Prime Integers Research Articles

Integer Programming Formulations For The Frobenius Problem

The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set of positive integers. Our first formulation is used to check if a given positive integer is the Frobenius number of a given set of positive integers. The second formulation aims at finding the Frobenius number directly. The third formulation involves the residue classes with respect to the least member of the given set of positive integers, where a residue table is computed comprising all values modulo that least member, and the Frobenius number is obtained from there. Based on the same approach underlying the third formulation, we propose our fourth formulation which produces the Frobenius number directly. We demonstrate how to use our formulations with several examples. For illustrative purposes, some computa-tional analysis is also presented.

The Intriguing Structure of Stripes in GRACE Geopotential Models

Geopotential models derived from Gravity Recovery and Climate Experiment (GRACE) mission measurements are significantly obscured by the presence of a systematic artifact, known as longitudinal stripes. Based on our previous work (Peidou and Pagiatakis, 2020) we provide an in-depth analysis of the latitudinal sampling characteristics of GRACE and we reveal the intriguing sampling mechanism that creates sub-Nyquist artifacts (stripes). Because the sub-Nyquist artifacts are poorly understood, we provide a simple simulation example to elucidate the mechanism of the sub-Nyquist artifact generation. Subsequently, we randomly select June 2009 daily GPS precise science orbits for GRACE-A to produce ground tracks to sample the low frequency disturbing potential (geoid) along the parallel of ϕ=10° N. The sampled geoid is then deinterlaced in space to produce a monthly data sequence whose detailed analysis shows that the sub-Nyquist artifacts (stripes) are produced from a critical sampling rate of the low degree gravitational field that is related to the ratio m/n of two mutually prime integers, where m is the number of days it takes to have a nearly repeat orbit and n is the number of complete orbits in one day. We perform extensive analyses of GRACE Level-2 data over a period of eight years to show the variability in the orbital characteristics that are directly linked to the orbit resonances (via integers m and n). It turns out that during short repeat cycle resonances the stripes are amplified. Finally, to minimize the presence of stripes in Level-2 data products, it is recommended that orbits of future missions should be designed to avoid the critical m/n ratios while appropriately monitoring and adjusting them during the mission. For completed missions, or missions that are already active, force modelling the latitudinal low frequency disturbing potential may be a viable and most preferred approach to filtering.

The 4d superconformal index near roots of unity and 3d Chern-Simons theory

We consider the S3×S1 superconformal index ℐ(τ) of 4d mathcal{N} = 1 gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential τ coupled to a combination of angular momenta on S3 and the U(1) R-charge. We develop the all-order asymptotic expansion of the index as q = e2πiτ approaches a root of unity, i.e. as overset{sim }{tau } ≡ mτ+n → 0, with m, n relatively prime integers. The asymptotic expansion of log ℐ(τ) has terms of the form overset{sim }{tau } k, k = −2, −1, 0, 1. We determine the coefficients of the k = −2, −1, 1 terms from the gauge theory data, and provide evidence that the k = 0 term is determined by the Chern-Simons partition function on S3/ℤm. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the S1. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the overset{sim }{tau } → 0 expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index.

Сравнение чисел и анализ переполнения в модулярной арифметике

The method of modular arithmetic consists in operating not with a number, but with its remainders after division by some integers. In the modular number system or the number system in the residual classes, a multi-bit integer in the positional number system is represented as a sequence of several positional numbers. These numbers are the remainders (residues) of dividing the original number into some modules that are mutually prime integers. The advantage of the modular representation is that it is very simple to perform addition, subtraction and multiplication operations. In parallel execution of operations, the use of modular arithmetic can significantly reduce the computation time. However, there are drawbacks to modular representation that limit its use. These include a slow conversion of numbers from modular to positional representation; the complexity of comparing numbers in modular representation; the difficulty in performing the division operation; and the difficulty of determining the presence of an overflow. The use of modular arithmetic is justified if there are fast algorithms for calculating a number from a set of remainders. This article describes a fast algorithm for converting numbers from modular representation to positional representation based on a geometric approach. The review is carried out for the case of a comparison system with two modules. It is also shown that as a result of increasing numbers in positional calculus, they successively change in a spiral on the surface of a two-dimensional torus. Based on this approach, a fast algorithm for comparing numbers and an algorithm for detecting an overflow during addition and multiplication of numbers in modular representation were developed. Consideration for the multidimensional case is possible when analyzing a multidimensional torus and studying the behavior of the turns on its surface.

Additive operations between connective K-theory and Chow theory

We determine all additive operations, stable and unstable, between connective K-theory and Chow theory modulo a prime integer p. It is proved the module of all stable operations is free of rank p−1 over the reduced Steenrod algebra.

Cyclic codes over a non-chain ring Re,q and their application to LCD codes

Let Fq be a finite field of order q, a prime power integer, such that q=et+1 where t≥1,e≥2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring Re,q=Fq[u]/〈ue−1〉. We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal Fq-linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd⁡(n,q)≠1 and gcd⁡(n,q)=1, respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image φ(C) is an LCD code of length en over Fq. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over Re,q.

On monogenity of certain pure number fields defined by xpr − m

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

Unit groups of some multiquadratic number fields and 2-class groups

Let pequiv -q equiv 5pmod 8 be two prime integers. In this paper, we investigate the unit groups of the fields L_1 =mathbb {Q}(sqrt{2}, sqrt{p}, sqrt{q}, sqrt{-1} ) and L_1^+=mathbb {Q}(sqrt{2}, sqrt{p}, sqrt{q} ). Furthermore , we give the second 2-class groups of the subextensions of L_1 as well as the 2-class groups of the fields L_n =mathbb {Q}( sqrt{p}, sqrt{q}, zeta _{2^{n+2}} ) and their maximal real subfields.

