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Copyright of Environmental & Social Management Journal / Revista de Gestão Social e Ambiental is the property of Environmental & Social Management Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of GeSec: Revista de Gestao e Secretariado is the property of Sindicato das Secretarias e Secretarios do Estado de Sao Paulo (SINSESP) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Allowing the inversion of the operations of summation and integration for trigonometric Fourier series we present the solution by Fourier method of the generalized mixed problem for the homogeneous wave equation with zero initial velocity and fixed ends boundary conditions. The solution has the form of a series converging at an exponential rate. This series converges the classical solution if the latter equists. The results of the article reinforce the previously obtained results.

Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time‐Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high‐order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier–Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time‐series integration even when resummation techniques like the Borel–Padé–Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher‐order FEs. Our approach involves the incorporation of artificial diffusion terms on the left‐hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators. [ABSTRACT FROM AUTHOR]

We investigate the relations between the rings E , G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E -functions), 0 (analytic continuations of G -functions) or 1 (renormalization of divergent series solutions at ∞ of E -operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D , and is the limit at infinity of some E -function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E -functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E -functions) and the conjecture D ∩ E = Q ‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E -functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E. [ABSTRACT FROM AUTHOR]

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series is nowadays beyond dispute. The Weniger transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better-known resummation techniques, like, for instance, Padé approximants. However, our theoretical understanding of Levin-type transformations is still far from being satisfactory and is particularly bad as far as the decoding of factorially divergent series is concerned. The Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the converging factor of any order of typical Stieltjes series can be expressed as an inverse factorial series, whose terms are analytically retrieved through a simple recursive algorithm. A few examples of applications are presented, in order to show the effectiveness and implementation ease of the algorithm itself. We believe that further investigations of the asymptotic forms of the remainder terms, encoded within the converging factors, could eventually lead toward a more general theory of the asymptotic behavior of the Weniger transformation when it is applied to Stieltjes series in high transformation orders. It is a rather ambitious project, which should be worthy of being pursued in the future. [ABSTRACT FROM AUTHOR]

Brazilian university students show interest and engage in entrepreneurial activities, but little is known about the nature, outcomes, and challenges of their ventures. Most studies in the field are predominantly quantitative and do not unveil the details of the entrepreneurial process. This research aimed to understand how the entrepreneurial process unfolded among students in the Administration program at the Federal University of Minas Gerais (UFMG). A qualitative and descriptive approach was employed in this study with four Administration students from UFMG. Data were collected through semi-structured interviews and analyzed using the Organizational Learning 4i Model. As a result, it was possible to observe that all the entrepreneurs in question engaged in entrepreneurship opportunistically, having never undertaken entrepreneurial activities before. However, they were exposed to entrepreneurship within their families and sought training along their journey. Their businesses had at least two founders, operated in the service sector, were financially independent from public and private organizations, generated revenue for at least one year, and most had late or no formalization. This study fulfilled the role of elucidating details and nuances of entrepreneurial initiatives among the students of Administration at the Federal University of Minas Gerais, by revealing a series of common and divergent characteristics among the processes, entrepreneurs, and companies in question. It also facilitated the understanding of these initiatives and can serve as a source of information for the development of future actions to encourage entrepreneurship. [ABSTRACT FROM AUTHOR]

The article is devoted to two volumes of Leonhard Euler’s correspondence with mathematicians and other scientists. The first of these volumes (in two parts) is devoted to correspondence with Christian Goldbach. We consider selected topics from this correspondence reflecting various branches of mathematics and demonstrate, where possible, the connection of the ideas and results presented there with modern mathematical research. The second of these two volumes contains Euler’s correspondence with scholars associated with the University of Halle. These letters, with a small exception, have less mathematical content, but allow to create an impression of academic life at that time.

