Nilpotent Operators Research Articles

Topological Quantum Field Theory on non-Abelian gerbes

The infinitesimal symmetries of a fully decomposed non-Abelian gerbe can be generated in terms of a nilpotent BRST operator, which is here constructed. The appearing fields find a natural interpretation in terms of the universal gerbe, a generalisation of the universal bundle. We comment on the construction of observables in the arising Topological Quantum Field Theory. It is also shown how the BRST operator and the trace part of a suitably truncated set of fields on the non-Abelian gerbe reduce directly to the coboundary operator and the pertinent cochains of the underlying Čech–de Rham complex.

Reductions in binary search trees

We analyze two bottom-up reduction algorithms over binary trees that represent replaceable data within a certain system, assuming the binary search tree (BST) probabilistic model. These reductions are based on idempotent and nilpotent operators, respectively. In both cases, the average size of the reduced tree, as well as the cost to obtain it, is asymptotically linear with respect to the size of the original tree. Additionally, the limiting distributions of the size of the trees obtained by means of these reductions satisfy a central limit law of Gaussian type.

An eigenvalue problem for derogatory matrices

A matrix A is called derogatory if there is more than one Jordan submatrix associated with an eigenvalue λ . In this paper, we are concerned with the eigenvalue problem of this type of matrices. The singularities of the resolvent of A : R ( z ) = ( A - zI ) - 1 are exactly the eigenvalues of A. Let us consider the Laurent series of R expanded at λ and denote its coefficients c k ( - ∞ ⩽ k ⩽ ∞ ) . D ≔ c - 2 is the nilpotent operator, that is, there exists the order l of λ such that D l ≔ c - l - 1 = 0 ( l ⩾ 1 ) . Additionally, for an arbitrary vector z, D l - 1 z is an eigenvector of λ . Then λ is computed from the corresponding eigenvector D l - 1 z . In order to estimate the integral representation of D k z , we apply the trapezoidal rule on the circle enclosing λ but excluding other eigenvalues of A. It is our result that, so far as related linear equations are solved with necessary precision, the eigenvalues of derogatory matrices can be computed numerically as exactly as we want and so are corresponding (generalized) eigenvectors, too.

BRST extension of the Faddeev model

The Faddeev model is a second class constrained system. Here we construct its nilpotent BRST operator and derive the ensuing manifestly BRST invariant Lagrangian. Our construction employs the structure of Stuckelberg fields in a nontrivial fashion.

Radiative decays of quarkonium states, momentum operator expansion and nilpotent operators

We present the method of calculation of the radiative decays of composite quark–antiquark systems, with different J PC: . The method is relativistic invariant; it is based on the double dispersion relation integrals over the masses of the composite mesons; it can be used for high spin particles and provides us with the gauge invariant transition amplitudes. We apply this method to the case when the photon is emitted by a constituent in the intermediate state (additive quark model). We perform the momentum operator expansion of the spin amplitudes for the decay processes. The problem of the nilpotent spin operators is discussed.

Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems

In this paper, we prove Slodkowski version of the infinite-dimensional spectral mapping theorem and Cartan–Slodkowski version of the finite-dimensional spectral mapping theorem for nilpotent operator Lie subalgebras with respect to the various noncommutative functional calculi.

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a (real or complex) Banach space with dimension greater than 2 and let ℬ0(X) be the subspace of ℬ(X) spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Φ on ℬ0(X) which preserve nilpotent operators in both directions. In particular, if X is infinite–dimensional, we prove that Φ has the form either Φ(T) = cATA−1 or Φ(T) = cAT'A−1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T' denotes the adjoint of T. As an application of these results, we show that every additive surjective map on ℬ(X) preserving spectral radius has a similar form to the above with |c| = 1.

Réflexivité d’une extension d’un opérateur normal par un opérateur nilpotent

Let T be an extension of a one to one normal operator A by a nilpotent operator N. In this paper we calculate the defect of reflexivity of T. We give a necessary and sufficient condition to insure the reflexivity of such extensions. In particular, it is shown that T is reflexive when N is reflexive.

