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Continuing the line of thought of an earlier work, we provide the first infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(5)$, the (unique) smallest nonsolvable group for which…
We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in…
We bound the condition number of the Jacobian in pseudo arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton GMRES solve. In pseudo arclength continuation one…
L\"uroth series, like regular continued fractions, provide an interesting identification of real numbers with infinite sequences of integers. These sequences give deep arithmetic and measure-theoretic properties of subsets of numbers…
This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily…
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field…
We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant…
Every Laurent series in $\mathbb{F}_q\left(\left(t^{-1}\right)\right)$ has a continued fraction expansion whose partial quotients are polynomials. De Mathan and Teuli\'e proved that the degrees of the partial quotients of the left-shifts of…
In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to…
We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate…
We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and…
We prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a p-adic variable satisfies a non-trivial…
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued…
In this paper, a Fourier series in fractional dimensional space is introduced for an arbitrarily periodic function $f(t;\alpha)$. We call it fractional Fourier series of the order $\alpha$. Extending the basis functions of the linear space…
In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,...,x_{r},x_{r+1}^{\pm 1},..., x_n^{\pm 1}]$, $k$ algebraicaly…
Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued…
Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…
We prove that the leading and penultimate leading coefficients in $u_3$ of the ``error" terms of NRS(2) applied to a cubic polynomial $f(z) =\sum_{i=0}^3 a_i z^i=\prod_{i=1}^3 (1-u_iz)$ with starting point $(-\frac{a_1}{a_2},…
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…