Differential Version Research Articles

Indirect adaptive control of nonlinearly parameterized nonlinear dissipative systems

AbstractIn this note we address the problem of indirect adaptive (regulation or tracking) control of nonlinear, input affine dissipative systems. It is assumed that the supply rate, the storage and the internal dissipation functions may be expressed as nonlinearly parameterized regression equations where the mappings (depending on the unknown parameters) satisfy a monotonicity condition—this encompasses a large class of physical systems, including passive systems. We propose to estimate the system parameters using the “power‐balance” equation, which is the differential version of the classical dissipation inequality, with a new estimator that ensures global, exponential, parameter convergence under the very weak assumption of interval excitation of the power‐balance equation regressor. To design the indirect adaptive controller we make the standard assumption of existence of an asymptotically stabilizing controller that depends—possibly nonlinearly—on the unknown plant parameters, and apply a certainty‐equivalent control law. The benefits of the proposed approach, with respect to other existing solutions, are illustrated with examples.

Bending analysis of nanobeams based on the integral form of nonlocal elasticity using the numerical Rayleigh-Ritz technique

The nonlocal theory is commonly applied for nanomaterials due to its capability in considering size influences. Available studies have shown that the differential version of this theory is not suitable for some problems such as bending of cantilever nanobeams, and the integral version must be used to avoid obtaining inconsistent results. Therefore, an attempt is made in this paper to propose an efficient variational formulation based on the integral nonlocal model for the analysis of nanobeams. The formulation is developed in a general form so that it can be used for arbitrary kernel functions. The nanobeams are modeled using the Bernoulli-Euler beam theory, and their bending behavior is analyzed. Derivation of governing equations is performed according to an energy-based approach. Also, a numerical approach based on the Rayleigh-Ritz method is developed for the solution of problem. Moreover, the results of integral and differential models are compared. It is revealed that by the proposed numerical solution, the paradox in the behavior of nanocantilever is resolved.

Optimizing Tandem Speaker Verification and Anti-Spoofing Systems

As automatic speaker verification (ASV) systems are vulnerable to spoofing attacks, they are typically used in conjunction with spoofing countermeasure (CM) systems to improve security. For example, the CM can first determine whether the input is human speech, then the ASV can determine whether this speech matches the speaker's identity. The performance of such a tandem system can be measured with a tandem detection cost function (t-DCF). However, ASV and CM systems are usually trained separately, using different metrics and data, which does not optimize their combined performance. In this work, we propose to optimize the tandem system directly by creating a differentiable version of t-DCF and employing techniques from reinforcement learning. The results indicate that these approaches offer better outcomes than finetuning, with our method providing a 20% relative improvement in the t-DCF in the ASVSpoof19 dataset in a constrained setting.

Identification of Nonlinearly Parameterized Nonlinear Dissipative Systems

In this note, we address the problem of parameter identification of nonlinear, input affine dissipative systems. It is assumed that the supply rate, the storage and the internal dissipation functions may be expressed as nonlinearly parameterized regression equations where the mappings (depending on the unknown parameters) satisfy a monotonicity condition—this encompasses a large class of physical systems, including passive systems. We propose to estimate the system parameters using the “power-balance” equation, which is the differential version of the classical dissipation inequality, with a new estimator that ensures global, exponential, parameter convergence under the very weak assumption of interval excitation of the power-balance equation regressor. The benefits of the proposed approach are illustrated with an example.

The impact of role reversal on the dynamics of predator-prey model with stage structure

In this paper, we investigate the dynamics of a predator-prey model with role reversal and its corresponding stochastic differential equation version. For the deterministic model, the existence and stability of the equilibrium are investigated and the occurrences of transcritical bifurcation and Hopf bifurcation are studied. For the stochastic model, we investigate the existence of a unique global positive solution and establish sufficient conditions for the existence of a unique ergodic stationary distribution. Finally, numerical simulations are carried out to verify our analytical results, which indicate that both the intensity of white noise and the predation of the juvenile predators have a significant relationship with the dynamics of the predator-prey model and expand the theoretical analyses.

Dissecting the collinear structure of quark splitting at NNLL

We explore the collinear limit of final-state quark splittings at order {alpha}_s^2 . While at general NLL level, this limit is described simply by a product of leading-order 1 → 2 DGLAP splitting functions, at the NNLL level we need to consider 1 → 3 splitting functions. Here, by performing suitable integrals of the triple-collinear splitting functions, we demonstrate how one may extract {mathrm{mathcal{B}}}_2^q(z) , a differential version of the coefficient {mathrm{mathcal{B}}}_2^q that enters the quark form factor at NNLL and governs the intensity of collinear radiation from a quark. The variable z corresponds to the quark energy fraction after an initial 1 → 2 splitting, and our results yield effective higher-order splitting functions, which may be considered as a step towards the construction of NNLL parton showers. Further, while in the limit z → 1 we recover the standard soft limit results involving the CMW coupling with scale kt, the z dependence we obtain also motivates the extension of the notion of a physical coupling beyond the soft limit.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

Blowing-up Solutions for 2nd-Order Critical Elliptic Equations: The Impact of the Scalar Curvature

Abstract Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet’s condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in \{4,5\}$, our proof requires to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$.