Control of Affine Nonlinear Systems with Nilpotent Structure in Singular Problems

The minimum singular functional control problem is analyzed for a class of multi-input affine nonlinear systems under the hypothesis that the associated Lie algebra is nilpotent. The optimal control corresponding to the first, second, and third order nilpotent operators is determined. We develop an algorithm for solving the singular problem that is applicable whether or not singular subarcs exist in the optimal control.

On nilpotent operators

We give several necessary and sufficient conditions in order that a bounded linear operator on a Banach space be nilpotent. We also discuss three necessary conditions for nilpotency. Furthermore, we construct an infinite family (in one-to-one corresponden

On the optimal control of affine nonlinear systems

The minimization control problem of quadratic functionals for the class of affine nonlinear systems with the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control corresponding to the first‐, second‐, and third‐order nilpotent operators is determined. In this paper, we have considered the minimum fuel problem for the multi‐input nilpotent control and for a scalar input bilinear system for such systems. For the multi‐input system, usually an analytic closed‐form solution for the optimal control is not possible and it is necessary to use numerical integration for the set of m nonlinear coupled second‐order differential equations. The optimal control of bilinear systems is obtained by considering the Lie algebra generated by the system matrices. It should be noted that we have obtained an open‐loop control depending on the initial value of the state x0.

Nilpotent deformations ofN=2 superspace

We investigate deformations of four-dimensional N=(1,1) euclidean superspace induced by nonanticommuting fermionic coordinates. We essentially use the harmonic superspace approach and consider nilpotent bi-differential Poisson operators only. One variant of such deformations (termed chiral nilpotent) directly generalizes the recently studied chiral deformation of N=(1/2,1/2) superspace. It preserves chirality and harmonic analyticity but generically breaks N=(1,1) to N=(1,0) supersymmetry. Yet, for degenerate choices of the constant deformation matrix N=(1,1/2) supersymmetry can be retained, i.e. a fraction of 3/4. An alternative version (termed analytic nilpotent) imposes minimal nonanticommutativity on the analytic coordinates of harmonic superspace. It does not affect the analytic subspace and respects all supersymmetries, at the expense of chirality however. For a chiral nilpotent deformation, we present non(anti)commutative euclidean analogs of N=2 Maxwell and hypermultiplet off-shell actions.

On minimum quadratic functional control of affine nonlinear systems

The minimization control problem of the quadratic functionals for the class of affine nonlinear systems in the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control correspondent to the first-, second- and third-order nilpotent operator is determined. H. Bourdache-Siguerdinljane has applied the method of Lie algebras to the study of optimal control regulation of satellites. S. Banks and M. Yew have studied the optimal control of energy consumption minimization for a class of bilinear systems and J.S. Liu et al., have generalized this result for the class of affine nonlinear systems. The optimal control determination of nonlinear system with a nilpotent structure minimizing the quadratic functionals generalizes the results of J.S. Liu, K. Yuan, W.S. Lin, and S.P. Banks, and M.K. Yew regarding the energy minimization of the affine nonlinear and bilinear systems.

Rank-1 nilpotency linear preservers on nest algebras

Let N and M be two nests on Banach spaces X and Y, respectively, and let Φ be a linear map between nest algebras Alg N and Alg M of which the range contains all rank-1 nilpotent operators in Alg M . We give some structure theorems for Φ to be rank-1 nilpotency preserving in both directions, and obtain particularly a characterization of such Φ if it is bounded and X and Y are Hilbert spaces with dim X⩾6.

