The explicit formulas and evaluations of Ramanujan's theta-function ψ

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

Domains of validity for some of Ramanujan's continued fraction formulas

Ramanujan, who loved continued fractions, recorded many of his formulas in his two notebooks. The 25th entry in Chapter 12 of Notebook II is a continued fraction involving a quotient of gamma function products. We are going to give a new proof of Entry 25, which, incidently, is the least difficult of all Ramanujan's continued fraction formulas involving quotients of gamma function products. In our proof, we make use of a hypergeometric formula of his.

On Ramanujan's continued fraction

Let k be an imaginary quadratic field, h the complex upper half plane, and let <TEX>$\tau{\in}h{\cap}k$</TEX>, <TEX>$q=e^{{\pi}i\tau}$</TEX>. We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.

This paper proposes a rational approximation-based approach to find positive real parameters for the extended Debye model (EDM), aimed at condition assessment of insulation systems of power transformers. The EDM can model the slow and fast polarization phenomenon, including relaxation mechanisms with different relaxation times within a composite dielectric material. In the proposed approach, the complex permittivity of the transformer’s composite insulation is approximated via rational functions, as given by the vector fitting (VF) software tool, and the EDM parameters are identified from the obtained poles/residues. To guarantee positive real parameters, i.e., a physically realizable circuit, VF is internally modified to calculate the final residues of the rational approximation via a constrained linear least-squares problem without resorting to further post-processing algorithms, as in existing methods, hence without affecting fitting accuracy. The effectiveness of the parametrized EDM is demonstrated in two ways: (a) by reconstructing frequency domain spectroscopy (FDS) curves provided via measurements in new oil-immersed power transformers and (b) by the comparison of the calculated polarization current given by EDM versus real measurements in time domain. The achieved fitting accuracy in most of the cases is above 99 percent for the reconstructed FDS curves, while the polarization current waveform is reproduced with good agreement.

We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_{1},a_{2},\ldots ,a_{n}]$ we associate a snake graph ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ such that the continued fraction is the quotient of the number of perfect matchings of ${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$ and ${\mathcal{G}}[a_{2},\ldots ,a_{n}]$. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.

In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8})) \equiv 3^{\alpha +1} \ ({\rm mod}\ 5)$ for α ⩾ 0.

Infinitely Many Critical Points of Non-Differentiable Functions and Applications to a Neumann-Type Problem Involving the p-Laplacian

This paper is intended to investigate a fractional telegraph equation of the form with positive real parameters a, b and c. Here , and are operators of the Riemann–Liouville fractional derivative, where 0<α≤1 and 0<β≤1. A symbolic operational form of the solutions in terms of the Mittag–Leffler functions is exhibited. Using the Banach fixed point theorem, the existence and uniqueness of solutions are studied for this kind of fractional differential equations.

We define and study a [0, 1] m -valued process depending on three positive real parameters p, q, β that specializes for β = 1, 2 to the eigenvalues process of the real and complex matrix Jacobi processes on the one hand and that has the distribution of the β -Jacobi ensemble as stationary distribution on the other hand. We first prove that this process, called β -Jacobi process, is the unique strong solution of the stochastic differential equation defining it provided that β &gt; 0, p ∧ q &gt; m – 1 + 1/ β . When specialized to β = 1, 2, our results actually improve well known results on eigenvalues of matrix Jacobi processes. While proving the strong uniqueness, the generator of the β -Jacobi process is mapped into the radial part of the Dunkl–Cherednik Laplacian associated with the non reduced root system of type BC . The transformed process is then valued in the principal Weyl alcove and this allows to define the Brownian motion in the Weyl alcove corresponding to all multiplicities equal one. Second, we determine, using stochastic calculus and a comparison theorem, the range of β, p, q for which the m components of the β -Jacobi process first collide, the smallest one reaches 0 and the largest one reaches 1. This is equivalent to the first hitting time of the boundary of the principal Weyl alcove by the transformed process. Finally, we write down its semi group density.

