Related papers: A brief introduction to p-adic numbers

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new…

Smoothed analysis of complexity bounds and condition numbers has been done, so far, on a case by case basis. In this paper we consider a reasonably large class of condition numbers for problems over the complex numbers and we obtain…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree--like behavior…

In this paper we study some properties of the fermionic p-adic integrals on Zp arising from the umbral calculus

Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of…

This article studies the first-order $p$-adic deformations of classical weight one newforms, relating their fourier coefficients to the $p$-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois…

We introduce $p$-adic Kummer spaces of continuous functions on $\mathbb{Z}_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and certain decompositions.…

What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much.…

We establish an algorithm for a criterion of the diagonalisability of a matrix over a local field by a unitary matrix. For this sake, we define the notion of normality of a $p$-adic operator, and give several criteria for the normality. We…

In this note we present a survey on some classical and modern approaches on Pythagorean triples. Some questions are also posed in direction of some materials under review. In particular some non commutative and operator theoretical…

We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…

We review definitions and basic properties of operads, PROPs and algebras over these structures.

For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $\nu_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $\nu_{p}(Q(n))$ is shown to be either periodic or unbounded. The first…

We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.

We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this…

In this note we give a detailed proof of certain results on geometry of numbers in the $S$-adic case. These results are well-known to experts, so the aim here is to provide a convenient reference for the people who need to use them.

Hypercomplex numbers are unital algebras over the real numbers. We offer a short demonstration of the practical value of hypercomplex analytic functions in the field of partial differential equations.