Factorization Of Polynomials Research Articles

Inapproximability of Matrix \(\boldsymbol{p \rightarrow q}\) Norms

We study the problem of computing the norm of a matrix , defined as . This problem generalizes the spectral norm of a matrix and the Grothendieck problem ( , ) and has been widely studied in various regimes. When , the problem exhibits a dichotomy: constant factor approximation algorithms are known if , and the problem is hard to approximate within almost polynomial factors when . The regime when , known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with and was studied by Barak et al. [Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 2012, pp. 307–326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any . We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any with , is hard to approximate within assuming . En route to the above result, we also prove almost tight results for the case when with .

A Generalization of Eisenstein-Schönemann’s Irreducibility Criterion

Abstract The Eisenstein criterion is a particular case of the Schönemann’s irreducibility criterion stated in 1846. In 1906, based on Newton polygon techniques, Dumas gave a generalization of the Eisenstein criterion. In this paper, we extend this last generalization. Some applications on factorization of polynomials, and prime ideal factorization will be given, too.

Analytical forms of the coefficients of the factored characteristic polynomials for zigzag SWCNT graphs

Factors of characteristic polynomials (CP) of zigzag single walled carbon nanotube (SWCNT) graphs have been constructed in generalized forms using rotational symmetry with respect to the tube axis. The CP coefficients in the factors have been expressed in analytical forms involving derivatives of different powers of pj, where pj=(1+ωj)(1+ωj*), ωj=0,1,2,…,(R−1), are the R-th roots of unity and R is the number of fused hexagonal rings per belt in the SWCNT graph. Some important results regarding total π-electron energy per electron, HOMO-LUMO energy gap and number of Kekulé structures for the SWCNTs have been derived from an analysis of the CPs and graph eigenspectra.

Research into the Dynamics of a System of Two Connected Bodies Moving in the Plane of a Circular Orbit by Applying Computer Algebra Methods

Computer algebra methods are used to determine the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves in a central Newtonian force field on a circular orbit under the action of gravitational torque. Primary attention is given to the study of equilibrium orientations of the two-body system in the plane of the circular orbit. By applying symbolic differentiation, differential equations of motion are derived in the form of Lagrange equations of the second kind. A method is proposed for transforming the system of trigonometric equations determining the equilibria into a system of algebraic equations, which in turn are reduced by calculating the resultant to a single algebraic equation of degree 12 in one unknown. The roots of the resulting algebraic equation determine the equilibrium orientations of the two-body system in the circular orbit plane. By applying symbolic factorization, the algebraic equation is decomposed into three polynomial factors, each specifying a certain class of equilibrium configurations. The domains with an identical number of equilibrium positions are classified using algebraic methods for constructing a discriminant hypersurface. The equations for the discriminant hypersurface determining the boundaries of domains with an identical number of equilibrium positions in the parameter space of the problem are obtained via symbolic computations of the determinant of the resultant matrix. By numerical analysis of the real roots of the resulting algebraic equations, the number of equilibrium positions of the two-body system is determined depending on the parameters.

On the factorization of lacunary polynomials

On the factorization of lacunary polynomials

Utilization of the ExactMPF package for solving a discrete analogue of the nonlinear Schrödinger equation by the inverse scattering transform method

We are revisiting the problem of solving a discrete nonlinear Schrödinger equation by the inverse scattering transform method, by use of the recently developed ExactMPF package within MAPLE Software. ExactMPF allows for an exact Wiener–Hopf factorization of matrix polynomials regardless of the partial indices of the matrix. The package can be widely used in various problems, where Wiener–Hopf factorization as one of the effective mathematical tools is required, as its code has already been disclosed. The analysis presented in this paper contains not only numerical examples of its use, but is also supported by appropriate and accurate a priori estimations. The procedure itself guarantees that the ExactMPF package produces all computations arithmetically exactly , and a detailed numerical analysis of various aspects of the computational algorithm and approximation strategies is provided in the case of a finite initial impulse.

