Extremal Vectors Research Articles

Extremal Vectors for a Class of Operators

Extremal Vectors for a Class of Operators

О структуре пространства весов голосующих процедур

The structure of the self-dual threshold functions space of weights used in voting decision-making procedures has been studied for dimensions 2-6. Extremal vectors of polyhedral cones representing these threshold functions in the space of weights have been found.

The Burge correspondence and crystal graphs

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first step towards an analogue of Schensted’s result for the RSK insertion which states that the length of the longest increasing subword of a word is the length of the largest row of the tableau under the RSK correspondence. Furthermore, we give a crystal structure on simple graphs of hook shape. The extremal vectors in this crystal are precisely the simple graphs whose degree sequence are threshold and hook-shaped.

Crouzeix’s Conjecture and Related Problems

In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.

Extremal elements of a sublattice of the majorization lattice and approximate majorization

Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball with p ⩾ 1 formed by the probability vectors whose ℓ p -norm distance to the center x is less than or equal to a radius ϵ. Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls with 1 < p < ∞. On the other hand, we show that is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the ℓ 1-norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

Maximizing the expected range from dependent observations under mean–variance information

In this article we derive the best possible upper bound for under given means and variances on n random variables . The random vector is allowed to have any dependence structure, provided and , . We provide an explicit characterization of the n-variate distributions that attain the equality (extremal random vectors), and the tight bound is compared to other existing results.

Extremal vectors of the Verma modules of the Lie algebra B 2 in Poincaré-Birkhoff-Witt basis

The aim of this work is to present a full set of expressions for the extremal vectors in a Verma module over the B2 complex semisimple Lie algebra in the Poincare-Birkhoff-Witt basis.

EXTREMAL VECTORS FOR VERMA TYPE REPRESENTATION OF B2

Starting from the Verma modules of the algebra &lt;em&gt;B&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt; we explicitly construct factor representations of the algebra &lt;em&gt;B&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt; which are connected with the unitary representation of the group SO(3, 2). We find a full set of extremal vectors for representations of this kind. So we can explicitly resolve the problem of the irreducibility of these representations.

Extremal vectors for Verma-type representations of su(2, 2)

Starting from the Verma modules of the algebra sl(4, ℂ) we explicitly construct factor representations of the algebra su(2, 2) which are connected with unitary representation of group SU(2, 2). We find a full set of extremal vectors for this kind of representations, so we can solve explicitly the problem of irreducibility of these representations.

Indecomposable representations and oscillator realizations of the exceptional Lie algebra G2

In this paper various representations of the exceptional Lie algebra G2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the Verma module are investigated in detail to construct the Dyson-Maleev-like realization of G2. After obtaining explicit forms of all twelve extremal vectors of the Verma module with the dominant highest weight $\Lambda $ , representations with their respective highest weights related to $\Lambda$ are systematically discussed. Moreover, this method can be used to construct fermion realizations from the irreducible representations, without referring to the Lie algebra chains, and a three-fermion realization is constructed as an example.

What are extremal Kerr Killing vectors up to?

In the extremal Kerr spacetime the horizon Killing vector field is null on a timelike hypersurface crossing the horizon at a fixed latitude and spacelike on both sides of the horizon in the equatorial plane. We explain in some detail how this behavior is consistent with the existence of timelike Killing vectors everywhere off the horizon, and how it arises in a limit from the Kerr spacetime where there is a similar hypersurface strictly outside the horizon.

Extremal vectors for Verma type factor-representations of U q (sl(3, ℂ))

To analyze the reducibility of the Verma modules one often needs to find the extremal vectors of the given representations. On the example of algebra Uq(sl(3, ℂ)) we study how the set of extremal vectors is affected when we factorize the original representation and give their explicit formula.

Extensions of Fisher Information and Stam's Inequality

We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed.

Order isomorphisms in cones and a characterization of duality for ellipsoids

We study order isomorphisms in finite-dimensional ordered vector spaces. We generalize theorems of Alexandrov, Zeeman, and Rothaus (valid for “non-angular” cones) to wide classes of cones, including in particular polyhedral cones, using a different and novel geometric method. We arrive at the following result: whenever the cone has more than n generic extremal vectors, an order isomorphism must be affine. In the remaining case, of precisely n extremal rays, the transform has a restricted diagonal form. To this end, we prove and use a new version of the well-known Fundamental theorem of affine geometry. We then apply our results to the cone of positive semi-definite matrices and get a characterization of its order isomorphisms. As a consequence, the polarity mapping is, up to a linear map, the only order-reversing isomorphism for ellipsoids.

