Nonzero Integer Research Articles

A New Type of Inclusion Exclusion Principle for Sequences and Asymptotic Formulas for ζ( k)

Let { a( n)} ∞ n = 0 be a sequence defined by the recurrence a( n + 2) = Aa( n + 1) + Ba( n), where a(0) = 0 and a(1), A, B are non-zero integers satisfying A 2 + 4 B ≠ 0. Assume that the ratio two roots of x 2 − Ax − B is not a root of unity. It is proved, for each positive integer l, that [formula] where [formula] and κ is a constant, depends only on A and B. This formula is an improvement and a generalization of the results of S. Akiyama (1990, J. Number Theory 36, 328-331). In the proof of this formula, we propose a new type of inclusion exclusion principle in a multiplicative sense which itself is of interest.

Quadratic forms in normal open induction

AbstractModels of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + b Y2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a model of NOI.We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.

On generalized quadratic equations in three prime variables

Leta 1,a 2 anda 3 be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation $$a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b$$ is insoluble in primesp j. We obtain CardE(X)≤X 1-e. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation $$x_1^2 + x_2^2 + x_3^k = b$$ in integral variablesx j with the above bound for CardE(X) better than that in [2].

Units differing by rationals in a cyclotomic field*

The problem of classifying the integers that can occur as the difference of two units in the cyclotomic field of prime level p > 3 is considered. It is shown that if d is such a nonzero integer, then (d,p) = 1.

On solving the Diophantine equation 𝑥³+𝑦³+𝑧³=𝑘 on a vector computer

Recently, the first author has proposed a new algorithm for solving the Diophantine equation x 3 + y 3 + z 3 = k {x^3} + {y^3} + {z^3} = k , where k is a given nonzero integer. In this paper we present the detailed versions of this algorithm for some values of k given below, and we describe how we have optimized and run the algorithm on a Cyber 205 vector computer. A vectorized version of the Euclidean algorithm was written, which is capable of solving the equations w i x i ≡ 1 mod n i , i = 1 , 2 , … {w_i}{x_i} \equiv 1 \bmod {n_i},i = 1,2, \ldots , at vector speed. Moreover, the basic double-precision arithmetic operations ( + , − , × , / ) ( + , - , \times ,/) were vectorized. The following cases were implemented and run: k = 3 , 30 , 2 , 20 , 39 k = 3,30,2,20,39 , and 42. For k = 3 k = 3 a range was searched which includes the cube | x | , | y | , | z | ≤ 10 8 |x|,|y|,|z| \leq {10^8} ; this considerably extends an earlier search in the cube | x | , | y | , | z | ≤ 2 16 |x|,|y|,|z| \leq {2^{16}} . No solutions were found apart from the known ones (1, 1, 1) and ( 4 , 4 , − 5 ) (4,4, - 5) . For k = 30 k = 30 , which probably is, with k = 3 k = 3 , the case which has attracted most attention in the literature, no solution was found. It is the smallest case for which no solution is known and for which one has not been able to find a proof that no solution exists. For k = 2 k = 2 a parametric form solution is known, but we also found one which does not belong to this parametric form, viz., ( 1214928 , 3480205 , − 3528875 ) (1214928,3480205, - 3528875) . For k = 20 k = 20 , several new large solutions were found in addition to several known ones; this case served as a (partial) check of the correctness of our program. Finally, for k = 39 k = 39 we found the first solution: ( 134476 , 117367 , − 159380 ) (134476,117367, - 159380) . Hence, this case can be dropped from the list of values of k ( k = 30 , 33 , 39 , 42 , … ) (k = 30,33,39,42, \ldots ) for which no solution is known (yet).

Planar Oscillation of the Satellite Near Parametric Resonance in the Elliptic Orbit

AbstractThis paper is devoted to the study of motion of an artificial satellite relative to its centre of mass near parametric resonance in elliptic orbit. It is well known fact that the satellite, of the form of an ellipsoid with three unequal axes, while moving about the central planet, oscillates about the stable position of equilibrium (the longest axis of the satellite coinciding with the radius vector of its centre of mass). The oscillation of the satellite about this position of equilibrium in the orbital plane of its centre of mass is described by a well known second order nonlinear differential equation with a periodic sine force. Naturally there will be resonance cases (main as well as parametric) for such a systems. In the previous author’s work, it was discovered a series of parametric resonances for the system, which corresponds to n = ½k where k is a non-zero integer and n is a parameter depending on the shape of the satellite. The parametric resonance, for k = 1, has been considered here. The first approximate solution of the equation of motion has been obtained by Eogoliubov-Krilov method with e (the eccentricity of the orbit of the centre of mass of the satellite) as the small parameter. This method enables us to visualise the oscillation of the satellite for the resonance case as well as near the resonance. Three stationary values of the amplitudes and phase of oscillation have been obtained, out of which only one is stable near this particular parametric resonance. At the resonance there appear only one stationary regime of oscillation with a very small amplitude.

