Coefficients Of Series Expansion Research Articles

An Euler-Like Product with Fibonacci Exponents

The pentagonal theorem for partitions is a consequence of the expansion of Euler’s famous product (1-y)(1-y2)(1-y3)(1-y4)(1-y5)⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (1-y) (1-y^2) (1-y^3)(1-y^4)(1-y^5) \\cdots $$\\end{document} We investigate the nature of the coefficients of the series expansion of (1-y)(1-y2)(1-y3)(1-y5)(1-y8)⋯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (1-y) (1-y^2) (1-y^3)(1-y^5)(1-y^8) \\cdots $$\\end{document}, in which the sequence of exponents is the Fibonacci numbers. As a part of the study of the combinatorial properties of the development of this product, we show that the series expansion coefficients are from {-1,0,1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\{ -1, 0, 1\\}$$\\end{document}, and their behavior is determined by a monoid of twenty-five 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} matrices.

CLASS NUMBER FORMULAS FOR CERTAIN BIQUADRATIC FIELDS

We consider the class numbers of imaginary quadratic extensions F (√ −p), for certain primes p, of totally real quadratic fields F which have class number one. Using seminal work of Shintani, we obtain two elementary class number formulas for many such fields. The first expresses the class number as an alternating sum of terms that we generate from the coefficients of the power series expansions of two simple rational functions that depend on the arithmetic of F and p. The second makes use of expansions of 1/p, where p is a prime such that p ≡ 3 (mod 4) and p remains inert in F. More precisely, for a generator εF of the totally positive unit group of OF , the base-εF expansion of 1/p has period length ℓF,p, and our second class number formula expresses the class number as a finite sum over disjoint cosets of size ℓF,p.

A matrix solver approach for fracture flow simulation by Analytic Element Method

The manuscript presents a study on 2D groundwater flow through porous and fractured media employing Analytic Element Method. The approximate solution based on a series expansion is presented for the problem with multiple fractures. Influence of each fracture can be expressed in a series satisfying Laplace’s equation perfectly. The unknown coefficients in series expansion are determined by using the discharge potentials of all the elements linked to them. Dimension, location and conductivities were randomly selected for the inhomogeneities. Furthermore, a matrix method as a direct solver for AEM, obtained by implementing the intern boundary conditions as well as convergence analysis of iterative method of Gauss–Seidel type are also discussed.

Mach number analysis of hydrogen flow in labyrinth passage under high pressure gradient

This work delves into a novel concept that harnesses the potential of labyrinth passages for hydrogen decompression, offering the advantage of dispersing the mainstream into multiple channels, thereby reducing hydrogen mass and energy. Compared to traditional hydrogen decompression devices, this approach demonstrates adaptability to high pressure gradients and boasts a simple structure. For the industry, the perennial challenge has been controlling energy loss and enhancing aerodynamic performance during hydrogen decompression. Hence, our primary focus lies in addressing the key challenges related to labyrinth passages in this field. Within this study, a detailed design of the labyrinth passage is proposed, and the correlation between Mach number and energy loss is studied. Additionally, an in-depth study is conducted on the effects of throttling parameters on Mach number in the labyrinth passage, aiming to achieve an optimized design. The results reveal a linear correlation between Mach number and energy loss. Furthermore, the research highlights that different throttling parameters have distinct impacts on energy loss. By utilizing the maximum average Mach number (Ma‾max) as a metric to evaluate different designs, we have determined the range (Ŕ) of factor values for series number (ni/n0), widths (wi/w0), expansion coefficients (ri/r0), interstage heights (hi/h0) and depths (di/d0) to be 0.687, 0.482, 0.228, 0.075, and 0.036, respectively. The range (Ŕ) directly signifies the influence that each factor has on the Mach number. As a result, it has been deduced that the parameter effects on controlling energy loss follow a descending order: series number (ni/n0), widths (wi/w0), expansion coefficients (ri/r0), interstage heights (hi/h0) and depths (di/d0). Moreover, for each factor, the preferred level ranking consistently remains as Ⅳ, Ⅲ, Ⅱ, and Ⅰ. Therefore, considering the need to control manufacturing costs, it is recommended to opt for a larger series number, expansion coefficient and a smaller width and interstage height.

