Arbitrary Degree Research Articles

Electromagnetic Multi–Gaussian Speckle

Generalizing our prior work on scalar multi-Gaussian (MG) distributed optical fields, we introduce the two-dimensional instantaneous electric-field vector whose components are jointly MG distributed. We then derive the single-point Stokes parameter probability density functions (PDFs) of MG-distributed light having an arbitrary degree and state of polarization. We show, in particular, that the intensity contrast of such a field can be tuned to values smaller or larger than unity. We validate our analysis by generating an example partially polarized MG field with a specified single-point polarization matrix using two different Monte Carlo simulation methods. We then compute the joint PDFs of the instantaneous field components and the Stokes parameter PDFs from the simulated MG fields, while comparing the results of both Monte Carlo methods to the corresponding theory. Lastly, we discuss the strengths, weaknesses, and applicability of both simulation methods in generating MG fields.

Multivariable variable‐gain super‐twisting control via output feedback for systems with arbitrary relative degrees

SummaryAn output‐feedback super‐twisting algorithm (STA) with variable gains is developed for multiple‐input and multiple‐output plants with arbitrary relative degrees. Global or semi‐global finite‐time exact tracking can be guaranteed for a class of uncertain systems with matched nonlinear time‐varying disturbances, possibly dependent on the unmeasured states. This represents a significant generalization of recently introduced versions of the STAs to a multivariable setup with arbitrary relative degrees, output‐feedback, and variable gains. The construction of such controller is based on a higher‐order sliding‐mode multivariable differentiator with dynamic gains. The gain adaptation for the controller and differentiator employs state‐norm observers to overcome the lack of the full‐state measurement. The stability properties of the proposed control system are demonstrated by means of input‐to‐state stability tools and a Lyapunov function‐based analysis. The theoretical results are verified through numerical and academic examples.

Hardware-In-The-Loop Assessment of Fuzzy and Neural Network Fault Diagnosis Schemes for a Wind Turbine Model

The fault diagnosis of wind turbines includes extremely challenging aspects that motivate the research issues considered in this paper. In particular, this work studies fault diagnosis solutions that are considered in a viable way and used as advanced techniques for condition monitoring of dynamic processes. To this end, the work proposes the design of fault diagnosis techniques that exploits the estimation of the fault by means of data-driven approaches. These fuzzy and neural network structures are integrated with auto-regressive with exogenous input regressors, thus making them able to approximate unknown nonlinear dynamic functions with arbitrary degree of accuracy. The capabilities of fault diagnosis schemes are validated by using a real-time simulator of a wind turbine system. Moreover, at this stage the benchmark is also useful to analyse the robustness and the reliability characteristics of the developed tools in the presence of model-reality mismatch and modelling error effects featured by the wind turbine simulator. This realistic simulator relies on a hardware-in-the-loop tool that is finally implemented for verifying and validating the performance of the developed fault diagnosis strategies in an actual environment.

A generalized isogeometric analysis of elliptic eigenvalue and source problems with an interface

The C0 stable generalized finite element methods (SGFEM) were recently developed for the elliptic source and eigenvalue problems with interfaces. This paper generalizes the SGFEM construction from its underlying C0 finite element basis to isogeometric analysis (IGA) with Cp−1 B-spline basis. The main challenge is how to construct globally (except where at the interfaces) Cp−1 enriched functions while restraining the resulting condition number from faster growth. A technique based on transformations between the B-splines and the Bernstein–Bézier polynomials is applied to meet the Cp−1 continuity requirement for enriched functions of arbitrary degree, and ensure good conditioning when the underlying IGA space is linear or quadratic. We establish the optimal error convergence of the approximate solutions for the elliptic source and eigenvalue problems with an interface for arbitrary degree. We verify our theoretical findings in various examples including both source and eigenvalue problems. We also make comparisons of the method with the SGFEM on computational time efficiency, scaled condition numbers(SCN), spectrum approximation and error convergences.

Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

In 2018, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron–Frobenius operators L_s , an idea known in varying degrees of generality for many years. Although L_s is not compact in the setting we consider, it possesses a strictly positive C^m eigenfunction v_s with eigenvalue R(L_s) for arbitrary m and all other points z in the spectrum of L_s satisfy |z| \le b for some constant b < R(L_s) . Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which R(L_s) =1 . This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree r . Using an extension of the Perron theory of positive matrices to matrices that map a cone K to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s , we give rigorous upper and lower bounds for the Hausdorff dimension s_* , and these bounds converge rapidly to s_* as the mesh size decreases and/or the polynomial degree increases.

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.

