Arbitrary Degree Research Articles

Arrayed multi-layer piezoelectric sensor based on electrospun P(VDF-TrFE)/ZnO with enhanced piezoelectricity

Flexible pressure sensors can maintain good pressure sensing capabilities even under arbitrary degrees of deformation, which are widely used in fields such as robotic perception, physiological signal detection and wearable electronics. Currently, piezoelectric-based flexible pressure sensors face significant limitations in terms of large-area, high-sensitivity, and high-resolution pressure detection. Targeted research efforts primarily focus on enhancing the performance of piezoelectric sensors by optimizing the properties of the sensitive layer piezoelectric material and modulating device structural design. This study introduces a novel multilayer array design for flexible composite piezoelectric sensors. A high-voltage electrospinning selective deposition process is proposed to fabricate arrayed nanofiber mats. Patterned electrospun films can be automatically patterned by patterned AgNW electrodes, which allows manual alignment of patterns for subsequent stacking. Compared to sensor based on P(VDF-TrFE) prepared under the same conditions, the sensitivity of the multi-layer P(VDF-TrFE)/ZnO sensor increased from 2.82 mV/kPa to 8.30 mV/kPa, providing an effective approach for highly sensitivity wearable devices with short response time (∼5 ms).

Smooth Gowdy-symmetric generalised Taub–NUT solutions with polynomial initial data

Abstract We consider smooth Gowdy-symmetric generalised Taub–NUT solutions, a class of inhomogeneous cosmological models with spatial three-sphere topology. They are characterised by existence of a smooth past Cauchy horizon and, with the exception of certain singular cases, they also develop a regular future Cauchy horizon. Several examples of exact solutions were previously constructed, where the initial data (in form of the initial Ernst potentials) are polynomials of low degree. Here, we generalise to polynomial initial data of arbitrary degree. Utilising methods from soliton theory, we obtain a simple algorithm that allows us to construct the resulting Ernst potential with purely algebraic calculations. We also derive an explicit formula in terms of determinants, and we illustrate the method with two examples.

On the thermodynamic invariance of fine‐grain and coarse‐grain fluid models

AbstractThermodynamics is an essential ingredient of fine‐grain and coarse‐grain models of oceanic and atmospheric flows. Thermodynamic functions and conservative variables, for a given set of observable quantities such as pressure, temperature, and composition, may be defined up to a certain degree of arbitrariness, in the sense that the predictions of the fine‐grain model are insensitive to, for example, some reference enthalpies, entropies, or pressures. Since the compressible Navier–Stokes model, regarded as a “mother” fine‐grain model, is invariant with respect to arbitrary changes in reference enthalpies and entropies, restricted only by phase change, any coarse‐grain model obtained from it, even conceptually, must be invariant at least to the same extent. A framework is devised that enables the systematic examination of this invariance. It is found that the dependence of usual conservative variables on a reference pressure propagates to their fluxes and gradients, and to downgradient closures based on them. To resolve this issue, it is shown how to construct invariant “reduced” gradients and fluxes of enthalpy and entropy. Closure relationships between such “reduced” fluxes and gradients are then guaranteed to be invariant. More work is required to address the invariance of more sophisticated closures, especially shallow and deep convective closures.

Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration.

The deep unfolding approach has attracted significant attention in computer vision tasks, which well connects conventional image processing modeling manners with more recent deep learning techniques. Specifically, by establishing a direct correspondence between algorithm operators at each implementation step and network modules within each layer, one can rationally construct an almost "white box" network architecture with high interpretability. In this architecture, only the predefined component of the proximal operator, known as a proximal network, needs manual configuration, enabling the network to automatically extract intrinsic image priors in a data-driven manner. In current deep unfolding methods, such a proximal network is generally designed as a CNN architecture, whose necessity has been proven by a recent theory. That is, CNN structure substantially delivers the translational symmetry image prior, which is the most universally possessed structural prior across various types of images. However, standard CNN-based proximal networks have essential limitations in capturing the rotation symmetry prior, another universal structural prior underlying general images. This leaves a large room for further performance improvement in deep unfolding approaches. To address this issue, this study makes efforts to suggest a high-accuracy rotation equivariant proximal network that effectively embeds rotation symmetry priors into the deep unfolding framework. Especially, we deduce, for the first time, the theoretical equivariant error for such a designed proximal network with arbitrary layers under arbitrary rotation degrees. This analysis should be the most refined theoretical conclusion for such error evaluation to date and is also indispensable for supporting the rationale behind such networks with intrinsic interpretability requirements. Through experimental validation on different vision tasks, including blind image super-resolution, medical image reconstruction, and image de-raining, the proposed method is validated to be capable of directly replacing the proximal network in current deep unfolding architecture and readily enhancing their state-of-the-art performance. This indicates its potential usability in general vision tasks.

Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.

Quasi-likelihood analysis for Student-Lévy regression

We consider the quasi-likelihood analysis for a linear regression model driven by a Student-t Lévy process with constant scale and arbitrary degrees of freedom. The model is observed at high frequency over an extending period, under which we can quantify how the sampling frequency affects estimation accuracy. In that setting, joint estimation of trend, scale, and degrees of freedom is a non-trivial problem. The bottleneck is that the Student-t distribution is not closed under convolution, making it difficult to estimate all the parameters fully based on the high-frequency time scale. To efficiently deal with the intricate nature from both theoretical and computational points of view, we propose a two-step quasi-likelihood analysis: first, we make use of the Cauchy quasi-likelihood for estimating the regression-coefficient vector and the scale parameter; then, we construct the sequence of the unit-period cumulative residuals to estimate the remaining degrees of freedom. In particular, using full data in the first step causes a problem stemming from the small-time Cauchy approximation, showing the need for data thinning.

Combinatorics of linear stability for Hamiltonian systems in arbitrary dimension

We address the general problem of studying linear stability and bifurcations of periodic orbits for Hamiltonian systems of arbitrary degrees of freedom. We study the topology of the GIT sequence introduced by the first author and Urs Frauenfelder in [7], in arbitrary dimension. In particular, we note that the combinatorics encoding the linear stability of periodic orbits is governed by a quotient of the associahedron. Our approach gives a topological/combinatorial proof of the classical Krein–Moser theorem, and refines it for the case of symmetric orbits.

Modified sine-Gordon theory with static multikinks

We construct a modified non-Bogomol’nyi-Prasad-Sommerfield sine-Gordon theory which supports stable static kinks of arbitrary topological degree N. We use this toy model to study problems that are interesting for higher-dimensional soliton theories supporting multisolitons. We construct a 2-kink collective coordinate model and use it to generate scattering trajectories, which are compared to full-field dynamics. We find that the approximation works well, but starts to fail as radiation becomes more important, due to our model becoming less Bogomol’nyi-Prasad-Sommerfield or when the initial kink velocities are large. We also construct the quantum 2-kink using various approximations and consider how these quantum corrections affect the binding energy of the 2-kink. Published by the American Physical Society 2024

Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

A semilinear initial–boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order 3−α, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.

Probabilistic analysis of the steady state of weakly perturbed linear oscillators subject to a class of Gaussian inputs

This paper aims to probabilistically study a class of nonlinear oscillator subject to weak perturbations and driven by stationary zero-mean Gaussian stochastic processes. For the sake of generality in the analysis, we assume that the perturbed term is a polynomial of arbitrary degree in the spatial position, that contains, as a particular case, the important case of the Duffing equation. We then take advantage of the so-called stochastic equivalent linearization technique to construct an equivalent linear model so that its behavior consistently approximates, in the mean-square sense, that of the nonlinear oscillator. This approximation allows us to take extensive advantage of the probabilistic properties of the solution of the linear model and its first mean-square derivative to construct reliable approximations of the main statistical moments of the steady state. From this key information, we then apply the principle of maximum entropy to construct approximations of the probability density function of the steady state. We illustrate the superiority of the equivalent linearization technique over the perturbation method through some examples.

A generalized Bogdanov-Takens system with arbitrary degree

The aim of this paper is to give the bifurcation diagram and all global phase portraits in the Poincaré disc of a generalized Bogdanov-Takens system with arbitrary degree. Moreover, the system has abundant nonlinear phenomena, such as double limit cycle bifurcation, Hopf bifurcation, homoclinic bifurcation, heteroclinic bifurcation, and so on. Notice that the nonlinear phenomena of the system are more abundant and more complex than the classic Bogdanov-Takens system.

