Arbitrary Degree Research Articles

Moduli of rank 2 Higgs sheaves on elliptic surfaces

We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces, (semi)stable with respect to suitable polarizations in the sense of Friedman and O’Grady. We prove that slope-semistability of a Higgs sheaf on the surface implies semistability on the generic fiber. In the case of Higgs sheaves of odd fiber degree on elliptic surfaces in characteristic \(\ne 2\), we prove that any moduli space of Higgs sheaves with fixed numerical invariants splits canonically as the product of the moduli space of ordinary sheaves (with the same invariants), and the space of global regular 1-forms on the surface. For elliptic surfaces with section in characteristic zero, and in the case arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman spectral curve, or is regular on a general fiber with non-reduced spectral cover, then its Higgs field takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface.

Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties

The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states (LDOSs). For random graphs with a negative binomial degree distribution, we show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the LDOSs at the centre of the spectrum displays a power-law tail controlled by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs.

Fast calculation of gravitational effects using tesseroids with a polynomial density of arbitrary degree in depth

Fast and accurate calculation of gravitational effects on a regional or global scale with complex density environment is a critical issue in gravitational forward modelling. Most existing significant developments with tessroid-based modelling are limited to homogeneous density models or polynomial ones of a limited order. Moreover, the total gravitational effects of tesseroids are often calculated by pure summation in these methods, which makes the calculation extremely time-consuming. A new efficient and accurate method based on tesseroids with a polynomial density up to an arbitrary order in depth is developed for 3D large-scale gravitational forward modelling. The method divides the source region into a number of tesseroids, and the density in each tesseroid is assumed to be a polynomial function of arbitrary degree. To guarantee the computational accuracy and efficiency, two key points are involved: (1) the volume Newton’s integral is decomposed into a one-dimensional integral with a polynomial density in the radial direction, for which a simple analytical recursive formula is derived for efficient calculation, and a surface integral over the horizontal directions evaluated by the Gauss–Legendre quadrature (GLQ) combined with a 2D adaptive discretization strategy; (2) a fast and flexible discrete convolution algorithm based on 1D fast Fourier transform (FFT) and a general Toepritz form of weight coefficient matrices is adopted in the longitudinal dimension to speed up the computation of the cumulative contributions from all tesseroids. Numerical examples show that the gravitational fields predicted by the new method have a good agreement with the corresponding analytical solutions for spherical shell models with both polynomial and non-polynomial density variations in depth. Compared with the 3D GLQ methods, the new algorithm is computationally more accurate and efficient. The calculation time is significantly reduced by 3 orders of magnitude as compared with the traditional 3D GLQ methods. Application of the new algorithm in the global crustal CRUST1.0 model further verifies its reliability and practicability in real cases. The proposed method will provide a powerful numerical tool for large-scale gravity modelling and also an efficient forward engine for inversion and continuation problems.

Strict Quantization of Polynomial Poisson Structures

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on {mathbb {R}}^d, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of hbar , giving strict quantizations on the space of analytic functions on {mathbb {R}}^d with infinite radius of convergence. We also address further questions such as continuity of the classical limit hbar rightarrow 0, compatibility with ^*-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as ^*-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

Computing Abductive Explanations

We study the computation of constrained explanations in the framework of abductive logic programming. A general characteristic of abductive reasoning is the existence of multiple abductive explanations. Therefore, identifying a subclass of “preferred explanations” is a relevant problem. A typical approach is to “prefer” explanations that are, in some sense, simple. Several concepts of simplicity were considered in the literature, most notably those based on minimality with respect to inclusion and cardinality. We adopt, as a measure of the quality of an explanation, its degree of arbitrariness that can be briefly described as the number of arbitrary assumptions that have been made to derive the explanation. The more arbitrary the explanation, the less appealing it is, with explanations having no arbitrariness, called constrained, being the preferred ones. In this article, we present a technique that, for a special class of theories, computes constrained explanations. It is based on a rewriting of the theory and the observation into a disjunctive logic program with negation so that the constrained explanations correspond to a subset of its stable models. The proposed technique lays the foundation for using ASP solvers to compute constrained explanations.

Extensively Explored and Evaluated Actor-Critic With Expert-Guided Policy Learning and Fuzzy Feedback Reward for Robotic Trajectory Generation

Trajectory generation for redundant manipulators based on inverse kinematics (IK) still faces some restraints, as it lacks universal IK calculation or specific trajectory generation methods that are suitable for robots with arbitrary degrees of freedom. In this article, the IK-free trajectory generation for robot manipulators is formulated as a Markov decision process and implemented by a general method based on deep reinforcement learning. First, an extensively explored and evaluated actor-critic (E3AC) algorithm that can make diverse action explorations and comprehensive evaluations is designed to solve the trajectory generation problem. Second, a dual-memory structure with expert-guided policy learning strategy is proposed to further enhance the performance of the algorithm in the early training period by additional successful experiences and performing an increasingly unbiased data sampling. Third, a fuzzy feedback reward mechanism that can directly establish a mapping from the abundant state variables to the self-tuning reward is constructed, instead of puzzling out an explicit function to feature the complex relations among the control objects. Finally, the comparative experimental results show that, the proposed approach is more efficient in algorithm convergence and reward calculation, and is more qualified for complex tasks with strong randomness.

