Explicit Series Research Articles

Action-angle Variables for Axisymmetric Potentials via Birkhoff Normalization

We describe a method for calculating action-angle (AA) variables in axisymmetric galactic potentials using Birkhoff normalization, a technique from Hamiltonian perturbation theory. An advantageous feature of this method is that it yields explicit series expressions for both the forward and inverse transformations between the AA variables and position–velocity data. It also provides explicit expressions for the Hamiltonian and dynamical frequencies as functions of the action variables. We test this method by examining orbits in a Milky Way model potential and compare it to the popular Stäckel approximation method. When vertical actions are not too large, the Birkhoff normalization method achieves fractional errors smaller than a part in 103 and outperforms the Stäckel approximation. We also show that the range over which Birkhoff normalization provides accurate results can be extended by constructing Padé approximants from the perturbative series expressions developed with the method. Numerical routines in Python for carrying out the Birkhoff normalization procedure are made available.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Homotopy analysis of explicit series solutions in a modified Nosé-Hoover oscillator

Homotopy analysis of explicit series solutions in a modified Nosé-Hoover oscillator

Duan Rach approach solution of MHD flow of ternary hybrid Casson magnetized nanofluid in converging/diverging geometry with velocity slip effects

This study explores the impact of ternary hybrid nanofluid and velocity slip on the Non-Newtonian fluid flow between two nonparallel plates. Additional influence of an applied magnetic field is considered with Casson fluid (Blood) as a base fluid. Both analytic and numerical techniques are employed to comprehensively analyze the system’s behavior under these combined effects. The governing partial differential equations (PDEs) governing continuity and momentum are simplified into a system of ordinary differential equations (ODEs) through similarity transformations. To obtain an explicit series solution for the resulting problem, we employ the Duan Rach approach (DRA). Subsequently, we conduct a thorough analysis of the solution’s convergence to ensure its reliability and accuracy. The same problem is also solved by using the Runge-Kutta-Fehlberg 4th–5th order approach featuring shooting technique and an excellent agreement is observed between the two sets of results. A comparison was made between the results obtained from this investigation in particular cases and the results obtained via the HAM-based Mathematica package for validation. Influence of various parameters of practical importance on the velocity and temperature profiles is studied and portrayed graphically. Values of skin friction coefficient are tabulated by assigning different values to various emerging parameters. It is found that the velocity slip parameter has a positive value in a converging channel, it implies that there is a slip between the fluid and the channel wall, and the slip velocity is directed in the same direction as the flow. Hence velocity increases. Also, results obtained reveal that for positive value of β + velocity gets high in converging case and lower velocity can be seen in diverging geometry. Same behavior is seen for β − .

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Explicit and approximate series solutions for nonlinear fractional wave-like differential equations with variable coefficients

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Algebraicity modulo p of generalized hypergeometric series [formula omitted

Let f(z)=nFn−1(α,β) be the hypergeometric series with parameters α=(α1,…,αn) and β=(β1,…,βn−1,1) in (Q∩(0,1])n, let dα,β be the least common multiple of the denominators of α1,…,αn, β1,…,βn−1 written in lowest form and let p be a prime number such that p does not divide dα,β and f(z)∈Z(p)[[z]]. Recently in [11], it was shown that if for all i,j∈{1,…,n}, αi−βj∉Z then the reduction of f(z) modulo p is algebraic over Fp(z). A standard way to measure the complexity of an algebraic power series is to estimate its degree and its height. In this work, we prove that if p>2dα,β then there is a nonzero polynomial Pp(Y)∈Fp(z)[Y] having degree at most p2nφ(dα,β) and height at most 5n(n+1)!p2nφ(dα,β) such that Pp(f(z)modp)=0, where φ is the Euler's totient function. Furthermore, our method of proof provides us a way to make an explicit construction of the polynomial Pp(Y). We illustrate this construction by applying it to some explicit hypergeometric series.

Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators

The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ∆h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ∆h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ∆h are also proved by demonstrating that the ∆h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ∆h bivariate Appell polynomials is provided, and symmetric identities for the ∆h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ∆h bivariate Appell polynomials. Additionally, generating relations for the ∆h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials.

