Arithmetical Point Of View Research Articles

On the ascent of atomicity to monoid algebras

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra R[M] of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

On the dynamics of n-circle inversion

The article deals with singular perturbation of polynomial maps where is a complex parameter and n is the degree, which is a particular case of the family of rational maps known as McMullen maps. Our main result shows that even when the geometric limit of the Julia set converges to the unit circle or the annulus for a.e. Lebesgue , as n tends to infinity, the measure of maximal entropy always converges to the Lebesgue measure supported on the unit circle.Additionally we describe the dynamics on the Julia set and show that is related to a quotient of a shift of n symbols by an equivalence relation. Finally we prove that the Thurston pull-back map associated with a particular four-circle inversion is a ramified Galois covering. From the arithmetical point of view we prove that each n-circle inversion can be defined over its field of moduli.

Congruences modulo 9 for singular overpartitions

In a recent work, Andrews introduced the new combinatorial objects called singular overpartitions. He proved that these singular overpartitions can be enumerated by the partition function [Formula: see text] which denotes the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. In this paper, we consider the function [Formula: see text] from an arithmetical point of view. We establish a number of Ramanujan-like congruences and a congruence relation modulo [Formula: see text] for [Formula: see text] by employing some generating function dissection techniques. With the aid of these congruences and the relation, we obtain two new infinite families of congruences modulo [Formula: see text] satisfied by [Formula: see text].

A natural extension of the Young partition lattice

Abstract Recently Andrews introduced the concept of signed partition: a signed partition is a finite sequence of integers ak, . . . , a1, a−1, . . . , a−l such that ak ≥ ... ≥ a1 > 0 > a−1 ≥ ... ≥ a−l. So far the signed partitions have been studied from an arithmetical point of view. In this paper we first generalize the concept of signed partition and we next use such a generalization to introduce a partial order on the set of all the signed partitions. Furthermore, we show that this order has many remarkable properties and that it generalizes the classical order on the Young lattice.

An infinite family of congruences modulo $$3$$ 3 for $$13$$ 13 -regular bipartitions

Let \(B_{13}(n)\) denote the number of \(13\)-regular bipartitions of \(n\). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function \(p(n)\). In particular, we shall prove an infinite family of congruences: for \(\alpha \ge 2\) and \(n\ge 0\), $$\begin{aligned} B_{13}(3^{\alpha }n+2\cdot 3^{\alpha -1}-1)\equiv \ 0\ (\mathrm{mod\ }3). \end{aligned}$$ In addition, we will also give an alternative proof of one infinite family of congruences for \(b_{13}(n)\), the number of \(13\) regular partitions of \(n\), due to Webb.

Arithmetic of the 7-regular bipartition function modulo 3

In this paper, we study the function B l (n) which counts the number of l-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0, $$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$ In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B 13(n) modulo 3.

Arithmetic properties of partitions with even parts distinct

In this work, we consider the function ped(n), the number of partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n): We prove a number of results for ped(n) including the following: For all n 0; ped(9n + 4) 0 (mod 4) and

Arithmetic properties of partitions with odd parts distinct

In this work, we consider the function pod(n), the number of partitions of an integer n wherein the odd parts are distinct (and the even parts are unrestricted), a function which has arisen in recent work of Alladi. Our goal is to consider this function from an arithmetic point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). We prove a number of results for pod(n) including the following infinite family of congruences: for all α≥0 and n≥0, $$\mathrm{pod}\biggl(3^{2\alpha+3}n+\frac{23\times3^{2\alpha+2}+1}{8}\biggr)\equiv 0\ (\mathrm{mod}\ 3).$$

Series of reciprocal products with factors from linear recurrence sequences

In this article we study the values of power series ∑ z^n/\prod^j_{i = 0} W_{(n + im)k} at certain points of their domain of meromorphy from the arithmetical point of view. (W_n) is a sequence of non-zero integers satisfying a recurrence W_{n + 1} = pW_n + qW_{n − 1} with non-zero integers p, q such that the discriminant Δ = p^2 + 4q is positive but not a square. The main interest is to characterize the situations, where these values lie in the real quadratic number field ℚ(√Δ) or even in ℚ , but we also include some transcendence problems.