The foundation of "inner-outer" expansions and allied methods is examined.Two limit definitions (both yielding Kaplun's extension theorem) are compared in regard to double limit asymptotics. Introduction. The great practical usefulness ([l]-[4]) of "innerouter" or "matched asymptotic" approximations and allied methods has created interest ([5]-[7]) in the mathematical concept of Kaplun limits.The following is an attempt to clarify the concept and some difficulties associated with it.Two definitions ( 2) will be compared: that of [6] and a broader one which may, perhaps, be closer to that envisioned by Kaplun [5].It is first established that Kaplun's extension theorem holds for the latter, so that the definition is consistent with the techniques of matched asymptotic expansions.The applicability of the limit process is not limited to matched asymptotic expansions, however; it has been applied to nonlinear problems of physical importance ([6], [8]) involving asymptotic approximations to solutions valid for large values of a variable and small values of a parameter.The limit definition presented here is formulated in order to establish a useful connection between such double limits and suitable families of Kaplun limits in which the variable and parameter tend to their respective limits simultaneously.The connection is found to necessitate the understanding of the sense of approximation in an appropriate rr.anner.2. Kaplun limits.Let e be a positive real parameter, and x a variable in Rn with Euclidean norm |jc|.Let F be a function, defined for e e (0, 1] and on some domain of x-space with pointwise norm ||F||.Typically, F might be a vector-valued function defined by a system of differential equations, and our interest is in the limit e|0.This limit may be singular; for example, it may result in the loss of the highest order derivatives in a

This paper studies one class of real functions, which we call Takagi power functions. Such functions have one positive real parameter; they are continuous, but nowhere differentiable, and are given on a real line using functional series. These series are similar to the series defining the continuous, nowhere differentiable Takagi function described in 1903. For each parameter value, we derive a functional equation for functions related to Takagi power functions. Then, using this equation, we obtain an accurate two-sides estimate for the functions under study. Next, we prove that for parameter values not exceeding 1, Takagi power functions satisfy the Holder logarithmic condition, and find the smallest value of the constant in this condition. As a result, we get the usual Hölder condition, which follows from the logarithmic Hölder condition. Moreover, for parameter values ranging from 0 to 1, we investigate the behavior of Takagi power functions in the neighborhood of their global maximum points. Then we show that the functions under study reach a strict local minimum on the real axis at binary-rational points, and only at them. Finally, we describe the set of points at which our functions reach a strict local maximum. The benefit of our research lies in the development of methods applicable to continuous functions that cannot be differentiated anywhere. This can significantly expand the set of functions being studied.

Existence of solutions for p-Laplacian discrete equations

In this note we shall describe the results obtained by applying the decomposition principle established in [2], [3] to poly-linking systems. In order to state those results in a logically self-consistent way, we shall introduce a new notion of 'minors' of generalized polymatroids. Firstly, it is observed that the subsystems defined from a poly-linking system through our decomposition method are not in general poly-linking systems any more. This difficulty can be overcome by considering a poly-linking system as a special case of generalized polymatroids. In fact, it can be shown by an easy calculation that a poly-linking system is equivalent to a special case of generalized polymatroids. And when a poly-linking system is considered as a generalized polymatroid, its resultant subsystems through our decomposition method are seen to be the 'minors' of this generalized polymatroid. The notion of a 'minor' of a generalized polymatroid is first introduced in this paper. Hence from the point of view of our decomposition principle, the notion of 'poly-linking system' is not self-consistent, and it should be treated as a special case of generalized polymatroids. Secondly, as a direct consequence of the results in [2], [3], a direct-sum decomposition of the optimal solutions of the intersection problems on poly-linking systems is induced. Lastly, we shall investigate the parametrized type of the intersection problem on poly-linking systems where the rank functions are multiplied by positive real parameters.