We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind. [ABSTRACT FROM AUTHOR]

The role of a parent is important in promoting the development of character, behaviour, discipline, resilience, academic success, social skills, emotional skills, and coping skills in children. Positive parenting styles have far-reaching benefits for children, families, communities and society, and negative styles have reverse effects. This research paper conducts a comparative analysis of the parenting styles depicted in Veronica Roth's Divergent series, focusing on Marcus as an authoritarian parent of Four and Natalie as an authoritative parent of Tris. Through the lens of Diana Baumrind's theory of parenting styles, this study examines the behaviours, interactions, and effects of these contrasting parenting approaches on the characters' development, relationships, and overall well-being. By examining the impact of these parenting styles on the characters' development, this research paper provides insights into the influence of authoritarian and authoritative parenting within the framework of Baumrind's theory. It contributes to a deeper understanding of the complex interplay between parenting styles and individual outcomes. [ABSTRACT FROM AUTHOR]

This experimental research investigates the creativity of anonymous video‐based groups through a series of collaborative divergent and convergent thinking stages of innovation. A sample of 113 undergraduate students participated in small groups of three or four to complete a creative problem‐solving task in either anonymous or identified conditions. The findings revealed that although anonymity did not positively impact the divergent thinking task of brainstorming, it significantly enhanced creativity in the convergent thinking process of collaborative idea refinement. Furthermore, the ideas in the refinement stage were higher in originality, elaboration, overall quality, and good quality compared with the generation and selection task. The participants' attentional focus was critical in creating high‐quality ideas by mediating the path from collaborative ideation to refinement. The current study contributes to the creativity literature by highlighting the importance of anonymous collaborative idea refinement as the teams proceed through multiple stages of convergent and divergent thinking processes. [ABSTRACT FROM AUTHOR]

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series is nowadays beyond dispute. The Weniger transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better-known resummation techniques, like, for instance, Padé approximants. However, our theoretical understanding of Levin-type transformations is still far from being satisfactory and is particularly bad as far as the decoding of factorially divergent series is concerned. The Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the converging factor of any order of typical Stieltjes series can be expressed as an inverse factorial series, whose terms are analytically retrieved through a simple recursive algorithm. A few examples of applications are presented, in order to show the effectiveness and implementation ease of the algorithm itself. We believe that further investigations of the asymptotic forms of the remainder terms, encoded within the converging factors, could eventually lead toward a more general theory of the asymptotic behavior of the Weniger transformation when it is applied to Stieltjes series in high transformation orders. It is a rather ambitious project, which should be worthy of being pursued in the future.

We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension forces compete to affect the interface shape as characterized by the Bond number. Here, we provide a convergent power series solution for interface shapes that rise above or fall below the horizontal pool as a function of contact angle and Bond number. We find that the power series solution expressed in terms of the radial distance from the wall is divergent, and thus rewrite the divergent series as a new power series expressed as powers of an Euler transformed variable; this series is modified to match the large distance asymptotic behaviour of the meniscus. The Euler transformation maps non-physical singularities to locations that do not restrict series convergence in the physical domain, while the asymptotic modification increases the rate of convergence of the series overall. We demonstrate that when the divergent series coefficients are used to implement the Euler transformation, finite precision errors are incurred, even for a relatively small number of terms. To avoid such errors, the independent variable in the governing differential equation is changed to that of the Euler transform, and the power series is developed directly without using the divergent series. The resulting power series solution is validated by comparison with a numerical solution of the interface shape and the small Bond number matched asymptotic solution for the height of the interface along the cylinder developed by Lo (1983, J. Fluid Mech. 132, p.65-78). The convergent power series expansion has the ability to exceed the accuracy of the matched asymptotic solution for any Bond number given enough terms, and the recursive nature of the solution makes it straightforward to implement. [ABSTRACT FROM AUTHOR]

We devise a prescription to use a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations (Galapon EA. 2017 Proc. R. Soc A473, 20160567) to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments, which we treated as Hadamard's finite part integrals. The result allowed us to compute the ground state energy of the quartic, sextic anharmonic oscillators as well as the PT symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behaviour of the spectra. [ABSTRACT FROM AUTHOR]

We investigate a puzzle involving the winning probabilities in a duel of two players. The problem of calculating limiting probabilities leads to the summation of a divergent infinite series. The solution admits a generalization that applies to a wide class of duels. [ABSTRACT FROM AUTHOR]

We make use of Borel resummation to extract the exact time dependence from the divergent series found in the context of stochastic inflation. Correlation functions of self-interacting scalar fields in de Sitter spacetime are known to develop secular IR divergences via loops, and the first terms of the divergent series have been consistently computed both with standard techniques for curved spacetime quantum field theory and within the framework of stochastic inflation. We show that Borel resummation can be used to interpret the divergent series and to correctly infer the time evolution of the correlation functions. In practice, we adopt a method called Borel-Padé resummation where we approximate the Borel transformation by a Padé approximant. We also discuss the singularity structures of Borel transformations and mention possible applications to cosmology. [ABSTRACT FROM AUTHOR]

