A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Perturbation of resonances in quantum mechanics

Schrödinger operators on L²($\text{R}$³) of the form ?Δ + V_λ with potentials V_λ real-analytic in λ are discussed. The analytic structure in V_λ and k (with k² the energy variable) of the resolvent kernel, the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions. The lower order expansion coefficients are computed explicitly. The resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions. This gives asymptotic low energy information about Schrödinger Hamiltonians with short range potentials. The perturbation theory of resonances, eigenvalues and of the corresponding functions for Hamiltonians describing n point interactions perturbed by a potential is also given.     

Unimodular eigenvalues, uniformly distributed sequences and linear dynamics

We study increasing sequences of positive integers (nk)_k?1 with the following property: every bounded linear operator T acting on a separable Banach (or Hilbert) space with supk?1‖Tnk‖<∞ has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (nkα)_k?1, α∈R, or on the growth of nk+1/nk. Counterexamples to some conjectures in linear dynamics are given. For instance, a Hilbert space operator which is frequently hypercyclic, chaotic, but not topologically mixing is constructed. The situation of C₀-semigroups is also discussed.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Laguerre polynomial expansions in indefinite inner product spaces

When ?j ? 1 < α < ?j, where j is a positive integer, the Laguerre polynomials {L_n^(α)}_{n = 0}^∞ form a complete orthogonal set in a nondegenerate inner product space H which is defined by employing an appropriate regularized linear functional on H^(j)[[0, ∞); x^{α + j}e^?x]. Expansions in terms of these Laguerre polynomials are exhibited. The Laguerre differential operator is shown to be self-adjoint with real, discrete, integer eigenvalues. Its spectral resolution and resolvent are exhibited and discussed.     

The level polynomials of the free distributive lattices

We show that there exist a set of polynomials {L_k?k = 0, 1?} such that L_k(n) is the number of elements of rank k in the free distributive lattice on n generators. L₀(n) = L₁(n) = 1 for all n and the degree of L_k is k?1 for k?1. We show that the coefficients of the L_k can be calculated using another family of polynomials, P_j. We show how to calculate L_k for k = 1,…,16 and P_j for j = 0,…,10. These calculations are enough to determine the number of elements of each rank in the free distributive lattice on 5 generators a result first obtained by Church [2]. We also calculate the asymptotic behavior of the L_k's and P_j's.     

U(n) Wigner coefficients,the path sum formula,and invariant G-functions

We prove the path sum formula for computing the U(n) invariant denominator functions associated to stretched U(n) Wigner operators. A family of U(n) invariant polynomials G_[λ]⁽ⁿ⁾ is then defined which generalize the _μG_q⁽ⁿ⁾ polynomials previously studied. The G_[λ]⁽ⁿ⁾ polynomials are shown to satisfy a number of difference equations and have symmetry properties similar to the _μG_q⁽ⁿ⁾ polynomials. We also give a direct proof of the important transposition symmetry for the G_[λ]⁽ⁿ⁾ polynomials. To enable the non-specialist to understand the foundations for these remarkable polynomials, we provide an exposition of the boson calculus and the construction of the multiplicity-free U(n) Wigner operators.     

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators H₀, H and discuss the spectral projections of these operators corresponding to the interval (−∞,λ). The purpose of the paper is to study the spectral properties of the difference D(λ) of these spectral projections. We completely describe the absolutely continuous spectrum of the operator D(λ) in terms of the eigenvalues of the scattering matrix S(λ) for the operators H₀ and H. We also prove that the singular continuous spectrum of the operator D(λ) is empty and that its eigenvalues may accumulate only at “thresholds” in the absolutely continuous spectrum.     

Some questions in the theory of inverse problems for differential operators

For the two operatorsLy=y ⁿ +σ _k=0 ⁿ⁻² p _k (x)y( ^k ) and Ry=yⁿ+σ _k=0 ⁿ⁻² p_k(x)y(^k) with a common set of boundary conditions we establish a connection between p_k(x) and P_k(x) in the case where the weight numbers coincide and a finite number of the eigenvalues do not coincide, in terms of the eigenfunctions of these operators corresponding to the noncoincident eigenvalues and the derivatives of these functions. This enables us to recover the operator L from the operator R by solving a system of nonlinear ordinary differential equations. For Sturm-Liouville operators an analogous relation is proved for the case where infinitely many eigenvalues do not coincide. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 151–160, February, 1977. I wish to express my thanks to my scientific adviser V. A. Sadovnich.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