The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach

This work scrutinizes the well-known nonlinear non-classical Sobolev-type wave model which addresses the fluid flow via fractured rock, thermodynamics and many other fields of modern sciences. The nonlinear non-classical Sobolev-type wave model provides a more comprehensive and accurate description of wave phenomena in a wide range of fields. By incorporating both nonlinearity and the complexities of dispersive waves, these models enhance our understanding of natural phenomena and enable more precise predictions and applications in various scientific and engineering disciplines. Therefore, this study is investigating it. Prior to this study, no previous research has performed Lie symmetry analysis and achieved invariant solutions of this kind. The symmetry generators are taking into account the Lie invariance criteria. The suggested approach produces the three dimensional Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy, respectively and the other symmetries are scaling or dilation. The nonlinear non-classical Sobolev-type wave partial differential equation is transformed into a system of highly nonlinear ordinary differential equations by employing appropriate similarity transformations through using Lie group methodology. The power series technique is used to generate exact wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The graphical behaviour of certain solutions is demonstrated in 3-D and 2-D for particular quantities of the physical factors in the investigated equation.

Thermal form-factor expansion of the dynamical two-point functions of local operators in integrable quantum chains

Evaluating a lattice path integral in terms of spectral data and matrix elements pertaining to a suitably defined quantum transfer matrix, we derive form-factor series expansions for the dynamical two-point functions of arbitrary local operators in fundamental Yang–Baxter integrable lattice models at finite temperature. The summands in the series are parameterised by solutions of the Bethe Ansatz equations associated with the eigenvalue problem of the quantum transfer matrix. We elaborate on the example of the XXZ chain for which the solutions of the Bethe Ansatz equations are sufficiently well understood in certain limiting cases. We work out in detail the case of the spin-zero operators in the antiferromagnetic massive regime at zero temperature. In this case the thermal form-factor series turn into series of multiple integrals with fully explicit integrands. These integrands factorize into an operator-dependent part, determined by the so-called Fermionic basis, and a part which we call the universal weight as it is the same for all spin-zero operators. The universal weight can be inferred from our previous work. The operator-dependent part is rather simple for the most interesting short-range operators. It is determined by two functions ρ and ω for which we obtain explicit expressions in the considered case. As an application we rederive the known explicit form-factor series for the two-point function of the magnetization operator and obtain analogous expressions for the magnetic current and the energy operators.

The computation of Lie point symmetry generators, modulational instability, classification of conserved quantities, and explicit power series solutions of the coupled system

The well-known Chaffee–Infante reaction hierarchy is examined in this article along with its reaction–diffusion coupling. It has numerous variety of applications in modern sciences, such as electromagnetic wave fields, fluid dynamics, high-energy physics, ion-acoustic waves in plasma physics, coastal engineering, and optical fibres. The physical processes of mass transfer and particle diffusion might be expressed in this way. The Lie invariance criteria is taken into consideration while we determine the symmetry generators. The suggested approach produces the six dimensional Lie algebra, where translation symmetries in space and time are associated to mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. Additionally, similarity reductions are performed, and the optimal system of the sub-algebra should be quantified. There are an enormous number of exact solutions can construct for the traveling waves when the governing system is transformed into ordinary differential equations using the similarity transformation technique. The power series approach is also utilized for ordinary differential equations to obtain closed-form analytical solutions for the proposed diffusive coupled system. The stability of the model under the limitations is ensured by the modulation instability analysis. The reaction diffusion hierarchy’s conserved vectors are calculated using multiplier methods using Lie Backlund symmetries. The acquired results are presented graphically in 2-D and 3-D to demonstrate the wave propagation behavior.

Walks avoiding a quadrant and the reflection principle

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.We expect a similar algebraicity phenomenon to hold for the seven Weyl step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.

