Wavelet functions have been successfully used in many problems as the activation function of feedforward neural networks ZB92],,STK92], PK93]. In this paper, a family of polynomial wavelets generated from powers of sigmoids is described which provides a robust way for designing neural network architectures. It is shown, through experimentation, that function members of this family can present a...

Numerous statistical and graphical problems have been reported for different versions of Microsoft Excel, including the newest version (Excel 2007). We report newly discovered problemswith Excel 2007when generating polynomial trend line equations, having a userspecified (forced) intercept, from graphed data. We also remind users of Excel’s Trendline function of problems with Excel 2003 that hav...

In the context of second-order polynomial-time computability, we prove that there is no general function space construction. We proceed to identify restrictions on the domain or the codomain that do provide a function space with polynomial-time function evaluation containing all polynomial-time computable functions of that type. As side results we show that a polynomial-time counterpart to admi...

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions. Keywords—Approximation Theory, Chebyshev Polynomial, Computable Functions, Comp...

Let S ⊂ R denote a compact set with infinite cardinality and C(S) the set of real continuous functions on S. We investigate the problem of polynomial and orthogonal polynomial bases of C(S). In case of S = {s0, s1, s2, . . .} ∪ {σ}, where (sk)k=0 is a monotone sequence with σ = limk→∞ sk, we give a sufficient and necessary condition for the existence of a so-called Lagrange basis. Furthermore, ...

In Computable Analysis, elements of uncountable spaces, such as the real line R, are represented by functions on strings and fed to Turing machines as oracles; or equivalently, they are represented by infinite strings and written on the tapes of Turing machines [Wei00, BHW08]. To obtain reasonable notions of computability and complexity, it is hence important to choose the “right” representatio...

This is a collection of definite integrals involving the logarithmic and polynomial functions in terms special fundamental constants. All results this work are new.

‎this paper obtains the exact solutions of the wave equation as a second-order partial differential equation (pde)‎. ‎we are going to calculate polynomial and non-polynomial exact solutions by using lie point symmetry‎. ‎we demonstrate the generation of such polynomial through the medium of the group theoretical properties of the equation‎. ‎a generalized procedure for polynomial solution is pr...

For a hypergraphic polytope there is weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines f-polynomial. This invariant of hypergraphs extends the Stanley chromatic symmetric simple graphs. We consider certain combinatorial Hopf algebra show that universal morphism to functions coincides with this enumerator function. calculate f-polynomial u...

the omega polynomial(x) was recently proposed by diudea, based on the length of stripsin given graph g. the sadhana polynomial has been defined to evaluate the sadhana index ofa molecular graph. the pi polynomial is another molecular descriptor. in this paper wecompute these three polynomials for some infinite classes of nanostructures.