Cotton gravity is not predictive

It is well known that the theory of Cotton gravity proposed by Harada is trivially solved by all isotropic and homogeneous cosmologies. We show that this under-determination is more general. More precisely, the degree of arbitrariness in the solutions increases with the degree of symmetry. The inverse problem of finding the matter source generating a given spacetime geometry is similarly indeterminate in the case of highly symmetric configurations. We give two simple examples. The first is that of static spherically symmetric solutions, which depend on an arbitrary function of the radial coordinate. The second is that of anisotropic cosmologies, which depend on an arbitrary function of time.

Size-Dependent Ab Initio Atomistic Thermodynamics from Cluster to Bulk: Application to Hydration of Titania Nanoparticles.

Ab initio atomistic thermodynamics (AIAT) has become an indispensable tool to estimate Gibbs free energy changes for solid surfaces interacting with gaseous species relative to pressure (p) and temperature (T). For such systems, AIAT assumes that solid vibrational contributions to Gibbs free energy differences cancel out. However, the validity of this assumption is unclear for nanoscale systems. Using hydrated titania nanoparticles (NPs) as an example, we estimate the vibrational contributions to the Gibbs free energy of hydration (ΔGhyd(T,p)) for arbitrary NP size and degree of hydration. Comparing ΔGhyd(T,p) phase diagrams for NPs when considering these contributions (AIATnano) relative to a standard AIAT approach reveals significant qualitative and quantitative differences, which only become negligible for large systems. By constructing a size-dependent ΔGhyd(T,p) phase diagram, we illustrate how our approach can provide deeper insights into how nanosytems interact with their environments, with many potential applications (e.g., catalytic nanoparticles, biological colloids, nanoparticulate pollutants).

Hodge-to-singular correspondence for reduced curves

We study the summands of the decomposition theorem for the Hitchin system for \operatorname{GL}_{n} , in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology of hypertoric quiver varieties. In contrast to the case of meromorphic Higgs fields, the intersection cohomology groups of moduli spaces of regular Higgs bundles depend on the degree. We describe this dependence.

Dynamics and sorting of run-and-tumble particles in fluid flows with transport barriers

We investigate the dynamics of individual run-and-tumble particles in a convective flow which is a prototype of fluid flows with transport barriers. We consider the most prevalent case of swimmers denser than the background fluid. As a result of gravity and the effects of the carrying flow, in the absence of swimming the particles either sediment or remain in a convective cell. When run-and-tumble also takes place, the particles may move to upper convective cells. We derive analytically the probability of uprise. Since that probability in a given fluid flow can vary strongly across species, our findings inspire a purely dynamical mechanism for species extraction in the dilute regime. Numerical simulations support our analytical predictions and demonstrate that a judicious choice of the fluid flow’s parameters can lead to particle sorting with an arbitrary degree of purity.

Is order-2 proportionality good enough for approximating the most likely path flow in user equilibrium traffic assignment?

The proportionality condition is a standard approach to dealing with the non-uniqueness issue in the user equilibrium (UE) traffic assignment problems (TAP). Although the proportionality condition can reduce the degree of arbitrariness, it remains unclear how much arbitrariness remains and whether it can meaningfully affect model outcomes and relevant decisions that depend on them. The answers to these questions are impeded by the lack of an efficient algorithm that can find the exact maximum entropy UE path flow solution for networks of practical size. In this paper, we fill this gap by developing a high-performance augmented Lagrangian algorithm that effectively exploits the special problem structure. Our numerical results reveal that there are a considerable number of links with non-negligible arbitrariness in the solution generated by the proportionality condition, and that this problem becomes worse if the level of congestion increases in the network. Since about a decade ago, many practitioners have relied on state-of-the-art traffic assignment tools based on the proportionality condition to perform select link analysis, among other applications. The results reported herein are a reminder that their toolbox may need reevaluation and perhaps an upgrade.

Geometric properties of generalized Bessel function of arbitrary order and degree

Geometric properties of generalized Bessel function of arbitrary order and degree

Inf-sup stabilized Scott–Vogelius pairs on general shape-regular simplicial grids for Navier–Stokes equations

This paper considers the discretization of the time-dependent Navier–Stokes equations with the family of inf-sup stabilized Scott–Vogelius pairs recently introduced in [John/Li/Merdon/Rui, Math. Models Methods Appl. Sci., 2024] for the Stokes problem. Therein, the velocity space is obtained by enriching the H1-conforming Lagrange element space with some H(div)-conforming Raviart–Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used.In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a Pk−P0-like system for arbitrary velocity polynomial degree k. Numerical studies verify the theoretical findings.

Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

Residue sums of Dickson polynomials over finite fields

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.