Adic Integers Research Articles

Universal dynamical function behind all genetic codes: [formula omitted]-adic attractor dynamical model

The genetic code is a map which gives the correspondence between codons in DNA and amino acids. In the attractor dynamical model (ADM), genetic codes can be described as the sets of the cyclic attractors of discrete dynamical systems - the iterations of functions acting in the ring of 2-adic integers Z2. This ring arises from representation of nucleotides by binary vectors and hence codons by triples of binary vectors. We construct a Universal Function B such that the dynamical functions for all known genetic codes can be obtained from B by simple transformations on the set of codon cycles - the “Addition” and “Division” operations. ADM can be employed for study of phylogenetic dynamics of genetic codes. One can speculate that the “common ancestor genetic code” was caused by B. We remark that this function has 24 cyclic attractors which distribution coincides with the distribution for the hypothetical pre-LUCA code. This coupling of the Universal Function with the pre-LUCA code assigns the genetic codes evolution perspective to ADM. All genetic codes are generated from B through the special chains of the “Addition” and “Division” operations. The challenging problem is to assign the biological meaning to these mathematical operations.

On Wiener and Levy Type Theorems for System of Characters of the Ring of $$p$$-Adic Integers

On Wiener and Levy Type Theorems for System of Characters of the Ring of $$p$$-Adic Integers

A note on the admissibility of smooth simple $RG$-modules

Let G be a p -adic reductive group and R be a noetherian Jacobson algebra over the ring \mathbb{Z}_{l} of l -adic integers with l\neq p . In this note, we show that every smooth irreducible R -linear representation of G is admissible using the finiteness result of Dat, Helm, Kurinczuk and Moss for Hecke algebras over R .

Almost everywhere convergence of varying-parameter setting Cesaro means of Fourier series on the group of 2-adic integers

In this paper we prove that the maximal operator of Cesaro-means for one-dimensional Fourier series on the group of 2-adic integers is of weak type (L^{1}, L^{1}). Moreover, we prove the almost everywhere convergence of Cesaro means of integrable functions.

Integrality of $v$-adic Multiple Zeta Values

In this article, we prove the integrality of v -adic multiple zeta values (MZVs). For any index \mathfrak{s}\in\mathbb{N}^r and finite place v\in A := \mathbb{F}_q[\theta] , Chang and Mishiba introduced the notion of the v -adic MZVs \zeta_A(\mathfrak{s})_v , which is a function field analogue of Furusho's p -adic MZVs. By estimating the v -adic valuation of \zeta_A(\mathfrak{s})_v , we show that \zeta_A(\mathfrak{s})_v is a v -adic integer for almost all v . This result can be viewed as a function field analogue of the integrality of p -adic MZVs, which was proved by Akagi–Hirose–Yasuda and Chatzistamatiou.

Approximation by matrix transform means with respect to the character system of the group of 2-adic integers

Abstract In this paper, we discuss the rate of the approximation by matrix transform means of Fourier series with respect to the character system of the group of 2-adic integers in L p ⁢ ( I ) {L^{p}(I)} spaces ( 1 ≤ p < ∞ {1\leq p<\infty} ) and in C ⁢ ( I ) {C(I)} . Moreover, we give an application for functions in Lipschitz classes Lip ⁢ ( α , p , I ) {\mathrm{Lip}(\alpha,p,I)} ( α > 0 {\alpha>0} , 1 ≤ p < ∞ {1\leq p<\infty} ) and Lip ⁢ ( α , C ⁢ ( I ) ) {\mathrm{Lip}(\alpha,C(I))} ( α > 0 {\alpha>0} ).

Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections

We compute the expectation of the number of linear spaces on a complete intersection in $p$-adic projective space. Here random means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a complete intersection tends to $1$. In the case of the number of lines on a cubic in three-space and on the intersection of two quadrics in four-space, we give an explicit formula for this expectation.

On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents

We consider a differential system x\frac{d}{dx} Y=GY , where G is a m\times m matrix whose coefficients are power series which converge and are bounded on the open unit disc D(0,1^-) . Assume that G(0) is a diagonal matrix with p -adic integer coefficients. Then there exists a solution matrix of the form Y=F \exp(G(0)\log x) at x=0 if all differences of exponents of the system are p -adically non-Liouville numbers. We give an example where F is analytic on the p -adic open unit disc and has log-growth greater than m . Under some conditions, we prove that if a solution matrix at a generic point has log-growth \delta , then F has log-growth \delta .