Estimates of Coefficients for Bi-Univalent Functions in the Subclass H_∑ (n,γ,φ)

Considering that finding the bounds for the coefficients of the Taylor-Maclaurin series expansion of bi-univalent functions is one of the important subjects in geometric function theory that has attracted the attention of many researchers in the last few decades, we also take a step in this direction. Finding such bounds is the main focus or, more clearly, the main problem of our work. In this article, we study the subclass H_∑ (n,γ,φ) of bi-univalent functions which is defined in the open unit disk D. Furthermore, we obtained the upper bounds estimates for the first coefficients |a_2 | and |a_3 | of the functions in this category by using subordination method. From the main result of the article (Theorem 2.1), special cases have been derived that improve some the results of previous articles.

Fourier Series Approximation in Besov Spaces

Defined on the top of classical L p -spaces, the Besov spaces of periodic functions are good at encoding the smoothness properties of their elements. These spaces are also characterized in terms of summability conditions on the coefficients in trigonometric series expansions of their elements. In this paper, we study the approximation properties of 2 π -periodic functions in a Besov space under a norm involving the seminorm associated with the space. To achieve our results, we use a summability method presented by a lower triangular matrix with monotonic rows.

Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients.

Physics-informed neural network algorithm for solving forward and inverse problems of variable-order space-fractional advection–diffusion equations

A new physics-informed neural network (PINN) algorithm is proposed to solve variable-order space-fractional partial differential equations (PDEs). For the forward problem, PINN algorithm based on a power series expansion is established to solve the space-fractional advection–diffusion equations with variable coefficients in one and two dimensions. The loss function is constructed by taking the coefficients in the power series expansion as the output of the network. The learning rate range is also analyzed to ensure that the error is reduced with respect to the training time. For the inverse problem, a novel dual-network architecture based on neural network parallelism is designed to estimate the order of variable fractional operator. One of the networks outputs the coefficients of the power series expansion, which is a function that depends only on time variables, and the other is fitted to the order of variable fractional operator, which is a function that is related to both space and time variables. Compared with the forward problem, we make more comparisons between the approximated solution and the exact solution at some interior points when constructing the loss function to improve the fitting ability of the inverse problem. The exact solutions are easy to find and can be readily obtained with a fine mesh. Several numerical examples are illustrated with graphs. These results confirm the effectiveness of our method in solving variable-order space-fractional partial differential equations.

On 𝑞-analogue of Euler–Stieltjes constants

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q q -analogue of the Euler constant and proved the irrationality of certain numbers involving q q -Euler constant. In this paper, we improve their results and prove the linear independence result involving q q -analogue of the Euler constant. Further, we derive the closed-form of a q q -analogue of the k k -th Stieltjes constant γ k ( q ) \gamma _k(q) . These constants are the coefficients in the Laurent series expansion of a q q -analogue of the Riemann zeta function around s = 1 s=1 . Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series ∑ n ≥ 1 σ 1 ( n ) / t n \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1 t > 1 . Finally, we study the transcendence nature of some infinite series involving γ 1 ( 2 ) \gamma _1(2) .

Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again

The fundamental lemma by Jan C. Willems and co-workers is deeply rooted in behavioral systems theory and it has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions – in particular Polynomial Chaos Expansions (PCE) of L2-random variables, which date back to Norbert Wiener’s seminal work – enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the L2-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear time-invariant systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts.

Mixed Mode Fracture Investigation of Rock Specimens Containing Sharp V-Notches.

This work aims to assess both experimentally and analytically the fracture behavior of rock specimens containing sharp V-notches (SV-notches) subjected to mixed mode I/II loading. To this end, firstly, several mixed mode fracture tests were conducted on Brazilian disk specimens weakened by an SV-notch (SVNBD sample), performed in their corresponding center and with various notch opening angles. Secondly, the fracture resistance of the tested samples was predicted using a criterion named MTS-FEM. This approach is based on the maximum tangential stress (MTS) criterion, in which the tangential stress is determined from the finite element method (FEM). Additionally, in the present research, the required critical distance is calculated directly from finite element analyses performed on cracked samples. Comparing the experimental results and the analytical predictions, it is shown that the fracture curves obtained from the MTS-FEM criterion are in agreement with the experimental results. These results are achieved without the need for the calculation of stress series expansion coefficients, as an additional advantage of the proposed approach.