This paper is devoted to mixed problems for the heat equation and the Schrödinger equation of the simplest form, involving divergent series in the sense of Euler. In addition, we consider two divergent series of cosines associated with the indicated mixed problems. The sums of these series are obtained using the theory of generalized functions, and thus open up new ways of using generalized functions in mixed problems. [ABSTRACT FROM AUTHOR]

The article titled "Mathematical Singularities at the Edge of the Universe (and a Brief Approach to its History)" by Emilio Elizalde, published in the Spanish Journal of Physics, addresses the importance of avoiding sensational terms and demystifying concepts such as "infinity" or "singularity" when describing the Cosmos. The singularity of the Big Bang is mentioned as an unfathomable concept but more precise than the biblical version. The text deals with the mathematical study of continuous change and the dense distribution of real numbers in mathematical continuity. It also discusses the abstraction of mathematical concepts such as points, lines, and planes, and questions the existence of point particles in physics. The importance of the zeta function in obtaining precise results in the physical world from divergent infinite series is highlighted. Additionally, the importance of regularization and renormalization in quantum field theories is mentioned. On the other hand, it discusses the origin and evolution of the universe, mentioning the Big Bang theory, inflation, and dark energy. The formation of galaxies and cosmic structures over time is also mentioned, and current and future missions that will help understand the nature of the cosmos, such as the Planck mission and the Euclid mission, are mentioned. [Extracted from the article]

Resummation methods using continued functions are implemented to converge divergent series appearing in perturbation problems related to continuous phase transitions in field theories. In some cases, better convergence properties are obtained using continued functions than diagonal Padé approximants, which are extensively used in literature. We check the reliability of critical exponent estimates derived previously in universality classes of O(n)-symmetric models (classical phase transitions) and Gross–Neveu–Yukawa models (quantum phase transitions) using new methods. [ABSTRACT FROM AUTHOR]

Recent Continental European scholarship has identified a problematic Anglocentric bias running through the field of comparative plural policing studies. It has sought to counter this bias by emphasizing a series of divergent plural policing trajectories between the more market-friendly countries in the Anglosphere and the more state-centric countries in Continental Europe. While acknowledging the significance of this corrective, we argue that it tends to overemphasize the levels of divergence between these two regions. We substantiate this claim by examining the rise of the private security industry and its regulation by the state in the UK (representing the Anglosphere) and Belgium (representing Continental Europe). Interpreting historical and contemporary data through Sabatier and Weible's advocacy coalition framework, which focuses on the cut and thrust of democratic politics, we observe how in both countries this important dimension of the plural policing landscape is characterized not by counterposed market-friendly and state-centric trajectories, but rather by a complex mix of state–market interactions. In other words, the dynamics of private security regulation are more state-centric in the UK and more market-friendly in Belgium than recent Continental European scholarship suggests. Moreover, we illustrate how, under conditions of post-financial crisis austerity, the overarching pattern is, if anything, one of convergence towards a common set of political dynamics. This is an important finding that not only makes an original contribution towards private security regulation scholarship but also encourages us to question the nature of Anglocentric bias within comparative plural policing studies. [ABSTRACT FROM AUTHOR]

Previous attempts to the resummation of divergent power series by means of analytic continuation are improved applying the Cauchy integral formula for complex functions. The idea is tested on divergent Møller-Plesset perturbation expansions of the electron correlation energy. In particular, the potential curve of the LiH molecule is computed from single reference MPn results which are divergent for bond distances larger than 3.6 Å. Preliminary results for the Hartree-Fock molecule are also tabulated. [ABSTRACT FROM AUTHOR]

With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series which corresponds to the simplest mixed problem with a summable initial function. This result helps to get the solution of the generalized mixed problem for a non-homogeneous equation under the assumption that non-homogeneity is characterized by a locally summable function. As an application, the mixed problem with a non-zero potential is considered, in which the differential equation is treated quite formally but the mixed problem itself is no longer a generalized one: instead of the formal solution of the separating variables method we get an integral equation which can be solved by the successive substitutions method. Thus we essentially simplify the arguments.

