Point Λ0 Research Articles

Noncollinear Equilibrium Points in CRTBP with Yukawa-Like Corrections to Newtonian Potential under an Oblate Primary Model

This study is about the effects of Yukawa-like corrections to Newtonian potential on the existence and stability of noncollinear equilibrium points in a circular restricted three-body problem when bigger primary is an oblate spheroid. It is observed that ∂x0/∂λ = 0 = ∂y0/∂λ at λ0 = 1/2, so we have a critical point λ0 = 1/2 at which the maximum and minimum values of x0 and y0 can be obtained, where λ ∈ (0, ∞) is the range of Yukawa force and (x0, y0) are the coordinates of noncollinear equilibrium points. It is found that x0 and y0 are increasing functions in λ in the interval 0 < λ < λ0 and decreasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α+. On the other hand, x0 and y0 are decreasing functions in λ in the interval 0 < λ < λ0 and increasing functions in λ in the interval λ0 < λ < ∞ for all α ∈ α−, where α ∈ (−1, 1) is the coupling constant of Yukawa force to gravitational force. The noncollinear equilibrium points are found linearly stable for the critical mass parameter β0, and it is noticed that ∂β0/∂λ = 0 at λ ∗ = 1/3; thus, we got another critical point which gives the maximum and minimum values of β0. Also, ∂β0/∂λ > 0 if 0 < λ < λ ∗ and ∂β0/∂λ < 0 if λ ∗ < λ < ∞ for all α ∈ α−, and ∂β0/∂λ < 0 if 0 < λ < λ ∗ and ∂β0/∂λ > 0 if λ ∗ < λ < ∞ for all α ∈ α+. Thus, the local minima for β0 in the interval 0 < λ < λ ∗ can also be obtained.

Root vectors of polynomial and rational matrices: Theory and computation

The notion of root polynomials of a polynomial matrix P(λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R(λ), possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R(λ), without any need to assume that R(λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R(λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ0. If λ0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations.

On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders

We consider a general elliptic operator in an infinite multi-dimensional cylinder with several distant perturbations; this operator is obtained by “gluing” several single perturbation operators H(k), k=1,…,n, at large distances. The coefficients of each operator H(k) are periodic in the outlets of the cylinder; the structure of these periodic parts at different outlets can be different. We consider a point λ0∈R in the essential spectrum of the operator with several distant perturbations and assume that this point is not in the essential spectra of middle operators H(k), k=2,…,n−1, but is an eigenvalue of at least one of H(k), k=1,…,n. Under such assumption we show that the operator with several distant perturbations possesses finitely many resonances in the vicinity of λ0. We find the leading terms in asymptotics for these resonances, which turn out to be exponentially small. We also conjecture that the made assumption selects the only case, when the distant perturbations produce finitely many resonances in the vicinity of λ0. Namely, as λ0 is in the essential spectrum of at least one of operators H(k), k=2,…,n−1, we do expect that infinitely many resonances emerge in the vicinity of λ0.

A new approach to nonlinear singular integral operators depending on three parameters

Abstract In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: T λ ( f ; x , y ) = ∬ R 2 K λ ( t − x , s − y , f ( t , s ) ) d s d t , ( x , y ) ∈ R 2 , λ ∈ Λ , $${T_\lambda }(f;x,y) = \iint\limits_{{\mathbb{R}^2}}K_\lambda {(t - x,s - y,f(t,s))dsdt,\;(x,y) \in {\mathbb{R}^2},\lambda \in \Lambda ,}$$ where Λ is a set of non-negative numbers with accumulation point λ0.

On k-quasi-paranormal operators

For a positive integer k, an operator T ∈ B(H ) is called k-quasi-paranormal if � T k+1 x� 2 � T k+2 x�� T k xfor all x ∈ H , which is a common generalization of paranormal and quasi-paranormal. In this paper, firstly we prove some inequalities of this class of oper- ators; secondly we give a necessary and sufficient condition for T to be k-quasi-paranormal. Using these results, we prove that: (1) ifT n+1 � = � Tn+1 for some positive integer n k, then a k-quasi-paranormal operator T is normaloid; (2) if E is the Riesz idempotent for an isolated point λ0 of the spectrum of a k-quasi-paranormal operator T , then (i) if λ0 � 0, then EH = ker(T − λ0); (ii) if λ0 = 0, thenEH = ker(T k+1 ).

