arXiv subjects

Jorge L. deLyra

Publications and source records attributed to Jorge L. deLyra.

At least 19 recordsLinked to original sources

arXiv2021

Exact Solution of the Einstein Field Equations for a Spherical Shell of Fluid Matter

We determine the exact solution of the Einstein field equations for the case of a spherically symmetric shell of liquid matter, characterized by an energy density which is constant with the Schwarzschild radial coordinate $r$ between two values $r_{1}$ and $r_{2}$. The solution is given in three regions, one being the well-known analytical Schwarzschild solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined mostly analytically but partially numerically, within the matter region. The solutions for the temporal coefficient of the metric and for the pressure within this region are given in terms of a non-elementary but fairly straightforward real integral. We show that in this solution there is a singularity at the origin, and give the parameters of that singularity in terms of the geometrical and physical parameters of the shell. This does not correspond to an infinite concentration of matter, but in fact to zero energy density at the center. It does, however, imply that the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This gravitational field has the effect of stabilizing the geometrical configuration of the matter, since any particle of the matter that wanders out into the vacuum regions tends to be brought back to the bulk of the matter by the gravitational field.

gr-qc↗

arXiv2021

Energetic Stability of the Solutions of the Einstein Field Equations for Spherically Symmetric Liquid Shells

We interpret the exact solutions previously obtained for spherically symmetric shells of liquid fluid in General Relativity in terms of the energies involved. We show that a certain parameter that was introduced into the solutions by the interface boundary conditions is related to the binding energies of the gravitational systems. We then use this fact in order to discuss the energetic stability of those solutions. We include in the stability discussion the well-known interior Schwarzschild solution for a liquid sphere, which can be obtained as a specific limit of the solutions that we previously obtained for the liquid shells. We show that this solution turns out to be a maximally unstable one, from the energetic point of view discussed here. We also perform a numerical exploration of the energetic stability criterion of the solutions, and show that indeed there is a particular subset of the solutions which are energetically stable. All these solutions have the form of shells with non-vanishing internal radii. This reduces the original three-parameter family of solutions to a two-parameter family of energetically stable solutions.

gr-qc↗

arXiv2019

Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions

A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the restriction of an analytic function to the unit circle, including functions which are non-differentiable, discontinuous or unbounded. An explicit construction of the analytic functions from the corresponding real functions is given. The complex-analytic structure can be understood as an universal regulator for analytic operations on real functions.

math.CV↗

arXiv2019

Complex Analysis of Real Functions II: Singular Schwartz Distributions

In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that singular Schwartz distributions can be represented within that same structure, so long as one defines the limits involved in an appropriate way. In that previous paper it was shown that essentially all integrable real functions can be represented within the complex-analytic structure. The infinite collection of singular objects which we analyze here can thus be represented side by side with those real functions, thus allowing all these objects to be treated in a unified way.

math.CV↗

arXiv2019

Complex Analysis of Real Functions III: Extended Fourier Theory

In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that the complete Fourier theory of integrable real functions is contained within that structure, that is, within the structure of the space of inner analytic functions on the open unit disk. We then extend the Fourier theory beyond the realm of integrable real functions, to include for example singular Schwartz distributions, and possibly other objects.

math.CV↗

arXiv2019

Complex Analysis of Real Functions IV: Non-Integrable Real Functions

In the context of the complex-analytic structure within the unit disk centered at the origin of the complex plane, that was presented in a previous paper, we show that a certain class of non-integrable real functions can be represented within that same structure. In previous papers it was shown that essentially all integrable real functions, as well as all singular Schwartz distributions, can be represented within that same complex-analytic structure. The large class of non-integrable real functions which we analyze here can therefore be represented side by side with those other real objects, thus allowing all these objects to be treated in a unified way.

math.CV↗

arXiv2019

Complex Analysis of Real Functions V: The Dirichlet Problem on the Plane

In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that correspondence lead to very general proofs of existence of solutions of the Dirichlet problem on the plane. At first, this establishes the existence of solutions for almost arbitrary integrable real functions on the unit circle, including functions which are discontinuous and unbounded. The proof of existence is then generalized to a large class of non-integrable real functions on the unit circle. Further, the proof of existence is generalized to real functions on a large class of other boundaries on the plane, by means of conformal transformations.