Edge of Chaos Kernel and neuromorphic dynamics of a locally-active memristor

Edge of Chaos Kernel (EOCK), the crown jewel of edge of chaos, is a nature’s optimal mechanism for creating action potentials, which can be exploited to uncover the neuromorphic dynamics of memristors. A memristor blessed with an EOCK, which can emulate the neuromorphic behaviors of biological neurons, is a natural candidate to be applied in neuromorphic computing. This paper proves that a first-order memristor blessed with an EOCK has a pole in the open left half plane (LHP), and presents a simple method for testing if a generic voltage-controlled first-order memristor has an EOCK using only its first-order Taylor series expansion coefficients. This paper presents, for the first time, that locally active voltage-controlled generic memristors contain an EOCK. Also, this paper designs a theoretical memristor model blessed with an EOCK, and constructs a memristive circuit for observing the relationship between the EOCK and neuromorphic behaviors. The memristor blessed with an EOCK generates some well-known neuromorphic behaviors, further verify the innate ability of Edge of Chaos Kernel to exhibit the complex phenomena. This paper pave the way towards further expounding the generating mechanism of the action potential from the perspective of circuit.

On general-n coefficients in series expansions for row spin–spin correlation functions in the two-dimensional Ising model

We consider spin–spin correlation functions for spins along a row, R n = ⟨σ 0,0 σ n,0⟩, in the two-dimensional Ising model. We discuss a method for calculating general-n expressions for coefficients in high-temperature and low-temperature series expansions of R n and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for a nonlinear ordinary differential equation whose solution would determine R n , analogous to the known nonlinear ordinary differential equation whose solution determines the diagonal correlation function ⟨σ 0,0 σ n,n ⟩ in this model.

Proofs of five conjectures on matching coefficients of Baruah, Das and Schlosser by an algorithmic approach

In this paper, we present an algorithm on vanishing coefficients with arithmetic progressions in the sum of two generalized eta-quotients by using the theory of modular forms, and utilize our algorithm to prove five conjectures on matching coefficients in series expansions of certain q-products and their reciprocals. One of which was provided by Baruah and Das, and the others were found by Schlosser. For instance, we prove that for any n≥0,λ12(10n+r)=λ12′(10n+r−6),r∈{7,9}, where the sequences {λ12(n)}n≥0 and {λ12′(n)}n≥0 are defined by∑n=0∞λ12(n)qn=1R(q)R(q2)R(q4)R(q8)=(∑n=0∞λ12′(n)qn)−1, and where R(q) is the Rogers–Ramanujan continued fraction, which has the following celebrated product representation:R(q)=∏n=0∞(1−q5n+1)(1−q5n+4)(1−q5n+2)(1−q5n+3). Finally, we find that this phenomenon also exists in other infinite q-series expansions.

Temperature Extrapolation of Henry's Law Constants and the Isosteric Heat of Adsorption.

Computational screening of adsorbent materials often uses the Henry's law constant (KH) (at a particular temperature) as a first discriminator metric due to its relative ease of calculation. The isosteric heat of adsorption in the limit of zero pressure (qst∞) is often calculated along with the Henry's law constant, and both properties are informative metrics of adsorbent material performance at low-pressure conditions. In this article, we introduce a method for extrapolating KH as a function of temperature, using series-expansion coefficients that are easily computed at the same time as KH itself; the extrapolation function also yields qst∞. The extrapolation is highly accurate over a wide range of temperatures when the basis temperature is sufficiently high, for a wide range of adsorbent materials and adsorbate gases. Various results suggest that the extrapolation is accurate when the extrapolation range in inverse-temperature space is limited to |β - β0 | < 0.5 mol/kJ. Application of the extrapolation to a large set of materials is shown to be successful provided that KH is not extremely large and/or the extrapolation coefficients converge satisfactorily. The extrapolation is also able to predict qst∞ for a system that shows an unusually large temperature dependence. The work provides a robust method for predicting KH and qst∞ over a wide range of industrially relevant temperatures with minimal effort beyond that necessary to compute those properties at a single temperature, which facilitates the addition of practical operating (or processing) conditions to computational screening exercises.

Metric approach to a mathrm{T}overline{mathrm{T}} -like deformation in arbitrary dimensions

We consider a one-parameter family of composite fields — bi-linear in the components of the stress-energy tensor — which generalise the mathrm{T}overline{mathrm{T}} operator to arbitrary space-time dimension d ≥ 2. We show that they induce a deformation of the classical action which is equivalent — at the level of the dynamics — to a field-dependent modification of the background metric tensor according to a specific flow equation. Even though the starting point is the flat space, the deformed metric is generally curved for any d > 2, thus implying that the corresponding deformation can not be interpreted as a coordinate transformation. The central part of the paper is devoted to the development of a recursive algorithm to compute the coefficients of the power series expansion of the solution to the metric flow equation. We show that, under some quite restrictive assumptions on the stress-energy tensor, the power series yields an exact solution. Finally, we consider a class of theories in d = 4 whose stress-energy tensor fulfils the assumptions above mentioned, namely the family of abelian gauge theories in d = 4. For such theories, we obtain the exact expression of the deformed metric and the vierbein. In particular, the latter result implies that ModMax theory in a specific curved space is dynamically equivalent to its Born-Infeld-like extension in flat space. We also discuss a dimensional reduction of the latter theories from d = 4 to d = 2 in which an interesting marginal deformation of d = 2 field theories emerges.

