On the Approximation of the Generalized Cut Function of Degree p+1 By Smooth Sigmoid Functions

We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p + 1 (GCFP). We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the logistic and the shifted logistic functions. The limiting case of the interval-valued Heaviside step function is also discussed which imposes the use of Hausdorff metric. Numerical examples are presented using CAS MATHEMATICA.

On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials

In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell and Gordon and show how to compute (2) the coefficients of an Euclidean remainder — obtained in finding in Q[x], the greatest common divisor of f, g ∈ Z[x] of degree n — in terms of the determinants (5) of the corresponding submatrices of sylvester1, Sylvester’s widely known and used matrix of 1840. (1) See the link http://en.wikipedia.org/wiki/Anna_Johnson_Pell_Wheeler for her biography (2) Both for complete and incomplete sequences, as defined in the sequel. (3) Also known as modified subresultants. (4) Using determinants Sylvester and Van Vleck were able to compute the coefficients of Sturmian remainders only for the case of complete sequences. (5) Also known as (proper) subresultants.

Lp extremal polynomials. Results and perspectives

2000 Mathematics Subject Classification: 30C40, 30D50, 30E10, 30E15, 42C05.

A Generalized Quasi-Likelihood Estimator for Nonstationary Stochastic Processes−Asymptotic Properties and Examples

2010 Mathematics Subject Classification: 62F12, 62M05, 62M09, 62M10, 60G42.

Approximation Properties of Schurer-Stancu Type Polynomials

MSC 2010: 41A25, 41A35

On Thom Polynomials for A4(−) via Schur Functions

2000 Mathematics Subject Classification: 05E05, 14N10, 57R45.

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

Some Coefficient Estimates for Polynomials on the Unit Interval

2000 Mathematics Subject Classification: 26C05, 26C10, 30A12, 30D15, 42A05, 42C05.

Generalized Backscattering and the Lax-Phillips Transform

2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.

An Iterative Procedure for Solving Nonsmooth Generalized Equation

2000 Mathematics Subject Classification: 47H04, 65K10.

Steffensen Methods for Solving Generalized Equations

2000 Mathematics Subject Classification: 65G99, 65K10, 47H04.

Generalized D-Symmetric Operators I

2000 Mathematics Subject Classification: Primary: 47B47, 47B10; secondary 47A30.

On Generalized Models and Singular Products of Distributions in Colombeau Algebra G(R)

MSC 2010: 46F30, 46F10

Predegree Polynomials of Plane Configurations in Projective Space

2000 Mathematics Subject Classification: 14N10, 14C17.

Letter to the Editor. Remarks on Some Inequalities for Polynomials

MSC 2010: 30A10, 30C10, 30C80, 30D15, 41A17.