The topological group of the $p$-adic integers

The topological group of the $p$-adic integers

Cohesive groups and $p$-adic integers

Cohesive groups and $p$-adic integers

Measuring the Preservation and Ergodicity of 1-Lipschitz Functions on the Ring of 3-Adic Integers in Terms of Coordinate Functions

Measuring the Preservation and Ergodicity of 1-Lipschitz Functions on the Ring of 3-Adic Integers in Terms of Coordinate Functions

An Estimate of the Maximal Operator of the Nörlund Logarithmic Means with Respect to the Character System of the Group of 2-adic Integers on the Hardy Space $$H_{1}$$

An Estimate of the Maximal Operator of the Nörlund Logarithmic Means with Respect to the Character System of the Group of 2-adic Integers on the Hardy Space $$H_{1}$$

On the Cantor and Hilbert cube frames and the Alexandroff-Hausdorff theorem

The aim of this work is to give a point-free description of the Cantor set . It can be shown (see, e.g. [18] ) that the Cantor set is homeomorphic to the p -adic integers Z p : = { x ∈ Q p : | x | p ≤ 1 } for every prime number p . To give a point-free description of the Cantor set, we specify the frame of Z p by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of Z p and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the Cantor set L ( Z p ) . A topological characterization of it is given by Brouwer's Theorem [7] : The Cantor set is the unique totally disconnected, compact metric space with no isolated points. We prove that L ( Z p ) is a spatial frame whose space of points is homeomorphic to Z p . In particular, we show with point-free arguments that L ( Z p ) is 0-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Moreover, we show that the frame L ( Z p ) satisfies c b d L ( Z p ) ( 0 ) = 0 , where c b d L ( Z p ) : L ( Z p ) → L ( Z p ) , defined by c b d L ( Z p ) ( a ) = ⋀ { x ∈ L ( Z p ) | x ≤ a and ( x → a ) = a } is the Cantor-Bendixson derivative (see, e.g. [19] ). It follows that a frame L is isomorphic to L ( Z p ) if and only if L is a 0-dimensional compact regular metrizable frame with c b d L ( 0 ) = 0 . Finally, we give a point-free counterpart of the Hausdorff-Alexandroff Theorem which states that every compact metric space is a continuous image of the Cantor space (see, e.g. [1] and [12] ). We prove the point-free analogue: if L is a compact metrizable frame, then there is an injective frame homomorphism from L into L ( Z 2 ) .

On some classes of 1-Lipschitz measure-preserving ergodic functions on ℤp

We study some classes of [Formula: see text]-Lipschitz measure-preserving functions on the group of [Formula: see text]-adic integers [Formula: see text] and then establish necessary and sufficient conditions for their ergodicity by means of their van der Put coefficients.

A LeVeque-type inequality on the ring of p-adic integers

We derive an inequality on the discrepancy of sequences on the ring of [Formula: see text]-adic integers [Formula: see text] using techniques from Fourier analysis. This is a [Formula: see text]-adic analogue of the classical LeVeque inequality on the circle group [Formula: see text]. Some applications of the inequality are also given.

Three pairs of congruences concerning sums of central binomial coefficients

Recently the first author proved a congruence proposed in 2006 by Adamchuk: [Formula: see text] for any prime [Formula: see text]. In this paper, we provide more examples (with proofs) of congruences of the same kind [Formula: see text] where [Formula: see text] is a prime such that [Formula: see text], [Formula: see text] is a fraction in [Formula: see text] and [Formula: see text] is a [Formula: see text]-adic integer. The key ingredients are the [Formula: see text]-adic Gamma function [Formula: see text] and a special class of computer-discovered hypergeometric identities.

Almost everywhere convergence of Riesz means of one-dimensional Fourier series on the group of 2-adic integers

Almost everywhere convergence of Riesz means of one-dimensional Fourier series on the group of 2-adic integers

Arithmetic of weighted Catalan numbers

In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the 2-adic valuations of the weighted Catalan numbers are equal to the 2-adic valuations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to 2-adic integers. These methods are also extended to q-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to the specific case of the number of combinatorial types of Morse links. Many open questions are mentioned.

SOME ALGEBRAS IN TERMS OF DIFFERENTIAL OPERATORS

Let $C$ be a commutative ring and $C[x_1,x_2,\ldots]$ the polynomial ring in a countable number of variables $x_i$ of degree 1. Suppose that the differential operator $d^1=\sum_i x_{i} \partial_{i} $ acts on $C[x_1,x_2,\ldots]$. Let $\mathbb{Z}_p$ be the $p$--adic integers, $K$ the extension field of the $p$--adic numbers $\mathbb{Q}_p$, and $\mathbb{F}_2$ the 2-element filed. In this article, first, the $C$-algebra $\mathcal{A}_1(C)$ of differential operators is constructed by the divided differential operators $(d^1)^{\vee k}/k!$ as its generators, where $\vee$ stands for the wedge product. Then, the free Baxter algebra of weight $1$ over $\varnothing$, the $\lambda$--divided power Hopf algebra $\mathcal{A}_\lambda$, the algebra $C(\mathbb{Z}_p,K)$ of continuous functions from $\mathbb{Z}_p$ to $K$, and the algebra of all $\mathbb{F}_2$--valued continuous functions on the ternary Cantor set are represented in terms of the differential operators algebra $\mathcal{A}_1(C)$.

Diophantine tuples over $\mathbb {Z}_p$

For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\ldots,a_m)$ of elements of $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ is a perfect square in $R$. In this article, we compute and estimate the measures of the sets of $D(r)$ $m$-tuples in the ring $\mathbb{Z}_p$ of $p$-adic integers, as well as its residue field $\mathbb{F}_p$.