Explicit Exact Solutions of Nonlinear Transient Thermal Models of a Porous Moving Fin using Laplace transform - Exp-function method

The present paper develops explicit and non-power series exact solutions to the nonlinear heat transfer models of conductive-radiative-convective moving-porous fin using Laplace transform - Exp-function method, is presented. The developed solutions are employed for investigation of the included parameters on the transient and steady states studies of the moving-porous fin. The results submitted that the fin temperature is augmented with time increase due to increase heat transfer rate as time progresses. However, thermal parameter of the fin reduces from the its base to its tip. As the porosity, moving, convective-conductive-radiative parameters are increased, the fin temperature are decreased due to increased heat transfer rate. The opposite trend is displayed for the conductive-radiative number. It can be stated that present work will be useful in the analysis of the device.

Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Approximate symmetry of time-fractional partial differential equations with a small parameter

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.

AVERAGED CIRCULAR SPATIAL RESTRICTED THREE-BODY PROBLEM: INTERNAL CASE, NEW RESULTS

We investigate the spatial restricted circular three-body problem in the nonresonant case. Namely, we apply Gaussian averaging to obtain averaged equations in terms of osculating elements and then investigate them. Keplerian ellipse with a focus in the main body (the Sun) is taken as an unperturbed orbit assuming the semi-major axis of the ellipse to be less than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula we have derived the twice-averaged perturbed force function of the problem in the form of an explicit analytical series with coefficients expressed in terms of the Gauss and Clausen hypergeometric functions. An investigation of averaged force function along it’s curves of non-analyticity showed that the series are asymptotic by Poincaré. For a reduced system with one degree of freedom, phase portraits of oscillations in the plane of the Keplerian elements \(e\), \(\omega \) are constructed in the second and fourth approximations. It is shown the topology of the phase portrait is more complicated in fourth approximation then in first and second approximations provided that the constant \({{c}_{1}}\) of the Lidov–Kozai integral belongs to the interval (0, 0.382).

Linking Biodiversity and Human Wellbeing in Systematic Conservation Assessments of Working Landscapes

Systematic land use planning to address environmental impacts does not typically include human health and wellbeing as explicit inputs. We tested the effects of including issues related to human health, ecosystem services, and community wellbeing on the outputs of a standard land use planning process which is primarily focused on environmental variables. We consulted regional stakeholders to identify the health issues that have environmental links in the Sacramento, California region and to identify potential indicators and datasets that can be used to assess and track these issues. Marxan planning software was used to identify efficient land use patterns to maximize both ecological conservation and human health outcomes. Outputs from five planning scenarios were compared and contrasted, resulting in a spatially explicit series of tradeoffs across the scenarios. Total area required to meet imputed goals ranged from 10.4% to 13.4% of the total region, showing somewhat less efficiency in meeting biodiversity goals when health outcomes are included. Additionally, we found 4.8% of residential areas had high greening needs, but this varied significantly across the six counties. The work provides an example of how integrative assessment can help inform management decisions or stakeholder negotiations potentially leading to better management of the production landscapes in food systems.

Symmetry group analysis of several coupled fractional partial differential equations

Under investigations were several coupled time–space fractional integrable equations which are called the (1+1)-dimensional KdV-type equations, the (2+1)-dimensional Burgers equations and the (3+1)-dimensional unsteady Euler equations of gas dynamics. In this article, these above considered time–space fractional models through the symmetry analysis approach were explored. As the most basic results, symmetries of these researched equations, were obtained. With the help of the multiple-parameters Erdélyi-Kober fractional operators, they can be reduced into lower dimensional systems. We noticed that this reduction process was somewhat different from the integer case. Besides, we also obtained the explicit power series solution and analyze the convergence to it. By considering the new fractional conservation theorem, conserved quantity of time and space of these time–space fractional systems, were also obtained.