Differential Weil descent

In this note a differential version of the classical Weil descent is established in all characteristics. This yields a ready-to-deploy tool of differential restriction of scalars for differential varieties over finite differential field extensions.

Monocular 3D Detection With Geometric Constraint Embedding and Semi-Supervised Training

In this work, we propose a novel one-stage and keypoint-based framework for monocular 3D object detection using only RGB images, called KM3D-Net. 2D detection only requires a deep neural network to predict 2D properties of objects, as it is a semanticity-aware task. For image-based 3D detection, we argue that the combination of a deep neural network and geometric constraints are needed to synergistically estimate appearance-related and spatial-related information. Here, we design a fully convolutional model to predict object keypoints, dimension, and orientation, and combine these with perspective geometry constraints to compute position attributes. Further, we reformulate the geometric constraints as a differentiable version and embed this in the network to reduce running time while maintaining the consistency of model outputs in an end-to-end fashion. Benefiting from this simple structure, we propose an effective semi-supervised training strategy for settings where labeled training data are scarce. In this strategy, we enforce a consensus prediction of two shared-weights KM3D-Net for the same unlabeled image under different input augmentation conditions and network regularization. In particular, we unify the coordinate-dependent augmentations as the affine transformation for the differential recovering position of objects, and propose a keypoint-dropout module for network regularization. Our model only requires RGB images, without synthetic data, instance segmentation, CAD model, or depth generator. Extensive experiments on the popular KITTI 3D detection dataset indicate that the KM3D-Net surpasses state-of-the-art methods by a large margin in both efficiency and accuracy. And also, to the best of our knowledge, this is the first application of semi-supervised learning in monocular 3D object detection. We surpass most of the previous fully supervised methods with only 13% labeled data on KITTI.

Reconstructing random heterogeneous media through differentiable optimization

Microstructure reconstruction is a key enabler of process-structure–property linkages, a central topic in materials engineering. Revisiting classical optimization-based reconstruction techniques, they are recognized as a powerful framework to reconstruct random heterogeneous media, especially due to their generality and controllability. The stochasticity of the available approaches is, however, identified as a performance bottleneck. In this work, reconstruction is approached as a differentiable optimization problem, where the error of a generic prescribed descriptor is minimized under consideration of its derivative. As an exemplary descriptor, a suitable differentiable version of spatial correlations is formulated, along with a multigrid scheme to ensure scalability. The applicability of differentiable optimization realized through this descriptor is demonstrated using a wide variety of heterogeneous media, achieving exact statistical equivalence with errors as low as 0% in a short time. We conclude that, while still in an early stage of development, this approach has the potential to significantly alleviate the computational effort currently associated with reconstructing general random heterogeneous media.

An Interactive Jupyter Notebook for Teaching the Foundational Mathematical Basis of Compartmental Pharmacokinetics

Pharmacokinetics is often taught as two different lessons, the foundational mathematical basis and the clinical application. Due to the lack of hands-on exercises, pharmacokinetics can be difficult to teach and students report struggling with the topic. Therefore, I have developed an interactive Jupyter notebook that demonstrates the mathematical basis behind one- and two-compartment pharmacokinetic models. This notebook uses a combination of “markup” cells, which display relevant equations and text, and “code” cells, which hold Python code that executes, models, and graphs the various pharmacokinetic equations. Students benefit from exposure to this aspect of computer science because the code remains visible. With the code pre-written, students are able to see the implementations of programming in drug pharmacokinetic modeling without having to construct the code themselves. This notebook includes both the differential and algebraic versions of one- and two- compartment models. The code also generates interactive graphs where students can use sliders for each parameter (half-life, inter-compartmental distribution rates, and dose of drug) to explore the effects of the equations parameters on the output of drug concentration over time. The conversion between macro constants and micro constants is also demonstrated and explained. All software used is freely available, so students can download the notebook onto their personal computers. No computer science experience is needed to execute this tool. However, instructors can easily adapt the notebook to their particular needs, adding or removing cells as desired. I successfully used this notebook in my undergraduate upper-division biology elective, “Foundations of Pharmacology”, and in a guest lectured Master's level course on the practical application of pharmacokinetics. Overall, this tool will be useful for teaching the mathematical basis of basic pharmacokinetics while allowing students to gain useful exposure to computer data science.

