Prime Integers Research Articles

Fast arbitrary resizing of images in the discrete cosine transform domain

This study presents a method to resize images by a rational factor of P/Q in the discrete cosine transform (DCT) domain, where P and Q are relatively prime integers larger than 1. Our method extends on the prior work of Mukherjee and Mitra, which utilises the spatial relationship of DCT coefficients between a block and its sub-blocks. To resize images by a factor of P/Q, the images are first up-sampled by a factor of P and then down-sampled by a factor of Q. Although this method produces resized images with good visual quality, it requires high computational cost. In this study, the authors generalise an observation found in the spatial relationship of the DCT coefficients between a block and its sub-blocks. Subsequently, a sparse matrix representation is derived from this observation to reduce the computational cost of the proposed method. To further reduce computational cost of the proposed method, a subset of up-sampled DCT coefficients is used in the down-sampling operation. From various experiments, the authors have determined the lowest number of up-sampled DCT coefficients to be used in the down-sampling operation without affecting the visual quality of the resized images. As compared to existing methods, the proposed method requires lower computational cost and produces resized images of good visual quality.

RESONANCES OF THE SD OSCILLATOR DUE TO THE DISCONTINUOUS PHASE

Resonance phenomena of a harmonically excited system with mul-tiple potential well play an important role in nonlinear dynamics research. In this paper, we investigate the resonant behaviours of a discontinuous dynamical system with double well potential derived from the SD oscillator to gain better understanding of the transition of resonance mechanism. Firstly, the time dependent Hamiltonian is obtained for a Duffing type discontinuous system modelling snap-through buckling. This system comprises two subsystems connected at x = 0, for which the system is discontinuous. We construct a series of generating functions and canonical transformations to obtain the canonical form of the system to investigate the complex resonant behaviours of the system. Furthermore, we introduce a composed winding number to explore complex resonant phenomena. The formulation for resonant phenomena given in this paper generalizes the formulation of n Omega0 = m Omega used in the regular perturbation theory, where n and m are relative prime integers, Omega 0 and Omega are the natural frequency and external frequencies respectively. Understanding the resonant behaviour of the SD oscillator at the discontinuous phase enables us to further reveal the vibrational energy transfer mechanism between smooth and discontinuous nonlinear dynamical systems

A lower bound on the essential dimension of simple algebras

Let p be a prime integer and F a field of characteristic different from p. We prove that the essential p-dimension edpCSA(pr) of the class CSA(pr) of central simple algebras of degree pr is at least (r−1)pr+1. The integer edpCSA(pr) measures complexity of the class of central simple algebras of degree pr over field extensions of F.

Bounds on generalized Frobenius numbers

Let N ≥ 2 and let 1 < a 1 < ⋯ < a N be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as ∑ i = 1 N a i x i where x 1 , … , x N are nonnegative integers. More generally, the s -Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this. We use techniques from the geometry of numbers to give upper and lower bounds on the s -Frobenius number for any nonnegative integer s .

Exceptional cases of Terai’s conjecture on Diophantine equations

Let a, b, c be relatively prime positive integers such that a p + b q = c r for fixed integers p, q, r ≥ 2. Terai conjectured that the equation a x + b y = c z in positive integers has only the solution (x, y, z) = (p, q, r) except for specific cases. In this paper, we consider the case q = r = 2 and give some results related to exceptional cases.

On the embedding problem for number fields in the case of an elementary Abellan kernel

The Galois embedding problem is considered in the case of number fields and an elementary Abelian kernel. New cases are discovered, in which the consistency condition is sufficient for the existence of a solution of the embedding problem. In particular, it is true when the order of the kernel is the cube of a prime integer.

Quantum phase measurement and Gauss sum factorization of large integers in a superconducting circuit

We study the implementation of quantum phase measurement in a superconducting circuit, where two Josephson phase qubits are coupled to the photon field inside a resonator. We show that the relative phase of the superposition of two Fock states can be imprinted in one of the qubits. The qubit can thus be used to probe and store the quantum coherence of two distinguishable Fock states of the single-mode photon field inside the resonator. The effects of dissipation of the photon field on the phase detection are investigated. We find that the visibilities can be greatly enhanced if the Kerr nonlinearity is exploited. We also show that the phase measurement method can be used to perform the Gauss sum factorization of numbers (${\geq} 10^4$) into a product of prime integers, as well as to precisely measure both the resonator's frequency and the nonlinear interaction strength. The largest factorizable number is mainly limited by the coherence time. If the relaxation time of the resonator were to be ${\sim} 10$ $\mu$s (${\sim} 1$ ms), then the largest factorizable number can be ${\geq} 10^4N$ (${\geq} 10^{7}N$), where $N$ is the number of photons in the resonator.