Explicit solutions for fractional Chaffee–Infante reaction–diffusion coupled hierarchy system with conservation laws

This research investigates the explicit series solutions of the fractional Chaffee–Infante reaction–diffusion coupled hierarchy system utilizing Lie symmetry analysis in two‐dimensional space. This fractional system is a prominent model in fluid dynamics to characterize the surface‐stripe wave evolution of liquids in light of the theory of conservation laws. The governing fractional system is analytically solved by the well‐defined technique, called the power series method, so that the total derivative used is based on the Riemann–Liouville derivative. Simulations are also provided in this direction. The acquired results are systematically validated through a series of test cases. The results show that the proposed method is robust, accurate, and flexible in gaining more solutions, and it is effective in treating various fractional physical phenomena within the framework of conservation laws as well.

Interior Whiteness: Race and the "Rise of the Novel"

Interior Whiteness:Race and the "Rise of the Novel" Sarah Vandegrift Eldridge Critical whiteness studies in the German context has thus far not examined the literature of the Goethezeit with the same detail as the philosophy of the period or literature and culture of later eras. This discipline also frequently draws on foundational work from the Anglo-American context; Julia Roth remarks that "most German texts on Critical Whiteness relate themselves (at least on the margins) to Toni Morrison's publications."1 Indeed, although Toni Morrison's 1992 Playing in the Dark: Whiteness and the Literary Imagination is concerned specifically with how an "Africanist presence" emerges in and defines literature by white American authors in the nineteenth and twentieth centuries, she poses a series of questions and calls for further examination that I see as entirely applicable to other (pre)national literatures and periods as well.2 In her preface, Morrison questions the relationship between race and notions of humanistic or universal literature: "How is 'literary whiteness' and 'literary blackness' made, and what is the consequence of that construction? How do embedded assumptions of racial (not racist) language work in the literary enterprise that hopes and sometimes claims to be 'humanistic?'" (xii–xiii) Later in the text, she offers an explicit series of calls for projects that she views as essential to understanding literary whiteness. Morrison asks, "In what ways does the imaginative encounter with Africanism enable white writers to think about themselves?"3 She then elaborates on this question by outlining multiple areas for future study: We need studies of the ways in which an Africanist character is used to limn out and enforce the invention and implications of whiteness. We need studies that analyze the strategic use of black characters to define the goals and enhance the qualities of white characters. … We need to analyze the manipulation of the Africanist narrative … as a means of meditation—both safe and risky—of one's own humanity.4 Although she couches these projects in terms of a specifically Africanist presence in American literature, I take these needs to apply to racial others in white cultural traditions in other linguistic contexts. I also take them to require attention from white scholars, of whom I am one, as well as BIPOC scholars inside and outside academia—following the lead of Black feminist writers in multiple national traditions, white scholars need to turn a careful eye on white scholarly positioning and practices, as well as white disciplinary [End Page 125] histories.5 In this reflection I outline one avenue for such work in German studies: the conceptions of selfhood emerging in the eighteenth century and still shaping the genre and scholarship of the novel today. In the German context, the discipline of philosophy has undertaken consideration of the question of race in the eighteenth century, perhaps because philosophy makes some of the most explicit claims to universality of the humanistic disciplines.6 Emmanuel Chukwudi Eze traces how Kant insists throughout his anthropological writings that the inherent nature of nonwhite races makes them incapable of full, rational humanity. That is, "for Kant, then, skin color encodes and codifies the 'natural' human capacity for reason and rational talents," and skin color, for Kant is not "merely a physical characteristic," but rather "evidence of an unchanging and unchangeable moral quality."7 Kant's overall conception of human nature means that racial categories and their moral implications are absolute and essential.8 Robert Bernasconi, Jennifer Mensch, and Susan Shell have all built on Eze's work to discuss further details of Kant's racialized anthropology, the relationship between Kant and Johann Friedrich Blumenbach, and the connection between Kant's anthropology and his aesthetics.9 Peggy Piesche also points out that Kant's racial anthropology prompts him to de facto remove whites from his racial hierarchies "and place them in a sort of neutrality"—a supposed neutrality that also marks the white protagonists of literary texts of the later eighteenth century but is still contested in its earlier decades.10 Susan Buck-Morss's Hegel, Haiti, and Universal history argues that in formulating his master-slave relation, Hegel was directly influenced by the Haitian Revolution.11 While...