Arbitrary Degree Research Articles

A pressure-robust HHO method for the solution of the incompressible Navier–Stokes equations on general meshes

Abstract In a recent work (Castanon Quiroz & Di Pietro (2020) A hybrid high-order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. Comput. Math. Appl., 79, 2655–2677), we have introduced a pressure-robust hybrid high-order method for the numerical solution of the incompressible Navier–Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper, we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for arbitrary polynomial degrees $k\geq 0$ and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples.

Distributed Adaptive Output Consensus of Unknown Heterogeneous Non-Minimum Phase Multi-Agent Systems

This article addresses the leader-following output consensus problem of heterogeneous linear multi-agent systems with unknown agent parameters under directed graphs. The dynamics of followers are allowed to be non-minimum phase with unknown arbitrary individual relative degrees. This is contrary to many existing works on distributed adaptive control schemes where agent dynamics are required to be minimum phase and often of the same relative degree. A distributed adaptive pole placement control scheme is developed, which consists of a distributed observer and an adaptive pole placement control law. It is shown that under the proposed distributed adaptive control scheme, all signals in the closed-loop system are bounded and the outputs of all the followers track the output of the leader asymptotically. The effectiveness of the proposed scheme is demonstrated by one practical example and one numerical example.

A note on the differentiability of discrete Palmer's linearization

In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are $C^2$ diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.

Equitable sharing of deep-sea mining benefits: More questions than answers

The International Seabed Authority is tasked to develop rules for “equitable sharing of financial and other economic benefits” from deep-sea mining activities in the seabed area beyond national jurisdiction. Without this element of the legal regime, the ISA cannot meet its stated aim of ensuring deep-sea mining activities are undertaken for the ‘benefit of [hu]mankind as a whole’, with particular consideration to the interests and needs of developing States. This paper examines proposals made at the ISA to date. It demonstrates, using modelled revenue estimates, that the direct distribution of funds would lead to States receiving economically insignificant benefits. It also examines formulae for dividing benefits between member States, noting a degree of arbitrariness in current proposals. The authors note the merit of an alternative proposal for pooling mining revenue into a ‘Seabed Sustainability Fund’, but question whether, as is currently proposed, the fund should be narrowly focused on deep-sea mining related activities, including activities that should be funded by miners, and/or before mining commences. In addition, aspects of the fund’s proposed governance are examined critically against international best practice. The paper finally raises the importance of the decision-making process for equitable benefit-sharing arrangements meeting the highest standards of process legitimacy, including transparency and consultation. The paper notes that these are decisions about the common heritage of [hu]mankind, which include value questions on which there may be diverse positions, and that set a precedent for other global common resources in the future.

Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise.

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.

The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

Accuracy of high-order, discrete approximations to the lifting-line equation

AbstractThe accuracy of several numerical schemes for solving the lifting-line equation is investigated. Circulation is represented on discrete elements using polynomials of varying degree, and a novel scheme is introduced based on a discontinuous representation that permits arbitrary polynomial degrees to be used. Satisfying the Helmholtz theorems at inter-element boundaries penalises the discontinuities in the circulation distribution, which helps ensure the solution converges towards the correct, continuous behaviour as the number of elements increases. It is found that the singular vorticity at the wing tips drives the leading-order error of the solution. With constant panel widths, numerical schemes exhibit suboptimal accuracy irrespective of the basis degree; however, driving the width of the tip panel to zero at a rate faster than the domain average enables improved accuracy to be recovered for the quadratic-strength elements. In all cases considered, higher-order circulation elements exhibit higher accuracy than their lower-order counterparts for the same total degrees of freedom in the solution. It is also found that the discontinuous quadratic elements are more accurate than their continuous counterparts while also being more flexible for geometric representation.