Envelopes are Solving Machines for Quadratics and Cubics and Certain Polynomials of Arbitrary Degree

Summary Everybody knows from school how to solve a quadratic equation of the form x 2 − p x + q = 0 graphically. To solve more than one equation this method can become tedious, as for each pair (p, q) a new parabola has to be drawn. Stunningly, there is one single curve that can be used to solve every quadratic equation by drawing tangent lines through a given point (p, q) to this curve. In this article we derive this method in an elementary way and generalize it to equations of the form x n − p x + q = 0 for arbitrary n ≥ 2 . Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. In concluding the article, we point out connections to the duality of points and lines in the plane and to the concept of Legendre transformation.

On the Implementation of an Adaptive Multirate Framework for Coupled Transport and Flow

This paper presents a multirate in time approach for coupled flow and transport problems combined with goal-oriented error control based on the Dual Weighted Residual (DWR) method. The focus is on the implementation of the multirate concept regarding different time scales for the underlying subproblems. Key ingredients are an arbitrary degree discontinuous Galerkin time discretization, an a posteriori error representation for the transport problem coupled with flow and the concept of space-time slabs based on tensor-product spaces. From the latter, a space-time mesh adaptation with highly economical grids is developed. The performance of the approach is studied carefully by numerical convergence examples as well as an example of physical interest for convection-dominated transport.

Waring Ranks of Sextic Binary Forms via Geometric Invariant Theory

We determine the Waring ranks of all sextic binary forms with complex coefficients using a Geometric Invariant Theory approach. Using the five basic invariants for sextic binary forms, our results give a rapid method to determine the Waring rank of any given sextic binary form. In particular, we shed new light on a claim by E. B. Elliott at the end of the nineteenth century concerning the binary sextics with Waring rank 3. We show that for binary forms of arbitrary degree the cactus rank, a.k.a. scheme rank, is determined by the corresponding Waring rank. Finally, we determine the border ranks of all binary sextics.

Bl-homogeneous sliding-mode observers for SISO linear time-varying systems with unknown input

In this article, a family of state observers with bl- homogeneous properties is proposed for a class of linear systems with time-varying coefficients, in presence of an unknown input and with an arbitrary relative degree. The proposed observers converge globally and at least asymptotically to the true states for systems that satisfy the structural property of strong observability. With an adequate selection of gains and parameters the convergence is global and theoretically exact, in finite time or even more, in fixed time.

Images of graded polynomials on matrix algebras

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra Mn(K) over a field K endowed with its canonical Zn-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field Q of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on n×n matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over Mn(K) when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on M2(K).

Quantum Level Instability of Transverse Excitation in Electron Flow

In current research, we use the effective Schrödinger-Poisson model to study a new kind of quantum-level instability in an infinite-wall slab electron flow. We use the Madelung fluid representation along with the conventional eigenvalue problem techniques in order to solve the linearized coupled differential equations representing the linear transverse collective excitations in the electron gas of arbitrary degree of degeneracy having a constant perpendicular drift. It is shown that the energy levels of collective electrostatic excitations are doubly quantized due to mutual interactions between single electron oscillations, analogous to the problem of a particle in a box, and collective Langmuir oscillations, which are modulated over single electron quantum state. We also report the transverse excitation instability of plasmon energy level in electron slab flow due to the interplay between the wave-like dispersion and the destabilizing perpendicular electron drift momentum. We further study in detail the parametric dependence of such instability in terms of different aspects of the many-electron system. Such a quantum-level instability may have important applications in characteristic behavior of plasmonic devices and their frequency response. Parametric quantization of drifting electron fluid may also have broad applications in nanoscale quantum device calibration and quantum measurements.

Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation

Abstract We present quadrature schemes to calculate matrices where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when the modified Hilbert transformation is used for the variational setting. This work provides the calculation of these matrices to machine precision for arbitrary polynomial degrees and non-uniform meshes. The proposed quadrature schemes are based on weakly singular integral representations of the modified Hilbert transformation. First, these weakly singular integral representations of the modified Hilbert transformation are proven. Second, using these integral representations, we derive quadrature schemes, which treat the occurring singularities appropriately. Thus, exponential convergence with respect to the number of quadrature nodes for the proposed quadrature schemes is achieved. Numerical results, where this exponential convergence is observed, conclude this work.

Born-Padé approach to electromagnetic scattering in complex one-dimensional inhomogeneous slabs

Perturbation theory is applied to one-dimensional scattering systems consisting of a general class of inhomogeneous and isotropic slabs having size $L$ described by the relative permittivity $\varepsilon(z) = 1 + \alpha \chi(z)$, where $\chi(z)$ is the electric susceptibility and $\alpha$ the perturbation parameter. The transmitted and reflected amplitudes are shown to be written as Born series in powers of $\alpha$ and Pad\'e approximants are used to obtain analytical results with a high and arbitrary degree of accuracy. The approach is very general and can handle oblique incidence. Examples are given for the transmission and reflection amplitudes of plane waves interacting with Hermitian and non-Hermitian structures with known and unknown exact solutions.

