Arbitrary Degree Research Articles

Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

We formulate and analyze a goal-oriented adaptive finite element method for a semilinear elliptic PDE and a linear goal functional. The discretization is based on finite elements of arbitrary (but fixed) polynomial degree and involves a linearized dual problem. The linearization is part of the proposed algorithm, which employs a marking strategy different to that of standard adaptive finite element methods. Moreover, unlike the well-known dual-weighted residual strategy, the analyzed error estimators are completely computable. We prove linear convergence and, for the first time in the context of goal-oriented adaptivity for nonlinear PDEs, optimal algebraic convergence rates. In particular, the analysis does not require a sufficiently fine initial mesh.

Entire functions of slow growth whose Julia set contains any infinite collection of quasiconformal copies of polynomial Julia sets

We prove that for any infinite collection of polynomial Julia sets of arbitrary degree at least two, there exists a transcendental entire function of arbitrarily slow growth such that its Julia set contains quasiconformal copies of the given polynomial Julia sets.

Solution of the initial-value problem of first-order nonlinear recursions with homogeneous right-hand sides

In this paper, we identify systems of an arbitrary number N of first-order recursions with nonlinear right-hand sides homogeneous of arbitrary degree M, which feature simple explicit solutions. A novelty of these findings is to consider systems of recursions characterized by constraints involving their parameters and/or their initial data. Simple extensions of these findings to first-order recursions with nonhomogeneous right-hand sides are also provided.

Trivial cup products in bounded cohomology of the free group via aligned chains

Abstract We prove that the cup product of Δ-decomposable quasimorphisms, Brooks quasimorphisms or Rolli quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial.

Output feedback control with prescribed performance via funnel pre-compensator

We study output reference tracking of systems with high relative degree via output feedback only; this is, tracking where the output derivatives are unknown. To this end, we prove that the conjunction of the funnel pre-compensator with a minimum phase system of arbitrary relative degree yields a system of the same relative degree which is minimum phase as well. The error between the original system’s output and the pre-compensator’s output evolves within a prescribed performance funnel; moreover, the derivatives of the funnel pre-compensator’s output are known explicitly. Therefore, output reference tracking with prescribed transient behaviour of the tracking error is possible without knowledge of the derivatives of the original system’s output; via funnel control schemes for instance.

Back to sources – the role of losses and coherence in super-resolution imaging revisited

Photon losses are intrinsic for any translationally invariant optical imaging system with a non-trivial Point Spread Function, and the relation between the transmission factor and the coherence properties of an imaged object is universal – we demonstrate the rigorous proof of this statement, based on the principles of quantum mechanics. The fundamental limit on the precision of estimating separation between two partially coherent sources is then derived. The careful study of the role of photon losses allows to resolve conflicting claims present in previous works. We compute the Quantum Fisher Information for the generic model of optical 4f imaging system, and use prior considerations to validate the result for a general, translationally invariant imaging apparatus. We prove that the spatial-mode demultiplexing (SPADE) measurement, optimal for non-coherent sources, remains optimal for an arbitrary degree of coherence. Moreover, we show that some approximations, omnipresent in theoretical works about optical imaging, inevitably lead to unphysical, zero-transmission models, resulting in misleading claims regarding fundamental resolution limits.

PRIMITIVE ELEMENT PAIRS WITH A PRESCRIBED TRACE IN THE CUBIC EXTENSION OF A FINITE FIELD

AbstractWe prove that for any prime power$q\notin \{3,4,5\}$, the cubic extension$\mathbb {F}_{q^{3}}$of the finite field$\mathbb {F}_{q}$contains a primitive element$\xi $such that$\xi +\xi ^{-1}$is also primitive, and$\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$for any prescribed$a\in \mathbb {F}_{q}$. This completes the proof of a conjecture of Guptaet al.[‘Primitive element pairs with one prescribed trace over a finite field’,Finite Fields Appl.54(2018), 1–14] concerning the analogous problem over an extension of arbitrary degree$n\ge 3$.