We consider transverse shear flow over a $2L$ -periodic array of air-filled grooves, which are separated by solid slats. We address the scenario where the liquid partially invades the grooves. Following Crowdy's solution of the longitudinal problem (J. Fluid Mech. vol. 925, 2021, R2), we focus upon the simplified geometry of infinitely narrow slats. For flat menisci, the ratio $\lambda$ of the slip length to $L$ depends upon a single geometric parameter, namely the ratio $H$ of the invasion depth to $L$. We analyse the singular limit $H\ll 1$ using matched asymptotic expansions, with an outer region on the scale of the period and an inner region on the scale of a single slat. In the outer region, we employ the complex-variable formulation of Stokes flows, obtaining the complex velocity as a divergent series. This formal representation is transformed to an admissible closed-form expression. In the inner region, the problem is equivalent to that of Stokes flow past a finite line segment. Asymptotic matching yields $\lambda = [\ln (2/{\rm \pi} H)-1]/{\rm \pi}$. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized discrete Lotka-Volterra equation and explore its connections with symmetric orthogonal polynomials, Hankel determinants and convergence acceleration algorithms. Firstly, we extend the fully discrete Lotka-Volterra equation to a generalized one with a sequence of given constants { u 0 (n) } and derive its solution in terms of Hankel determinants. Then, it is shown that the discrete equation of motion is transformed into a discrete Riccati system for a discrete Stieltjes function, hence leading to a complete linearization. Besides, we obtain its Lax pair in terms of symmetric orthogonal polynomials by generalizing the Christoffel transformation for the symmetric orthogonal polynomials. Moreover, a generalization of the famous Wynn's ε-algorithm is also derived via a Miura transformation to the generalized discrete Lotka-Volterra equation. Finally, the numerical effects of this generalized ε-algorithm are discussed by applying to some linearly, logarithmically convergent sequences and some divergent series. [ABSTRACT FROM AUTHOR]

We introduce a novel class of non-linear transformations of sequences that can accelerate both linear and logarithmic convergence and perform the summation of divergent series. Those transformations naturally arise when the ratio of successive forward differences of the original sequence are approximated by rational functions. The transformed sequences are thus expressible in terms of generalized hypergeometric functions, whose parameters depend on the original sequence. We explicitly derive analytical expressions for the lowest order transformations, present an alternative derivation using continued fractions, investigate convergence and accelerative properties, and derive the kernels of the transformations. The lowest order transformation is equivalent to Aitken's Δ2 process, while, in the following order, the transformation specialized to logarithmic convergence is equivalent to Lubkin-W transformation. Numerical examples that illustrate the power and limitations of iterations or combinations of hypergeometric transformations are also investigated. [ABSTRACT FROM AUTHOR]

In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Even if the case that really matters physically is an analytic inhomogeneity, we systematically examine here the cases where the inhomogeneity is s-Gevrey for any s ⩾ 0 , in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy: for any s ⩾ 1 , the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < 1 , the formal solutions are generically 1-Gevrey. In the latter case, we give in particular an explicit example in which the formal solution is s ′ -Gevrey for no s ′ < 1 , that is exactly 1-Gevrey. Then, we give a necessary and sufficient condition under which the formal solutions are 1-summable in a given direction arg (t) = θ. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results. [ABSTRACT FROM AUTHOR]

This article presents a pedagogical introduction to resurgent transseries in physics. Transseries are asymptotic series that describe non-perturbative expansions generated by asymptotically divergent series. The article discusses the formal definition of an asymptotic series, the Borel summation process for asymptotically divergent series, and the phenomenon of Stokes, which gives rise to transseries. Additionally, an example of a resurgent transseries is presented in the calculation of the non-perturbative energy spectrum of a particle confined to a double quantum well potential. [Extracted from the article]

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. [ABSTRACT FROM AUTHOR]

The two most famous quotes about divergent series are Abel's "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever," and Heaviside's "This series is divergent, therefore we may be able to do something with it." Today a lot more is known about divergent series than in either's day, so we can say now that, on balance, Heaviside wins, and we now have plenty of license to use divergent series. This article talks about some "well-known" methods (that is, well-known to experts) to do so, and in particular talks about some of the devilishly good features of evalf/Sum, long one of my favourite tools in Maple. But Abel had a point, too, and we'll see some "shameful" things, which will give the reader some necessary caution to go along with their license. [ABSTRACT FROM AUTHOR]