Winning property of badly approximable points on curves

We prove that badly approximable points on any analytic nondegenerate curve in Rn are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among various consequences of our main result is a solution to Bugeaud’s problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.

On Free Velocity Cones of Narrow Passages of High-DoF Kinematic Chains

Narrow passages in free space pose great challenges to many sampling-based motion planners. In problems with high-dimensional configuration spaces (C-spaces), narrow passages in free space (C-free) can be hard to find and navigate and, sometimes, even counterintuitive. In this article, we present an algorithm to construct cones of available instantaneous velocities (“free velocity cones” or, briefly, “free cones”) on the boundary of C-free to facilitate finding and navigating narrow passages. This is accomplished by first developing free cones and associated measures of local C-space narrowness for a single rigid link. These results are then extended to kinematic chains (open and closed) of arbitrary degrees of freedom. It turns out that the degeneracy of the free cones dictates the existence of C-space narrow passages, and the locations and orientations of the links in the workspace relate intimately to the corresponding connected component of C-free. This observation leads us to a modified probabilistic roadmap (M-PRM) algorithm and a modified rapidly exploring random vine (M-RRV) algorithm of combining the enumeration of topological components and random sampling of the free cones. Experimental results from applying our algorithms to several challenging examples show that our new algorithms are more efficient than some variants of the PRM algorithms, and several variants of the rapidly exploring random tree (RRT) algorithms, such as RRT-CONNECT and RRV.

Explicitly solvable systems of first-order ordinary differential equations with homogeneous right-hand sides, and their periodic variants

In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data.

A scheme for the analysis of primal stationary boundary value problems based on FE/FD multi-method

We present a non-iterative scheme for solving general linear stationary primal boundary value problems with an arbitrary number of degrees of freedom. The main goal is to combine the finite element method performance for domains with arbitrary geometry with the finite difference method efficiency related to its low computational cost. We apply a domain decomposition scheme to define rectangular subdomains (2D case) or rectangular hexahedral subdomains (3D case). A finite element mesh is generated in such a way that the elements in the regular regions are squares (2D case) or cubes (3D case). A suitable overlap scheme allows the finite element method to be used in regions of non-regular elements, while the finite difference method can be used in regions with regular elements (squares for 2D case or cubes for 3D case). We present six computational simulations showing the best performance in the CPU time of the proposed scheme compared to the classic finite element method while maintaining solution accuracy. • Reduction in the computational cost of assembling the matrices. • Solution time of the system of equations is preserved. • Feasible for problems with arbitrary degrees of freedom and non-homogeneous media. • Able to deal with several regular finite difference subdomains.

A fast and accurate DE-formula algorithm to evaluate Ambartsumian-Chandrarasekhar $H$-function for isotropic scattering

In the present work, a compact, fast, and yet accurate algorithm is developed to calculate the numerical values of the Ambartsumian-Chandrasekhar’s \(H\)-function for isotropic scattering and its moments on the basis of the double exponential (DE) formula of Takahashi and Mori (RIMS, Kyoto Univ., 9:721, 1974). The main improvement made in the new method is an elimination of the iterative procedure for automatic adjustment of the step-size of integrations carried out with the DE-formula. Instead, a set of optimal values for the upper limit of integration \(T_{\text{max}}\) and the number of division points \(N_{\text{T}}\) to specify the step-size of the quadrature is predetermined for calculations of the \(H\)-function with a 15-digit accuracy, and also another for the evaluations of the moments with an accuracy of 14-digits or better. FORTRAN90 subroutines HFISCA for the \(H\)-function and HFMOMENT for the moments of arbitrary degrees are subsequently constructed (their source codes and a driver together with a sample set of output are shown in Appendix of this paper). Tables of sample calculations of the \(H\)-function and its moments of degree −1 through 6 carried out by these programs are also presented. The routines HFISCA and HFMOMENT should prove useful not only in astrophysical applications but also in other disciplines of science such as the electron transports in condensed matter and remote-sensing data analyses. A request for a copy of the Fortran 90 source code of the program can be made by writing to kawabata@rs.kagu.tus.ac.jp.

Linear Forms in Polylogarithms

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $\Phi_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic function. Let $\alpha_1, \cdots, \alpha_m$ be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with $0<|\alpha_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion for the linear independence, over an algebraic number field containing $\mathbb{Q}(\alpha_1, \cdots, \alpha_m)$, of all the $rm+1$ numbers : $\Phi_1(x,\alpha_1)$, $\Phi_2(x,\alpha_1), $ $\cdots , \Phi_r(x,\alpha_1)$, $\Phi_1(x,\alpha_2)$, $\Phi_2(x,\alpha_2), $ $\cdots , \Phi_r(x,\alpha_2), \cdots, \cdots, \Phi_1(x,\alpha_m)$, $\Phi_2(x,\alpha_m)$, $\cdots , \Phi_r(x,\alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [10], together with a proof in detail on the non-vanishing Wronskian of Hermite type.