Mathematics

In this note, it is shown that the Ramanujan Master Theorem (RMT) when n is a positive integer can be obtained, as a special case, from a new integral formula. Furthermore, we give a simple proof of the RMT when n is not an integer.

In this note we obtain a new convergence result for the Adomian decomposition method.

I discuss the nature of a Fractional Discrete Fourier Transform (FrDFT) described algorithmically by a combination of chirp transforms and ordinary DFTs. The transform is shown to be consistent with a continuous two-dimensional rotation between the time and frequency domains. I further present a new closed-form expression for the transformation matrix and some preliminary analysis of its properties.

Recently we have reported a new method of rational approximation of the sinc function obtained by sampling and the Fourier transforms. However, this method involves a trigonometric multiplier e 2πiνa =cos(2πνa)+isin(2πνa), where ν is the argument and a is the shift constant. In this work we show how to avoid this trigonometric multiplier in order to represent the Fourier transform of function f(t) in explicit form as a rational approximation of kind F{f(t)}(ν)= ∫ ∞ −∞ f(t) e −2πiνt dν≈ P(ν) Q(ν) , where P(ν) and Q(ν) are polynomials. A MATLAB code showing algorithmic implementation of the proposed method for rational approximation of the Fourier transform is presented.

We generalize several integrals studied by Glaisher. These ideas are then applied to obtain an analog of an integral due to Ismail and Valent.

This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the solution free from such coefficients. In addition, the usual substitution needed to reduce the cubic equation is implicit in that the final solution is expressed in terms only of the function and derivative values of the given cubic polynomial at a single point. The proposed methodology simplifies the solution to cubic equations making them easy to remember and solve.

In this work we show how to perform a rapid computation of the Voigt/complex error over a single domain by vectorized interpolation. This approach enables us to cover the entire set of the parameters x,y∈R required for the HITRAN-based spectroscopic applications. The computational test reveals that within domains x∈[0,15]∩y∈[ 10 −8 ,15] and x∈[0,50000]∩y≥ 10 −8 our algorithmic implementation is faster in computation by factors of about 8 and 3 , respectively, as compared to the fastest known C/C++ code for the Voigt/complex error function. A rapid MATLAB code is presented.

We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method with elementary tools in order to be understood and useful by students and educators. For this reason, we provide a self contained approach. Our method is a variation of the well known method of resultant, that has its origin back to Euler. Our goal, in the present paper, is in the spirit of calculus and secondary school mathematics. An extensive discussion of the theory of zeros of polynomials and extremal problems for polynomials the reader can find in the books 10 and 13 in our references.

Through the hands-on creation of two sashiko pieces of work - a counted thread kogin bookmark and a single running stitched hitomezashi sampler - participants will explore not only the living cultural history of this traditional Japanese needlework but will also experience the mathematics of sashiko in a tangible form, and will take away with them items of simple beauty.

In this paper, we determine the complex-valued solutions of the functional equation f(xσ(y))+f(τ(y)x)=2f(x)f(y) for all x,y∈M , where M is a monoid, σ : M⟶M is an involutive automorphism and τ : M⟶M is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of 2 -dimensional irreducible representations of M .