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EN
Let En = {xi = 1; xi + xj = xk; xi · xj = xk : i; j; k ∈ {1,...,n}}. We conjecture that if a system S ⊆ En has only finitely many solutions in integers x1,...,xn, then each such solution (x1,...,xn) satisfies |x1|,...,|xn| ≤ 22n−1. Assuming the conjecture, we prove: (1) there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set, (2) if a set M Í \mathbbN is recursively enumerable but not recursive, then a finite-fold Diophantine representation of M does not exist.
2
Regular octahedra in {0, 1, ..., n} 3
EN
In this paper we describe a procedure for calculating the number of regular octahedra, RO(n), which have vertices with coordinates in the set {0,1,...,n}. As a result, we introduce a new sequence in The Online Encyclopedia of Integer Sequences (A178797) and list the first one hundred terms of it. We improve the method appeared in [12] which was used to find the number of regular tetrahedra with coordinates of their vertices in {0,1,..., n}. A new fact proved here helps increasing considerably the speed of all programs used before. The procedure is put together in a series of commands written for Maple and it is included in an earlier version of this paper in the matharxiv. Our technique allows us to find a series of cubic polynomials ...[wzór]
EN
In 1970 a negative solution to the tenth Hilbert problem, concerning the determination of integral solutions of diophantine equations, has been published by Y. W. Matiyasevich (see Matiyasevich, 1970). Despite this result, we can present algorithms to compute integral solutions (roots) for a wide class of quadratic diophantine equations of the form q(x) = d, where q : Zn → Z is a homogeneous quadratic form. We will focus on the roots of one (i.e., d = 1) of quadratic Euler forms of selected posets from Loupias list (see Loupias, 1975). In particular, we will describe the roots of positive definite quadratic forms and the roots of quadratic forms that are principal (see Simson, 2010a). The algorithms and results we present here are successfully used in the representation theory of finite groups and algebras.
EN
In this paper, the adaptive control based on symbolic solution of Diophantine equation is used to suppress circular plate vibrations. It is assumed that the system to be regulated is unknown. The plate is excited by a uniform force over the bottom surface generated by a loudspeaker. The axially-symmetrical vibrations of the plate are measured by the application of the strain sensors located along the plate radius, and two centrally placed piezoceramic discs are used to cancel the plate vibrations. The adaptive control scheme presented in this work has the ability to calculate the error sensor signals, to compute the control effort and to apply it to the actuator within one sampling period. For precise identification of system model the regularized RLS algorithm has been applied. Self-tuning controller of RST type, derived for the assumed system model of the 4th order is used to suppress the plate vibration. Some numerical examples illustrating the improvement gained by incorporating adaptive control are demonstrated.
5
Rational points on certain hyperelliptic curves over finite fields
EN
Let K be a field, a, b ∈ K and ab ≠ 0. Consider the polynomials g1 (x) = xn + ax + b, g2(x) = xn + ax2 + bx, where n is a fixed positive integer. We show that for each k ≥ 2 the hypersurface given by the equation S[...], i = 1, 2 contains a rational curve. Using the above and van de Woestijne's recent results we show how to construct a rational point different from the point at infinity on the curves C1 : y2 = gi(x), (i = 1,2) defined over a finite field, in polynomial time.
6
On the diophantine equation x2 + 52k = yn
EN
In this paper we prove that the title equation where k >0 and n > 3, may have a solution in integers (x, y, k, n) only if 5\x and p \ k where p any odd prime dividing n, by using a recent result of Bilu, Hanrot and Voutier [3].
EN
We show how to determine the k-th bit of Chaitin's algorithmically random real number W by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of W to determining whether a certain Diophantine equation with two parameters, k and N, has solutions for an odd or an even number of values of N. We also demonstrate two further examples of W in number theory: an exponential Diophantine equation with a parameter k which has an odd number of solutions iff the k-th bit of W is 1, and a polynomial of positive integer variables and a parameter k that takes on an odd number of positive values iff the k-th bit of W is 1.
PL
W niniejszej publikacji omawiane jest równanie diofantyczne (...). Jest ono uogólnieniem równania Perrine'a. Pokazano, że wszystkie rozwiązania tego równania mogą być ułożone w kształt drzewa lub też kilku drzew, w zależności od ilości rozwiązań prostych. Także dla ustalonych wartości L, k, b i n mamy nieskończenie wiele ciągów k-elementowych, takich, że równanie powyższego typu ma rozwiązanie.
9
Suboptimal Nonlinear Predictive Controllers
EN
Predictive control based on linear models has become a mature technology in the last decade. Many successful real-time applications can be found in almost every sector of industry. Nonlinear predictive control can further increase the performance of this easy-to-understand control strategy. One of the main problems of implementing nonlinear predictive control is the computational aspect, which is of most importance in real-life applications. In this paper, suboptimal nonlinear predictive control strategies are proposed and compared. The nonlinear predictors are built based on neural identification methods or by white modelling. The use of diophantine equations, which is a common technique to calculate the optimal contribution of the noise model, is avoided by using a more natural method. The comparison between the control algorithms is made based on a simulated discrete multivariable nonlinear system and a continuous stirred tank reactor.
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