Weighted bootstrap for two-sample [formula omitted]-statistics

In this paper, we introduce weighted bootstrap algorithms for both non-degenerate and degenerate two-sample U-statistics with arbitrary degrees. For the non-degenerate case, weighted bootstrap with dependent weights is introduced as a generalization of Efron’s conventional bootstrap. In addition, two weighted bootstrap procedures with independent productive weights and independent additive weights are proposed under non-degeneracy. More importantly, we extend the weighted bootstrap method to two-sample U-statistics under the degeneracy of order 2 with a novel construction of random weights. Theoretical supports of the proposed weighted bootstrap procedures under non-degeneracy and degeneracy of order 2 are established. Numerical studies illustrate that the proposed approaches are feasible and effective for statistical inferences.

Taking stock of long-horizon predictability tests: Are factor returns predictable?

This study provides a critical assessment of long-horizon return predictability tests using highly persistent regressors. We show that the commonly used statistics are typically oversized, leading to spurious inference. Instead, we propose a Wald statistic, which accommodates multiple predictors of (unknown) arbitrary persistence degree within the I(0)-I(1) range. The test statistic, based on an adaptation of the IVX procedure to a long-horizon regression framework, is shown to have a standard chi-squared asymptotic distribution (regardless of the stochastic properties of the regressors used as predictors) and to exhibit excellent finite-sample size and power properties. Employing this test statistic, we find evidence of predictability for “old” and “new” pricing factors with monthly returns, but this becomes weaker as the predictive horizon increases. The predictability evidence substantially weakens with annual data. Overall, we question the incremental value of using long-horizon predictive regressions.

Time-optimal construction of overlay networks

This article shows how to construct an overlay network of constant degree and diameter O(log n) in O(log n) time starting from an arbitrary weakly connected graph. We assume a synchronous communication network in which nodes can send messages to nodes they know the identifier of, and new connections can be established by sending node identifiers. Suppose the initial network’s graph is weakly connected and has constant degree. In that case, our algorithm constructs the desired topology with each node sending and receiving only O(log n) messages in each round in O(log n) time w.h.p., which beats the currently best O(log ^{3/2} n) time algorithm of Götte et al. (International colloquium on structural information and communication complexity (SIROCCO), Springer, 2019). Since the problem cannot be solved faster than by using pointer jumping for O(log n) rounds (which would even require each node to communicate Omega (n) bits), our algorithm is asymptotically optimal. We achieve this speedup by using short random walks to repeatedly establish random connections between the nodes that quickly reduce the conductance of the graph using an observation of Kwok and Lau (Approximation, randomization, and combinatorial optimization. Algorithms and techniques (APPROX/RANDOM 2014), Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014). Additionally, we show how our algorithm can be used to efficiently solve graph problems in hybrid networks (Augustine et al. in Proceedings of the fourteenth annual ACM-SIAM symposium on discrete algorithms, SIAM, 2020). Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the initial edges is unrestricted, whereas only polylogarithmically many messages can be sent over edges that have been established throughout an algorithm’s execution. For an (undirected) graph G with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in O(log n) time w.h.p. Furthermore, we show how to compute an MIS in O(log d + log log n) time w.h.p., where d is the initial degree of G.

Linked cluster expansion on trees.

The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.

Two families of two‐dimensional finite element interpolation functions of general degree with general continuity

AbstractThis work presents a generalization of the functions used in Fraeijs de Veubeke's quadrilateral (FVQ) element and in Hsieh, Clough and Tocher's (HCT) triangle, allowing for an arbitrary degree of the approximation, , and for arbitrary order of continuity, . These functions can be used to define finite elements that lead by direct assembly to approximations. Their determination for each element is procedural, generalizing the reasoning used by Zienkiewicz to explain the FVQ element. The completeness of each basis is confirmed by comparing its dimension with that of a broken basis, defined primitive element wise, upon which the continuity constraints are successively imposed. The order of continuity at the vertices that is required to enable the independent definition of the normal derivatives at the sides is confirmed numerically. The corresponding interpolation functions are symbolically derived using Mathematica, and simple matlab codes using the data thus obtained are made available, to illustrate their application.

