scholarly journals Finding all S-Diophantine quadruples for a fixed set of primes S

2021 ◽  
Author(s):  
Volker Ziegler
Keyword(s):  
Prime Divisors ◽  
Fixed Set ◽  
Finite Set ◽  
Practical Algorithm ◽  
Diophantine Quadruples

AbstractGiven a finite set of primes S and an m-tuple $$(a_1,\ldots ,a_m)$$ ( a 1 , … , a m ) of positive, distinct integers we call the m-tuple S-Diophantine, if for each $$1\le i < j\le m$$ 1 ≤ i < j ≤ m the quantity $$a_ia_j+1$$ a i a j + 1 has prime divisors coming only from the set S. For a given set S we give a practical algorithm to find all S-Diophantine quadruples, provided that $$|S|=3$$ | S | = 3 .

scholarly journals Modular forms of degree n and representation by quadratic forms IV

1987 ◽  
Vol 107 ◽  
pp. 25-47
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Quadratic Form ◽  
Finite Index ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Prime Divisors ◽  
Positive Definite Quadratic Form ◽  
Finite Set ◽  
Quadratic Lattice

Let M be a quadratic lattice with positive definite quadratic form over the ring of rational integers, M’ a submodule of finite index, S a finite set of primes containing all prime divisors of 2[M: M’] and such that Mp is unimodular for p ∉ S. In [2] we showed that there is a constant c such that for every lattice N with positive definite quadratic form and every collection (fp)p∊s of isometries fp: NP → MP there is an isometry f: N → M satisfyingf ≡ fp mod M′p for every p |[M: M],f(Np) is private in Mp for every p ∉ S,provided the minimum of N ≥ c and rank M ≥ 3 rank N + 3.

scholarly journals A multi-robot allocation model for multi-object based on Global Optimal Evaluation of Revenue

2021 ◽  
Vol 18 (6) ◽  
pp. 172988142110606
Author(s):  
Xun Li ◽  
Zhi Zhang ◽  
Dan-Dan Wu ◽  
Michel Medema ◽  
Alexander Lavozik
Keyword(s):  
Numerical Simulation ◽  
Task Characteristics ◽  
Allocation Model ◽  
Object Based ◽  
Optimal Value ◽  
Fixed Set ◽  
Finite Set ◽  
Global Optimal ◽  
Optimal Evaluation ◽  
Multi Robot

The problem of global optimal evaluation for multi-robot allocation has gained attention constantly, especially in a multi-objective environment, but most algorithms based on swarm intelligence are difficult to give a convergent result. For solving the problem, we established a Global Optimal Evaluation of Revenue method of multi-robot for multi-tasks based on the real textile combing production workshop, consumption, and different task characteristics of mobile robots. The Global Optimal Evaluation of Revenue method could traversal calculates the profit of each robot corresponding to different tasks with global traversal over a finite set, then an optimization result can be converged to the global optimal value avoiding the problem that individual optimization easy to fall into local optimal results. In the numerical simulation, for fixed set of multi-object and multi-task, we used different numbers of robots allocation operation. We then compared with other methods: Hungarian, the auction method, and the method based on game theory. The results showed that Global Optimal Evaluation of Revenue reduced the number of robots used by at least 17%, and the delay time could be reduced by at least 16.23%.

On the Selmer Group of Twists of Elliptic Curves with Q-Rational Torsion Points

1988 ◽  
Vol 40 (3) ◽  
pp. 649-665 ◽  
Author(s):  
G. Frey
Keyword(s):  
Algebraic Number ◽  
Finite Extension ◽  
Algebraic Number Field ◽  
Prime Numbers ◽  
Selmer Group ◽  
Prime Divisors ◽  
Torsion Points ◽  
Root Of Unity ◽  
Finite Set ◽  
Image Position

(1) The symbols p and q stand for prime numbers and throughout the paper we assume that p is fixed and contained in {3, 5, 7}. Let L be an algebraic number field (i.e., L is a finite extension of Q). Then prime divisors of L dividing p (resp. q) are denoted by (resp. ). The completion of L with respect to is denoted by . Let S be a finite set of prime numbers, and let M/L be a Galois extension with abelian Galois group of exponent p.Definition. M/L is said to be little ramified outside S if for primes q ∉ S and all one haswith k ∊ N and . Here ζp is a pth root of unity, u1, …, uk are elements in and is the normed valuation belonging to . In particular M/L is unramified at all divisors of primes q ∉ S ∪ {p}.

scholarly journals Extremal sequences of polynomial complexity

2013 ◽  
Vol 155 (2) ◽  
pp. 191-205 ◽  
Author(s):  
KEVIN G. HARE ◽  
IAN D. MORRIS ◽  
NIKITA SIDOROV
Keyword(s):  
Polynomial Complexity ◽  
Exponential Growth Rate ◽  
Maximal Rate ◽  
Joint Spectral Radius ◽  
Bounded Set ◽  
Fixed Set ◽  
Finite Set ◽  
Dimension 2 ◽  
Subword Complexity ◽  
Real Matrices

AbstractThe joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer p ≥ 1, there exists a pair of square matrices of dimension 2p(2p+1 − 1) for which every extremal sequence has subword complexity at least 2−p2np.

scholarly journals On prime divisors of large powers of elements in Noether lattices

