Summability Research Articles

Multidimensional gauge theory via summability methods

Kurzweil and Henstock presented the notion of Gauge integral, independently. Using their definition Savas and Patterson examined the relationship between Gauge integral and Summability theory. Because of the esoteric of both Gauge and Summability theory, the body of literature is limited. As such the only accessible notion to both theories is Pringsheim limits. The goal of this paper is to present a natural multidimensional extension of Gauge theory via Summability methods. To accomplish this we examine double measurable real-valued functions of the type of f (x, y) in the Gauge sense on (1,?)?(1,?). Additionally, we introduce the definition of double ?2 ?strongly summable to L with respect to Gauge and present inclusion theorems.

Solving the Inverse Initial Value Problem for the Heat Conductivity Equation by Using the Picard Method

In this work, the inverse initial value problem IVP for the heat equation is formulated and solved. Initial temperature (initial condition) distribution is unknown in this problem, and instead, the temperature spreading at period t= T 0 is assumed. Among mathematical problems, a class of problems is singled out, the solutions of which are unstable to minor variations in the initial information. It is well identified that this problem is ill-posed. In order to solve the direct problem, we has used the separation of variables way. Note that the method of separation of variables is completely inapplicable for solving IVP, since it principals to rather errors, also divergent series. Ivanov V.K. noticed that if the inverse problem IP is solved by the method separation of variables, and then the resulting series is changed by a incomplete sum of the series, in which the term number is depending on δ, N=N(δ), then as a result we obtain a stable approximate solution. The Picard method customs a regularizing family of operators that map space to same space.

Функциональное суммирование рядов

We consider a summation technique which reduces summation of a series to the solution of some linear functional equations. Partial sums of a series satisfy an obvious difference equation. This equation is transformed to the functional equation on the interval [0,1] for the continuous argument. Then this equation is either solved explicitly (to within an arbitrary constant) or an asymptotic expansion of the solution is computed at the origin. The sum of the original series is determined uniquely as a constant needed for the matching of the asymptotic series with partial sums of the original series. The notion of a limit is not involved in this computational technique, which allows summation of divergent series as well.

Simple renormalization schemes for multiple scattering series expansions.

A number of renormalization schemes for improving the convergence of multiple scattering series expansions are investigated. Numerical tests on a small Cu(111) cluster demonstrate their effectiveness, for example increasing the rate of convergence by up to a factor 2 or by transforming a divergent series into a convergent one. These techniques can greatly facilitate multiple scattering calculations, especially for spectroscopies such as photoelectron diffraction, Auger electron diffraction, low energy electron diffraction etc., where an electron propagates with a kinetic energy of hundreds of eV in a cluster of hundreds of atoms.

Direct numerical evaluation of multi-loop integrals without contour deformation

We propose a method for computing numerically integrals defined via i epsilon deformations acting on single-pole singularities. We achieve this without an explicit analytic contour deformation. Our solution is then used to produce precise Monte Carlo estimates of multi-scale multi-loop integrals directly in Minkowski space. We corroborate the validity of our strategy by presenting several examples ranging from one to three loops. When used in connection with four-dimensional regularization techniques, our treatment can be extended to ultraviolet and infrared divergent integrals.

High resolution bolt pre-load looseness monitoring using coda wave interferometry

This paper proposes a new concept, named the Detectable Resolution of Bolt Pre-load (DRBP), by using the coda wave interferometry (CWI) to quantitatively measure the pre-load looseness at a high resolution. Due to its characteristics of roughness, irregularity, and randomly distributed asperities, the contact surface of the bolted components can function as a natural interferometer to scatter the propagation waves. The multiply-scattered coda waves can amplify the slight changes in the travel path and show the visible perturbation in the time domain. By calculating the time-shifted correlation coefficient of coda waves before and after the slight pre-load looseness, the tiny pre-load changes can be clearly revealed. To evaluate the feasibility of the proposed method, a theoretical model considering the time shifts of coda waves and the variations of pre-load is established. Based on the acoustoelastic effect and the wave path summation theory of coda wave interferometry, the model shows that the time shifts of coda waves change linearly with the variations of pre-load. Verification experiments are conducted, and the results show that the R-square values of the fitting curves are larger than 0.9216. In addition, the proposed approach has the feature of high resolution. The ultimate pre-load resolution of the proposed approach is 0.331%, that is, when the variation of pre-load is larger than 0.331%, it can be detected. Therefore, theoretical analysis and experimental results prove that the CWI-based pre-load detection approach holds great potential for the detection of bolt pre-load looseness, especially during the initial stage.

Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

On Green’s function for Laplace’s equation in a rigid tube

A classical problem from potential theory (a point source inside a long rigid tube) is revisited. It has an extensive literature but its resolution is not straightforward: standard approaches lead to divergent integrals or require the discarding of infinite constants. We show that the problem can be solved rigorously using classical methods.

Exact penalties for decomposable convex optimization problems

We consider a general decomposable convex optimization problem. By using right-hand side allocation technique, it can be transformed into a collection of small dimensional optimization problems. The master problem is a convex non-smooth optimization problem. We propose to apply the exact non-smooth penalty method, which gives a solution of the initial problem under some fixed penalty parameter and provides the consistency of lower level problems. The master problem can be solved with a suitable non-smooth optimization method. The simplest of them is the custom subgradient projection method using the divergent series step-size rule without line-search, whose convergence may be, however, rather low. We suggest to enhance its step-size selection by using a two-speed rule. Preliminary results of computational experiments confirm efficiency of this technique.

