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This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…

While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero distance between their absolute values) does not seem to have been studied much. We…

Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a<q$ be a pair of relatively prime integers. It is proved that there is a prime primitive root $u\ll (\log p)(\log \log p)^5$ such that $u\equiv a\bmod q$ in the prime…

In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner…

$ $[This paper is a (self contained) chapter in a new book, Mathematics and Computation, whose draft is available on my homepage at https://www.math.ias.edu/avi/book ]. We survey some concrete interaction areas between computational…

In this paper, we will fully describe the representations of the crystallographic rank two affine Hecke algebras using elementary methods, for all possible values of q. The focus is on the case when q is a root of unity of small order.

Some fractional Newton methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper we introduce a fractional Newton method with order $\alpha+1$ and compare with another fractional…

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$…

The first-order model theory of modules has been studied for decades. More recently, the model theoretic study of nonelementary classes of modules--especially Abstract Elementary Classes of modules--has produced interesting results. This…

This article introduces an intuitive function MY that simplifies solving cubic equations without venturing into the complex space. To many, it's quite strange that cubic root(s) are expressed using trigonometric functions in the…

Suppose $ m,n\geq 2 $ are co prime integers. We prove certain new symmetries of the base $ n $ representation of $ 1/m $, and in particular characterize the subgroup generated by $ n $ inside $ (\mathbb{Z}/m\mathbb{Z})^\times $. As an…

Although $S_\infty$ (the group of all permutations of $\mathbb{N}$) is size continuum, both it and its closed subgroups can be presented as the set of paths through a countable tree. The subgroups of $S_\infty$ that can be presented this…

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a…

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…

Let $\delta(p)$ tend to zero arbitrarily slowly as $p\to\infty$. We exhibit an explicit set $\mathcal{S}$ of primes $p$, defined in terms of simple functions of the prime factors of $p-1$, for which the least primitive root of $p$ is $\le…

We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…

A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one…

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…

We consider the sums of the form $$ S=\sum_{x=1}^{N} \exp\big((ax+b_1g_1^x+... +b_rg_r^x)/p \big) $$, where $p$ is prime and $g_1,..., g_r$ are primitive roots $\pmod p$. An almost forty years old problem of L. J. Mordell asks to find a…

Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence;…