Size of the peripheral point spectrum under power or resolvent growth conditions

We characterize Jamison sequences, that is sequences (nk) of positive integers with the following property: every bounded linear operator T acting on a separable Banach space with supk‖Tnk‖<+∞ has a countable set of peripheral eigenvalues. We also discuss partially power-bounded operators acting on Banach or Hilbert spaces having peripheral point spectra with large Hausdorff dimension. For a Lavrentiev domain Ω in the complex plane, we show the uniform minimality of some families of eigenvectors associated with peripheral eigenvalues of operators satisfying the Kreiss resolvent condition with respect to Ω. We introduce and study the notion of Ω-Jamison sequence, which is defined by replacing the partial power-boundedness condition supk‖Tnk‖<+∞ by , where is the nth Faber polynomial of Ω. A characterization of Ω-Jamison sequences is obtained for domains with sufficiently smooth boundary.     

Hilbertian Jamison sequences and rigid dynamical systems

A strictly increasing sequence (nk)_k?0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that supk?0‖Tnk‖<+∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (nk)_k?0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.     

On the number of coprime integral solutions of y2 = x3 + k

Let N′(k) denote the number of coprime integral solutions x, y of y² = x³ + k. It is shown that lim sup_k→∞N′(k) ≥ 12.     

A theorem on the factorization of matrix polynomials

A similarity condition is developed for the factorization of monic matrix polynomials L(λ) into the forms L(λ) = L_k(λ) … L₁(λ), wihtout any restriction on the spectrum of factors L_j(λ).     

Spectral synthesis of diagonal operators and representing systems for the space of entire functions

In this paper, we study continuous linear operators on spaces of functions analytic on disks in the complex plane having as eigenvectors the monomials zn whose associated eigenvalues λn are distinct. In particular, we show that under mild conditions, such a diagonal operator has non-spectral invariant subspaces (that is, closed invariant subspaces which are not the closed linear span of collections of monomials) if and only if every entire function of a particular growth rate is representable as a generalized Dirichlet series .     

Spectral asymptotic behavior of elliptic systems

One investigates a first-order elliptic self-adjoint pseudodifferential operator A (x,D) acting in sections of a Hermitian vector bundle over a compactn-dimensional manifold x. It is assumed that the principal symbol A(x, ξ) of the operator is locally diagonalizable and that its eigenvaluesaj(x, ξ) have a variable multiplicity and that {a _i,a _k}≠0 whenevera _i=a _k. Under indicated conditions one constructs an expansion of the fundamental solution of the hyperbolic system \(i\frac{{\partial u}}{{\partial t}} = A(x,D)u\) and one investigates the asymptotic properties of the spectrum of the operator A (x,D). For the distribution functionN(λ) of the eigenvalues one establishes that . Under further assumptions on the properties of the bicharacteristic of the symbolsaj(x, ξ) one establishes a stronger estimate of Duistermaat-Guillemin type:N(λ)=Cλ ⁿ +C′λ ^n?i +0(γ ^n?1 )     

Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.     

Characterization of similarities between two n-frames of projectors

An n-frame on a Banach space $\text{X}$ is E=(E₁,?, E_n) where the E_j's are bounded linear operators on $\text{X}$ such that E_j≠0,

$\text{∑}\text{j=1}\text{n}\text{E}{}_{\mathrm{j}}$

, and E_jE_k=δ_jkE_k (j, k=1,?, n). It is known that if two n-frames E and F are sufficiently close to each other, then they are similar, that is, F_j=TE_jT^-1 with T a bounded linear operator. Among the operators which realize the similarity of the two frames, there is the balanced transformation U(F, E)=(Σⁿ_j=1F_jE_j)(Σⁿ_j=1E_jF_jE_j)^{$\text{-1}\text{2}$}. One of our main results is a local characterization of the balanced transformation. Another operator which implements the similarity between E and F is the direct rotation R(F, E). It comes up in connection with the study of the set of all n-frames as a Banach manifold with an affine connection. Finally, it is shown that for quite a large set of pairs of 2-frames, the direct rotation has a global characterization.     

Starlikeness and subordination properties of a linear operator

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator L_k^α. For functions in this class H_k^α(p,λh) we estimate the coefficients. Furthermore, some subordination properties related to the operator L_k^α are also derived.     

Integers divisible by sums of powers of their prime factors

For each positive integer j, let βj(n):=_∑p|npj. Given a fixed positive integer k, we show that there are infinitely many positive integers n having at least two distinct prime factors and such that βj(n)|n for each j∈{1,2,…,k}.