Diophantine System Research Articles

Stable transitivity of Heisenberg group extensions of hyperbolic systems

We consider skew-extensions with fibre the standard real Heisenberg group of a uniformly hyperbolic dynamical system.We show that among the Cr extensions (r > 0) that avoid an obvious obstruction, those that are topologically transitive contain an open and dense set. More precisely, we show that an -extension is transitive if and only if the -extension given by the Abelianization of is transitive.A new technical tool introduced in the paper, which is of independent interest, is a diophantine approximation result. We show, under general conditions, the existence of an infinite set of approximate positive integer solutions for a diophantine system of equations consisting of a quadratic indefinite form and several linear equations. The set of approximate solutions can be chosen to point in a certain direction. The direction can be chosen from a residual subset of full measure of the set of real directions solving the system of equations exactly.Another contribution of the paper, which is used in the proof of the main result, but it is also of independent interest, is the solution of the so-called semigroup problem for the Heisenberg group. We show that for a subset , which avoids any maximal semigroup with non-empty interior, the closure of the semigroup generated by S is actually a group.

Diophantine Approach to Blind Interference Alignment of Homogeneous K-User 2x1 MISO Broadcast Channels

The sufficient condition for a blindly interference-aligned 2-user 2 × 1 broadcast channel (BC) in homogeneous block fading to achieve its maximal 4/3 degree-of-freedom (DoF) is well understood. However, the condition for the general case, i.e., a homogeneous K-user 2 × 1 BC, to achieve the corresponding optimal 2K/2+K-1 DoF remains unsolved and is, thus, the focus of this paper. Here, a K-user 2×1 BC is said to be BIA-feasible if it achieves the optimal DoF via blind interference alignment (BIA). In this paper, we cast this general BIA-feasibility problem in the framework of finding solutions for a system of linear Diophantine equations. Then by studying the solvability of the Diophantine system, we derive the necessary and sufficient conditions on the K users' fading block offsets to ensure the BIA feasibility of the K-user BC. Using the results above, we also examine a 2 × 1 broadcast network with M homogeneous users, in which their offsets are independent of one another and are uniformly distributed over a coherence block. We show by simulations that when the network contains M = 11 homogeneous users, there is more than 95% chance of finding 3 users to form a BIA-feasible 3-user 2 × 1 BC. Furthermore, increasing M to infinity ensures a virtual multiple-input multiple-output (MIMO) channel to be formed from the DoF perspective.

Solving Rank One Perturbed Linear Diophantine Systems Using the Hermite Normal Form

We show how we can obtain the general solution of rank one perturbed linear Diophantine systems (A + uv T )x = b using only information from the application of the Hermite normal form algorithm to the corresponding linear Diophantine system Ax = b. The empirical results show that use of the proposed algorithm may result in saving considerable computing time.

Complementaries to Kummer's degree seven reciprocity law and a Dickson Diophantine system

Let ${\mathbb Q}(\zeta)$ be the cyclotomic field obtained from ${\mathbb Q}$ by adjoining a primitive seventh root of unity $\zeta$. Normalized primary elements of this field are characterized and related to Jacobi sums and to solutions of a system of quadratic Diophantine equations of Dickson type involving a rational prime $p\equiv 1\pmod{7}$. These objects and their connection are then used to give another formulation of the complementary laws to Kummer's reciprocity law of degree seven.

A symmetric diophantine system concerning fifth powers

A symmetric diophantine system concerning fifth powers

The multidimensional Frobenius problem

Consider the problem of determining maximal vectors g such that the Diophantine system Mx = g has no solution. We provide a variety of results to this end: conditions for the existence of g, conditions for the uniqueness of g, bounds on g, determining g explicitly in several important special cases, constructions for g, and a reduction for M.

EULER'S CRITERION FOR SEPTIC NONRESIDUES

Let p be a prime ≡ 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity ( mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D ( mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.

A kaleidoscope of solutions for a Diophantine system

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

On the diophantine system a 2 + b 2 = c 3 and a x + b y = c z for b is an odd prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a2 + b2 = c3 and b is an odd prime, then the equation ax + by = cz has only the positive integer solution (x, y, z) = (2, 2, 3).

Formula omitted]-extension of linear codes

We construct new linear codes with high minimum distance d . In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n , k . Among these new codes there is an optimal ternary [ 88 , 8 , 54 ] 3 code. We develop an algorithm, which starts with already good codes C , i.e. codes with high minimum distance d for given length n and dimension k over the field G F ( q ) . The algorithm is based on the newly defined ( l , s ) -extension. This is a generalization of the well-known method of adding a parity bit in the case of a binary linear code of odd minimum weight. ( l , s ) -extension tries to extend the generator matrix of C by adding l columns with the property that at least s of the l letters added to each of the codewords of minimum weight in C are different from 0 . If one finds such columns the minimum distance of the extended code is d + s provided that the second smallest weight in C was at least d + s . The question whether such columns exist can be settled using a Diophantine system of equations.

A simple (inductive) proof for the non-existence of 2-cycles of the [formula omitted] problem

A 2-cycle of the 3 x + 1 problem has two local odd minima x 0 and x 1 with x i = a i 2 k i − 1 . Such a cycle exists if and only if an integer solution exists of a diophantine system of equations in the coefficients a i . We derive a numerical lower bound for a 0 ⋅ a 1 , based on Steiner's proof for the non-existence of 1-cycles. We derive an analytical expression for an upper bound for a 0 ⋅ a 1 as a function of K and L (the number of odd and even numbers in the cycle). We apply a result of de Weger on linear logarithmic forms to show that these lower and upper bounds are contrary. The proof does not use exterior lower bounds for numbers in a cycle and for the cycle length.