A robust optimization approach for dynamic input allocation

This paper addresses the problem of dynamic input allocation in the presence of plant uncertainties. The current state of the art shows how to design an Allocator as the cascade of an Optimizer and an Annihilator to achieve steady-state input optimality and output invisibility simultaneously. This work proposes a novel algorithm based on polynomial factorization to design a dynamic Annihilator. The critical aspect of this approach lies in the assumption of the perfect plant knowledge, making the Annihilator not robust to uncertainties. A robustification process is introduced by optimizing its design parameters. This approach is formulated as a model-matching problem aiming to reduce the output mismatch induced by the allocation scheme while maintaining steady-state optimality. As the numerical simulations highlight, this method applies to linear and nonlinear allocation problems.

Expansion in supercritical random subgraphs of the hypercube and its consequences

It is well known that the behaviour of a random subgraph of a d-dimensional hypercube, where we include each edge independently with probability p, undergoes a phase transition when p is around 1d. More precisely, standard arguments show that just below this value of p all components of this graph have order O(d) with probability tending to one as d→∞ (whp for short), whereas Ajtai, Komlós and Szemerédi (Combinatorica 2 (1982) 1–7) showed that just above this value, in the supercritical regime, whp there is a unique “giant” component of order Θ(2d). We show that whp the vertex expansion of the giant component is inverse polynomial in d. As a consequence, we obtain polynomial in d bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollobás, Kohayakawa and Łuczak (Random Structures and Algorithms 5 (1994) 627–648) and of Pete (Electron. Commun. Probab. 13 (2008) 377–392). Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of p, which are tight up to polynomial factors in d.

Modified Boltzmann Factor on Rupture Probability

Fission Products are data that is very much needed for developing nuclear technology. Considering that the experimental results of nuclear data are minimal, theoretical modeling and calculations are needed. One of the theoretical models is the "multimodal random neck-rupture model. (M-RNRM)" However, although it has completed the fission product data, it needs closer to the reference value. For this reason, the idea of modifying the Boltzmann factor on rupture probability was developed. This modification is in the form of adding a polynomial factor to the Boltzmann factor. This work has succeeded in showing better fission product calculation results closer to the reference value.

Comparative asymptotics for discrete semiclassical orthogonal polynomials

We study the ratio [Formula: see text] asymptotically as [Formula: see text], where the polynomials [Formula: see text] are orthogonal with respect to a discrete linear functional and [Formula: see text] denote the falling factorial polynomials. We give recurrences that allow the computation of high order asymptotic expansions of [Formula: see text] and give examples for most discrete semiclassical polynomials of class [Formula: see text]. We show several plots illustrating the accuracy of our results.

A multi-Bennett 8R mechanism obtained from factorization of bivariate motion polynomials

We present a closed-loop 8R mechanism with two degrees of freedom whose motion exhibits curious properties. In any point of a two-dimensional component of its configuration variety it is possible to fix every second joint while retaining one degree of freedom. This shows that the even and the odd axes, respectively, always form a Bennett mechanism. In this mechanism, opposite distances and angles are equal and all offsets are zero. The 8R mechanism has four “totally aligned” configurations in which the common normals of any pair of consecutive axes coincide.

Maximizing Convergence Time in Network Averaging Dynamics Subject to Edge Removal

We consider the consensus interdiction problem (CIP), in which the goal is to maximize the convergence time of consensus averaging dynamics subject to removing a limited number of network edges. We first show that CIP can be cast as an effective resistance interdiction problem (ERIP), in which the goal is to remove a limited number of network edges to maximize the effective resistance between a source node and a sink node. We show that ERIP is strongly NP-hard, even for bipartite graphs of diameter three with fixed source/sink edges, and establish the same hardness result for the CIP. We then show that both ERIP and CIP cannot be approximated up to a (nearly) polynomial factor assuming the exponential time hypothesis. Subsequently, we devise a polynomial-time $mn$-approximation algorithm for the ERIP that only depends on the number of nodes $n$ and the number of edges $m$ but is independent of the size of edge resistances. Finally, using a quadratic program formulation for the CIP, we devise an iterative approximation algorithm to find a first-order stationary solution for the CIP and evaluate its good performance through numerical experiments.