Conditional bounds for small prime solutions of linear equations

Leta 1,a 2,a 3 be non-zero integers with gcd(a 1 a 2,a 3)=1 and letb be an arbitrary integer satisfying gcd (b, a i,a j) =1 fori≠j andb≡a 1+a 2+a 3 (mod 2). In a previous paper [3] which completely settled a problem of A. Baker, the 2nd and 3rd authors proved that ifa 1,a 2,a 3 are not all of the same sign, then the equationa 1 p 1+a 2 p 2+a 3 p 3=b has a solution in primesp j satisfying $$\mathop {\max }\limits_{1 \leqslant j \leqslant 3} p_j \leqslant 3\left| b \right| + (3\mathop {\max }\limits_{1 \leqslant j \leqslant 3} \left| {a_j } \right|)^A $$ whereA>0 is an absolute constant. In this paper, under the Generalized Riemann Hypothesis, the authors obtain a more precise bound for the solutionsp j . In particular they obtainA<4+∈ for some ∈>0. An immediate consquence of the main result is that the Linnik's courtant is less than or equal to 2.

Distances from differences of roots of polynomials to the nearest integers

We present tight bounds for distances from differences of roots of the polynomial f(x)∈ R[x] over a discrete normed commutative ring R without zero divisors to the nearest element of R . In the case most interesting for applications, R= Z , an algorithm for the determination of the existence of nonzero integers among these differences in time ( n 4log( H( f)+1)) 1+ ε is given. This problem arises when constructing algorithms for solving some systems of ODE.

Some remarks on the eigenvalue problem for [formula omitted]n symmetric vertex and face models

The eigenvalue solution of the transfer matrix of Belavin's Z n symmetric vertex model is further extended to the case of the model with the crossing parameter ω = 1 / L, here L is a nonzero positive integer. Then the so-called cyclic IRF model is defined and the eigenvalue solution of its transfer matrices is found. The eigenvalue spectrum of the IRF model at critical limit is shown to be the same as that of a modified many component six-vertex model, therefore the spectrums of both models are analysed with the quantum group SU q ( n).

MULTIPLICATIVE FUNCTIONS ON THE SET OF SHIFTED PRIME NUMBERS

This paper is concerned with the behavior of multiplicative functions on the set , where is a prime and is a nonzero integer. Several results are obtained in which either the average value of a multiplicative function on this set is estimated or its asymptotic behavior is determined. As one application a nontrivial estimate of is found, where , is a sufficiently large prime, and is a character of degree greater than 4.

Accurate current amplification by using dynamic current copier

A current-mode circuit consisting of dynamic current copiers is presented. The circuit can perform the sum of two input currents weighted by any two integer numbers. Based on this circuit, an accurate current amplifier with an amplification factor equal to the ratio of two arbitrary positive nonzero integers can be built.

On the quasi-splitting of exact sequences

Perhaps the most significant difference between the theories of torsion and torsion-free abelian groups is the large number of splitting results in the former, and the lack of such in the latter. For instance, Szele’s theorem [F, Vol. 1, Proposition 27.11 has no obvious equivalent in the theory of torsion-free abelian groups, although it is one of the widely used results in the discussion of torsion and mixed groups: Let B g QJI Z/p’7 be a subgroup of an abelian group G. Then B is a direct summand of G if and only if B np”G = 0. The author was able to extend Szele’s theorem to valuated p-groups in [A2, Satz 6.81 using methods similar to those which proved useful in the discussion of Baer’s lemma in [AL], [A3], [A4]. It is the goal of this paper to prove a result which can be considered the equivalent of Szele’s theorem in the quasi-category of torsion-free abelian groups. Before we can describe the result, it is necessary to introduce some notation. We consider an abelian group A. The class of A-projective groups of finite A-rank is obtained by closing {A} with respect to direct summands and finite direct sums. If G is an abelian group, then the A-radical of G, which is denoted by R,(G), is R,(G) = n {ker 41 #E Hom(G, A)). Finally, consider modules B and C over a ring R. A homomorphism ~1: B + C quasi-splits if there are a non-zero integer n and a map /3~Hom,(C, B) with /?cr=n.id,, where id, denotes the identity map on B. If the additive group of B is torsion-free, then a quasi-splitting homomorphism is one-to-one [Jl, 52, Re]. We say that an abelian group A has the radical-splitting property if every exact sequence 0 + PA G of torsion-free abelian groups, in which P is a quasi-summand of an A-projective group of finite A-rank and u(P) n R,(G) = 0, quasi-splits. Szele’s original result guarantees the splitting of the sequence in the case in which A z Z/p’7 for some n < w. Moreover, if A z Z/p”Z, then R,(G) = p”G. Theorem 3.1 of this paper establishes that every strongly indecomposable group of finite rank has the