On the efficient global dynamics of Newton’s method for complex polynomials

We investigate Newton’s method as a root finder for complex polynomials of arbitrary degrees. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical mathematics, numerics, computer graphics and physics, known methods may have excellent theoretical complexity but cannot be used in practice, or are practically efficient but lack a successful theory behind them. We provide precise and strong upper bounds for the theoretical complexity of Newton’s method and show that it is near-optimal with respect to the known set of starting points that find all roots. This theoretical result is complemented by a recent implementation of Newton’s method that finds all roots of various polynomials of degree more than a billion, significantly faster than our upper bounds on the complexity indicate, and often much faster than established fast root finders. Newton’s method thus stands out as a method that has strong merits both from the theoretical and from the practical point of view. Our study is based on the known explicit set of universal starting points, for each degree d, that are guaranteed to find all roots of polynomials of degree d (appropriately normalized). We show that this set contains d points that converge very quickly to the d roots: the expected total number of Newton iterations required to find all d roots with precision ɛ is , which can be further improved to . The key argument shows that many root finding orbits are ‘R-central’ in the sense that they stay forever in a disk of radius R, and each iteration ‘uses up’ an explicit amount of area within this disk.

Photo-plasmonic effect as the hot electron generation mechanism

Based on the effective Schrödinger–Poisson model a new physical mechanism for resonant hot-electron generation at irradiated half-space metal–vacuum interface of electron gas with arbitrary degree of degeneracy is proposed. The energy dispersion of undamped plasmons in the coupled Hermitian Schrödinger–Poisson system reveals an exceptional point coinciding the minimum energy of plasmon conduction band. Existence of such exceptional behavior is a well-know character of damped oscillation which in this case refers to resonant wave–particle interactions analogous to the collisionless Landau damping effect. The damped Schrödinger–Poisson system is used to model the collective electron tunneling into the vacuum. The damped plasmon energy dispersion is shown to have a full-featured exceptional point structure with variety of interesting technological applications. In the band gap of the damped collective excitation,depending on the tunneling parameter value, there is a resonant energy orbital for which the wave-like growing of collective excitations cancels the damping of the single electron tunneling wavefunction. This important feature is solely due to dual-tone wave-particle oscillations, characteristics of the collective excitations in the quantum electron system leading to a resonant photo-plasmonic effect, as a collective analog of the well-known photo-electric effect. The few nanometer wavelengths high-energy collective photo-electrons emanating from the metallic surfaces can lead to a much higher efficiency of plasmonic solar cell devices, as compared to their semiconductor counterpart of electron–hole excitations at the Fermi energy level. The photo-plasmonic effect may also be used to study the quantum electron tunneling and electron spill-out at metallic surfaces. Current findings may help to design more efficient spasers by using the feature-rich plasmonic exceptional point structure.

Generalized de Boor–Cox Formulas and Pyramids for Multi-Degree Spline Basis Functions

The conventional B-splines possess the de Boor–Cox formula, which relates to a pyramid algorithm. However, for multi-degree splines, a de Boor–Cox-type evaluation algorithm only exists in some special cases. This paper considers any multi-degree spline with arbitrary degree and continuity, and provides two generalized de Boor–Cox-type relations. One uses several lower degree polynomials to build a combination to evaluate basis functions, whose form is similar to using the de Boor–Cox formula several times. The other is a linear combination of two functions out of the recursive definition, which keeps the combination coefficient polynomials of degree 1, so it is more similar to the de Boor–Cox formula and can be illustrated by several pyramids with different heights. In the process of calculating the recursions, a recursive representation using the Bernstein basis is used and numerically analyzed.