Extreme Points of Gram Spectrahedra of Binary Forms

The Gram spectrahedron mathrm {Gram}(f) of a form f with real coefficients is a compact affine-linear section of the cone of psd symmetric matrices. It parametrizes the sum of squares decompositions of f, modulo orthogonal equivalence. For f a sufficiently general positive binary form of arbitrary degree, we show that mathrm {Gram}(f) has extreme points of all ranks in the Pataki range. We also calculate the dimension of the set of rank r extreme points, for any r. Moreover, we determine the pairs of rank two extreme points for which the connecting line segment is an edge of mathrm {Gram}(f). The proof of the main result relies on a purely algebraic fact of independent interest: Whenever d,rge 1 are integers with left( {begin{array}{c}r+1 2end{array}}right) le 2d+1, there exists a length r sequence f_1,dots ,f_r of binary forms of degree d for which the left( {begin{array}{c}r+1 2end{array}}right) pairwise products f_if_j, ile j, are linearly independent.

How to prove optimal convergence rates for adaptive least-squares finite element methods∗

Abstract The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53 (1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.

Adaptive regularization minimization algorithms with nonsmooth norms

AbstractAn adaptive regularization algorithm (AR$1p$GN) for unconstrained nonlinear minimization is considered, which uses a model consisting of a Taylor expansion of arbitrary degree and regularization term involving a possibly nonsmooth norm. It is shown that the nonsmoothness of the norm does not affect the ${\mathcal {O}}(\epsilon _1^{-(p+1)/p})$ upper bound on evaluation complexity for finding first-order $\epsilon _1$-approximate minimizers using $p$ derivatives, and that this result does not hinge on the equivalence of norms in $\mathbb {R}^n$. It is also shown that, if $p=2$, the bound of ${\mathcal {O}}(\epsilon _2^{-3})$ evaluations for finding second-order $\epsilon _2$-approximate minimizers still holds for a variant of AR$1p$GN named AR2GN, despite the possibly nonsmooth nature of the regularization term. Moreover, the adaptation of the existing theory for handling the nonsmoothness results is an interesting modification of the subproblem termination rules, leading to an even more compact complexity analysis. In particular, it is shown when the Newton’s step is acceptable for an adaptive regularization method. The approximate minimization of quadratic polynomials regularized with nonsmooth norms is then discussed, and a new approximate second-order necessary optimality condition is derived for this case. A specialized algorithm is then proposed to enforce first- and second-order conditions that are strong enough to ensure the existence of a suitable step in AR$1p$GN (when $p=2$) and in AR2GN, and its iteration complexity is analyzed. A final section discusses how practical approximate curvature measures may lead to weaker second-order optimality guarantees.

EFIE analyses of 3-D scattering from an object above arbitrarily rough periodic surfaces by current decomposition method

This paper addresses a three-dimensional scattering problem from a PEC object above a periodic conductive rough surface by using a decomposition method based on the separation of equivalent current density on the periodic surface as induced and perturbed current densities. The infinite solution domain is forcibly bounded by assuming that the perturbed current density caused by the object is intensively dominant in only a limited area on the periodic surface. The proposed decomposition method achieves considerable accuracy while providing an effortless computation without attempting a half-space Green's function or exercising a tapered wave illumination. This method provides a significantly superior solution to the tapered wave approach in illumination at low-grazing angles, although a plane wave is harnessed. Also, this proposed method can handle an arbitrary degree of roughness since the solution method does not contain an approximation of the surface roughness.

Adaptive Asymptotic Tracking With Global Performance for Nonlinear Systems With Unknown Control Directions

This article presents a global adaptive asymptotic tracking control method, capable of guaranteeing prescribed transient behavior for uncertain strict-feedback nonlinear systems with arbitrary relative degree and unknown control directions. Unlike most existing funnel controls that are built upon time-varying feedback gains, the proposed method is derived from a tracking error-dependent normalized function and a barrier function, together with a time-varying scaling transformation, leading to an improved prescribed performance control solution with the following features: 1) the developed control is embedded with normalized error based adaptive parameter tuning and is able to ensure asymptotic tracking; 2) given transient performance is guaranteed in that the tracking error preserves in the prescribed boundary for <inline-formula><tex-math notation="LaTeX">$\forall t\geq 0$</tex-math></inline-formula>; and 3) it is able to cope with nonlinear systems with arbitrary relative degree, mismatched uncertainties, and unknown control directions. Both theoretical analysis and numerical simulations verify the effectiveness and benefits of the proposed method.