math.CV↗

arXiv2019

Complex Analysis of Real Functions VI: On the Convergence of Fourier Series

We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms of Dirichlet integrals. The expression for the remainder is written in terms of a similar type of integral. Since the asymptotic limit of the remainder being zero is a necessary and sufficient condition for the convergence of the series, this same condition on the asymptotic behavior of the corresponding integrals constitutes such a necessary and sufficient condition.

math.CV↗

arXiv2019

Complex Analysis of Real Functions VII: A Simple Extension of the Cauchy-Goursat Theorem

In the context of the complex-analytic structure within the open unit disk, that was established in a previous paper, here we establish a simple generalization of the Cauchy-Goursat theorem of complex analytic functions. We do this first for the case of inner analytic functions, and then generalize the result to all analytic functions. We thus show that the Cauchy-Goursat theorem holds even if the complex function has isolated singularities located on the integration contour, so long as these are all integrable ones.

math.CV↗

arXiv2015

Real Functions for Physics

A new classification of real functions and other related real objects defined within a compact interval is proposed. The scope of the classification includes normal real functions and distributions in the sense of Schwartz, referred to jointly as "generalized functions". This classification is defined in terms of the behavior of these generalized functions under the action of a linear low pass-filter, which can be understood as an integral operator acting in the space of generalized functions. The classification criterion defines a class of generalized functions which we will name "combed functions", leaving out a complementary class of "ragged functions". While the classification as combed functions leaves out many pathological objects, it includes in the same footing such diverse objects as real analytic functions, the Dirac delta "function", and its derivatives of arbitrarily high orders, as well as many others in between these two extremes. We argue that the set of combed functions is sufficient for all the needs of physics, as tools for the description of nature. This includes the whole of classical physics and all the observable quantities in quantum mechanics and quantum field theory. The focusing of attention on this smaller set of generalized functions greatly simplifies the mathematical arguments needed to deal with them.

math-ph↗

arXiv2015

Fourier Theory on the Complex Plane V: Arbitrary-Parity Real Functions, Singular Generalized Functions and Locally Non-Integrable Functions

A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set of all integrable real functions is then extended to include a set of singular "generalized functions" by the side of the integrable real functions. A general definition of these generalized functions is proposed and explored. The generalized functions are introduced loosely in the spirit of the Schwartz theory of distributions, and include the Dirac delta "function" and its derivatives of all orders. The inner analytic functions corresponding to this infinite set of singular real objects are given by means of a recursion relation. The set of inner analytic functions is then further extended to include a certain class of non-integrable real functions. The concept of integral-differential chains is used to help to integrate both the normal functions and the singular generalized functions seamlessly into a single structure. It does the same for the class of non-integrable real functions just mentioned. This extended set of generalized functions also includes arbitrary real linear combinations of all these real objects. An interesting connection with the Dirichlet problem on the unit disk is established and explored.

math.CV↗

arXiv2015

Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients

The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends the results obtained there to a wider class of real functions. The other two relate to a closer examination of the issue of the representability of real functions by their Fourier coefficients. As was shown in the previous paper, in many cases one can recover the real function from its Fourier coefficients even if the corresponding Fourier series diverges almost everywhere. In such cases we say that the real function is still representable by its Fourier coefficients. Here we establish a very weak condition on the Fourier coefficients that ensures the representability of the function by those coefficients. In addition to this, we show that any real function that is absolutely integrable can be recovered almost everywhere from, and hence is representable by, its Fourier coefficients, regardless of whether or not its Fourier series converges. Interestingly, this also provides proof for a conjecture proposed in the previous paper.

math.CV↗

arXiv2015

Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities

The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of convergence of the Taylor series of the inner analytic function is the open unit disk. An essentially complete classification, in terms of the singularity structure of the corresponding inner analytic functions, of the modes of convergence of a large class of DP Fourier series, is established. Given a weakly convergent Fourier series of a DP real function, it is shown how to generate from it other expressions involving trigonometric series, that converge to that same function, but with much better convergence characteristics. This is done by a procedure of factoring out the singularities of the corresponding inner analytic function, and works even for divergent Fourier series. This can be interpreted as a resummation technique, which is firmly anchored by the underlying analytic structure.