1988 ◽  
Vol 30 (3) ◽  
pp. 359-367 ◽  
Author(s):  
Linda Becerra
Keyword(s):  
Noetherian Ring ◽  
Integral Closure ◽  
Noetherian Rings ◽  
Prime Divisors ◽  
Commutative Noetherian Ring ◽  
Fixed Element ◽  
Fixed Set ◽  
Decomposition Theorems ◽  
Lattice Of Ideals ◽  
Multiplicative Lattices

In [4], R. P. Dilworth introduced the concept of a Noether lattice as an abstraction of the lattice of ideals of a Noetherian ring and he showed that many important properties of Noetherian rings, such as the Noether decomposition theorems, also hold for Noether lattices. It was later shown, in [1], that every Noether lattice is not the lattice of ideals of any Noetherian ring, yet many studies have successfully been undertaken to relate other concepts between Noetherian rings and Noether lattices as had been begun by Dilworth. (See [3], [5], and [6].) In this paper we undertake such a study and show that some results of M. Brodmann in [2] and L. Ratliff in [7] concerning prime divisors of large powers of a fixed element of a commutative Noetherian ring may be generalized and extended to the setting of a Noether lattice. It is shown (Theorem 2.8) that if A is an element of a Noether lattice then all large powers of A have the same prime divisors and (Corollary 3.8) included among this fixed set of primes are those primes that are prime divisors of the integral closure of Ak for some k≧l. We note that the ring proof of this latter result does not generalize directly since it uses the notion of transcendence degree which to our knowledge has no analogue in multiplicative lattices.

scholarly journals Asymptotic prime divisors over complete intersection rings

2016 ◽  
Vol 160 (3) ◽  
pp. 423-436 ◽  
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Algebraic Geometry ◽  
Complete Intersection ◽  
Finitely Generated ◽  
Prime Divisors ◽  
Intersection Ring ◽  
Finite Set ◽  
Image Position

AbstractLet A be a local complete intersection ring. Let M, N be two finitely generated A-modules and I an ideal of A. We prove that $$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$ is a finite set. Moreover, we prove that there exist i0, n0 ⩾ 0 such that for all i ⩾ i0 and n ⩾ n0, we have $$\begin{linenomath}\begin{subeqnarray*} {\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\ {\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \end{subeqnarray*}\end{linenomath}$$ We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity cxA(M, N/InN) is constant for all sufficiently large n.

scholarly journals Inhomogeneous Diophantine Approximation on M0-Sets with Restricted Denominators

2020 ◽  
Author(s):  
Andrew D Pollington ◽  
Sanju Velani ◽  
Agamemnon Zafeiropoulos ◽  
Evgeniy Zorin
Keyword(s):  
Probability Measure ◽  
Uniform Distribution ◽  
Diophantine Approximation ◽  
Fundamental Theorem ◽  
Type Theorem ◽  
Quantitative Result ◽  
Prime Divisors ◽  
Points Of Interest ◽  
Finite Set ◽  
Almost All

Abstract Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu $ with the property that $ |\widehat{\mu }(t)| \ll (\log |t|)^{-A}$ for some constant $ A&gt; 0 $. Let $\mathcal{A}= (q_n)_{n\in{\mathbb{N}}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A&gt;2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (1) the points of interest are restricted to $F$ and (2) the denominators of the “shifted” rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx \textrm{mod} 1)_{n\in{\mathbb{N}}} $ is uniformly distributed for $\mu $ almost all $x \in F$. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences $\mathcal{A}$ for which the prime divisors are restricted to a finite set of $k$ primes and $A&gt; 2k$. Loosely speaking, for such sequences, our result can be viewed as a quantitative refinement of the fundamental theorem of Davenport, Erdös, and LeVeque (1963) in the theory of uniform distribution.

Development of a Jacobian Module for Advanced Pulp and Paper Simulation and Control

TAPPI Journal ◽  
2009 ◽  
Vol 8 (1) ◽  
pp. 4-11
Author(s):  
MOHAMED CHBEL ◽  
LUC LAPERRIÈRE
Keyword(s):  
Control System ◽  
Nonlinear Behavior ◽  
Simulation Software ◽  
Pulp And Paper ◽  
Pid Controllers ◽  
Tuning Parameters ◽  
Pid Tuning ◽  
Fixed Set ◽  
And Control ◽  
Operating Points

Pulp and paper processes frequently present nonlinear behavior, which means that process dynam-ics change with the operating points. These nonlinearities can challenge process control. PID controllers are the most popular controllers because they are simple and robust. However, a fixed set of PID tuning parameters is gen-erally not sufficient to optimize control of the process. Problems related to nonlinearities such as sluggish or oscilla-tory response can arise in different operating regions. Gain scheduling is a potential solution. In processes with mul-tiple control objectives, the control strategy must further evaluate loop interactions to decide on the pairing of manipulated and controlled variables that minimize the effect of such interactions and hence, optimize controller’s performance and stability. Using the CADSIM Plus™ commercial simulation software, we developed a Jacobian sim-ulation module that enables automatic bumps on the manipulated variables to calculate process gains at different operating points. These gains can be used in controller tuning. The module also enables the control system designer to evaluate loop interactions in a multivariable control system by calculating the Relative Gain Array (RGA) matrix, of which the Jacobian is an essential part.

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