Number Of Nonzero Terms Research Articles

Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics

Constraints in power consumption and computational power limit the skill of operational numerical weather prediction by classical computing methods. Quantum computing could potentially address both of these challenges. Herein, we present one method to perform fluid dynamics calculations that takes advantage of quantum computing. This hybrid quantum–classical method, which combines several algorithms, scales logarithmically with the dimension of the vector space and quadratically with the number of nonzero terms in the linear combination of unitary operators that specifies the linear operator describing the system of interest. As a demonstration, we apply our method to solve the advection–diffusion equation for a small system using IBM quantum computers. We find that reliable solutions of the equation can be obtained on even the noisy quantum computers available today. This and other methods that exploit quantum computers could replace some of our traditional methods in numerical weather prediction as quantum hardware continues to improve.

Sparse polynomial interpolation based on derivatives

We propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program (SLP). Both of our algorithms work over any finite fields Fq with large characteristic.The first algorithm is randomized of the Monte Carlo type. Its bit complexity is linear in the number of non-zero terms T and the number of variables n of f. Let D be the partial degree bound of f. If q is in O((nTD)(1)), our algorithm has better complexity than other existing algorithms.The second algorithm is deterministic; it has better complexity than any existing deterministic algorithms in the field with large characteristic. Its bit complexity is quadratic in n,T,log⁡D.

Lower Bounds of Complexity for Polarized Polynomials over Finite Fields

We obtain an efficient lower bound of complexity for n-ary functions over a finite field of arbitrary order in the class of polarized polynomials. The complexity of a function is defined as the minimal possible number of nonzero terms in a polarized polynomial realizing the function.

The closure in a Hilbert space of a preHilbert space Chebyshev set that fails to be a Chebyshev set

In 1987 the author gave an example of a non convex Chebyshev set S in the incomplete inner product space E consisting of the vectors in l2 which have at most a finite number of non zero terms. In this paper, we show that the closure of S in the Hilbert space completion l2 of E is not Chebyshev in l2.

Series misdemeanors

Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series -- perhaps generalized to allow coefficients containing functions of the series variable that are dominated by any power of that variable, such as logarithms and nested logarithms of the series variable. Some computer-algebra systems also offer functions that can compute more-general truncated recursive hierarchical series. However, for all of these kinds of truncated series there are important implementation details that haven't been addressed before in the published literature and in current implementations. For implementers this article contains ideas for designing more convenient, correct, and efficient implementations or improving existing ones. For users, this article is a warning about some of these limitations. More specifically, this article discusses issues such as avoiding unnecessary restrictions such as prohibiting negative or fractional requested orders, the pros and cons of displaying results with explicit infectious error terms of the form o (...), O (...), and/or θ(...), efficient data structures, and algorithms that efficiently give users exactly the order or number of nonzero terms they request. . Most of the ideas in this article have been implemented in the computer-algebra within the TI-Nspire calculator, Windows and Macintosh products.

Lacunary formal power series and the Stern–Brocot sequence

Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-\lambda_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $\lambda_{n+1}/\lambda_n > 2$. It is known that the denominators $Q_n(X)$ of the convergents of its continued fraction expansion are polynomials with coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is the $n$th term of the Stern-Brocot sequence. We show that replacing the index $n$ by any 2-adic integer $\omega$ makes sense. We prove that $Q_{\omega}(X)$ is a polynomial if and only if $\omega \in {\mathbb Z}$. In all the other cases $Q_{\omega}(X)$ is an infinite formal power series, the algebraic properties of which we discuss in the special case $\lambda_n = 2^{n+1} - 1$.

Computing the torsion points of a variety defined by lacunary polynomials

We present an algorithm for computing the set of torsion points satisfying a given system of multivariate polynomial equations. Its complexity is quasilinear in the logarithm of the degree of the input equations and exponential in their number of non zero terms and variables.

A modification of Fitzgerald's characterization of primitive polynomials over a finite field

A characterization of primitive polynomials, among irreducible polynomials, over a finite field is established. Such characterization is determined by the number of nonzero terms in certain quotient of polynomials. The proof, which is a modification of the one discovered by Fitzgerald in 2003 yet revealing more general structure, is based principally on counting the number of occurrences of elements in a linear recurring sequence over a finite field.

Cancellation in cyclic consecutive systems

We consider the structure and number of non-zero terms in the reliability polynomials for cyclic consecutive systems. We explain the large amount of cancellation, the fact that all but one of the coefficients are 0, 1 or −1, and show that the number of non-zero coefficients is asymptotic to α k , where α is the largest root of 2+ x r − x r+1 =0.

Symbolic and Numeric Methods for Exploiting Structure in Constructing Resultant Matrices

Resultants characterize the existence of roots of systems of multivariate nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse (or toric) resultant, which takes into account the sparse structure of the polynomials. The construction of sparse resultant, or Newton, matrices is the critical step in the computation of the multivariate resultant and the solution of a nonlinear system. We reveal and exploit the quasi-Toeplitz structure of the Newton matrix, thus decreasing the time complexity of constructing such matrices by roughly one order of magnitude to achieve quasi-quadratic complexity in the matrix dimension. The space complexity is also decreased analogously. These results imply similar improvements in the complexity of computing the resultant polynomial itself and of solving zero-dimensional systems. Our approach relies on fast vector-by-matrix multiplication and uses the following two methods as building blocks. First, a fast and numerically stable method for determining the rank of rectangular matrices, which works exclusively over floating point arithmetic. Second, exact polynomial arithmetic algorithms that improve upon the complexity of polynomial multiplication under our model of sparseness, offering bounds linear in the number of variables and the number of non-zero terms.