Extremal vectors and rectifiability

The concept of extremal vectors of a linear operator with a dense range but not onto on a Hilbert space was introduced by P. Enflo in 1996 as a new approach to study invariant subspaces. Following this, there were several studies on analytic and geometric properties of backward minimal vectors and their applications to construction of invariant subspaces. In this paper, we consider one other property: rectifiability. We show that in general curves that map numbers to backward minimal vectors are not rectifiable.

Friedel-oscillations-induced surface magnetic anisotropy

We present detailed numerical studies of the magnetic anisotropy energy of a magnetic impurity near the surface of metallic hosts (Au and Cu) that we describe in terms of the tight-binding surface Green's-function technique. We study the case when spin-orbit coupling originates from the $d$ band of the host material and we also investigate the case of a strong local spin-orbit coupling on the impurity itself. The splitting of the impurity's spin states is calculated to leading order in the exchange interaction between the impurity and the host atoms using the diagrammatic Green's-function technique. The magnetic anisotropy constant is an oscillating function of the separation $d$ from the surface. It asymptotically decays as $\ensuremath{\sim}1/{d}^{2}$ and its oscillation period is determined by the extremal vectors of the host's Fermi surface. Our results clearly show that the host-induced magnetic anisotropy energy is by several orders of magnitude smaller than the anisotropy induced by the local mechanism, which provides sufficiently large anisotropy values to explain the size dependence of the Kondo resistance observed experimentally.

Surface-Induced Magnetic Anisotropy of Impurities

We present theoretical and numerical studies of the magnetic anisotropy energy of an atomic-like impurity near the surface of a metallic host (Au). The valence band of the host metal is described in terms of a realistic tight-binding surface Green's function technique. We compare two models: (i) when spin-orbit coupling is taken into account in the d-band of the host and (ii) when the impurity's d-level experiences strong spin-orbit splitting. The level splitting of the impurity's spin-states is calculated in leading (first or second) order of the exchange interaction between the impurity and the host atoms. It is shown that the magnetic anisotropy constant is an oscillating function of the distance d of the impurity from the surface. For large distances, an asymptotic analysis implies that the period of these oscillations is determined by the extremal vectors of the host's Fermi Surface and the amplitude decays as 1/d2. Our numerical results clearly suggest that the host-induced magnetic anisotropy energy is by several orders smaller in magnitude than the one originating from a strong local spin-orbit coupling.

Hyperinvariant subspaces for quasinilpotent operators on Hilbert spaces

In this paper, we employ the model theory due to C. Foias and C. Pearcy and the notion of Enflo's extremal vectors of quasinilpotent operators to study the hyperinvariant subspace problem for quasinilpotent operators. Our main result is that if a quasinilpotent quasiaffinity T has a sequence of “ c-eigenvectors” x n of T n T ∗ n such that the set cl { x n : n ∈ N } is compact, then T has a nontrivial hyperinvariant subspace.

Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let $$({\mathcal{CRQ}})$$ denote the class of quasinilpotent quasiaffinities Q in $${\mathcal{L}}(H)$$ such that Q * Q has an infinite dimensional reducing subspace M with Q * Q| M compact. It was known that if every quasinilpotent operator in $$({\mathcal{CRQ}})$$ has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in $$({\mathcal{CRQ}})$$ . The purpose of this paper is to provide sufficient conditions for elements in $$({\mathcal{CRQ}})$$ to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.

Three kinds of extremal weight vectors fixed by a diagram automorphism

We define actions of a diagram automorphism on certain crystals, and then study three kinds of “extremal” elements of them that are fixed by a diagram automorphism. We will show that such extremal elements can be identified with extremal elements of the corresponding crystal for the orbit Lie algebra. As a consequence, we prove that there exists a canonical injection from the elements of the crystal base of an extremal weight module fixed by a diagram automorphism into the elements of the crystal base of the corresponding extremal weight module for the orbit Lie algebra.