Newton's Identities Research Articles

Dual Feature Interaction-Based Graph Convolutional Network

Graphs are widely used to model various practical applications. In recent years, graph convolution networks (GCNs) have attracted increasing attention due to the extension of convolution operation from traditional grid data to graph one. However, the representation ability of current GCNs is undoubtedly limited because existing work fails to consider feature interactions. Toward this end, we propose a Dual Feature Interaction-based GCN. Specifically, it models feature interaction in the aspects of 1) node features where we use Newton's identity to extract different-order cross features implicit in the original features and design an attention mechanism to fuse them; and 2) graph convolution where we capture the pairwise interactions among nodes in the neighborhood to expand a weighted sum operation. We evaluate the proposed model with graph data from different fields, and the experimental results on semi-supervised node classification and link prediction demonstrate the effectiveness of the proposed GCN. The data and source codes of this work are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/ZZY-GraphMiningLab/DFI-GCN</uri> .

Quantum algebra of multiparameter Manin matrices

Multiparametric quantum semigroups Mqˆ,pˆ(n) are generalization of the one-parameter general linear semigroups Mq(n), where qˆ=(qij) and pˆ=(pij) are 2n2 parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir's identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also lifted to multiparameter q-Yangians.

Application of Newton Identities in Solving Selective Harmonic Elimination Problem With Algebraic Algorithms

Algebraic algorithms are powerful methods in solving the selective harmonic elimination (SHE) problem, which can find all exact solutions without the requirements of choosing initial values. However, the huge computational burden and long solving time limit the solving capability of algebraic algorithms. This article presents a novel Newton’s identifies-based method to simplify the SHE equations including the order reduction and the variable elimination, thereby reducing the computational burden and the solving time of algebraic algorithms or in other words improving the solving capability of the algebraic algorithms. Compared with existing simplification methods, the proposed method significantly improves the efficiency of solving SHE equations. With the proposed method, the degree of reduction is no longer the bottleneck of solving the SHE equations by using algebraic algorithms. By using the proposed method, the SHE equations with ten switching angles are completely solved with the algebraic algorithm for the first time. The simulation and experimental results indicate that the proposed method is effective and correct.

Partitions with k sizes from a set

Let $n$ and $k$ be two positive integers and let $A$ be a set of positive integers. We define $t_A(n,k)$ to be the number of partitions of $n$ with exactly $k$ sizes and parts in $A$. As an implication of a variant of Newton's product-sum identities we present a generating function for $t_A(n,k)$. Subsequently, we obtain a recurrence relation for $t_A(n,k)$ and a divisor-sum expression for $t_A(n,2)$. Also, we present a bijective proof for the latter expression.

On the behavior of multiple zeta-functions with identical arguments on the real line

We study the behavior of r-fold zeta-functions of Euler-Zagier type with identical arguments ζr(s,s,…,s) on the real line. Our basic tool is an “infinite” version of Newton's classical identities. We carry out numerical computations, and draw graphs of ζr(s,s,…,s) for real s, for several small values of r. Those graphs suggest various properties of ζr(s,s,…,s), some of which we prove rigorously. When s∈[0,1], we show that ζr(s,s,…,s) has r asymptotes at ℜs=1/k (1≤k≤r), and determine the asymptotic behavior of ζr(s,s,…,s) close to those asymptotes. Numerical computations establish the existence of several real zeros for 2≤r≤10 (in which only the case r=2 was previously known). Based on those computations, we raise a conjecture on the number of zeros for general r, and gives a formula for calculating the number of zeros. We also consider the behavior of ζr(s,s,…,s) outside the interval [0,1]. We prove asymptotic formulas for ζr(−k,−k,…,−k), where k takes odd positive integer values and tends to +∞. Moreover, on the number of real zeros of ζr(s,s,…,s), we prove that there are exactly (r−1) real zeros on the interval (−2n,−2(n−1)) for any n≥2.

Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities

Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres–Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes’ rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit-qutrit, and qubit-four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit–qutrit states. Based on this numerical approach with the help of the Clauser–Horne–Shimony–Holt inequality and the Collins–Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.

Algebraic Decoding of Quasi-Reversible BCH Codes Using Band Matrices

A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,⋯,a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> in the defining set, where a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is a negative integer and a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is a positive integer. A matrix in band form presented in this article involves the use of the Newton identities and symmetric polynomial identities. This matrix is the coefficient matrix of a linear system. The solution to such a linear system is the coefficients of an error-locator polynomial for quasi-reversible BCH codes. Its computational complexity is significantly lower than BCH decoding using the linear system with coefficient matrix in column echelon form. This article also proposes a new partial syndrome matrix with radical locators and a lot of zeros. Such a matrix is still in band form. The expansion of its matrix determinant is exactly a radical-locator polynomial. Compared to the former radical-locator polynomials in the literature, the newly discovered radical-locator polynomial not only requires less storage memory but also results in computational benefits. Surprisingly, the resulting polynomials are much sparser than those for the narrow-sense BCH codes. Finally, a complete algebraic decoding algorithm for quasi-reversible BCH codes is provided. The error-locator and radical-locator polynomials of degree less than or equal to 5 are listed.