Copyright of Revista EntreLínguas is the property of Revista EntreLinguas and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We introduce a theory of probabilistic renormalization for series, the renormalized values being encoded in the expectation of a certain random variable on the set of natural numbers. We identify a large class of weakly renormalizable series of Dirichlet type, whose analysis depends on the properties of a (infinite order) difference operator that we call Bernoulli operator. For the series in this class, we show that the probabilistic renormalization is compatible with analytic continuation. The general zeta series for s ≠ 1 is found to be strongly renormalizable and its renormalized value is given by the Riemann zeta function. [ABSTRACT FROM AUTHOR]

In the paper, the theorems implying the existence of a Rademacher series, convergent to each real-valued function, piecewise continuous over (0, 1) on certain dense subsets of (0, 1), are announced. The set of all Rademacher series with the above-mentioned property is fully described. Among the elements of this set are both almost everywhere convergent and almost everywhere divergent Rademacher series. [ABSTRACT FROM AUTHOR]

Under minimal conditions on the initial data of the mixed problem for the telegraph equation, a generalized solution is obtained in two different ways in the form of a rapidly converging function series—an analog of the well-known d'Alembert formula. The first approach is based on the sequential method: a generalized solution of the problem is determined as the limit of classical solutions of a sequence of problems. The second approach is based on the axiomatic method. Classical solutions are not involved in constructing a generalized solution. Euler's theory of divergent series with an augmented system of axioms is used. The specific feature of the problem under consideration is the presence of a nonlocal boundary condition in which the value of the function at an interior point of the interval occurs. Both approaches to constructing a generalized solution lead to the same rapidly (exponentially) converging function series. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Chloropupukeananin is a natural product that inhibits HIV-1 replication and has antitumor activity. Its structure consists of a chlorinated tricyclo[4.3.1.03,7]decane core skeleton with an array of highly oxidized multifunctional groups. In the biosynthesis of chloropupukeananin, (+)-iso-A82775C and (−)-maldoxin are employed as biosynthetic precursors for the intermolecular Diels–Alder and carbonyl–ene reactions, followed by the migration of the p-orcellinate group. Chloropupukeanolides and chloropestolides are intermediates and isomers in biosynthesis; their unique chemical structures and biosynthetic pathways have attracted significant attention from synthetic chemists. In this review, I present the synthetic studies on chloropupukeananin and its related compounds that have been conducted thus far. [ABSTRACT FROM AUTHOR]

Oligomeric ellagitannins are challenging synthetic targets due to the need for an abundant supply of their composed monomeric ellagitannins and a synthetic methodology to connect them. This work focused on the divergent synthesis of the four monomeric ellagitannins from a common intermediate as a step toward the total synthesis of nobotanin K, a class of compounds that includes oligomeric ellagitannins and were isolated in plants belonging to the Melastomataceae family. Implementing our method, the four natural products could be easily supplied, suggesting that through this novel route, the total synthesis of nobotanin K could be achieved smoothly. [ABSTRACT FROM AUTHOR]

A new way of constructing a rapidly converging series is presented. The series is a generalized solution to a mixed problem for a telegraph equation considered in a half-strip. The case of an essentially non-self-adjoint operator with respect to the spatial variable is considered. To construct the solution, we apply the axiomatic model of A.P. Khromov based on the active participation of divergent series. Euler's ideas of assigning a sum to a divergent series are used. Variable separation, which usually results in slowly converging series, generally cannot be applied to the considered problem. An analog of Lebesgue's theorem on the integration of trigonometric Fourier series for the considered biorthogonal series is proved. [ABSTRACT FROM AUTHOR]

For the quantum quartic anharmonic oscillator with the Hamiltonian H = 1 2 (p 2 + x 2) + λ x 4 , which is one of the traditional quantum-mechanical and quantum-field-theory models, the summation of its factorially divergent perturbation series is studied on the basis of the proposed method of the averaging of the corresponding Padé approximants. Thus, applying proper averaging weight function, we are able for the first time to construct the Padé-type approximations that possess correct asymptotic behavior at infinity with a rise of the coupling constant λ. The approach gives very essential theoretical and applicatory-computational advantages in applications of the given method. The convergence of the utilized approximations is studied and the values for the ground state energy E 0 (λ) of the anharmonic oscillator are calculated by the proposed method for a wide range of variation of the coupling constant λ. In addition, we perform comparative analysis of the proposed method with the modern Weniger delta-transformation method and show insufficiency of the latter to sum the divergent perturbation series in the region of the superstrong coupling λ ≳ 5. [ABSTRACT FROM AUTHOR]

