相关论文: A Digital Binomial Theorem

We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney's embedding theorem allows us…

The semiring of discrete dynamical systems is a simple algebraic model for modularity in deterministic systems. The objects of the semiring are finite transformations (viewed as directed graphs and regarded up to isomorphism), the sum of…

In this article we present Pickands theorem and his double sum method. We follow Piterbarg's proof of this theorem. Since his proof relies on general lemmas we present a complete proof of Pickands theorem using Borell inequality and Slepian…

This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a…

We prove an extension of the Bourgain-Sarnak-Ziegler theorem and then apply it to bound certain polynomial exponential sums with modular coefficients.

The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…

We construct a counterexample to a well-known extension theorem for slice regular functions, which motivates us to develop a theory of Riemann slice-domains by introducing a new topology on quaternions. By some paths describing axial…

This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion…

We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.

We introduce an algorithm to describe Pieri's Rule for multiplication of Schubert polynomials. The algorithm uses tower diagrams introduced by the authors and another new algorithm that describes Monk's Rule. Our result is different from…

In this paper, we describe a class of elements in the ring of $\mathrm{SL}(V)$-invariant polynomial functions on the space of configurations of vectors and linear forms of a 3-dimensional vector space $V.$ These elements are related to one…

The binary signed-digit representation of integers is used for efficient computation in various settings. The Stern polynomial is a polynomial extension of the well-studied Stern diatomic sequence, and has itself has been investigated in…

This study applies the binomial, k-binomial, rising k-binomial and falling k-binomial transforms to the modified k-Fibonacci-like sequence. Also, the Binet formulas and generating functions of the above mentioned four transforms are newly…

We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.

Spivey presented a new approach to evaluate combinatorial sums by using finite differences. We present some closed forms for sums involving the binomial coefficients, Fibonacci and Lucas numbers in terms of the falling factorial.

The paper describes a system of rays declining at small angles in lasers. The correlation between a group of rays and binomial coefficients is shown. The correlation of distribution of rays in the system of numbers placed in a…

From an identity connecting a combinatorial sum and Legendre polynomials, we derive closed forms for a number of combinatorial sums. Some of them are obtained via results about the integrals of functions associated with Legendre…

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…

The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based…