In this thesis, we study sequences of functions of the form Fp → {0, 1} for varying n, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. One of the key tools we use is the recently developed theory of ‘higher order Fourier analysis’, where the characters of standard Fourier analysis are replaced with exponentials of...

We study an infinite class of sequences sparse polynomials that have binomial coefficients both as exponents and coefficients. This generalizes a sequence which arises in natural way graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity log-concavity, well identities involving derivatives. also prove upper lower bounds on the moduli zero...

Let G be a bounded simply-connected domain in the complex plane C, whose boundary Γ := ∂G is a Jordan curve, and let {pn} ∞ n=0 denote the sequence of Bergman polynomials of G. This is defined as the sequence pn(z) = λnz n + · · · , λn > 0, n = 0, 1, 2, . . . , of polynomials that are orthonormal with respect to the inner product

k=0 (−1)k 2k + 1 x } , the sequence of Taylor polynomials centered at 0 that converges to arctanx on [−1, 1]. Like the Taylor polynomials for several other classical functions, e.g., cosx, sinx, and ex, this sequence of polynomials is very easy to describe and work with; but unlike those Taylor sequences with factorials in the denominators of their coefficients, it does not converge rapidly for...

The establishing of a pairwise key between two nodes for encryption in a wireless sensor network is a challenging issue. To do this, we propose a new deterministic key pre-distribution scheme which has modified the multiple key space scheme (MKSS). In the MKSS, the authors define two random parameters to make better resilience than existing schemes. Instead of a random selection of these parame...

This work is based on the Euler-Seidel matrix method [11] which is related to algorithms, combinatorics and generating functions. This method is quite useful to investigate properties of some special numbers and polynomials. In this paper we consider the Euler-Seidel matrix method for some combinatorial numbers and polynomials. This method is relatively easier than the most of combinatorial met...

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco–Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.