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Angelo B. Mingarelli

Publications and source records attributed to Angelo B. Mingarelli.

At least 19 recordsLinked to original sources
arXiv2025

Complex oscillations of non-definite Sturm-Liouville problems, II

We correct and update a result of R.G.D. Richardson [13] dealing with the separation of zeros of the real and imaginary parts of non-real eigenfunctions of non-definite Sturm-Liouville eigenvalue problems. We then extend it to the case where the weight function is allowed to be identically zero on a subinterval that excludes the end-points and study the behavior of the zeros of the real and imaginary parts when the end-points are included. Examples are given illustrating the sharpness of the results along with open questions.

math.CA↗
arXiv2024

On the non-existence of oscillation numbers in Sturm-Liouville theory

We prove an old conjecture that relates the existence of non-real eigenvalues of Sturm-Liouville Dirichlet problems on a finite interval to the non-existence of oscillation numbers of its real eigenfunctions, [[6], p.104, Problems 3 and 5]. This extends to the general case, a previous result in [1], [2] where it was shown that the presence of even one pair of non-real eigenvalues implies the non-existence of a positive eigenfunction (or ground state). We also provide estimates on the Haupt and Richardson indices and Haupt and Richardson numbers thereby complementing the original Sturm oscillation theorem with the Haupt-Richardson oscillation theorem discovered over 100 years ago with estimates on the missing oscillation numbers of the real eigenfunctions observed.

math.CA↗
arXiv2022

On the basic theory of some generalized and fractional derivatives

We continue the development of the basic theory of generalized derivatives as introduced in \cite{JPA} and give some of their applications. In particular, we formulate versions of a weak maximum principle, Rolle's theorem, the Mean value theorem, the Fundamental theorem of Calculus, Integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which includes, as corollaries, the corresponding version for conformable fractional derivatives considered by \cite{kat}, \cite{kha} among many others. Finally, we show that for each $α> 1$ there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line.

math.CA↗
arXiv2022

Fractional Sturm-Liouville eigenvalue problems, II

We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {\it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $α$, $0<α<1$, there is a finite set of real eigenvalues and that, for $α$ near $1/2$, there may be none at all. As $α\to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.

math.CA↗
arXiv2022

On a converse of Sturm's comparison theorem

We show that Sturm's classical comparison theorem (SCT) on the interlacing of zeros of solutions of pairs of real second order two-term ordinary differential equations necessarily fails if the usual Sturmian-type conditions on the coefficients are violated. We also show that the conditions on the coefficients are, in some sense, best possible. We give a necessary and sufficient condition for the failure of SCT and we note that its converse is false, and that the comparison theorem may still hold with very mild hypotheses on the coefficients.

math.CA↗
arXiv2021

Uncertainty Principles in Krein Space

Uncertainty relations between two general non-commuting self-adjoint operators are derived in a Krein space. All of these relations involve a Krein space induced fundamental symmetry operator, $J$, while some of these generalized relations involve an anti-commutator, a commutator, and various other nonlinear functions of the two operators in question. As a consequence there exist classes of non-self-adjoint operators on Hilbert spaces such that the non-vanishing of their commutator implies an uncertainty relation. All relations include the classical Heisenberg uncertainty principle as formulated in Hilbert Space by Von Neumann and others. In addition, we derive an operator dependent (nonlinear) commutator uncertainty relation in Krein space.

quant-ph↗
arXiv2018

Proving the short-wavelength approximation in Pulsar Timing Array gravitational-wave background searches

A low-frequency gravitational-wave background (GWB) from the cosmic merger history of supermassive black holes is expected to be detected in the next few years by pulsar timing arrays. A GWB induces distinctive correlations in the pulsar residuals --- the expected arrival time of the pulse less its actual arrival time. Simplifying assumptions are made in order to write an analytic expression for this correlation function, called the Hellings and Downs curve for an isotropic GWB, which depends on the angular separation of the pulsar pairs, the gravitational-wave frequency considered, and the distance to the pulsars. This is called the short-wavelength approximation, which we prove here rigorously and analytically for the first time.

astro-ph.IM↗
arXiv2018

On generalized and fractional derivatives and their applications to classical mechanics

(Draft 3) A generalized differential operator on the real line is defined by means of a limiting process. These generalized derivatives include, as a special case, the classical derivative and current studies of fractional differential operators. All such operators satisfy properties such as the sum, product/quotient rules, chain rule, etc. We study a Sturm-Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm-Liouville Theory. As an application of the developments herein we find the general solution of a generalized harmonic oscillator. We also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse, and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift. The previous result on the generic nature of phase plane orbits is extended to the classical gravitational n-body problem of Newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law (modulo a time shift). Finally, restricting the generalized derivatives to a special class of fractional derivatives, we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter. The results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives.

math-ph↗
arXiv2017

Fractional Sturm-Liouville eigenvalue problems, I

We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<α<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $α\to 1^-$, and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as $α\to 1^-$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $α$.

math.CA↗
arXiv2014

The Mathematics of Fels Sculptures

We give a purely mathematical interpretation and construction of sculptures rendered by one of the authors, known herein as Fels sculptures. We also show that the mathematical framework underlying Ferguson's sculpture, {\it The Ariadne Torus}, may be considered a special case of the more general constructions presented here. More general discussions are also presented about the creation of such sculptures whether they be virtual or in higher dimensional space.

math.HO↗
arXiv2013

On non-definite Sturm-Liouville problems with two turning points

This is an inaugural study of the Dirichlet problem associated with a regular non-definite Sturm-Liouville equation in the case of two turning points. We give a priori lower bounds on the Richardson numbers associated with this problem thereby complementing pioneering results by Atkinson and Jabon (1984) in the one turning point case

math.CA↗
arXiv2013

Some conjectures in elementary number theory

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

math.NT↗
arXiv2012

Abstract factorials

A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number. Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational.

math.NT↗
arXiv2011

A note on Sturm-Liouville problems whose spectrum is the set of prime numbers

We show that there is no classical regular Sturm-Liouville problem on a finite interval whose spectrum consists of infinitely many distinct primes numbers. In particular, this answers in the negative a question raised by Zettl in his book on Sturm-Liouville theory. We also show that there {\it may} exist such a problem if the parameter dependence is nonlinear.

math.CA↗
arXiv2008

Remarks on an 1828 theorem of Clausen

We list some explicit calculations related to a theorem of Clausen originally published in 1828, more commonly known as the result that describes the linear third order differential equation satisfied by the squares and the product of any two solutions of a linear second order differential equation in the real domain. The case of the cube of a solution dates to Appell, 1880. Although not commonly known and perhaps even new to some extent we show that the fourth and fifth powers of such solutions actually satisfy a linear differential equation of order five and six respectively provided the coefficients are sufficiently smooth. Indeed, it is the case that given any solution of a linear second order equation its m-th power satisfies an effectively computable linear differential equation of order m+1.

math.CA↗
arXiv2008

Principal solutions of recurrence relations and irrationality questions in number theory

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of three-term linear recurrence relations we show that there exists a four-term linear recurrence relation whose solutions allow us to show that the number is a quadratic irrational if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher order recurrence relations and a transcendence criterion can also be formulated in terms of these principal solutions. When applied to the situation of powers of $ζ(3)$ it is not known whether the corresponding four term recurrence relation does or does not have such a principal solution, however the method does generate new series expansions of powers of $ζ(3)$ and $ζ(2)$ in terms of Apéry's now classic sequences.

math.NT↗