Infinite randomness with continuously varying critical exponents in the random XYZ spin chain

We study the antiferromagnetic XYZ spin chain with quenched bond randomness, focusing on a critical line between localized Ising magnetic phases. A previous calculation using the spectrum-bifurcation renormalization group, and assuming marginal many-body localization, proposed that critical indices vary continuously. In this work we solve the low-energy physics using an unbiased numerically exact tensor network method named the "rigorous renormalization group." We find a line of fixed points consistent with infinite-randomness phenomenology, with indeed continuously varying critical exponents for average spin correlations. A self-consistent Hartree-Fock-type treatment of the $z$ couplings as interactions added to the free-fermion random XY model captures much of the important physics including the varying exponents; we provide an understanding of this as a result of local correlation induced between the mean-field couplings. We solve the problem of the locally-correlated XY spin chain with arbitrary degree of correlation and provide analytical strong-disorder renormalization group proofs of continuously varying exponents based on an associated classical random walk problem. This is also an example of a line of fixed points with continuously varying exponents in the equivalent disordered free-fermion chain. We argue that this line of fixed points also controls an extended region of the critical interacting XYZ spin chain.

An Alternative Nutrient Rich Food Index (NRF-ai) Incorporating Prevalence of Inadequate and Excessive Nutrient Intake.

Most nutrient profiling models give equal weight to nutrients irrespective of their ubiquity in the food system. There is also a degree of arbitrariness about which nutrients are included. In this study, an alternative Nutrient Rich Food index was developed (NRF-ai, where ai denotes adequate intake) incorporating prevalence of inadequate and excessive nutrient intake among Australian adults. Weighting factors for individual nutrients were based on a distance-to-target method using data from the Australian Health Survey describing the proportion of the population with usual intake less than the Estimated Average Requirement defined by the Nutrient Reference Values for Australia and New Zealand. All nutrients for which data were available were included, avoiding judgements about which nutrients to include, although some nutrients received little weight. Separate models were developed for females and males and for selected age groups, reflecting differences in nutrient requirements and usual intake. Application of the new nutrient profiling models is demonstrated for selected dairy products and alternatives, protein-rich foods, and discretionary foods. This approach emphasises the need to identify foods that are rich in those specific nutrients for which intake is below recommended levels and can be used to address specific nutrient gaps in subgroups such as older adults. In addition, the new nutrient profiling model is used to explore other sustainability aspects, including affordability (NRF-ai per AUD) and ecoefficiency (NRF-ai/environmental impact score).

The Compact Support Neural Network.

Neural networks are popular and useful in many fields, but they have the problem of giving high confidence responses for examples that are away from the training data. This makes the neural networks very confident in their prediction while making gross mistakes, thus limiting their reliability for safety-critical applications such as autonomous driving and space exploration, etc. This paper introduces a novel neuron generalization that has the standard dot-product-based neuron and the radial basis function (RBF) neuron as two extreme cases of a shape parameter. Using a rectified linear unit (ReLU) as the activation function results in a novel neuron that has compact support, which means its output is zero outside a bounded domain. To address the difficulties in training the proposed neural network, it introduces a novel training method that takes a pretrained standard neural network that is fine-tuned while gradually increasing the shape parameter to the desired value. The theoretical findings of the paper are bound on the gradient of the proposed neuron and proof that a neural network with such neurons has the universal approximation property. This means that the network can approximate any continuous and integrable function with an arbitrary degree of accuracy. The experimental findings on standard benchmark datasets show that the proposed approach has smaller test errors than the state-of-the-art competing methods and outperforms the competing methods in detecting out-of-distribution samples on two out of three datasets.

The Art Gallery Problem is ∃ℝ-complete

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

New exact solutions of the sixth-order thin-film equation with complex method

By the Wiman–Valiron theory and the complex method, we prove that all meromorphic solutions of the fifth-order ODE u n u ′ ′ ′ ′ ′ − c u = 0 belong to W , and obtain several new solutions of the sixth-order thin-film equation comparing to the opening literature. At last we propose two unsolved problems for the readers. • New exact solutions of the sixth-order thin-film equation have been investigated. • The complex method has been used first time for PDEs with arbitrary degree n. • The obtained results are different from the founded ones in literature. • Two open questions are given for further research.

Analysis‐Suitable T‐Splines of arbitrary degree and dimension

AbstractThis paper defines analysis‐suitable T‐splines for arbitrary degree (including even and mixed degrees) and arbitrary dimension. We generalize the concept of anchor elements known from the two‐dimensional setting, extend two existing concepts of analysis‐suitability and justify their sufficiency for linear independence of the T‐spline basis.