On quaternion algebras over the composite of quadratic number fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

Triple product p-adic L-functions associated to finite slope p-adic families of modular forms

Let p be a positive prime integer. We construct p-adic families of de Rham cohomology classes and therefore p-adic families of nearly overconvergent elliptic modular forms. As an application we define triple product p-adic L-functions attached to three finite slope families of modular forms satisfying certain assumptions.

Prime numbers demystified

The paper is the ultimate prime numbers algorithm that gets rid of the unneccessary mystery about prime numbers. All the numerous arithmetic series patterns observed between various prime numbers are clearly explained with an elegant "pattern of remainders". With this algorithm we prove that odd numbers too can make an Ulam spiral contrary to current ""proofs". At the end of the paper this author proves the relationship between a simple arithmetic series pattern and the Riehmann's prime numbers distribution equation. This paper would be important for encryption too. As an example, prime integer 1979 is expressed as 1.2.4.5.10.3.7.3.1.7.26.18.11.1. This makes even smaller primes useful for encryption as well.

Elliptic curves over a nonlocal ring 𝔽2d[𝜀],𝜀2 = 𝜀

In this work, we came to present the elliptic curves over a nonlocal ring [Formula: see text] where [Formula: see text] is a finite field of characteristic 2, and [Formula: see text] is a prime integer. We consider an elliptic curve given by the Weierstrass equation of the form [Formula: see text] such that [Formula: see text] are in the ring [Formula: see text] and b is invertible in [Formula: see text] see Tadmori et al. [Normal form of the elliptic curves over the finite ring, J. Math. Syst. Sci. 4 (2014) 194–196]. More precisely, we give a classification of elements of the elliptic curve over this ring, and we construct the group law over it, which allows us to prove same properties of the elliptic group [Formula: see text] by showing the isomorphism [Formula: see text]

On the 2-rank and 4-rank of the class group of some real pure quartic number fields

Abstract Let K = ℚ ( p d 2 4 ) $\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p. In this work, we calculate r 2(K), the 2-rank of the class group of K, in terms of the number of prime divisors of d that decompose or remain inert in ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ), then we will deduce forms of d satisfying r 2(K) = 2. In the last case, the 4-rank of the class group of K is given too.

Predicting The Data For Security Using Rjb23 Algorithm

The current world is information world; without this information can’t make due in present stage. This information created more from web- based media; this media information is public information; This public information did not have well security; so we applying the proposed method and it has 3 steps; 1. Using prime numbers in quadratic equations; 2. Prime and non-negative integer number used to swap the numbers; 3. Column operations execution; The new method gives well security when contrast to Salsa method.

Singularities of the dual curve of a certain plane curve in positive characteristic

It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree $q^2 + q + 1$ for which the Gauss map is inseparable with inseparable degree $q$. As a special case, we show a relation between the dual curve of the Fermat curve of degree $q^2 + q + 1$ and the Ballico-Hefez curve.

Semi-clean group rings

Recall that a ring is said to be a clean ring if every element can be expressed as the sum of a unit and an idempotent . In one variant of this definition, a ring is said to be a semi-clean ring if every element can be expressed as the sum of a unit and a periodic element. Ye's Theorem [12] states that the group ring Z ( p ) [ C 3 ] is semi-clean, where p is a prime integer and C 3 is a cyclic group of order 3. In this article, we generalize Ye's Theorem by demonstrating that, if R is a local ring, then the group ring R [ G ] is semi-clean if and only if G is a torsion abelian group .

Shadowing and stability in p-adic dynamics

In this paper, we study dynamical properties as shadowing and structural stability for a class of functions on the p-adic metric spaces Zp and Qp where p≥2 is a prime integer.

English

Let K be a number field defined by an irreducible polynomial F(X) ∈ ℤ[X] and ℤK its ring of integers. For every prime integer p, we give sufficient and necessary conditions on F(X) that guarantee the existence of exactly r prime ideals of ℤK lying above p, where $$\overline F \left( X \right)$$ factors into powers of r monic irreducible polynomials in $${\mathbb{F}_p}\left[ X \right]$$ . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of ℤK lying above p. We further specify for every prime ideal of ℤK lying above p, the ramification index, the residue degree, and a p-generator.

Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

CAP-3 Coprime Array for DOA Estimation With Enhanced Uniform Degrees of Freedom and Reduced Mutual Coupling

Compared with nested array, coprime array can significantly alleviate the mutual coupling between array sensors due to the sparser array structure. However, plenty of holes in its difference co-array extremely reduce the achievable number of uniform degrees of freedom(uDOFs). In this letter, a coprime array with prime integer 3(CAP-3) is proposed, in contrast to several well-known nested arrays, which can offer comparable number of uDOFs, and alleviate mutual coupling simultaneously. Specifically, the CAP-3 consists of three appropriately placed subarrays, and it can eradicate most of the holes in difference co-array except two at both ends. Hence a large number of uDOFs can be obtained. Furthermore, CAP-3 possesses robustness against heavy level of mutual coupling due to the reduced number of sensor pairs with small inter-element spacing. Simulations reveal the advantage of the CAP-3 in uDOF, mutual coupling robustness, and direction of arrival estimation accuracy.