A Note on Quasi-paranormal Operators

Let T be a bounded linear operator on an infinite dimensional complex Hilbert space. In this paper, we introduce the new class, denoted \({{\mathcal{QP}}}\) , of operators satisfying \({{\|T^{2}x\|^{2}\leq \|T^{3}x\|\|Tx\|}}\) for all \({{x \in \mathcal{H}}}\). This class includes the classes of paranormal operators and quasi-class A operators. We prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a nonzero isolated point λ0 of the spectrum of \({{T \in \mathcal{QP}}}\) , then E is self-adjoint if and only if \({{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}\).

Bifurcation currents in holomorphic dynamics on

Potential theory has been introduced in one dimensional rational dynamics by Brolin and Tortrat ([4], [29]) but does not play a central role there. In higher dimension however, as the classical tools are not any longer efficient, pluri-potential theory has revealed itself to be essential. The fundamental works of Hubbard-Papadopol, Fornaess-Sibony, BedfordSmillie, Briend-Duval (see [26] for precise references) enlight the remarkable efficency of pluri-potential theory in holomorphic dynamics on Pk or Ck. It is therefore tempting to study the parameter spaces in a similar way. More precisely, one would like to relate the bifurcations of an holomorphic family {fλ}λ∈X of endomorphisms of Pk to the powers of a certain current on the parameter space X. Let us recall that in dimension k = 1, a bifurcation is said to occur at some point λ0 ∈ X if the Julia set of fλ does not move continuously around λ0. The famous work of Mane-SadSullivan [16], which is based on the λ-lemma and the Fatou-Cremer-Sullivan classification, relates the bifurcations with the instability of the critical orbits. It also establishes that the bifurcations concentrate on the complement of an open dense subset of X (for the quadratic family {z2 + λ}λ∈X=C the bifurcation locus is precisely the boundary of the Mandelbrot set).

Holomorphic functional calculus for operators on a locally convex space

One-variable holomorphic functional calculus is studied on the bornological algebra Lec(E) of all continuous linear oprators on a complete locally convex space E. It is proven that the following three basic notions of the theory are equivalent: (i) existence of projective resolvent of an operator T at a point λ0, (ii) strict regularity of λ0 for the operator T in the sense of [12, 13, 15], (iii) tamability of the operator (λ0 − T)−1 (T if λ0 = ∞), which means that there is a new equivalent system of seminorms on E, such that the operator is bounded in each of them.

Conditions for eigenvectors of a selfadjoint operator-function to form a basis

Consider a functionL(λ) defined on an interval Δ of the real axis whose values are linear bounded selfadjoint operators in a Hilbert spaceH. A point λ0 ∈ Δ and a vectorϕ0 ∈H(ϕ0 ≠ 0) are called eigenvalue and eigenvector ofL(λ) ifL(λ) ifL(λ0)ϕ0 = 0. Supposing that the functionL′(λ) can be represented as an absolutely convergent Fourier integral, the interval Δ is sufficiently small and the derivative will be positive at some point, it has been proved that all the eigenvectors of the operator-functionL(λ) corresponding to the eigenvalues from the interval Δ form an unconditional basis in the subspace spanned by them.

On the (sub)logarithmic property of the pole-, zero- and algebraic multiplicity of operator functions

This paper contains an extension of a result obtained by H. Bart, M. A. Kaashoek and D. C. Lay in (2). These authors studied the reduced algebraic multiplicity RM(A; λ0) of a meromorphic operator function at a point λ0 ∈ C. They proved that under certain conditions this quantity has logarithmic behaviour, i.e.,For more restricted cases such results had been proved by others, notably I. C. Gohberg and E. I. Sigal (see (4) and (5)). Here we shall prove that such a result also holds for a larger class of operator functions than the diagonable functions considered in (2).

Analytic renormalisation of the exponential interaction and weak interaction singularities

The parity non conserving interaction of a neutral vector meson with fermions is considered as a mathematical model suitable for investigating divergence problems of the weak interactions. Through the Stückelberg formalism and a canonical transformation the interaction is converted into an exponential form. The exponential interaction is studied, in the second order of perturbation theory, through the method of analytic renormalisation. Generalised amplitudes are introduced as localizable distributions depending on auxiliary complex parameters λ. It is shown that the distributions possess a nonisolated singularity at the physical point λ0. A method is developed for discarding the singularity thereby obtaining the physical amplitudes as localizable distributions which display a non-analytic dependence on the coupling constant.