Modified Newton identities

The sums of powers with identical exponents of natural, real, or complex numbers, considered as roots of algebraic equation, are expressed directly through the products of coefficients of that equation, starting from the well-known Newton identities. The final Eq. (6) includes the same power of sum of all numbers ± a sum over all partitions of the exponent. Each term of the last sum is the equation coefficients product with the net power keeping the “dimensionality” of the exponent and having a numerical factor, equal to a proper polynomial coefficient, built of exponents of equation coefficients entering the product. The revers Eq. (43) for equation coefficients is also a sum over all partitions of the same exponent with known numerical coefficients. The entering products are built of “commutators-anticommutators of power of sum and sum of powers” (C-A) of the initial sum addends. The numerous identities Eq. (44) for a C-A with an exponent, exceeding the number of C-A sum terms by 2, and similar C-A-s with lower exponents are established.

Trace formulas for the Casimir operators of the unextended Schrödinger algebra S(N)

Using the contraction of the centrally extended Schrödinger algebra S^(N) onto the Lie algebra S(N)⊕R in combination with the Newton identities associated with the characteristic polynomial of a matrix, we derive explicit expressions for the Casimir operators of the unextended Schrödinger algebra S(N) in terms of trace operators. It is shown that these operators can be defined independently of the contraction from which a direct method for the computation of the S(N)-invariants is deduced.

Generalized Chudnovsky Algorithm for Real-Time PWM Selective Harmonic Elimination/Modulation: Two-Level VSI Example

This article presents a novel approach to significantly reducing the mathematical complexity associated with the selective harmonic elimination (SHE), which allows for the real-time implementation of SHE with a reasonable utilization of the computational power of the digital controller. First, the transcendental equations of the optimization problem are converted to a set of algebraic equations using Chebyshev polynomials. Second, the set of algebraic equations is reduced to a single univalent polynomial using generalized Newton's identities. The roots of the polynomial hold a unique solution to the optimization problem. We show that the roots of the univalent polynomial can be solved in real time by direct substitution with no initial guess or iteration. The proposed method is a generalization of a Chudnovsky algorithm; it allows the modulation of the selected harmonics rather than eliminating them. Furthermore, the technique by which the roots of the polynomial are obtained enables one to implement the algorithm in real time with determinate execution time. This article lays the groundwork for presenting the generalized algorithm through a two-level voltage-source inverter drive. More application will follow in the near future.

Radical-Locator Polynomials and Row-Echelon Partial Syndrome Matrices With Applications to Decoding Cyclic Codes

Partial syndrome matrices and weak-locator polynomials have received considerable attention in recent years due to their applications in decoding cyclic codes. The technical contribution of this paper is threefold: 1) cyclic codes can be decoded by the newly proposed partial syndrome matrices, which generalize the previously known results on the determination of error positions; 2) a new type of polynomial associated with error locations, called radical-locator polynomial , is defined, which includes the weak-locator polynomial as a special case; and 3) a novel class of matrices with many zero entries, called row-echelon partial syndrome matrices , is presented, based on the Newton identities, for efficiently decoding cyclic codes. It is also shown that the radical-locator polynomials can be obtained from the determinants of the above-mentioned (row-echelon) partial syndrome matrices.

On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients

We present an efficient deterministic algorithm which outputs exact expressions in terms of n for the number of monic degree n irreducible polynomials over Fq of characteristic p for which the first l<p coefficients are prescribed, provided that n is coprime to p. Each of these counts is 1n(qn−l+O(qn/2)). The main idea behind the algorithm is to associate to an equivalent problem a set of Artin–Schreier curves defined over Fq whose number of Fqn-rational affine points must be combined. This is accomplished by computing their zeta functions using a p-adic algorithm due to Lauder and Wan. Using the computational algebra system Magma one can, for example, compute the zeta functions of the arising curves for q=5 and l=4 very efficiently, and we detail a proof-of-concept demonstration. Due to the failure of Newton's identities in positive characteristic, the l≥p cases are seemingly harder. Nevertheless, we use an analogous algorithm to compute example curves for q=2 and l≤7, and for q=3 and l=3. Again using Magma, for q=2 we computed the relevant zeta functions for l=4 and l=5, obtaining explicit formulae for these open problems for n odd, as well as for subsets of these problems for all n, while for q=3 we obtained explicit formulae for l=3 and n coprime to 3. We also discuss some of the computational challenges and theoretical questions arising from this approach in the general case and propose some natural open problems.

