Nonzero Integer Research Articles

Shewhart control schemes with supplementary 2‐of‐(h + 1) side‐sensitive runs‐rules under the Burr‐type XII distribution

AbstractControl schemes that require strict distributional assumptions are called parametric control schemes. When the distributional assumptions fail to hold, nonparametric control schemes and other schemes based on more flexible probability distributions are needed. The traditional Shewhart control scheme has been improved upon using various techniques, one being the addition of non–side‐sensitive (NSS) and side‐sensitive runs‐rules. Many authors showed that side‐sensitive runs‐rules schemes are significantly superior to NSS runs‐rules schemes. In this paper, we propose side‐sensitive Shewhart‐type schemes supplemented with 2‐of‐(h + 1) standard and improved runs‐rules (where h is a nonzero positive integer) under the violation of the normality assumption for monitoring the process mean. The zero‐state and steady‐state performances of the proposed schemes are investigated using a Markov chain approach. It is observed that the proposed schemes outperform the existing schemes in many situations and possess very interesting zero‐state and steady‐state properties under normal and non‐normal distributions. An illustrative example is provided to facilitate the design and implementation of the proposed schemes.

FDTD: Solving 1+1D delay PDE in parallel

We present a proof of concept for solving a 1+1D complex-valued, delay partial differential equation (PDE) that emerges in the study of waveguide quantum electrodynamics (QED) by adapting the finite-difference time-domain (FDTD) method. The delay term is spatially non-local, rendering conventional approaches such as the method of lines inapplicable. We show that by properly designing the grid and by supplying the (partial) exact solution as the boundary condition, the delay PDE can be numerically solved. In addition, we demonstrate that while the delay imposes strong data dependency, multi-thread parallelization can nevertheless be applied to such a problem. Our code provides a numerically exact solution to the time-dependent multi-photon scattering problem in waveguide QED. Program summaryProgram Title: FDTD: solving 1+1D delay PDEProgram Files doi:http://dx.doi.org/10.17632/mmyw3fgjxh.1Licensing provisions: MITProgramming language: C (C99)Nature of problem: This program solves an unconventional 1+1D delay PDE that emerges in the study of waveguide quantum electrodynamics. The delay PDE is complex-valued and has a non-local delay term, and the solution to it provides the full dynamics of the system consisting of a few 1D photons and a two-level system in front of a mirror.Solution method: The finite-difference time-domain (FDTD) method is adapted. Given the initial condition of the system, the corresponding boundary condition is generated, and then the FDTD solver marches through the entire spacetime grid. Multiple solvers are supported using either OpenMP (wavefront) or pthreads (swarm).Additional comments including restrictions and unusual features: 1. Depending on the input parameters the memory and disk usages of the program can be excessive, so the users should choose the parameters wisely (see main text). 2. The multi-thread support using OpenMP is turned on by default. See README for how to turn it off and switch to pthreads instead. 3. As a by-product, a numerical routine is provided for evaluating the incomplete Gamma function γ(n,z) with nonzero positive integers n≥1 and complex-valued z.

Pythagorean triples containing generalized Lucas numbers

Let $P$ and $Q$ be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: $U_{0}(P,Q)=0,U_{1}(P,Q)=1,$ and $ U_{n+1}(P,Q)=PU_{n}(P,Q)+QU_{n-1}(P,Q)$ for $n\geq 1$ and $ V_{0}(P,Q)=2,V_{1}(P,Q)=P,$ and $V_{n+1}(P,Q)=PV_{n}(P,Q)+QV_{n-1}(P,Q)$ for $n\geq 1,$ respectively. In this paper, we assume that $P$ and $Q$ are relatively prime odd positive integers and $P^{2}+4Q>0.$ We determine all indices $n$ such that $U_{n}=(P^{2}+4Q)x^{2}.$ Moreover, we determine all indices $n$ such that $(P^{2}+4Q)U_{n}=x^{2}.$ As a result, we show that the equation $V_{n}^{2}(P,1)+V_{n+1}^{2}(P,1)=x^{2}$ has solution only for $n=2,$ $P=1,$ $x=5$ and $V_{n+1}^{2}(P,-1)=V_{n}^{2}(P,-1)+x^{2}$ has no solutions. Moreover, we solve some Diophantine equations.

