Case Of Polynomials Research Articles

Lot sizing with minimum order quantity

We consider the single item lot sizing problem with minimum order quantity where each period has an additional constraint on minimum production quantity. We study special cases of the general problem from the algorithmic and mathematical programming perspective. We exhibit a polynomial case when capacity is constant and minimum order quantities are non-increasing in time. Linear programming extended formulations are provided for various cases with constant minimum order quantity.

Biorthogonal Approximation by Splines

We establish two-sided boundary biorthogonal approximations of twice continuously differentiable functions by Bφ-splines of the second order. We obtain an integral representation for the remainder of biorthogonal approximations by quadratic splines. Based on these results, we derive error estimates for some problems of approximation and interpolation of Lagrange type. These estimates are attained in the polynomial case. Bibliography: 5 titles.

Quantum partially observable Markov decision processes

We present quantum observable Markov decision processes (QOMDPs), the quantum analogues of partially observable Markov decision processes (POMDPs). In a QOMDP, an agent's state is represented as a quantum state and the agent can choose a superoperator to apply. This is similar to the POMDP belief state, which is a probability distribution over world states and evolves via a stochastic matrix. We show that the existence of a policy of at least a certain value has the same complexity for QOMDPs and POMDPs in the polynomial and infinite horizon cases. However, we also prove that the existence of a policy that can reach a goal state is decidable for goal POMDPs and undecidable for goal QOMDPs.

Difference equations of q-Appell polynomials

In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known classical (q=1) results. We also provide the recurrence relations and the q-difference equations for q-Bernoulli polynomials, q-Euler polynomials, q-Genocchi polynomials and for q-Hermite polynomials, as special cases of q-Appell polynomials.

Stable polynomials over finite fields

We use the theory of resultants to study the stability, that is, the property of having all iterates irreducible, of an arbitrary polynomial f over a finite field \mathbb{F}_q . This result partially generalizes the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for p=3 , we show that certain polynomials of degree three are not stable. We also use the Weil bound for multiplicative character sums to estimate the number of stable polynomials over a finite field of odd characteristic.

Generalized Christoffel functions for power orthogonal polynomials

In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.

Measurable diagonalization of positive definite matrices

In this paper we show that any positive definite matrix V with measurable entries can be written as V=UΛU∗, where the matrix Λ is diagonal, the matrix U is unitary, and the entries of U and Λ are measurable functions (U∗ denotes the transpose conjugate of U).This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.

Two-stage robust optimization, state-space representable uncertainty and applications

The present paper addresses the class of two-stage robust optimization problems which can be formulated as mathematical programs with uncertainty on the right-hand side coefficients (RHS uncertainty). The wide variety of applications and the fact that many problems in the class have been shown to be NP-hard, motivates the search for efficiently solvable special cases. Accordingly, the first objective of the paper is to provide an overview of the most important applications and of various polynomial or pseudo-polynomial special cases identified so far. The second objective is to introduce a new subclass of polynomially solvable robust optimization problems with RHS uncertainty based on the concept of state-space representable uncertainty sets . A typical application to a multi period energy production problem under uncertain customer load requirements is described into details, and computational results including a comparison between optimal two-stage solutions and exact optimal multistage strategies are discussed.

On the Discriminant Scheme of Homogeneous Polynomials

In this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of n − 1 homogeneous polynomials in $${n\geqslant 2}$$ variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring k, typically a domain, if $${2\neq 0}$$ in k. The case where 2 = 0 in k is also analyzed in detail.

Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients

Suppose that the first m m Fourier coefficients of a piecewise analytic function are given. Direct expansion in a Fourier series suffers from the Gibbs phenomenon and lacks uniform convergence. Nonetheless, in this paper we show that, under very broad conditions, it is always possible to recover an n n -term expansion in a different system of functions using only these coefficients. Such an expansion can be made arbitrarily close to the best possible n n -term expansion in the given system. Thus, if a piecewise polynomial basis is employed, for example, exponential convergence can be restored. The resulting method is linear, numerically stable and can be implemented efficiently in only O ( n m ) \mathcal {O} \left (n m\right ) operations. A key issue is how the parameter m m must scale in comparison to n n to ensure such recovery. We derive analytical estimates for this scaling for large classes of polynomial and piecewise polynomial bases. In particular, we show that in many important cases, including the case of piecewise Chebyshev polynomials, this scaling is quadratic: m = O ( n 2 ) m=\mathcal {O}\left (n^2\right ) . Therefore, with a system of polynomials that the user is essentially free to choose, one can restore exponential accuracy in n n and root-exponential accuracy in m m . This generalizes a result proved recently for piecewise Legendre polynomials.