Hydrodynamic attractor in a Hubble expansion

We analytically investigate hydrodynamic attractor solutions in both M\"{u}ller-Israel-Stewart (MIS) and kinetic theories in a viscous fluid system undergoing a Hubble expansion with a fixed expansion rate. We show that the gradient expansion for the MIS theory and the Chapman-Enskog expansion for the Boltzmann equation within the relaxation time approximation are factorially divergent and obtain hydrodynamic attractor solutions by applying the Borel resummation technique to those asymptotic divergent series. In both theories, we find that the hydrodynamic attractor solutions are globally attractive and only the first-order non-hydrodynamic mode exists. We also find that the hydrodynamic attractor solutions in the two theories disagree with each other when gradients become large, and that the speed of the attraction is different. Similarities and differences from hydrodynamic attractors in the Bjorken and Gubser flows are also discussed. Our results push the idea of far-from-equilibrium hydrodynamics in systems undergoing a Hubble expansion.

Two Forms of An Inverse Operator to the Generalized Bessel Potential

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor–Delsarte series.

Finitely convergent iterative methods with overrelaxations revisited

We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation parameters converge to zero, but with a rate slower than any geometric sequence. Unlike other works in this area, which require divergent series of overrelaxations, our approach allows us to consider some summable series. By employing quasi-Fejérian analysis in the latter case, we obtain additional asymptotic convergence guarantees, even when the interior of the solution set is empty.

Scalar one-loop four-point integral with one massless vertex in loop regularization

The scalar one-loop four-point function with one massless vertex is evaluated analytically by employing the loop regularization method. According to the method, a characteristic scale μ s is introduced to regularize the divergent integrals. The infrared divergent parts, which take the form of and as μ s → 0 where λ is a constant and expressed in terms of masses and Mandelstam variables, and the infrared stable parts are well separated. The result is shown explicitly via 44 dilogarithms in the kinematic sector in which our evaluation is valid.

Divergent Fourier series in function spaces near L1[0;1

Bochkariev's theorem states that for any uniformly bounded orthonormal system Φ, there is a Lebesgue integrable function such that the Fourier series of it with respect to the system Φ diverges on the set of positive measure. In this paper, we extended Bochkariev's theorem for some class of variable exponent Lebesgue spaces. We characterized the class of variable exponent Lebesgue spaces Lp(⋅)[0;1], 1<p(x)<∞ a.e. on [0;1], such that above mentioned Bochkarev's theorem is valid.

The Products of Hyperreal Series and the Limitations of Cauchy Products

Cauchy products are used to take the products of convergent series. Here, we show the limitations of this approach in divergent series, including those that can be analyzed through the BGN method. Alternative approaches and formulas for divergent series are suggested, as well as their benefits and drawbacks.

Theoretical description of the plaquette with exponential accuracy

We review recent studies of the operator product expansion of the plaquette and of the associated determination of the gluon condensate. One first needs the perturbative expansion to orders high enough to reach the asymptotic regime where the renormalon behavior sets in. The divergent perturbative series is formally regulated using the principal value prescription for its Borel integral. Subtracting the perturbative series truncated at the minimal term, we obtain the leading non-perturbative correction of the operator product expansion, i.e., the gluon condensate, with superasymptotic accuracy. It is then explored how to increase such precision within the context of the hyperasymptotic expansion. The results fully confirm expectations from renormalons and the operator product expansion.

Reduction of order, resummation, and radiation reaction

The Landau-Lifshitz equation is the first in an infinite series of approximations to the Lorentz-Abraham-Dirac equation obtained from `reduction of order'. We show that this series is divergent, predicting wildly different dynamics at successive perturbative orders. Iterating reduction of order ad infinitum in a constant crossed field, we obtain an equation of motion which is free of the erratic behaviour of perturbation theory. We show that Borel-Pad\'e resummation of the divergent series accurately reproduces the dynamics of this equation, using as little as two perturbative coefficients. Comparing with the Lorentz-Abraham-Dirac equation, our results show that for large times the optimal order of truncation typically amounts to using the Landau-Lifshitz equation, but that this fails to capture the resummed dynamics over short times.

Effect of temperature on red snow crab Chionoecetes japonicus (Crustacea, Decapoda, Majoidea) larval survival, development, and growth under laboratory conditions

ABSTRACT In the Sea of Japan, the mature red snow crab Chionoecetes japonicus is distributed at a depth of 500–2,700 m. However, its planktonic larvae occur at depths shallower than 300 m. To better understand the factors influencing larval dispersal and settlement of the red snow crab in its natural habitat, we tested the effects of temperatures ranging from 1°C to 18°C on the survival, developmental period, and growth of red snow crab zoeal and megalopal larvae. The mean number of days from hatching to second zoeae and megalopae and from megalopae to the first crab instar was significantly shorter at warmer temperatures. Moreover, the relationships between mean temperatures and larval periods were well described by heat summation theory equations and a nonlinear thermodynamic Sharpe-Schoolfield-Ikemoto (SSI) model. The higher survival rate, intrinsic optimum temperatures in the SSI model, and larger carapace width suggested that the optimum temperatures for the development of zoeal and megalopal stages of the red snow crab are 9–10°C and 7–8°C, respectively. A lower optimum temperature in the megalopal stage than in the zoeal stages was considered an adaptation to facilitate deeper vertical distributions in the megalopal stage.

Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories

We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized field theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum fields. In a first part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as unstable particles. Various problems (mostly open) are finally discussed in connection with concrete models.

Some Problems with the Dirac Delta Function: Divergent Series in Physics

Some Problems with the Dirac Delta Function: Divergent Series in Physics