Bericht zur Meldung nach § 21 TFG für das Jahr 2007

The present report contains the data collected in 2007, pursuant to Section 21 Transfusionsgesetz (German Transfusion Act), and an analysis of the supply situation over the past eight years. The recording of the data by online reporting is in the meantime well established and generally accepted. As in previous years, all blood donation centers located in Germany transmitted data on the collection, manufacture, import and export of blood components for transfusion, so that meaningful data are available. According to these data, a total of 6.7 million blood collections were performed in 2007. The number of whole blood donations was at the level of previous years, with 4.7 million, whereas the number of apheresis donations rose again, to 1.9 million. The portion of autologous blood collections accounts for only 1.1% and thus continues to decline. Since 2003, the number of red blood cell concentrates prepared has been a constant 4.5 million transfusion units. The decrease in the portion of decay of red blood cell concentrates on the user side is particularly good news. In 2000, it accounted for 5% and in 2007, it was just above 3%, referred to the total quantity of data reported as transfused and decayed. The manufacture of platelet concentrates rose from 366,000 to 480,000 transfusion units between 2003 and 2007. The production of therapeutic single plasmas also markedly increased in 2007 compared with previous years, accounting for 1.2 million transfusion units. In 2007, 2.2 million liters of plasma for fractionation were collected in Germany. This trend went hand in hand with the increasing number of apheresis donations that year. In addition, 1.0 million liters were imported, and, at the same time, 1.8 million liters were exported. The quantity available in Germany from a pure arithmetic point of view of 1.4 million liters was almost entirely allocated to basic fractionation, so that a sufficient plasma supply can be assumed. The assessment of the degree of self-sufficiency is made difficult because of the influence of imports and exports; however, the results show no deficit for plasma derivatives. Due to the fact that manufacturing capacities are still lacking in Germany, recombinant factors need to be imported in their entirety. Since 2003, Germany has by far been the leader in Europe with more than 20 liters of fractionation plasma collected per 1,000 inhabitants. Furthermore, regarding the manufacturing figures of red blood cell concentrates, platelet concentrates, and therapeutic single plasma, Germany is in the top third for all these products compared with other European countries. The manufacture of allogeneic stem cell products for hematopoietic reconstitution, obtained by apheresis, has continuously risen to 4,700 in the reporting year. A large portion of this, 1,810 transplants could be exported while only a small number, 179 preparations, had to be imported. The manufacture of autologous stem cell preparations from cord blood also rose drastically compared with 2006, to more than 10,000 in 2007. It must be emphasized that these products were entirely placed into stock; none were transplanted in the reporting year. The interest in the figures collected in compliance with Section 21, Transfusion Act remains high both in Germany and at the international level. Reliable data are available thanks to the evaluations of trends over years, above all on the availability of blood components for transfusion. In addition, the Paul Ehrlich Institute will continue to strive to meet the demands for high-quality information on the supply situation in the future.

Projective Surface Automorphisms of Positive Topological Entropy from an Arithmetic Viewpoint

Let X be a smooth projective surface over a number field K ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] C), and f : X → X an automorphism of positive topological entropy. In this paper, we show that there are only finitely many f -periodic curves on X . Then we define a height function ĥ D corresponding to a certain nef and big R-divisor D on X and transforming well relative to f , and deduce some arithmetic properties of f -periodic points and non f -periodic points.

Similarities between the trigonometric function and the lemniscate function from arithmetic view point

We show that the rational functions associated with the lemniscate functions and primary numbers of Gaussian integers, defined by Eisenstein, has a symmetrical relation with respect to the zeros and the poles. We also consider some polynomials associated with the trigonometric functions and odd integers, and point out a similar symmetry of the polynomials.

Illitization and silicification in Greater Alwyn: I. Assessing and synthesizing experimental data

AbstractSilicification and illitization due to diagenesis result in the modification of porosity and permeability in many North Sea reservoirs, including Greater Alwyn. A good estimation of the mineral composition is required as input/output data for geochemical modelling modes intended to reproduce these effects. Using sedimentological studies to determine laterally equivalent facies, three facies were chosen. ‘Initial states’ for modelling were thus defined as the present-day composition of less-evolved laterally-equivalent facies, ‘final states’ as the present-day composition in the most evolved part of the basin. Precise actual mineral compositions were calculated using a reconciliation program, DATREC. Average compositions calculated for each facies in each field provided data useful in constraining input/output values used in modelling, and gave insights into depositional variation. Arithmetic tests assuming isochemical transformations, constraining the chemical compositions of ‘initial states’ to fit actual mineralogical compositions of ‘final states’, were run using this software. They show that, from an arithmetical point of view, if the assumption of equivalent facies holds, isochemical reactions can occur in some, but not in all facies, thereby demonstrating the importance of checking equivalence of facies and diagenesis hypotheses by quantitative calculations.

Spectral Estimation by AFT Computation

Di Lecce, V., and Guerriero, A., Spectral Estimation by AFT Computation,Digital Signal Processing6(1996) 213–223.At the beginning of this century Bruns developed a method for computing the coefficients of the Fourier series of a periodic functiony(t) using the Möbius inversion formula. This idea for Fourier analysis was considered again by Wintner from an arithmetical point of view in 1945. In recent papers, many authors have shown that the arithmetic Fourier transform (AFT) computation is more convenient in signal processing, requiring a reduced computation load, than are fast Fourier transform and convolution algorithms. The data dependence in the AFT is not uniform (this algorithm requires nonequidistant inputs to produce equidistant spectral coefficients). To have a series of suitable values as AFT inputs, oversampling or interpolation is used. In these papers, bases on algorithms, evaluations of errors in the spectral coefficients computation using AFT, and the complexity of different hardware and software solutions for the AFT computation are proposed. The spectral coefficients computed via AFT and via discrete Fourier transform are compared in terms of accuracy. AFT computation proves to be an easy task but its software or hardware implementation is much more complex. Furthermore there is not a complete evaluation of AFT in any of the papers. Our aim is to provide a complete evaluation of this algorithm.