On the theory of superfluidity in two dimensions

The superfluid phase transition of the general vortex gas, in which the circulations may be any non-zero integer, is studied. When the net circulation of the system is not zero the absence of a superfluid phase is shown. When the net circulation of the vortices vanishes, the presence of off-diagonal long range order is demonstrated and the existence of an order parameter is proposed. The transition temperature for the general vortex gas is shown to be the Kosterlitz-Thouless temperature. An upper bound for the average vortex number density is established for the general vortex gas and an exact expression is derived for the Kosterlitz-Thouless ensemble.

On partition regular systems of equations

The theory of partition regular systems of linear equations over the set of integers has been developed by Rado and Deuber. In this paper the nonlinear situation is considered. In particular, we characterize with respect to partition regularity systems of homogeneous equations, where in each equation the variables are raised to the same exponent 1 k for nonzero integers k and in the whole system each variable is raised to exactly one exponent.

Catalan’s equation 𝑥^{𝑝}-𝑦^{𝑞}=1 and related congruences

We investigate solutions of Catalan’s equation x p − y q = 1 {x^p} - {y^q} = 1 in nonzero integers x, y, p, q. By use of class numbers and congruences p q ≡ p ( mod q 2 ) {p^q} \equiv p\;\pmod {q^2} we show the impossibility of the equation for a large number of prime exponents p, q. Both theoretical and computer results are included. We also study lower bounds on possible, hitherto unknown, solutions x , y &gt; 2 x,y &gt; 2 ; we especially wish to communicate the bound x , y ≥ 10 500 x,y \geq {10^{500}} .

Gauss polynomials and the rotation algebra

Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=γab with γ commuting witha andb, then the (generalized) binomial coefficient $$\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)_r $$ arising in the expansion $$\left( {a + b} \right)^n = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} _\gamma a^{n - k} b^k $$ (resulting from these relations) is equal to the value at γ of the Gaussian polynomial $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right] = \frac{{\left[ n \right]}}{{\left[ k \right]\left[ {n - k} \right]}}$$ where [m]=(1-x m )(1-x m−1)...(1-x). (This is of course known in the case γ=1.) From this it is deduced that in the (universal)C *-algebraA gq generated by unitariesu andv such thatvu=e 2πiθ uv, the spectrum of the self-adjoint element (u+v)+(u+v)* has all the gaps that have been predicted to exist-provided that either θ is rational, or θ is a Liouville number. (In the latter case, the gaps are labelled in the natural way-viaK-theory-by the set of all non-zero integers, and the spectrum is a Cantor set.)

Lehmer numbers and an asymptotic formula for π

Let { a n } n = 0 ∞ be an integer sequence defined by the non-degenerate binary linear recurrence a n = A a n − 1 + Ba n − 2 , where a 0 = 0, a 1 ≠ 0, and A, B are fixed non-zero integers. It is proved, for a certain constant κ, that 6(1−K) log|a 1a 2…a N| log[A 1,a 2,…,a N] 1 2 =π+0 1 logn , which is the generalization of the formula of P. Kiss and F. Mátyás.

Vortices in complex scalar fields

Consider a spatially extended field which evolves in time according to a PDE. The solutions contain particle-like defects whose motions are parts of the full dynamics. We inquire into the formulation of an asymptotic “particle + field” defect dynamics which derives from the full PDE. We carry out this program for two complex scalar field equations in two space dimensions, the nonlinear Schrödinger equation and the nonlinear heat equation. The topological defects are zeros of the complex scalar field with nonzero integer winding numbers, called vortices. Vortices evolving under the nonlinear Schrödinger equation behave like point vortices in ideal fluid. Pairs of vortices evolving under the nonlinear heat equation with like (opposite) winding numbers undergo a repulsive (attractive) interaction.

On almost-primes in arithmetic progressions

for any fixed non-zero integer a and almost-all moduli q with {q, a)=l. His argument based upon the weighted linear sieve and the Brun-Titchmarsh theorem on average, which are due to H. E. Richert [10] and C. Hooley [4], respectively. H. Iwaniec's fundamental works [6, 7], therefore, suggest possibilities of an improvement upon the above result. In this paper we present an estimation for P2. We shall prove the following

Perturbed isotropic harmonic oscillator

The isotropic harmonic oscillator is studied under the action of a central perturbative force n beta /rn+1, where n is a non-zero integer. The case n=2 is solved exactly. For other values of n a perturbative method that allows the determination of the speed ore precession and the polar equation of the orbit is developed; the method is also applied to the case of the charged isotropic oscillator in a constant magnetic field.