A heuristic technique for decomposing multisets of non-negative integers according to the Minkowski sum

We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.

A FINITE SUM INVOLVING GENERALIZED FALLING FACTORIAL POLYNOMIALS AND DEGENERATE EULERIAN POLYNOMIALS

The aim of this paper is two-fold. First, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the degenerate Bernoulli polynomials and the degenerate Frobenius–Euler polynomials. Second, we consider the degenerate Eulerian polynomials and deduce the generating function and a recurrence relation for them.

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

Computing splitting fields using Galois theory and other Galois constructions

We describe algorithms to compute splitting fields and towers of radical extensions in which a polynomial splits, without using polynomial factorisation in towers or constructing any field properly containing the constructed field, instead extending Galois group computations for this task. We also describe the computation of geometric Galois groups (monodromy groups) and their use in computing absolute factorizations.

Machine learning-based fault diagnosis for research nuclear reactor medium voltage power cables in fraction Fourier domain

Fault diagnosis of Medium Voltage power Cables (MVCs) research nuclear reactor, incredibly inaccessible/remote ones, has to be carefully identified, located, and fixed within a short time. Therefore, this paper proposes a perfect simultaneous fault diagnosis scheme based on Multiclass Support-Vector Machine (MCSVM) in the fractional Fourier domain. First, the three-phase sending currents are simulated under different conditions then their features are extracted using Discrete Fractional Fourier Transform (DFRFT). Afterward, the features reduction process occurs via the Singular Value Decomposition (SVD) approach. MCSVM scheme is used to diagnose faults (i.e., discover, categorize, and trace) using reduced features outcome from DFRFT and SVD stages. Alternating Transient Program/Electromagnetic Transient Program (ATP/EMTP) simulations have been carried out for 22 kV unreachable MVC. Different kernels of SVM, i.e., linear, quadratic, or polynomial, and diverse factors of DFRFT, i.e., α, are investigated in simulations to obtain the optimum performance (i.e., best α and kernel pair). Hence, performance analysis of the proposed diagnosis method under different conditions (i.e., various fault resistances, locations, and inception angles) concluded two highest accuracy and lowest time settings, which were found at α = 0.5 (for both) quadratic kernel, and linear kernel, respectively. Moreover, the linear kernel achieves 99.8% accuracy rate, the lowest execution time (10 ms), and fault tracing error rate of 0.525789%, which is proper for real-time applications. Besides, our proposed method is more reliable and accurate against variable operating conditions (fault resistances, distances, and inception angles), leading to more reliable power production systems.

Stirling-Dunkl numbers

The Appell sequences of polynomials can be extended to the Dunkl context, where the ordinary derivative is replaced by Dunkl operator on the real line, and the exponential function is replaced by the so-called Dunkl kernel. In a similar way, the discrete Appell sequences can be extended to the Dunkl context, where the role of the ordinary translation is played by the Dunkl translation, that is a much more intricate operator. In particular, this allows to define the falling factorial polynomials in the Dunkl context. Some numbers closely related to falling factorial are the so called Stirling numbers of the first kind and of the second kind, as well as the Bell numbers and the Bell polynomials. In this paper, we define these numbers and polynomials in the Dunkl context, and prove some of their properties.

Difference equation satisfied by the Stieltjes transform of a sequence

We study a class of generating functions related to the Stieltjes transform of a sequence of moments with respect to the basis of falling factorial polynomials. Given a recurrence relation for the coefficient sequence, it is shown how to compute the difference equation satisified by its generating function w.r.t. this basis. We give several examples from the class of discrete semiclassical orthogonal polynomials.

Universality for Low-Degree Factors of Random Polynomials over Finite Fields

Abstract We show that the counts of low-degree irreducible factors of a random polynomial $f$ over $\mathbb {F}_q$ with independent but nonuniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\leq \exp ({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis and rely on tools recently applied by Breuillard and Varjú to study the $ax+b$ process, which show equidistribution for $f(\alpha )$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.