We describe a Genetic Algorithm that can evolve complete programs. Using a variable length linear genome to govern the mapping of a Backus Naur Form grammar definition to a program, expressions and programs of arbitrary complexity may be evolved. Our system, Grammatical Evolution, has been applied to problems such as a Symbolic Regression problem, and finding Trigonometric Identities. In this p...

The function g (which contains the singular part of the free energy) must have a special symmetry that relates its values for dual arguments. (For example the function f(x) = x/(1 + x) equals f(x), establishing a duality between the arguments at x and x.) Eq.(VI.23) has the following properties: 1. Low temperatures are mapped to high temperatures, and vice versa. 2. The mapping connects pairs o...

This study investigates the effect of representational sequence on students’ understanding of mathematical concepts. Pilot studies were conducted with 129 high school students on solving inverse trigonometric identities and with 10 pre-service secondary teachers on representing Simpson’s Paradox. Structured activities with a variety of representations and representational sequences were used to...

Using Krattenthaler’s operator method, we give a new proof of Warnaar’s recent elliptic extension of Krattenthaler’s matrix inversion. Further, using a theta function identity closely related to Warnaar’s inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton-type. A special case yields a particular summation that was used by Warnaar to ...

In this study, base on traditional lapped transform’s properties, inherit from some of the special characteristic properties of discrete cosine transform, trigonometric identities and the structure of linear phase filter banks. We present some theoretic properties in the generalized structure of linear phase paraunitary filter bank and build some examples corresponding to these properties. Thes...

For a proper cone K ⊂ Rn and its dual cone K∗ the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K∗ }. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones,...

In this paper, many formulas and identities for computing the r-parametric Hermite type polynomials are given with help of generating functions. Using functions algebraic methods, a relation is also including these 2-variable Kampé de Fériet polynomials. Moreover, relations containing two parametric Apostol-Bernoulli higher order, Apostol-Euler Apostol-Genocchi order Dickson obtained. Finally, ...

In this article, we construct the generating functions for new families of special polynomials including two parametric kinds Bell-based Bernoulli and Euler polynomials. Some fundamental properties these are given. By using some identities, relations among trigonometric polynomials, Stirling numbers presented. Computational formulae obtained. Applying a partial derivative operator to functions,...

This paper presents a modified 32-bit ROM-based Direct Digital Frequency Synthesizer (DDFS). Maximum output frequency of the DDFS is limited by the structure of the accumulator used in the DDFS architecture. The hierarchical pipeline accumulator (HPA) presented in this paper has less propagation delay time rather than the conventional structures. Therefore, it results in both higher maximum ope...

We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting Ruijsenaars-Macdonald operators. prove that the Polychronakos freezing trick can be applied to these operators, thus providing set of Hamiltonians chain constructed by means Baxter-Belavin ${\rm GL}_M$ $R$-matrix. Namely, we show reduced a function ident...