Construction of Linear Codes With Large Minimum Distance

A natural goal in coding theory is to find a linear [n, k;q]-code such that the minimum distance d is maximal. In this paper, we introduce an algorithm to construct linear [n, k;q]-codes with a prescribed minimum distance d by constructing an equivalent structure, the so-called minihyper, which is a system of points in the (k - 1)-dimensional projective geometry P/sub k-1/(q) over the finite field F/sub q/ with q elements. To construct such minihypers we first prescribe a group of automorphisms, transform the construction problem to a diophantine system of equations, and then apply a lattice-point-enumeration algorithm to solve this system of equations. Finally, we present a list of parameters of new codes that we constructed with the introduced method. For example, there is a new optimal [80, 4; 8]-code with minimum distance 68.

An analogue of Van der Corput's A 5 ‐process for exponential sums

A diophantine system is studied from which is deduced an analogue of van der Corput's A5-process in order to bound analytic exponential sums of the form . The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a “ninth derivative test” for exponential sums which is essential for giving new exponent pairs in [3].

Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry

Mallard's law of observation states that known twins by reticular pseudomerohedry have low twin indexmand low obliquity δ between a lattice row and a lattice plane. Crystals in the twin are related by exact reflection in the lattice plane, exact rotation by π about the perpendicular to the lattice plane, or by exact rotation about the lattice row by 2kπ/q, withq= 2, 3, 4 or 6, and 1 ≤k<q. In known cases,mis up to five or six, and δ is up to five or six degrees. Mallard's converse problem is then about finding all pairs of indices for rows and planes leading to twin indices not larger thanmand to obliquities that are at most δ. Mallard's law is recast as the Diophantine pair constituted by the equalityh · u=nand the inequality |h × u| <n tan δ. If a primitive reference system is used, the integernis eitherm, 2mor 3m. A direct general solution of this system forugivenn,h, δ and lattice data is detailed. That straightforward solution involves a tiny two-dimensional grid foru, considerably reducing the number of permutations to be considered. If the primitive reference system is Buerger-reduced, then moduli for indices of solutionshcannot exceed 3m, thus establishing a simple way to produce complete solutions. A program calledOBLIQUEwas designed on such principles. An implementation is available for free execution at http://ylp.icpet.nrc.ca/oblique/.OBLIQUEis also an interactive tool in Toth Information Systems'Materials Toolkit(http://www.tothcanada.com/), an exploitation framework currently operating on CRYSTMET and ICSD crystal structure data. The example of quartz cell data is described withmmax= 3 and δmax= 6°.

On the complexity of recognizing the Hilbert basis of a linear diophantine system

The problem of computing the Hilbert basis of a homogeneous linear Diophantine system over nonnegative integers is often considered in automated deduction and integer programming. In automated deduction, the Hilbert basis of a corresponding system serves to compute the minimal complete set of associative-commutative unifiers, whereas in integer programming the Hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality. In this paper, we sharpen the previously known result that the problem, asking whether a given solution belongs to the Hilbert basis of a given system, is coNP-complete. We show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed, but it is coNP-complete in the strong sense if the given system is unbounded. This result is important in the scope of automated deduction, where the input is given in unary and therefore the previously known coNP-completeness result was unusable. Moreover, we show that, from the complexity standpoint, it is not important to know the underlying homogeneous linear Diophantine system when we ask whether a given set of vectors constitutes a Hilbert basis.

Renormalization group approach as an approximate solution of diophantine system of equations: Application to the Ising model

It is demonstrated that the real-space renormalization group approach is equivalent to an appropriate diophantine system of equations. The method is applied on several examples of the Ising model.

A Diophantine system

It is shown how to find all integers a, b such that a + b, a2 + b2 and a3 + b3 are simultaneously perfect squares.

Euler's Criterion for Quintic Nonresidues

Let e be an integer ≧ 2, and p a prime = 1 (mod e). Euler's criterion states that for D ∊ Z,(1.1)if and only if D is an e-th power residue (mod p). If D is not an e-th power (mod p), one has(1.2)for some e-th root α(≠1) of unity (mod p). Sometimes expressions for roots of unity (mod p) can be given in terms of quadratic partitions of p. For example,(1.3)are the four distinct fourth roots of unity (mod p) for a prime p ≡ 1 (mod 4) in terms of a solution (a, b) of the diophantine system(a, b unique), whereas for p ≡ 1 (mod 3), a solution (L, M) of the systemgives(1.4)as the three distinct cuberoots of unity (mod p).

Asymptotic expansion for the number of solutions of the diophantine system of Hilbert-Kamke with increasing number of variables

Asymptotic expansion for the number of solutions of the diophantine system of Hilbert-Kamke with increasing number of variables

Determination of a unique solution of the quadratic partition for primesp≡ 1 (mod7)

Let p be a rational prime = 1 (mod 7). Williams shows that a certain triple of a Diophantine system of quadratic equations has exactly six nontrivial solutions. We obtain here a congruence condition which uniquely fixes one of these six solutions. Further if 2 is not a seventh power residue (mod p) then we obtain a congruence (mod p) for 2 Cpl)/7 in terms of the above uniquely fixed solution.