math.CV↗

arXiv2015

Low-Pass Filters, Fourier Series and Partial Differential Equations

When Fourier series are used for applications in physics, involving partial differential equations, sometimes the process of resolution results in divergent series for some quantities. In this paper we argue that the use of linear low-pass filters is a valid way to regularize such divergent series. In particular, we show that these divergences are always the result of oversimplification in the proposition of the problems, and do not have any fundamental physical significance. We define the first-order linear low-pass filter in precise mathematical terms, establish some of its properties, and then use it to construct higher-order filters. We also show that the first-order linear low-pass filter, understood as a linear integral operator in the space of real functions, commutes with the second-derivative operator. This can greatly simplify the use of these filters in physics applications, and we give a few simple examples to illustrate this fact.

math-ph↗

arXiv2015

Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions

A correspondence between arbitrary Fourier series and certain analytic functions on the unit disk of the complex plane is established. The expression of the Fourier coefficients is derived from the structure of complex analysis. The orthogonality and completeness relations of the Fourier basis are derived in the same way. It is shown that the limiting function of any Fourier series is also the limit to the unit circle of an analytic function in the open unit disk. An alternative way to recover the original real functions from the Fourier coefficients, which works even when the Fourier series are divergent, is thus presented. The convergence issues are discussed up to a certain point. Other possible uses of the correspondence established are pointed out.

math.CV↗

arXiv2015

Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters

When Fourier series are employed to solve partial differential equations, low-pass filters can be used to regularize divergent series that may appear. In this paper we show that the linear low-pass filters defined in a previous paper can be interpreted in terms of the correspondence between Fourier Conjugate (FC) pairs of Definite Parity (DP) Fourier series and inner analytic functions, which was established in earlier papers. The action of the first-order linear low-pass filter corresponds to an operation in the complex plane that we refer to as "singularity splitting", in which any given singularity of an inner analytic function on the unit circle is replaced by two softer singularities on that same circle, thus leading to corresponding DP Fourier series with better convergence characteristics. Higher-order linear low-pass filters can be easily defined within the unit disk of the complex plane, in terms of the first-order one. The construction of infinite-order filters, which always result in $C^{\infty}$ real functions over the unit circle, and in corresponding DP Fourier series which are absolutely and uniformly convergent to these functions, is presented and discussed.

math.CV↗

arXiv2014

Gaussian-Perturbative Calculations with a Homogeneous External Source

We derive the equation of the critical curve and calculate the renormalized masses of the $SO(N)$-symmetric $λϕ^{4}$ model in the presence of a homogeneous external source. We do this using the Gaussian-Perturbative approximation on finite lattices and explicitly taking the continuum limit. No disabling divergences are found in the final results, and no renormalization is necessary. We show that the results give a complete description of the critical behavior of the model and of the phenomenon of spontaneous symmetry breaking, at the quantum-field-theoretical level. We show that the renormalized masses depend on the external source, and point out the consequences of that fact for the design of computer simulations of the model. We point out a simple but interesting consequence of the results, regarding the role of the $λϕ^{4}$ model in the Standard Model of high-energy particle physics. Using the experimentally known values of the mass and of the expectation value of the Higgs field, we determine uniquely the values of the bare dimensionless parameters $α$ and $λ$ of the model, which turn out to be small numbers, significantly less that one.

hep-th↗

arXiv2014

On the Sums of Inverse Even Powers of Zeros of Regular Bessel Functions

We provide a new, simple general proof of the formulas giving the infinite sums $σ(p,ν)$ of the inverse even powers $2p$ of the zeros $ξ_{νk}$ of the regular Bessel functions $J_ν(ξ)$, as functions of $ν$. We also give and prove a general formula for certain linear combinations of these sums, which can be used to derive the formulas for $σ(p,ν)$ by purely linear-algebraic means, in principle for arbitrarily large powers. We prove that these sums are always given by a ratio of two polynomials on $ν$, with integer coefficients. We complete the set of known formulas for the smaller values of $p$, extend it to $p=9$, and point out a connection with the Riemann zeta function, which allows us to calculate some of its values.

math-ph↗