Degree estimates of one class cut-offs for improper integrals

The paper considers a normalized non-integral integral of the first kind with a variable lower bound. In this case the integrand is a generalization of the standard Gaussian distribution density. Such integrals are often called cutoffs or incomplete functions. The purpose of this paper is to obtain power inequalities for this kind of integrals. The necessity of obtaining this type of estimations is due to the fact that incomplete functions have become widespread in applications and theoretical studies. The peculiarity of the results established in the article consists in the fact that arbitrary degrees of a given integral for any value of an argument are evaluated from above not by means, of the value of integrable functions at a certain point, but by the value of the integral in question at some point proportional to this argument. The coefficient of proportionality, a parameter, can take any value from some closed interval. The main difficulty in obtaining these inequalities is that the integrand is a logarithmically concave function, that is, its logarithm is a concave function. The paper also proves that both limits of the closed interval for the parameter cannot be extended. This shows that the obtained estimates are unimprovable.

MESHLESS SIMULATION OF HEAT CONDUCTION IN FUNCTIONALLY GRADED MATERIALS SUBJECTED TO TEMPERATURE-DEPENDENT HEAT SOURCES

In this paper, an efficient meshless local B-spline based finite difference (FD) method for analysis of functionally graded materials (FGMs) subjected to temperature-dependent heat sources is presented. The favourable properties of B-spline basis functions in having arbitrary degree for resolution of solution, partition of unity and the Kronecker delta properties are combined with high accuracy and low computational effort of differential quadrature method in approximating shape functions and their derivatives. In this study, the FGMs are assumed to have temperature-dependent materials properties that vary as a function of radial distance. The homogenized properties are evaluated with power-law mixture rule. The nonlinearities from material properties and heat source terms are handled by the predictor-corrector method along with the Crank-Nicolson scheme for time integration. Case of nonlinear 2D heat conduction in FGM due to temperature dependentheat sources is examined. The method is shown to be accurate and efficient for complex thermal analysis of FGMs taking into account temperature dependency of material properties and heat sources.

Which rational double points occur on del Pezzo surfaces?

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.

Epidemics with asymptomatic transmission: Subcritical phase from recursive contact tracing.

The challenges presented by the COVID-19 epidemic have created a renewed interest in the development of new methods to combat infectious diseases, and it has shown the importance of preparedness for possible future diseases. A prominent property of the SARS-CoV-2 transmission is the significant fraction of asymptomatic transmission. This may influence the effectiveness of the standard contact tracing procedure for quarantining potentially infected individuals. However, the effects of asymptomatic transmission on the epidemic threshold of epidemic spreading on networks have rarely been studied explicitly. Here we study the critical percolation transition for an arbitrary disease with a nonzero asymptomatic rate in a simple epidemic network model in the presence of a recursive contact tracing algorithm for instant quarantining. We find that, above a certain fraction of asymptomatic transmission, standard contact tracing loses its ability to suppress spreading below the epidemic threshold. However, we also find that recursive contact tracing opens a possibility to contain epidemics with a large fraction of asymptomatic or presymptomatic transmission. In particular, we calculate the required fraction of network nodes participating in the contact tracing for networks with arbitrary degree distributions and for varying recursion depths and discuss the influence of recursion depth and asymptomatic rate on the epidemic percolation phase transition. We anticipate recursive contact tracing to provide a basis for digital, app-based contact tracing tools that extend the efficiency of contact tracing to diseases with a large fraction of asymptomatic transmission.

Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces

The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We establish and prove the following results: to an arbitrary degree of accuracy, (a) location-scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set; and (b) for any finite location-scale mixtures of an essentially bounded PDF can approximate any PDF in in the norm.

Effects of Causes and Causes of Effects

We describe and contrast two distinct problem areas for statistical causality: studying the likely effects of an intervention (effects of causes) and studying whether there is a causal link between the observed exposure and outcome in an individual case (causes of effects). For each of these, we introduce and compare various formal frameworks that have been proposed for that purpose, including the decision-theoretic approach, structural equations, structural and stochastic causal models, and potential outcomes. We argue that counterfactual concepts are unnecessary for studying effects of causes but are needed for analyzing causes of effects. They are, however, subject to a degree of arbitrariness, which can be reduced, though not in general eliminated, by taking account of additional structure in the problem.

Construction of simplicial complexes with prescribed degree-size sequences.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s≥3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young etal., Phys. Rev. E 96, 032312 (2017)2470-004510.1103/PhysRevE.96.032312], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analyses of higher-order phenomena based on local structures.

Measuring Stokes parameters by means of unitary operations.

We propose an improved scheme to measure the Stokes polarization parameters by making the to-be-measured state evolve unitarily. An optical setup able to implement the scheme is presented theoretically and experimentally. Theoretical results indicate that at least two unitary operators should be employed to determine the Stokes parameters of light with arbitrary degrees of polarization. Compared with the results acquired by a commercial polarimeter, the proposed scheme is used to obtain the Stokes parameters of 13 representative states with degrees of polarization r=1 and r=0.5, experimentally. The fidelity between the two states gained by the above methods is in the range of 0.9802±0.0046 to 1.0000±0.0000.