Rees algebras of non-singular equimultiple prime ideals

In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”

Some new results on sets of type (m,n) in projective planes

All possible sets of type (l,n) in a finite projective plane are completely characterized from the arithmetical point of view. Furthermore, necessary existence conditions are proved for sets of type (m,n) which are stronger then the previously known ones and provide the unique possible arithmetical solutions whenever they are satisfied.

Linear extrapolation by rational functions, exponentials and logarithmic functions

In this paper linear extrapolation by rational functions with given poles is considered from an arithmetical point of view. It is shown that the classical interpolation algorithms of Lagrange, Neville-Aitken and Newton which are well known for polynomial interpolation can be extended in a natural way to this problem yielding recursive methods of nearly the same complexity. The proofs are based upon explicit representations of generalized Vandermonde-determinants which are calculated by the elimination method combined with analytical considerations. As an application a regularity criterion for certain linear sequence-transformations is given. Also, by the same method simplified recurrence relations for linear extrapolation by exponentials and logarithmic functions at special knots are derived.

Remarque sur l’arithmétique des 2-formes différentielles de deuxième espèce

We consider a complex smooth and proper surface X X and a prime number l l . We know that the invariant B 2 − ρ {B_2} - \rho represents the dimension of the vector space of classes of 2 2 -forms of the second kind on X X . Grothendieck has observed that B 2 − ρ {B_2} - \rho is an arithmetic invariant, and this leads naturally to interpret a 2 2 -form of the second kind from an arithmetic point of view. By using Grothendieck’s techniques we associate to each class ω \omega of a 2 2 -form of the second kind an element \[ ω 1 {\omega _1} \] of ω {\omega } -adic part of the Brauer group H 2 ( X et, O X et ∗ ) {H^2}({X_{{\text {et,}}}}\mathcal {O}_{{X_{{\text {et}}}}}^*) of X X . In the case where X X is a fibered space on a curve, if one subjects this fibration to some given conditions (Artin), ω l {\omega _l} is then also interpreted as an element of H 1 ( Γ et , i ∗ J ) ( l ) {H^1}{({\Gamma _{{\text {et}}}},{i^*}J)_{(l)}} or, in the equivalent manner, as an element of the Tate-Schafarievitch group II I 1 ( C ( Γ ) , J ) {\text {II}}{{\text {I}}^1}({\mathbf {C}}(\Gamma ),J) studied by Ogg, where J J is the Jacobian of the generic fiber of X X and C ( Γ ) {\mathbf {C}}(\Gamma ) is the functions field of Γ \Gamma . Reciprocally each element of the l l -divisible part of II I 1 ( C ( Γ ) , J ) ( l ) {\text {II}}{{\text {I}}^1}{({\mathbf {C}}(\Gamma ),J)_{(l)}} comes from a 2 2 -form of the second kind. This correspondence permits us to deduce some consequences on 2 2 -forms of the second kind.

Quantitative Aspect of the Hypothetical Incorporation of Injected Antigen in Resulting Antibody. II. Experimental

The authors have previously examined, from an arithmetical point of view, the Buchnerian hypothesis that antibody is a conjugate of body globulin and injected antigen. These speculations led us to undertake the following experimental investigation based upon the reactivity of artificial compound proteins in which the hapten chosen to confer the new specificity contains integrally an inorganic atom (As) for which a direct and very delicate chemical test is available. Two compounds, ovomucoid-diazo-arsanilic acid and casein-diazo-arsanilic acid, were prepared. The serum of a rabbit immunized against the first reacted with the second, showing the presence of an arsanilic-aeid-specific antibody. The optimal precipitation ratio by volumes of antiserum to casein-diazo-arsanilic acid was found to be about 150. Such a mixture is considered neutral— at least neither reagent is present in any large excess. It seems reasonable to suppose that within this zone, antibody and antigen enter respectively into the precipitate in the proportion of one molecule to one molecule, or one molecule to a small number of molecules. If it is assumed that the antibody is a globulin conjugate of the antigen, and contains a similar amount of arsenic (as would necessarily be true if there were no breaking up of the injected antigen), it can be shown from the amounts of arsenic and casein found in the casein-diazo-arsanilic acid solution, and the above ratio, that 10 cc. of antiserum should have contained about 20 mmg. (20 × 10-6 g.) of arsenic, an amount easily detectable. None was found. This would seem to cast doubt on the Buchnerian hypothesis.