An extendable Space-Time Curvature Mode model

Einstein calculated the curvature of space-time from a Field equation based on an equation of motion derived by the variation of a geodetic path and a second rank Metric tensor. In this study a Space-Time Curvature Mode (STCM) model is calculated from a series of coupled Field equations derived from the variation of a Taylor series expansion of a Lagrangian potential of the STCM. The Taylor series expansion coefficients are Metric tensors of increasing rank. The STCM model can be extended to any number of Field equations. The coupled Field equations of the STCM describe the warping of space-time by masses and energy fields, and the effect of the curved space-time on the motion of masses and energy fields. This results in a description of space coordinates that have locally positive and negative slopes as a function of the time coordinate and can describe a Universe with at times an accelerating expansion and at later times a contracting Universe. This study replaces the postulated concepts of Einstein’s Cosmological constant and postulated concepts of Dark Mass and Dark Energy. It is shown that the Hamiltonian is conserved.

Thermodynamic perturbation theory coefficients for hard spherocylinders and cylinders

Thermodynamic perturbation theories are indispensable in the development of equations of state based on Statistical Mechanics. Recently, a new statistical thermodynamic model has been proposed in which molecules are represented as cylindrically symmetric hard bodies (ellipsoids of revolution, spherocylinders, and cylinders) interacting with identical neighboring molecules through a spherically symmetric square-well potential. This novel equation of state embeds at least two approximations to the Onsager’s second virial theory: one restrains the equation of state to systems with an isotropic (disordered) orientation distribution, while the other decouples the translational and orientational degrees of freedom, being only valid at low densities. Using canonical Monte Carlo simulations, we investigated the feasibility of such approximations by determining the uniaxial nematic order parameter of the systems and calculating the first- and second-order perturbation coefficients of the high-temperature series expansion (HTSE) of the Helmholtz free energy. Our results indicate that both approximations are reasonable for a certain range of packing fractions and molecular anisotropies. We have also calculated the full Helmholtz free energy of the perturbed system, which can be used for determining high-order perturbation terms to improve the predictive capability of the equation of state.

Comparison between computational cost of fractals using line-doublets

In this work, we study the two-dimensional groundwater flow of fractured porous media with the help of analytic element method. A numerical solution formed by line elements called line-doublets based on a series expansion has been presented in the literature. In this solution, each fracture has an influence that may expand in a series that obeys Laplace’s equation exactly. The unknown coefficients are found from the discharge potentials of all other elements that are related to the expansion coefficients in series expansion. This work aims to make comparison between the computational cost for fractals by the analytic element method with different techniques like: Iterative and Matrix method.

Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion and their evaluation through molecular dynamics simulations

Molecular dynamics method and finite element analysis are applied for the analysis of the stress field in the neighborhood of the crack tip in a copper plate with a single edge notch. The molecular dynamics simulations implemented in a classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) are aimed at evaluating classical continuum linear elastic fracture mechanics parameters such as stress intensity factors, T-stresses and higher order coefficients of the Williams power series expansion of the near crack stress field for Mode I, Mode II and Mixed Mode (Mode I + Mode II) loadings of the cracked specimen in isotropic linear elastic materials. The key objective of the study is the comparison of continuum and atomistic approaches for the estimation of the near crack tip fields using the example of one of the most common cracked configurations. Stress intensity factors, T-stresses and higher order coefficients of the Williams series expansion for a copper plate with the single edge notch under Mode I and Mixed Mode loadings are evaluated by atomistic modeling and by finite element method. The wide class of the computations in LAMMPS is performed. The atomistic values of stress intensity factors and higher order terms of the Williams series expansion are compared with the values obtained from the numerical solutions given by finite element method. It is shown that the continuum fracture theory successfully describes fracture and the near crack tip fields even at extremely confined singular stress field of only several nanometers. The angular distributions of the stress components from atomistic modeling are retrieved and compared with the angular distributions of the stresses from continuum linear elastic fracture mechanics. The comparison shows good agreement between two approaches.