The Inverse Problem for Ellis–Gohberg Orthogonal Matrix Functions

This paper deals with the inverse problem for the class of orthogonal functions that for the scalar case was introduced by Ellis and Gohberg (J Funct Anal 109:155–198, 1992). The problem is reduced to a linear equation with a special right hand side. This reduction allows one to solve the inverse problem for square matrix functions under conditions that are natural generalizations of those appearing in the scalar case. These conditions lead to a unique solution. Special attention is paid to the polynomial case. A number of partial results are obtained for the non-square case. Various examples are given to illustrate the main results and some open problems are presented.

Integral Representations and Sharp Estimates of B φ -Splines

We derive an integral representation of second order coordinate B φ -splines. Based on this representation, we obtain sharp estimates for such splines and show that the estimates are attained in the polynomial case. Bibliography: 3 titles.

The uniqueness of the entire functions whose n-th powers share a small function with their derivatives

In this paper, we prove the following result: Let f(z) be a transcendental entire function, Q(z) ≢ 0 be a small function of f(z), and n ≥ 2 be a positive integer. If f n (z) and (f n (z))′ share Q(z) CM, then \(f(z) = ce^{\tfrac{1} {n}z}\), where c is a nonzero constant. This result extends Lv’s result from the case of polynomial to small entire function.

A ''Continuous'' Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials

A novel family of $-1$ orthogonal polynomials called the Chihara polynomials is characterized. The polynomials are obtained from a "continuous" limit of the complementary Bannai-Ito polynomials, which are the kernel partners of the Bannai-Ito polynomials. The three-term recurrence relation and the explicit expression in terms of Gauss hypergeometric functions are obtained through a limit process. A one-parameter family of second-order differential Dunkl operators having these polynomials as eigenfunctions is also exhibited. The quadratic algebra with involution encoding this bispectrality is obtained. The orthogonality measure is derived in two different ways: by using Chihara's method for kernel polynomials and, by obtaining the symmetry factor for the one-parameter family of Dunkl operators. It is shown that the polynomials are related to the big $-1$ Jacobi polynomials by a Christoffel transformation and that they can be obtained from the big $q$-Jacobi by a $q\rightarrow -1$ limit. The generalized Gegenbauer/Hermite polynomials are respectively seen to be special/limiting cases of the Chihara polynomials. A one-parameter extension of the generalized Hermite polynomials is proposed.

Polynomial cases of the economic lot sizing problem with cost discounts

In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified all-unit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.

Sums of products of the degenerate Euler numbers

The paper focuses on the degenerate Euler numbers, the degenerate Euler polynomials and the degenerate Bernoulli polynomials. By adopting the method of recurrences, two explicit expressions have been established for sums of products of the degenerate Euler polynomials and the degenerate Bernoulli polynomials. As a special case of the degenerate Euler polynomials, an expression can be obtained for . MSC:11B68, 11B65, 11B73.

On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.

The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

Locating sensors to observe network arc flows: Exact and heuristic approaches

The problem of optimally locating sensors on a traffic network to monitor flows has been an object of growing interest in the past few years, due to its relevance in the field of traffic management and control. Sensors are often located in a network in order to observe and record traffic flows on arcs and/or nodes. Given traffic levels on arcs within the range or covered by the sensors, traffic levels on unobserved portions of a network can then be computed. In this paper, the problem of identifying a sensor configuration of minimal size that would permit traffic on any unobserved arcs to be exactly inferred is discussed. The problem being addressed, which is referred to in the literature as the Sensor Location Problem (SLP), is known to be NP-complete, and the existing studies about the problem analyze some polynomial cases and present local search heuristics to solve it. In this paper we further extend the study of the problem by providing a mathematical formulation that up to now has been still missing in the literature and present an exact branch and bound approach, based on a binary branching rule, that embeds the existing heuristics to obtain bounds on the solution value. Moreover, we apply a genetic approach to find good quality solutions. Extended computational results show the effectiveness of the proposed approaches in solving medium-large instances.

A New Extended Padé Approximation and Its Application

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The importance of this extension is that the ordinary Pad&#xe9; approximation is a particular case of our extended Pad&#xe9; approximation. Also the parameterization (<svg style="vertical-align:-0.1638pt;width:9.5124998px;" id="M4" height="12.4375" version="1.1" viewBox="0 0 9.5124998 12.4375" width="9.5124998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D706"/></g> </svg> is the corresponding parameter) of new extended Pad&#xe9; approximation is an important subject which, obtaining the optimal value of this parameter, can be a good question for a new research.