Generalized confluent hypergeometric solutions of the Heun confluent equation

We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.

Sums of two cubes as twisted perfect powers, revisited

We sharpen earlier work (2011) of the first author, Luca and Mulholland, showing that the Diophantine equation A 3 + B 3 = q α C p , A B C ≠ 0 , gcd ( A , B ) = 1 , has, for “most” primes q and suitably large prime exponents p , no solutions. We handle a number of (presumably infinite) families where no such conclusion was hitherto known. Through further application of certain symplectic criteria, we are able to make some conditional statements about still more values of q ; a sample such result is that, for all but O ( x ∕ log x ) primes q up to x , the equation A 3 + B 3 = q C p . has no solutions in coprime, nonzero integers A , B and C , for a positive proportion of prime exponents p .

Completely p-primitive binary quadratic forms

Let f(x,y)=ax2+bxy+cy2 be a binary quadratic form with integer coefficients. For a prime p not dividing the discriminant of f, we say f is completely p-primitive if for any non-zero integer N, the diophantine equation f(x,y)=N always has an integer solution (x,y)=(m,n) with (m,n,p)=1 whenever it has an integer solution. In this article, we study various properties of completely p-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form f to be completely p-primitive.

A lower bound for the number of integral solutions of Mordell equation

For a nonzero integer $d$, a celebrated Siegel Theorem says that the number $N(d)$ of integral solutions of Mordell equation $y^2+x^3=d$ is finite. We find a lower bound for $N(d)$, showing that the number of solutions of Mordell equation increases dramatically. We also prove that for any positive integer $n$, there is an integer square multiply represented by Mordell equations, i.e., $k^2=y_1^2+x_1^3=y_2^2+x_2^3=\cdots =y_n^2+x_n^3$.

Construction of Period <inline-formula> <tex-math notation="LaTeX">$qp$ </tex-math> </inline-formula> PGISs With Degrees Equal to or Larger Than Four

The degree of a perfect Gaussian integer sequence (PGIS) is defined as the number of distinct nonzero Gaussian integers within one period of the sequence. This paper focuses on constructing PGISs with degrees equal to or larger than four and period of $N = qp$ , where $q$ and $p$ are distinct primes. The study begins with the partitioning of a ring $\mathbb {Z}_{N}$ into four subsets, after which degree-4 PGISs can be constructed from either the time or frequency domain. In these two approaches, nonlinear constraint equations are derived to govern the coefficients for the associative sequences to be perfect. By transforming nonlinear constraint equations into a system of linear equations, the construction of degree-4 PGISs becomes straightforward. To construct PGISs with degrees larger than four, further partitioning of $\mathbb {Z}_{N}$ should be carried out; here, two cases, the even period $N = 2p$ and the odd period $N = qp$ , are treated separately. We can adopt the Legendre sequences of the prime period $p$ to construct PGISs of period $2p$ with degrees larger than four. For the case of period $qp$ , we introduce the Jacobi symbols to partition $\mathbb {Z}_{N}$ into seven subsets and construct PGISs with more diverse degrees.

$S$-parts of values of univariate polynomials,\\ binary forms and decomposable forms at integral points

Let $S = \{p_1, \ldots , p_s\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m = p_1^{a_1} \ldots p_s^{a_s} b$, where $a_1, \ldots , a_s$ are non-negative integers and $b$ is an integer relatively prime to

A note on multiplicative functions on progressions to large moduli

Let f : ℕ → ℂ be a bounded multiplicative function. Let a be a fixed non-zero integer (say a = 1). Then f is well distributed on the progression n ≡ a (mod q) ⊂ {1,…, X}, for almost all primes q ∈ [Q, 2Q], for Q as large as X1/2+1/78−o(1).