One of the most often used methods of summing divergent series in physics is the Borel-type summation with control parameters improving convergence, which are defined by some optimization conditions. The well known annoying problem in this procedure is the occurrence of multiple solutions for control parameters. We suggest a method for resolving this problem, based on the minimization of cost functional. Control parameters can be introduced by employing the Borel–Leroy or Mittag–Leffler transforms. Also, two novel transformations are proposed using fractional integrals and fractional derivatives. New cost functionals are advanced, based on lasso and ridge selection criteria, and their performance is studied for a number of models. The developed method is shown to provide good accuracy for the calculated quantities. [ABSTRACT FROM AUTHOR]

A method is described for the extrapolation of perturbative expansions in powers of asymptotically small coupling parameters or other variables onto the region of finite variables and even to the variables tending to infinity. The method involves the combination of ideas from renormalization group theory, approximation theory, dynamical theory, and optimal control theory. The extrapolation is realized by means of self-similar factor approximants, whose control parameters can be uniquely defined. The method allows to find the large-variable behavior of sought functions knowing only their small-variable expansions. Convergence and accuracy of the method are illustrated by explicit examples, including the so-called zero-dimensional field theory and anharmonic oscillator. Strong-coupling behavior of Gell-Mann–Low functions in multicomponent field theory, quantum electrodynamics, and quantum chromodynamics is found, being based on their weak-coupling perturbative expansions. • A method of extrapolating small-variable expansions to their finite values is described. • The method allows for finding strong coupling limits from weak-coupling expansions. • The strong-coupling extrapolation of Gell-Mann–Low functions is accomplished. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is to propose a closed expression for the Ramanujan constant of alternating series, based on the Euler–Boole summation formula. Such an expression is not present in the literature. We also highlight the only choice for the parameter a in the formula proposed by Hardy for a series of positive terms, so the value obtained as the Ramanujan constant agrees with other summation methods for divergent series. Additionally, we derive the closed-formula for the Ramanujan constant of a series with the parameter chosen, under a natural interpretation of the integral term in the Euler–Maclaurin summation formula. Finally, we present several examples of the Ramanujan constant of divergent series. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour of the solution to a singularly perturbed model differential equation. The analytical solution is asymptotically represented by a formal power series in the perturbation parameter. This paper provides an explanation of the relation between the pre‐factor functions P(z) and Q(z) and the singulants of the expansion of the general second‐order singularly perturbed differential equation in higher levels of the asymptotic analysis. Exponential asymptotics typically relies upon the derivation of the singulant function as it is crucial to determine the location of the Stokes lines. By doing the careful analysis, we connect the subsequent order singulants with the previous order singulants and determine the asymptotics of the late coefficients. By this relation, one can predict and investigate the Stokes phenomenon and exponential smoothing of the equations in the form of the model differential equation. [ABSTRACT FROM AUTHOR]

Pascal's triangle is the simplest triangle containing binomial coefficients and possessing exclusive mathematical properties in it. There are several triangles of numbers exhibiting exotic mathematical oddities. In this paper, we will introduce two number triangles and determine Ramanujan summation related to their entries. In particular, we have proved new results related to Ramanujan summation values of series of odd and even powers of numbers considered from the two number triangles. [ABSTRACT FROM AUTHOR]

The classical problem on construction of continuous iterations of an analytical map is considered as a problem on construction of invariant curves of discrete dynamical systems. Such systems are often studied as reductions of continuous dynamical systems (Poincarè map). The existence of analytical invariant curves in a discrete dynamical system implies (locally) the existence of an additional analytical first integral in the continuous dynamical system. However, the proofs of existence of such integrals are extremely rare, since these proofs are usually based on convergence of formal power series representing these curves. We give some examples of discrete dynamical systems invariant curves of which are given by a fortiori divergent series but are analytical nonetheless. In particular, we give an example of an integrable discrete dynamical system which has chaotic trajectories. [ABSTRACT FROM AUTHOR]

The completeness relation for the eigenfunctions of a self-adjoint operator generally involves a divergent series or integral. In this paper, we show, using the eigenfunctions of the infinite square well as an example, that these divergent objects can be interpreted as distributions. This should be obvious since the right-hand side of these completeness relations is the Dirac delta function but the direct calculation of the right-hand side can be very laborious, but instructive. [ABSTRACT FROM AUTHOR]