The judge's discretion to compensate for an arbitrary divorce (Analytical Comparative Study): السلطة التقديرية للقاضي في التعويض عن الطلاق التعسفي (دراسة تحليلية مقارنة)

Divorce is the right of the husband, and is nullified by his unilateral will, and no one can prevent it in its infliction. However, it is a protection for the wife from the abuse and injustice of the husband. Compensation is established for her when his arbitrariness is proven in the divorce. The judge enjoys wide discretionary power in estimating compensation. This study aims to shed light on the most important issue which the judge has discretionary power, such us the issue of compensation for arbitrary divorce and the related provisions contained in the Iraqi Personal Status law No (188) and its amend application in the Kurdistan Region- Iraq. It is worth noting, the Iraqi and Kurdish legislators restricted the judge’s authority in order to assess the wife’s compensation for arbitrary divorce with three conditions: The first is that it must be proven that the husband was abusive in inflicting the divorce. The second is that this abuse causes damages to the wife and her submitting a request for compensation, and submitting a claim for compensation. We have followed the comparative law approach that contained in the two mentioned laws, in addition to following the analytical method for analyzing and commenting about legal texts, with referring to the judicial decisions. The study reached a number of conclusions and recommendations, including: 1- Arbitrary divorce is a divorce that the husband contradicts the purpose of the legislator in personal status law, and the husband does this divorce without reason, and the wife’s consent and her request or her misbehave. 2- It is possible to rely on the theory of abuse in the use of the right that contained in the Iraqi Civil law as a legal basis for the husband’s obligation to pay the compensation to his wife in the event of his abuse of divorce. According to one of the criteria of the mentioned theory, that the person uses his right that aims only to harm the others, the husband is reluctant to divorce his wife as long as he signs it without reason and intends to harm her; that means he deviates from the purpose for which it was initiated and causes him harm to others, he is obliged to pay compensation. This is because the Iraqi and Kurdish legislators have indicated that the wife is awarded compensation whenever the court finds that she has been harmed due to the husband’s arbitrariness in inflicting divorce. 3- We suggest to the Iraqi legislator to amend the text of third Paragraph of Article (39) of the Personal Status Law, and determining the maximum period of wife’s alimony and making it three years instead of two years, in order to protect the rights of the wife, We suggest to the Kurdistan legislator to amend the second Paragraph of Article (17) of the Amending law of the application of Iraqi Personal Status Law No (188) of 1959 in the Kurdistan Region – Iraq, and determining the same period instead of determining the immediate period, by leaving this issue according to the discretionary power of the judge, in order to estimate the compensation according to each case separately. To read the text in the two laws as follows: (If the husband divorced his wife and the court finds that the husband was abusive in her divorce and the wife suffered harm as a result of that; the court, based on the wife’s request, shall award compensation commensurate with his financial condition and degree of arbitrariness, estimated in total, provided that it does not exceed her maintenance for a period of three years in addition to her other fixed rights).

Robust gain-scheduled output feedback H2 controller synthesis with reduced conservativeness: An application to EMS system

This article investigates the design problem of gain-scheduled output feedback (GSOF) [Formula: see text] controller exploiting uncertain scheduling parameters for continuous-time linear parameter-varying (LPV) systems. The problem of coexistence of absolute and proportional uncertainties on the scheduling parameters is considered to address more practical situations. This challenging issue has been tackled by introducing a novel admissible region for the actual and measured scheduling parameters. Furthermore, all the system matrices of the LPV system are assumed to be polynomially dependent on the scheduling parameters with arbitrary degrees. Both Lyapunov and auxiliary matrices are considered to be parameter-dependent. The merit of the proposed method lies in its less conservativeness in comparison with the available approaches. The design method is presented in terms of solutions to a set of parameter-dependent linear matrix inequalities (LMIs) including parameter searches for two scalar values. The application of the provided method on an electromagnetic suspension (EMS) system is considered to demonstrate the applicability and benefits of the proposed approach for practical systems.

Entangled two-photon absorption with Brownian-oscillator fluctuations.

We theoretically investigate the two-photon absorption signals of a three-band (g, e, f) system diagonally coupled to an over-damped Brownian oscillator bath, which induces random Gaussian modulations of energy levels with an arbitrary degree of correlation. For fast modulation, extra 2ωeg and 2ωfe peaks may obscure the g-f transitions in the classical two-photon absorption (CTPA) spectra for nearly resonant e states. These peaks arise from one-photon resonant g-e or e-f transitions. In the slow modulation limit, these peaks vanish because of the short tails of the Gaussian line shape. CTPA strongly depends on the correlations between energy fluctuations. In entangled two-photon absorption, the extra peaks are eliminated because of the broad one-photon but narrow two-photon spectrum of the twin photons. The variation of the coherences between f states with the correlation between energy fluctuations is explored.