Techniques for exploiting structure in matrix formulae of the sparse resultant

Resultants characterize the existence of roots of systems of polynomial equations, while their matrix formulae reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse resultant, which takes into account the sparse structure of the polynomials, as expressed by the respective supports or Newton polytopes. Sparse resultant, or Newton, matrices generalize Macaulay’s matrix and define nontrivial multiples of the sparse resultant from which the latter can be recovered. We exploit their structure with the objective to decrease the complexity of constructing such matrices and of computing the exact resultant. Our present study of these structured matrices implies substantial progress in this direction. Subsequent application of fast matrix-by-vector multiplication shall lead to computational complexity bounds that are roughly quadratic in the matrix size, whereas the existing methods have cubic complexity. Our auxiliary techniques for testing exact and numerical nonsingularity of rectangular matrices may be of some independent interest. Finally, we establish complexity bounds, linear in the number of nonzero terms, for polynomial multiplication, under our model of sparseness.

Learning Sparse Multivariate Polynomials over a Field with Queries and Counterexamples

We consider the problem of learning a polynomial over an arbitrary fieldFdefined on a set of boolean variables. We present the first provably effective algorithm for exactly identifying such polynomials using membership and equivalence queries. Our algorithm runs in time polynomial inn, the number of variables, andt, the number of nonzero terms appearing in the polynomial. The algorithm makes at mostnt+2 equivalence queries, and at most (nt+1)(t2+3t)/2 membership queries. Our algorithm is equally effective for learning a generalized type of polynomial defined on certain kinds of semilattices. We also present an extension of our algorithm for learning multilinear polynomials when the domain of each variable is the entire fieldF.

Fractals related to Pascal's triangle

A precise definition of a fractalFpr1 derived from Pascal's triangle modulop r (p prime) is given. The number of nonzero terms in the firstp s lines of Pascal's triangle modulop r is computed. From this result the Hausdorff dimension and Hausdorff measure ofFpr1 are deduced. The nonself-similarty ofFpr1,r≥2, is also discussed.

The Strong ϕ Topology on Symmetric Sequence Spaces

The strong ϕ topology. Let S be a linear space of real sequences written in functional notationThere is a natural duality between S and the space ϕ of sequences which are eventually ϕ given by the equationThe series has only a finite number of nonzero terms since t is in ϕ.A subset B of ϕ is called S-bounded iffor each s in S.

A note on the bilinear transformation matrices for multivariable polynomials

Special symmetry of bilinear transformation matrices for multivariable polynomials is stated. It is shown that reduction in computation in existing algorithm can be achieved by making use of this symmetry. It has been brought out that the number of nonzero terms in the given polynomial of specified order plays a role in the computational efficiency of the ‘Rao-Aatre algorithm’.

Graphical rules for enzyme-catalysed rate laws.

Two graphical rules for calculating the reaction rates of steady-state enzyme-catalysed systems are presented. According to Rule 1, the number of non-zero terms can be easily calculated, which is useful to check the results and avoid mistakes. Rule 2 provides a new graphical method, which will be much more effective than the existing methods in dealing with complicated problems. In addition, during the process of deriving Rule 1, the mathematical principles of the Wong-Hanes structural rule have been naturally obtained too.

Spectrum Shaping with Alphabetic Codes with Finite Autocorrelation Sequence

In this paper we consider the problem of recoding binary data into ternary sequences for synchronous baseband data transmission systems. Our major objective is the shaping of the power spectral density of the channel process in order to combat the effects of intersymbol interference caused by the low-frequency cut-off of the channel. This motivates our interest in designing codes whose channel process has small Power in the low frequency range. Based on the MacLaurin expansion of the power density function, we use as a design heuristic the normalized second derivative Δ of the latter. To make computationally manageable the expression of Δ, we restrict ourselves to a class of codes whose autocorrelation sequence contains a finite number of nonzero terms, and present a corresponding code design algorithm. As examples, we exhibit two new codes which in many respects improve over the spectra of corresponding well-known codes.

On the Computation of Powers of Sparse Polynomials

This paper examines the time for computing integer powers of a class of polynomials in one or several variables. A member f of the class of “sparse” polynomials can be characterized by the number of nonzero terms, t, in the representation of f. Where, for power 4, the most obvious methods result in multiplications, a method is presented which requires fewer than multiplications.This algorithm requires, for t » n » 1 about tn/n! + O(tn − 1 /(2(n − 2)!)) multiplications, which is proved to be a lower‐bound on the computation.

A Combinatorial Problem and Congruences for the Rayleigh Function

Let $z$ be a positive integer and let $m$ be the number of nonzero terms in the base 2 expansion of $z$. Define $f(z,s)$ as the number of positive integers $r \leqq z/2$ such that the number of nonzero terms in the base 2 expansion of $r$ plus the number of nonzero terms in the base 2 expansion of $z - r$ is equal to $m + s$. We find formulas for $f(z,s)$ and show how these formulas can be used in proving congruences for the Rayleigh function.

Imbeddings of Real Projective Spaces

THEOREM II. Suppose n = 4k + 1, but k is not a power of 2. Then Pa can be imbedded in (2n 2)-space. In each case the result would be false for k a power of 2. Pa cannot be imbedded in (2n 1)-space for n = 2r, by a result of Stiefel or Thom, [11, p. 157, III. 16]. Pa cannot be imbedded in (2n 2)-space for n = 2r + 1, by a result of Levine [5] and Mahowald. There is a tentative conjecture, due to Atiyah, that Pn can be imbedded in (2n a(n) + 1)-space but not in (2n a(n))-space. Here a(n) is the number of non-zero terms in the dyadic expansion of n. Our results agree with the first part of this conjecture for n = 2r, 2r + 1, 2r + 28, 2r+1 + 28+1 + 1(r > s > 0).