invariant Subspaces for Algebras of Subnormal Operators. II

Invariant Subspaces for Algebras of Subnormal Operators. II

Invariant Subspaces for Algebras of Linear Operators and Amenable Locally Compact Groups

Let $G$ be a locally compact group. We prove in this paper that $G$ is amenable if and only if the group algebra ${L_1}\left ( G \right )$ (respectively the measure algebra $M\left ( G \right )$) satisfies a finite-dimensional invariant subspace property $T\left ( n \right )$ for $n$-dimensional subspaces contained in a subset $X$ of a separated locally convex space $E$ when ${L_1}\left ( G \right )$ (respectively $M\left ( G \right )$) is represented as continuous linear operators on $E$. We also prove that for any locally compact group, the Fourier algebra $A\left ( G \right )$ and the Fourier Stieltjes algebra $B\left ( G \right )$ always satisfy $T\left ( n \right )$ for each $n = 1,2, \ldots$.

Invariant subspaces for algebras of linear operators and amenable locally compact groups

Let G G be a locally compact group. We prove in this paper that G G is amenable if and only if the group algebra L 1 ( G ) {L_1}\left ( G \right ) (respectively the measure algebra M ( G ) M\left ( G \right ) ) satisfies a finite-dimensional invariant subspace property T ( n ) T\left ( n \right ) for n n -dimensional subspaces contained in a subset X X of a separated locally convex space E E when L 1 ( G ) {L_1}\left ( G \right ) (respectively M ( G ) M\left ( G \right ) ) is represented as continuous linear operators on E E . We also prove that for any locally compact group, the Fourier algebra A ( G ) A\left ( G \right ) and the Fourier Stieltjes algebra B ( G ) B\left ( G \right ) always satisfy T ( n ) T\left ( n \right ) for each n = 1 , 2 , … n = 1,2, \ldots .

Invariant subspaces for algebras of subnormal operators. II

We continue our study of hyperinvariant subspaces for rationally cyclic subnormal operators. We establish a connection between hyperinvariant subspaces and weak-star continuous point evaluations on the commutant.

Invariant subspaces for algebras of subnormal operators

Every rationally cyclic subnormal operator has a hyperinvariant subspace.

Invariant subspaces for algebras of analytic operators associated with a periodic flow on a finite von Neumann algebra

Soit M une algebre de Von Neumann avec un etat τ et {α t } t∈Π est une representation σ-faiblement continue du cercle unite, Π des *-automorphismes de M tel que τοα t =τ, t∈Π. Soit H ∞ (α) l'ensemble de tous les x∈M tels que Spα(x)⊆{n∈Z:n≥0}. On etudie les sous-espaces invariants de H ∞ (α)

Invariant subspaces for algebras generated by strongly reductive operators

Invariant subspaces for algebras generated by strongly reductive operators

Conditions for the Existence of Contractions in the Category of Algebraic Spaces

Artin’s conditions for the existence of a new contraction of an algebraic subspace are changed. The new conditions are more applicable in special cases, especially when the subspace has a conormal bundle.

Conditions for the existence of contractions in the category of algebraic spaces

Artin’s conditions for the existence of a new contraction of an algebraic subspace are changed. The new conditions are more applicable in special cases, especially when the subspace has a conormal bundle.