${\rm C}^*$-ALGEBRAS WITH DENSE NILPOTENT SUBALGEBRAS

In a beautiful result, Herrero (D. A. Herrero, ‘Normal limits of nilpotent operators’, Indiana Univ. Math. J. 23 (1973/74) 1097–1108) showed that a normal operator on $l^2$ lies in the closure of the set of nilpotent operators if and only if its spectrum is connected and contains zero. In the quest for an automatic continuity result for algebra homomorphisms between ${\rm C}^*$ -algebras, Dales showed that, if a discontinuous algebra homomorphism $\theta :{\cal A} \rightarrow {\cal U}$ exists between ${\rm C}^*$ -algebras ${\cal A}$ and ${\cal U}$ , and if $\theta({\cal A})$ is dense in ${\cal U}$ , then there is a ${\rm C}^*$ -algebra ${\cal U}_2$ with a dense subalgebra ${\cal N} \subset {\cal U}_2$ such that every $x \in {\cal N}$ is quasinilpotent (see p. 685 of H. G. Dales, Banach algebras and automatic continuity , London Mathematical Society Monographs 24, Oxford University Press, 2001). (A discontinuous homomorphism $\theta_2 :{\cal A}_2 \rightarrow {\cal U}_2$ can be defined with the same basic properties as $\theta$ , but the revised target space ${\cal U}_2$ has a dense subalgebra consisting of quasinilpotent elements.) As remarked by Dales, no such ${\rm C}^*$ -algebra was then known; but here we present one. Indeed, using the full power of Herrero's result, one may arrange that every $x \in {\cal N}$ is nilpotent. The ${\rm C}^*$ -algebra is constructed in a ‘neat’ way; it is most naturally constructed as a non-separable, concrete ${\rm C}^*$ -algebra of operators on a separable Hilbert space ${\cal K}$ ; but one can arrange that the algebra ${\cal U}$ itself be separable if desired.

SOLUTION ESTIMATES OF NONLINEAR VECTOR VOLTERRA–STIELTJES EQUATIONS

Supremum and L2-norms estimates of the solutions of nonlinear vector Volterra–Stieltjes equations in the complex Euclidean space [Formula: see text] are derived using recently established estimates for the norm of vector-valued functions and the resolvents of quasi–nilpotent operators. Such equations include both continuous time Volterra integral equations and discrete time Volterra difference equations as well as mixtures of these two basic types of time scales. The estimates are formulated in terms of the coefficients of the equations, which are separated into a linear part and a nonlinear part satisfying a local Lipschitz condition. Stability conditions are then derived from these estimates.

Nilpotent operators and discretely concave functions

Given a pair of submodules of a module over a discrete valuation ring, we construct a function of two integer variables. We show that this function is discretely concave and, conversely, every discretely concave function of two integer variables arises in this way.

Additive mappings that preserve rank one nilpotent operators

Let B( X) be the algebra of bounded operators on a complex infinite dimensional Banach space X, and F( X) be the subalgebra of all finite rank operators in B( X). A characterization of additive mappings on F( X) which preserve rank one nilpotent operators in both directions is given. As applications of this result, some additive preservers are described.

Prediction of mean annual flood (MAF) for ogun drainagebasin, Southwestern Nigeria

In this paper, we are motivated towards a natural problem to find classes of non-normal operators which satisfy one of the most important property of the spectrum that Reσ(T) = σ(ReT). Simple examples show that this remarkable relation does not hold for arbitrary operators. It is clear that for nilpotent operators or more generally for quasinilpotent operators, this relation fails to hold. However, if T is bounded operator and one of the following conditions holds, then the relation Reσ(T) = σ(ReT) is true: 1. T is hyponormal 2. The adjoint of T, T*, is hyponormal 3. Spectrum of T, σ(T ), is a spectral set of T 4. T has the G 1 -property and spectrum of T is connected.

How to generate spinor representations in any dimension in terms of projection and nilpotent operators

We present a technique to construct a spinor space basis as products of certain nilpotents and projections formed from γa for which we only need to know that they obey the Clifford algebra. The technique works for all dimensions and signatures. We use this technique to deliver a concrete choice of γ-matrices and Lorentz group generators for (fundamental representations of) spinors in a rather systematic and transparent way. We further develop the formalism by proposing the corresponding graphic presentation of basic states, which offers an easy way to see all the quantum numbers of states with respect to the generators of the Lorentz group, as well as transformation properties of the states under any element of the Clifford algebra.