Genuinely high-dimensional nonlocality optimized by complementary measurements

Qubits exhibit extreme nonlocality when their state is maximally entangled and this is observed by mutually unbiased local measurements. This criterion does not hold for the Bell inequalities of high-dimensional systems (qudits), recently proposed by Collins–Gisin–Linden–Massar–Popescu and Son–Lee–Kim. Taking an alternative approach, called the quantum-to-classical approach, we derive a series of Bell inequalities for qudits that satisfy the criterion as for the qubits. In the derivation each d-dimensional subsystem is assumed to be measured by one of d possible measurements with d being a prime integer. By applying to two qubits (d=2), we find that a derived inequality is reduced to the Clauser–Horne–Shimony–Holt inequality when the degree of nonlocality is optimized over all the possible states and local observables. Further applying to two and three qutrits (d=3), we find Bell inequalities that are violated for the three-dimensionally entangled states but are not violated by any two-dimensionally entangled states. In other words, the inequalities discriminate three-dimensional (3D) entanglement from two-dimensional (2D) entanglement and in this sense they are genuinely 3D. In addition, for the two qutrits we give a quantitative description of the relations among the three degrees of complementarity, entanglement and nonlocality. It is shown that the degree of complementarity jumps abruptly to very close to its maximum as nonlocality starts appearing. These characteristics imply that complementarity plays a more significant role in the present inequality compared with the previously proposed inequality.

Minimal Gröbner bases and the predictable leading monomial property

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Gröbner bases of modules in F[x]q, no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (=row reducedness), a terminology introduced by Forney in the 70’s. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type Zpr (where p is a prime integer and r is an integer >1) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Gröbner basis, a so-called “minimal Gröbner p-basis” that does have a PLM property. We demonstrate that minimal Gröbner p-bases lend themselves particularly well to derive minimal realization parametrizations over Zpr. Applications are in coding and sequences over Zpr.

A Polynomial Model for Logics with a Prime Power Number of Truth Values

This paper is concerned with a polynomial model (residue class ring) for a given q-valued propositional logic (where q is a power of a prime integer). This model allows to transfer logic problems into algebraic terms, resulting in an immediate computational approach to Knowledge Based Systems based on multi-valued logics. By means of this new approach, we have extended an already existent algebraic model to logics with a prime power number of truth values, while also getting more straightforward proofs and a more direct enunciation of the central theorem of this model.

Modular Las Vegas algorithms for polynomial absolute factorization

Let f(X,Y)∈Z[X,Y] be an irreducible polynomial over Q. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f, or more precisely, of f modulo some prime integer p. The same idea of choosing a p satisfying some prescribed properties together with LLL is used to provide a new strategy for absolute factorization of f(X,Y). We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to construct the algebraic extension containing one absolute factor of a polynomial of degree up to 400.

A binary linear recurrence sequence of composite numbers

Let ( a , b ) ∈ Z 2 , where b ≠ 0 and ( a , b ) ≠ ( ± 2 , − 1 ) . We prove that then there exist two positive relatively prime composite integers x 1 , x 2 such that the sequence given by x n + 1 = a x n + b x n − 1 , n = 2 , 3 , … , consists of composite terms only, i.e., | x n | is a composite integer for each n ∈ N . In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs ( a , ± 1 ) , computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where ( a , b ) = ( 1 , 1 ) .

Medial groupoid varieties defined by finite cyclic groups

Let G be the abelian group generated by α and β. We write Σ(G; α, β) for the set of groupoid identities that are satisfied in the group ring \({\mathbb{Z}[G]}\) when the binary operation is αx + βy. When the constant e, representing the zero of \({\mathbb{Z}}\) , is added to the type, we write Σe(G; α, β) for the corresponding set of identities. In this paper, we assume that G is an infinite cyclic group with generator δ. We write Σa,b for Σ(G; δa, δb) and \({\Sigma^{e}_{a,b}}\) for Σe(G; δa, δb). For each pair of relatively prime integers a and b, we determine whether Σa,b is finitely based and whether \({\Sigma^{e}_ {a,b}}\) is finitely based. One of our results implies that Σe(G; α, β) is finitely based whenever G is finite.

Essential 𝑝-dimension of 𝐏𝐆𝐋(𝐩²)

Letppbe a prime integer andFFa field of characteristic different frompp. We prove that the essentialpp-dimension of the groupPGLF⁡(p2)\operatorname {\mathbf {PGL}}_F(p^2)is equal top2+1p^2+1. This integer measures the complexity of the class of central simple algebras of degreep2p^2over field extensions ofFF.

The Frobenius Complex

Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers $\mathbb{Z}$, that is, for a sub-semigroup $\Lambda$ of the non-negative integers $(\mathbb{N},+)$, we define the order by $n \leq_{\Lambda} m$ if $m-n \in \Lambda$. When $\Lambda$ is generated by two relatively prime integers $a$ and $b$, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when $\Lambda$ is generated by the integers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, the order complex is homotopy equivalent to a wedge of spheres. Motivé par le problème de Frobenius classique, nous introduisons l'ensemble partiellement ordonné de Frobenius sur les entiers $\mathbb{Z}$, c.à.d. que pour un sous-semigroupe $\Lambda$ de les entiers non-négatifs $(\mathbb{N},+)$ nous définissons l'ordre par $n \leq_{\Lambda} m$ si $m-n \in \Lambda$. Quand le $\Lambda$ est engendré par deux nombres $a$ et $b$, relativement premiers entre eux, nous montrons que le complexe des chaînes d'un intervalle quelconque dans l'ensemble partiellement ordonné de Frobenius est soit contractible soit homotopiquement équivalent à une sphère. Nous montrons aussi que dans le cas où $\Lambda$ est engendré par les entiers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, le complexe des chaînes a le type de homotopie d'un bouquet de sphères.

