Newton's method for the eigenvalue problem of a symmetric matrix

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

Comment on: Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

Pedrosa et al.1 have recently used a [Formula: see text] symmetric linear invariant to study a unidimensional time-dependent [Formula: see text] symmetric harmonic oscillator with a complex time-dependent [Formula: see text] symmetric external force. We show in this comment that the normalization condition of the eigenfunctions of the invariant is not verified as claimed in Ref. 1. In order to obtain the normalization condition, we introduce a novel concept of the pseudoparity-time (pseudo-[Formula: see text]) symmetry.

The natural operators similar to the twisted courant bracket on couples of vector fields and p-forms

Given natural numbers m and p with m ? p + 2 ? 3, all Mfm-natural operators A sending closed (p+2)-forms H on m-manifolds M into R-bilinear operators AH transforming pairs of couples of vector fields and p-forms on M into couples of vector fields and p-forms on M are found. If m ? p + 2 ? 3, all Mfm-natural operators A (as above) such that AH satisfies the Jacobi identity in Leibniz form are extracted, and that the twisted Courant bracket [-,-]H is the unique Mfm-natural operator AH (as above) satisfying the Jacobi identity in Leibniz form and some normalization condition is deduced.

Parabolic conformally symplectic structures I; definition and distinguished connections

We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle {\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

Life expectancy and life energy according to Evo-SETI theory

This paper is profoundly innovative for the Evo-SETI (Evolution and SETI) mathematical theory. While this author's previous papers were all based on the notion of a b-lognormal, that is a probability density function in the time describing one's life between birth and ‘senility’ (the descending inflexion point), in this paper the b-lognormals range between birth and peak only, while a descending parabola covers the lifespan after the peak and down to death. The resulting finite curve in time is called a LOGPAR, a nickname for ‘b-LOGnormal and PARabola’. The advantage of such a formulation is that three variables only (birth, peak and death) are sufficient to describe the whole Evo-SETI theory and the senility is discarded forever and so is the normalization condition of b-lognormals: only the shape of the b-lognormals is kept between birth and peak, but not its normalization condition.In addition, further advantages exist:1) The notion of ENERGY becomes part of Evo-SETI theory. This is in addition to the notion of ENTROPY already contained in the theory as the Shannon Information Entropy of b-lognormals, as it was explored in this author's previous papers. Actually, the LOGPAR may now be regarded as a POWER CURVE, i.e. a curve expressing the power of the living being to which it refers. And this power is to be understood both in the strict sense of physics (i.e. a curve measured in Watts) and in the loose sense of ‘political power’ if the logpar refers to a Civilization.Then the integral in the time of this power curve is, of course, the ENERGY either absorbed or produced by the physical phenomenon that the LOGPAR is describing in the time. For instance, if the logpar shows the time evolution of the Sun over about 10 billion years, the integral of such a curve is the energy produced by the Sun over the whole of its lifetime. Or, if the logpar describes the life of a man, the integral is the energy that this man must use in order to live.2) The PRINCIPLE OF LEAST ENERGY, reminiscent of the Principle of Least Action, i.e. the key stone to all Physics, also enters now into the Evo-SETI Theory by virtue of the so-called LOGPAR HISTORY FORMULAE, expressing the b-lognormal's mu and sigma directly in terms of the three only inputs b, p, d. The optimization of the lifetime of a living creature, or of a Civilization, or of a star, is obtained by setting to zero the first derivative of the area under the logpar power curve with respect to sigma. That yields the best value of both mu and sigma fulfilling the Principle of Least Energy for Evo-SETI Theory.3) We also derive for the first time a few more mathematical equations related to the ‘adolescence’ (or ‘puberty’) time, i.e. the time when the living organism acquires the capability of producing offsprings. This time is defined as the abscissa of ascending inflection point of the b-lognormal between birth and peak. In addition, we prove that the straight line parallel to the time axis and departing from the puberty time comes to mean the ‘Fertility Span’ in between puberty and the EOF (End-Of-Fertility time), which is where the above straight line intersects the descending parabola. All these new results apply well to the description of Man as the living creature to which our Evo-SETI mathematical theory perfectly applies.In conclusion, this paper really breaks new mathematical ground in Evo-SETI Theory, thus paving the way to further applications of the theory to Astrobiology and SETI.

Energy-dependent harmonic oscillator in noncommutative space

In noncommutative quantum mechanics, the energy-dependent harmonic oscillator problem is studied by solving the Schrödinger equation in polar coordinates. The presence of the noncommutativity in space coordinates and the dependence on energy for the potential yield energy-dependent mass and potential. The correction of normalization condition is calculated and the parameter-dependences of the results are studied graphically.

Submaximally symmetric c-projective structures

[Formula: see text]-projective structures are analogues of projective structures in the almost complex setting. The maximal dimension of the Lie algebra of [Formula: see text]-projective symmetries of a complex connection on an almost complex manifold of [Formula: see text]-dimension [Formula: see text] is classically known to be [Formula: see text]. We prove that the submaximal dimension is equal to [Formula: see text]. If the complex connection is minimal (encoded as a normal parabolic geometry), the harmonic curvature of the [Formula: see text]-projective structure has three components and we specify the submaximal symmetry dimensions and the corresponding geometric models for each of these three pure curvature types. If the connection is non-minimal, we introduce a modified normalization condition on the parabolic geometry and use this to resolve the symmetry gap problem. We prove that the submaximal symmetry dimension in the class of Levi-Civita connections for pseudo-Kähler metrics is [Formula: see text], and specializing to the Kähler case, we obtain [Formula: see text]. This resolves the symmetry gap problem for metrizable [Formula: see text]-projective structures.