Control Barriers in Bayesian Learning of System Dynamics

This article focuses on learning a model of system dynamics online, while satisfying safety constraints. Our objective is to avoid offline system identification or hand-specified models and allow a system to safely and autonomously estimate and adapt its own model during operation. Given streaming observations of the system state, we use Bayesian learning to obtain a distribution over the system dynamics. Specifically, we propose a new matrix variate Gaussian process (MVGP) regression approach with an efficient covariance factorization to learn the drift and input gain terms of a nonlinear control-affine system. The MVGP distribution is then used to optimize the system behavior and ensure safety with high probability, by specifying control Lyapunov function (CLF) and control barrier function (CBF) chance constraints. We show that a safe control policy can be synthesized for systems with arbitrary relative degree and probabilistic CLF-CBF constraints by solving a second-order cone program. Finally, we extend our design to a self-triggering formulation, adaptively determining the time at which a new control input needs to be applied in order to guarantee safety.

An extension of Venkatesh’s converse theorem to the Selberg class

Abstract We extend Venkatesh’s proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree $2$ gamma factors of Selberg class type.

An Output-Feedback Design Approach for Robust Stabilization of Linear Systems With Uncertain Time-Delayed Dynamics in Sensors and Actuators

In this paper, we propose a control approach for the robust stabilization of linear time-invariant (LTI) systems with non-negligible sensor and actuator dynamics subject to time-delayed signals. Our proposition is based on obtaining an augmented model that encompasses the plant, sensor, and actuator dynamics and also the time-delay dynamic effect. We make use of the Padé Approximation for modeling the time-delay impact on the feedback loop. Since the actual plant state variables are not available for feedback, the sensor outputs, which represent a subset of the augmented system state variables, are used for composing a static output-feedback control law. The robust controller gains are computed by means of a two-stage strategy based on linear matrix inequalities (LMI). For obtaining less conservative conditions we consider the use of homogeneous-polynomial Lyapunov functions (HPLF)– and other decision variables– of arbitrary degree. In our proposition, we also take into account the inclusion of a minimum decay rate criterion in order to improve closed-loop system transient response. Disturbance rejection is also addressed through extensions to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathscr {H}_{2}$ </tex-math></inline-formula> guaranteed cost minimization. The effectiveness of the proposed strategy is attested in the design of a controller for the lateral axis dynamics of an aircraft and other academic examples.

Investigation of Field and Energy in a Weakly-Conducting Optical Fiber with an Arbitrary Degree of Refractive Index Profile

Introduction. We consider a weakly conductive gradient fiber in the single-mode regime and solve the equation for the electric field in the core of this fiber in a general form in the first approximation. The aim of this study is to study the field and energy in the core of a weakly conductive gradient fiber without taking into account the polarization in the single-mode regime in the case of a power-law (generally) refractive index profile. Materials and Methods. From Maxwell’s equations for dielectric media, there was derived an equation for the field in a fiber with gradient refractive index profile. Making the appropriate substitutions, replacing the zero-order Bessel function with a Gaussian function, and making the necessary approximation of the resulting equation, we arrive at an equation that we solve by the Wentzel – Kramers – Brillouin method and obtain analytical expressions for the field and energy inside waveguide for an arbitrary degree of the refractive index. Results. There was obtained a solution of the equation for the field in fiber with a powerlaw refractive index profile. Numerical calculations were carried out. A graph of the dependence of a dimensionless quantity – “normalized” energy – on the waveguide parameter for the first five powers of the profile (n = 1, 2, 3, 4, 5) was plotted. Discussion and Conclusion. It is shown that the energy increases faster for the profile with n = 1, and after this value, the energy for the profile with n = 1 increases sharply, and for n &gt; 1, the energy growth decreases with increasing n. The results obtained in this work can be used for creating an energy-efficient core, for carrying out a possible analysis of information transmission, and for designing waveguides taking into account specific applications.

Generalized trigonometry and Chebyshev functions in finite fields

Trigonometry in finite fields was introduced by de Souza et al. and further developed by Lima and Panario and others, giving functions with many properties similar to trigonometric functions over the reals. Those explorations used a degree-2 extension of a base field. While this corresponds most closely to trigonometry over the reals, in finite fields we can have extensions of other degrees. In this paper we generalize the definitions of trigonometric functions and their related Chebyshev polynomials to arbitrary degrees and explore their properties. Many familiar results carry over into the generalized setting.