Basic reproduction number for the SIR epidemic in degree correlated networks

The basic reproduction number R0 is an important indicator of the severity for an epidemic outbreak, but it may not be obtained easily for heterogeneous populations especially with biased mixing contacts. Usually, explicit formula for the basic reproduction number in degree correlated networks is difficult to get, thus it is very useful to understand the relationship with their uncorrelated counterparts. It seems that the assortative mixing increases the basic reproduction number of susceptible–infected–recovered (SIR) epidemic, whereas disassortative mixing decreases it, or that assortative mixing enhances the percolation, whereas disassortative mixing weakens it. This result is obtained for degree correlated networks based on a small deviation from the uncorrelated networks, and may not be universal. In this paper, we show rigorously that the result is true for degree correlated networks with arbitrary bimodal degree distribution and for assortative mixing networks with arbitrary degree distribution. However, this may not be always true for general disassortative mixing networks. We also introduce a numerical algorithm to construct an arbitrary joint degree distribution and generate a counterexample of disassortative mixing network that the result fails. Finally, a sufficient condition is given to guarantee that the SIR model in disassortative mixing networks yields smaller R0.

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

Distributed optimization for a class of uncertain nonlinear multi‐agent systems with arbitrary relative degree subject to exogenous disturbances

AbstractThis paper studies the distributed optimization problem on the framework of a class of arbitrary relative degree uncertain nonlinear multi‐agent systems with the presence of exogenous disturbances. Based on the internal model principle and the pseudo gradient approach, a novel continuous‐time distributed optimization control protocol is proposed to make all the agents reach optimal consensus by utilizing local cost functions information and neighbors' states information. The main advantage of our algorithm is that it can run on systems with more complex dynamics, while the existing algorithms apply only to integral‐type systems or unity relative degree nonlinear systems. Moreover, the convergence of the algorithm is proved by Lyapunov function and graph theory analysis. Finally, the distributed protocol is physically implemented by circuits and tested on a group of Chua's circuit systems to show the effectiveness of the theoretical results.

Delegation and ambiguity in correlated equilibrium

In the context of normal-form games with complete information, we introduce a notion of correlated equilibrium that allows partial delegation to a mediator and ambiguity in the correlation device. Without ambiguity, the sets of equilibrium action distributions are equivalent to those for coarse correlated equilibrium (Moulin and Vial, 1978). With correlation devices that incorporate ambiguity, any action distribution that Pareto dominates a coarse correlated equilibrium or a correlated equilibrium (Aumann, 1974), can be approximated with an arbitrary degree of precision using the proposed equilibrium notion. These approximations are attained in one-shot, static strategic interactions, and do not require repeated play. We also analyze such equilibria when the set of feasible posteriors is exogenously constrained, which yields, as a special case, a definition and characterization of an “ambiguous correlated equilibrium” that does not require delegation to the mediator.

The Design of Matched Balanced Orthogonal Multiwavelets

The theory of orthogonal multiwavelets offers enhanced flexibility for signal processing applications and analysis by employing multiple waveforms simultaneously, rather than a single one. When implementing them with polyphase filter banks, it has been recognized that balanced vanishing moments are needed to prevent undesirable artifacts to occur, which otherwise compromise the interpretation and usefulness of the multiwavelet analysis. In the literature, several such balanced orthogonal multiwavelets have been constructed and published; but however useful, their choice is still limited. In this work we present a full parameterization of the space of all orthogonal multiwavelets with two balanced vanishing moments (of orders 0 and 1), for arbitrary given multiplicity and degree of the polyphase filter. This allows one to search for matching multiwavelets for a given application, by optimizing a suitable design criterion. We present such a criterion, which is sparsity-based and useful for detection purposes, which we illustrate with an example from electrocardiographic signal analysis. We also present explicit conditions to build in a third balanced vanishing moment (of order 2), which can be used as a constraint together with the earlier parameterization. This is demonstrated by constructing a balanced orthogonal multiwavelet of multiplicity three, having three balanced vanishing moments, but this approach can easily be employed for arbitrary multiplicity.