On quasi-cyclic codes over $${\mathbb{Z}_q}$$

Quasi-cyclic (QC) codes are a remarkable generalization of cyclic codes. Many QC codes have been shown to be best for their parameters. In this paper, some structural properties of QC codes over the prime power integer residue ring $${\mathbb{Z}_q}$$are considered. An l-QC code of length lm over $${\mathbb{Z}_q}$$is viewed both as in the conventional row circulant form and also as a $${\frac{\mathbb{Z}_q[x]}{\langle x^m-1 \rangle}}$$-submodule of $${\frac{GR(q,l)[x]}{\langle x^m-1 \rangle}}$$, where GR(q, l) is the Galois extension ring of degree l over $${\mathbb{Z}_q}$$. A necessary and sufficient condition for cyclic codes over Galois rings to be free is obtained and a BCH type bound for them is also given. A sufficient condition for 1-generator QC codes to be $${\mathbb{Z}_q}$$-free is given and a formula to evaluate their ranks is derived. Some distance bounds for 1-generator QC codes are also discussed. The duals of QC codes over $${\mathbb{Z}_q}$$are also briefly discussed.

The lower bound of the 𝑤-indices of surface links via quandle cocycle invariants

The w w -index of a surface link F F is the minimal number of the triple points of surface braids representing F F . In this paper, for a given 3 3 -cocycle, we consider the minimal number of the w w -indices of surface links whose quandle cocycle invariants associated with f f are non-trivial, and denote it ω ( f ) \omega (f) . In particular, we show that ω ( θ 3 ) = 6 \omega (\theta _3)=6 and ω ( θ p ) ≥ 7 \omega (\theta _p)\geq 7 , where θ n \theta _n is Mochizuki’s 3 3 -cocycle of the dihedral quandle of order n n and p p is an odd prime integer ≠ 3 \not =3 . As a consequence, for a given non-negative integer g g , there are surface knots with genus g g with the w w -index 6 6 .

Accelerated Search for Gaussian Generator Based on Triple Prime Integers

Problem statement: Modern cryptographic algorithms are based on complexity of two problems: Integer factorization of real integers and a Discrete Logarithm Problem (DLP). Approach: The latter problem is even more complicated in the domain of complex integers, where Public Key Cryptosystems (PKC) had an advantage over analogous encryption-decryption protocols in arithmetic of real integers modulo p: The former PKC have quadratic cycles of order O (p2) while the latter PKC had linear cycles of order O(p). Results: An accelerated non-deterministic search algorithm for a primitive root (generator) in a domain of complex integers modulo triple prime p was provided in this study. It showed the properties of triple primes, the frequencies of their occurrence on a specified interval and analyzed the efficiency of the proposed algorithm. Conclusion: Numerous computer experiments and their analysis indicated that three trials were sufficient on average to find a Gaussian generator.

Generalized Reed–Muller codes over $${\mathbb{Z}_q}$$

We have given a generalization of Reed---Muller codes over the prime power integer residue ring $${\mathbb{Z}_q}$$ . These codes are analogs of generalized Reed---Muller (GRM) codes over finite fields. We mainly focus on primitive GRM codes, which are basically a generalization of Quaternary Reed---Muller (QRM) codes. We have also given a multivariate representation of these codes. Non-primitive GRM codes over $${\mathbb{Z}_q}$$ are also briefly discussed. It has been shown that GRM codes over $${\mathbb{Z}_q}$$ are free extended cyclic codes. A trace description of these codes is also given. We have obtained formulas for their ranks and also obtained expressions for their minimum Hamming distances.

ON THE CONJECTURE OF JEŚMANOWICZ CONCERNING PYTHAGOREAN TRIPLES

AbstractLeta,b,cbe relatively prime positive integers such thata2+b2=c2withbeven. In 1956 Jeśmanowicz conjectured that the equationax+by=czhas no solution other than (x,y,z)=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly dividesb. The main results of this paper include the case where 8 dividesb. One of our results treats the case whereahas no prime factor congruent to 1 modulo 4, which can be regarded as a relevant analogue of results due to Deng and Cohen concerning the prime factors ofb. Furthermore, we examine parities of the three variablesx,y,z, and give new triplesa,b,csuch that the conjecture holds for the case wherebis divisible by 8. In particular, to prove our results, we shall show an important result which asserts that ifx,y,zare all even, thenx/2,y/2,z/2 are all odd. Our methods are based on elementary congruence and several strong results on generalized Fermat equations given by Darmon and Merel.