On the number of distinct prime factors of a sum of super-powers

Given k,ℓ∈N+, let xi,j be, for 1≤i≤k and 0≤j≤ℓ, some fixed integers, and define, for every n∈N+, sn:=∑i=1k∏j=0ℓxi,jnj. We prove that the following are equivalent:(a)There are a real θ>1 and infinitely many indices n for which the number of distinct prime factors of sn is greater than the super-logarithm of n to base θ.(b)There do not exist non-zero integers a0,b0,…,aℓ,bℓ such that s2n=∏i=0ℓai(2n)i and s2n−1=∏i=0ℓbi(2n−1)i for all n. We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes S, an effective bound on the number of non-degenerate solutions of an S-unit equation in k variables over the rationals) and the other using only elementary methods.As a corollary, we find that, for fixed c1,x1,…,ck,xk∈N+, the number of distinct prime factors of c1x1n+⋯+ckxkn is bounded, as n ranges over N+, if and only if x1=⋯=xk.

Subgroup avoidance for primes dividing the values of a polynomial

For $f \in \mathbb{Q} [x]$, we say that a rational prime~$p$ is a prime divisor of $f$ if $p$ divides the numerator of $f(n)$ for some integer $n$. Let $\mathcal{P} (f)$ denote the set of prime divisors of~$f$. We present an elementary proof of the following theo\-rem, which generalizes results of Bauer and Brauer: fix a nonzero integer~$g$. Suppose that $f(x) \in \mathbb{Q} [x]$ is a nonconstant polynomial having a root in $\mathbb{Q} _p$ for every prime $p$ dividing $g$, and having a root in $\mathbb{R} $ if $g \lt 0$. Let $m$ be a positive integer coprime to~$g$, and let~$H$ be a subgroup of $(\mathbb{Z} /m\mathbb{Z} )^{\times }$ not containing $g\bmod {m}$. Then there are infinitely many primes $p \in \mathcal{P} (f)$ with $p\bmod {m} \notin H$.

Seifert fibrations of lens spaces

We classify the Seifert fibrations of any given lens space L(p, q). Starting from any pair of coprime non-zero integers $$\alpha _1^0,\alpha _2^0$$ , we give an algorithmic construction of a Seifert fibration $$L(p,q)\rightarrow S^2(\alpha |\alpha _1^0|,\alpha |\alpha _2^0|)$$ , where the natural number $$\alpha $$ is determined by the algorithm. This algorithm produces all possible Seifert fibrations, and the isomorphisms between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are isomorphic to certain standard models.

Effective resolution of Diophantine equations of the form $$u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}$$ u n + u m = w p 1 z 1 ⋯ p s z s

Let $$u_n$$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, w a fixed non-zero integer and $$p_1,p_2,\ldots ,p_s$$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $$u_n+u_m=w p_1^{z_1} \ldots p_s^{z_s}$$ and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for n, m and $$z_1, \ldots , z_s$$ . Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.

Small prime solutions of a nonlinear equation

Let a1,⋯,a4 be non-zero integers and n any integer. Suppose that a1,⋯,a4 and n satisfy some related conditions. In this paper we prove that(i)if aj are not all of the same sign, then the equation a1p1+a2p22+a3p32+a4p42=n has prime solutions satisfying max⁡{p1,p22,p32,p42}≪|n|+max⁡{|aj|}14+ε;(ii)if all aj are positive and n≫max⁡{|aj|}15+ε, then the equation a1p1+a2p22+a3p32+a4p42=n is soluble in primes pj.

Chiral topological excitons in a Chern band insulator

A family of semiconductors called as Chern band insulator are shown to host exciton bands with non-zero topological Chern integers and chiral exciton edge modes. Using a prototypical two-band Chern insulator model, we calculate a cross-correlation function to obtain the exciton bands and their Chern integers. The lowest exciton band acquires Chern integers such as $\pm 1$ and $\pm 2$ in electronic Chern insulator phase. The non-trivial topology can be experimentally observed both by non-local optoelectronic response of exciton edge modes and by a phase shift in the cross-correlation response due to the bulk mode. Our result suggests that magnetically doped HgTe, InAs/GaSb quantum wells and $\text{(Bi,Sb)}_{2} \text{Te}_{3}$ thin film are promising candidates for a platform of topological excitonics.