Braided Yangians

Yangian-like algebras associated with current R-matrices different from the Yang ones are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangian-like algebras are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, the quantum determinant, are introduced. It is proved that in any braided Yangian this determinant is always central, whereas, in general, this is not true for the Yangians of RTT type. Analogs of the Cayley–Hamilton–Newton identities in the braided Yangians are exhibited. A bosonic realization of the braided Yangians is performed.

Finding Zeros of Analytic Functions and Local Eigenvalue Analysis Using Contour Integral Method in Examples

A numerical method for computing zeros of analytic complex functions is presented. It relies on Cauchy's residue theorem and the method of Newton's identities, which translates the problem to finding zeros of a polynomial. In order to stabilize the numerical algorithm, formal orthogonal polynomials are employed. At the end the method is adapted to finding eigenvalues of a matrix pencil in a bounded domain in the complex plane. This work is based on a series of papers of Professor Sakurai and collaborators. Our aim is to make their work available by means of a systematic study of properly chosen examples.

Research of Fast Satellite Selection Algorithm for Multi‐constellation

We propose a fast satellite selection algorithm for multi-constellation, which is based on both Newton's identities for Geometric dilution of precision (GDOP) fast computation and optimal satellite geometric distribution for less cycle participation. An effective closed-form formula is proposed for GDOP approximation, avoiding conventional matrix inversion and complexity. We reduce computational cycles with auxiliary optimal satellite geometric distribution, which improves the efficiency of realtime positioning under the combined action of both. Simulation results indicate that we can save more than 44% and 99% computational complexity when selecting 7 satellites from 10 and 30 visible satellites respectively. The GDOP of proposed method is very close to optimal result, better than that of Quasi-Optimal algorithm.

An Improved Decoding Algorithm of the (71, 36, 11) Quadratic Residue Code Without Determining Unknown Syndromes

In this paper, a new algebraic method to decode the (71, 36, 11) QR code up to five errors is proposed. It completely avoids computing the unknown syndromes, and uses the previous scheme of decoding this QR code up to three errors, but corrects four and five errors with a new different method. In the four-error case, the new algorithm directly determines the coefficients of the error-locator polynomial by eliminating unknown syndromes in Newton identities. Subsequently, the shift-search algorithm can be utilized to decode the fifth error and the concept of bit reliability is also introduced to accelerate the decoding process. In other words, a weight-five-error pattern can be decoded in terms of the four-error case by inverting an incorrect bit of the received word in ascending order of reliability. Particularly, a threshold parameter $\gamma$ can be preset to limit the number of inverting bits one by one, and a corresponding upper bound of the probability that decoding fails is derived. Finally, simulation and analysis show that the proposed new decoding algorithm for the abovementioned QR code not only significantly reduces the decoding complexity in terms of CPU time but also saves a lot of memory while maintaining the same error-rate performance. Additionally, the introduction of $\gamma$ achieves a better tradeoff between the decoding performance and the computational complexity.

Fast Decoding of the (47, 24, 11) Quadratic Residue Code Without Determining the Unknown Syndromes

In this paper, a hard-decision (HD) scheme is presented to facilitate faster decoding of the (47, 24, 11) quadratic residue (QR) code. The new HD algorithm uses the previous scheme of decoding the (47, 24, 11) QR code up to three errors, but corrects four and five errors with new different methods. In the four-error case, the new algorithm directly determines the coefficients of the error-locator polynomial by eliminating unknown syndromes in Newton identities and simplifies the condition that exactly indicates the occurrence of four errors. Subsequently, the reliability-based shift-search algorithm can be utilized to decode weight-5 error patterns. In other words, a five-error case can be decoded in terms of a four-error case after inverting an incorrect bit of the received word. Simulation results show that the new HD algorithm not only significantly reduces the decoding complexity in terms of CPU time but also saves a lot of memory while maintaining the same error-rate performance.

Multiplicative relations between coefficients of logarithmic derivatives of 𝔽q-linear functions and applications

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from 𝔽q-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli–Carlitz fractions and power sums as well as their multivariable generalizations. Resulting understanding of their factorizations has arithmetic significance, as well as applications to function field zeta and multizeta values evaluations and relations between them. Using specialization/generalization arguments, we provide much more general identities on linear forms providing a switch between power sums for positive and negative powers.

Modular Macdonald functions and generalized Newton's identity

Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall–Littlewood functions.

A generalization of Newton's identity and Macdonald functions

A generalization of Newton's identity on symmetric functions is given. Using the generalized Newton identity we give a unified method to show the existence of Jack and Macdonald polynomials. We also give a simple proof of the Jing–Józefiak formula for two-row Macdonald functions.