An Accurate and Robust Geometrically Exact Curved Beam Formulation for Multibody Dynamic Analysis

An accurate and robust geometrically exact beam formulation (GEBF) is developed to simulate the dynamics of a beam with large deformations and large rotations. The undeformed configuration of the centroid line of the beam can be either straight or curved, and cross sections of the beam can be either uniform or nonuniform with arbitrary shapes. The beam is described by the position of the centroid line and a local frame of a cross section, and a rotation vector is used to characterize the rotation of the cross section. The elastic potential energy of the beam is derived using continuum mechanics with the small-strain assumption and linear constitutive relation, and a factor naturally arises in the elastic potential energy, which can resolve a drawback of the traditional GEBF. Shape functions of the position vector and rotation vector are carefully chosen, and numerical incompatibility due to independent discretization of the position vector and rotation vector is resolved, which can avoid the shear locking problem. Numerical singularity of the rotation vector with its norm equal to zero is eliminated by Taylor polynomials. A rescaling strategy is adopted to resolve the singularity problem with its norm equal to 2mπ, where m is a nonzero integer. The current formulation can be used to handle linear and nonlinear dynamics of beams under arbitrary concentrated and distributed loads. Several benchmark problems are simulated using the current formulation to validate its accuracy, adaptiveness, and robustness.

Triples which are D(n)-sets for several n's

For a nonzero integer n, a set of distinct nonzero integers {a1,a2,…,am} such that aiaj+n is a perfect square for all 1≤i<j≤m, is called a Diophantine m-tuple with the property D(n) or simply D(n)-set. D(1)-sets are known simply as Diophantine m-tuples. Such sets were first studied by Diophantus of Alexandria, and since then by many authors. It is natural to ask if there exists a Diophantine m-tuple (i.e. D(1)-set) which is also a D(n)-set for some n≠1. This question was raised by Kihel and Kihel in 2001. They conjectured that there are no Diophantine triples which are also D(n)-sets for some n≠1. However, the conjecture does not hold, since, for example, {8,21,55} is a D(1) and D(4321)-triple, while {1,8,120} is a D(1) and D(721)-triple. We present several infinite families of Diophantine triples {a,b,c} which are also D(n)-sets for two distinct n's with n≠1, as well as some Diophantine triples which are also D(n)-sets for three distinct n's with n≠1. We further consider some related questions.

Equation-regular sets and the Fox–Kleitman conjecture

Given k≥1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation ∑i=1k(xi−yi)=bis (2k−1)-regular. This is best possible, since Fox and Kleitman showed that for all b≥1, this equation is not 2k-regular. While the conjecture has recently been settled for all k≥2, here we focus on the case k=3 and determine the degree of regularity of the corresponding equation for all b≥1. In particular, this independently confirms the conjecture for k=3. We also briefly discuss the case k=4.

The size of algebraic integers with many real conjugates

In this paper we show that the relative normalised size with respect to a number field K of an algebraic integer α ≠ -1, 0, 1 is greater than 1 provided that the number of real embeddings s of K satisfies s ≥ 0.828 n , where n = [ K : Q ]. This can be compared with the previous much more restrictive estimate s ≥ n − 0.192 √ ( n /log n ) and shows that the minimum m ( K ) over the relative normalised size of nonzero algebraic integers α in such a field K is equal to 1 which is attained at α = ± 1. Stronger than previous but apparently not optimal bound for m ( K ) is also obtained for the fields K satisfying 0.639 ≤ s / n < 0.827469…. In the proof we use a lower bound for the Mahler measure of an algebraic number with many real conjugates.