A Comparative Study of Curvature-Based and Differential Versions of Dotter Raster-stereography Techniques

Conventional Line-based Raster-stereography has been a popular technique for 3-D surface topography. However, in its application for human face screening, the problem of line breaking was observed. In order to resolve this problem, there came up a new technique called dotted raster-stereography. The previously reported version of dotted raster-stereography extracted the curvature features of human face. This paper presents a modified version, viz. differential dotted raster-stereography in which instead of curvature, differences in straight line distances between adjacent points are calculated. A comparative picture of the two versions of dotted raster-stereography techniques is presented. Results suggest that this new differential version of dotted raster-stereography algorithm is faster in execution due to its simpler implementation in software, though lower in accuracy, as compared with the previously reported curvature-based version of dotted raster stereography.

Pathfollowing of high-dimensional hysteretic systems under periodic forcing

The dynamic response and bifurcations of high-dimensional systems endowed with hysteretic restoring forces in all degrees of freedom are investigated. Two types of hysteresis models are considered, namely the Bouc–Wen model and a differential version of the so-called exponential model of hysteresis. The numerical technique tailored for tackling high-dimensional hysteretic systems is based on an enhanced pathfollowing approach based on the Poincare map. In particular, a five-dof mass-spring-damper-like system, with each rheological element described by the Bouc–Wen or the exponential model of hysteresis enriched by cubic and quintic nonlinear elastic terms, is investigated and a rich variety of nonlinear responses and bifurcations is found and discussed.

A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

60-700 K CTAT and PTAT Temperature Sensors with 4H-SiC Schottky Diodes.

A SiC Schottky dual-diode temperature-sensing element, suitable for both complementary variation of VF with absolute temperature (CTAT) and differential proportional to absolute temperature (PTAT) sensors, is demonstrated over 60–700 K, currently the widest range reported. The structure’s layout places the two identical diodes in close, symmetrical proximity. A stable and high-barrier Schottky contact based on Ni, annealed at 750 °C, is used. XRD analysis evinced the even distribution of Ni2Si over the entire Schottky contact area. Forward measurements in the 60–700 K range indicate nearly identical characteristics for the dual-diodes, with only minor inhomogeneity. Our parallel diode (p-diode) model is used to parameterize experimental curves and evaluate sensing performances over this far-reaching domain. High sensitivity, upwards of 2.32 mV/K, is obtained, with satisfactory linearity (R2 reaching 99.80%) for the CTAT sensor, even down to 60 K. The PTAT differential version boasts increased linearity, up to 99.95%. The lower sensitivity is, in this case, compensated by using a high-performing, low-cost readout circuit, leading to a peak 14.91 mV/K, without influencing linearity.

ALTAIR: A Low-Noise, Low-Power, High-Speed and Wide Dynamic Range ASIC for X- and γ-Ray Spectroscopy With CdTe/CdZnTe Pixel Detectors

ALTAIR is an application-specific integrated circuit (ASIC) developed for the readout of CdTe/CdZnTe detectors for X-γ-ray spectroscopic imaging. ALTAIR is composed of 16 channels with analog and mixed-signal sections specifically designed for high energy resolution and high-rate applications with the constraints of a low power consumption and operation at room temperature. The ASIC has three selectable ranges for input charge signals covering photon energies up to 90, 160, or 300 keV, which allow to optimize the signal-to-noise ratio under different experimental conditions. Eight peaking times of the pulse shaper are selectable from 200 up to 2.3 μs. ALTAIR includes a 180 bits register and an integrated serial peripheral interface (SPI) interface to configure the signal-processing parameters globally or independently for each channel. The power consumption is 950 μW/channel or 1.35 mW/channel for single-ended or differential output versions, respectively. A minimum equivalent noise charge (ENC) of 37 electrons rms [386 eV full-width at half-maximum (FWHM) in CdTe] has been measured at 500 ns peaking time and 90-keV energy range. With a CdTe pixel detector connected to ALTAIR and operated at +24 °C, a pulser FWHM of 1.7 keV (160 el. rms) and 1.9-keV FWHM on 59.54 keV line of <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">241</sup> Am have been achieved at a peaking time of 500 ns.

Novel Expression-Template-Based Automatic Differentiation of Fortran Codes for Aerodynamic Optimization

An elegant and robust approach for discrete adjoint-based automatic differentiation of the Fortran codes is proposed in this work. The technique aims at bridging the gap within the Fortran programming language that currently lacks certain metaprogramming paradigms that are readily available at compile time in C/C++ codes. More specifically, the present work uses an expression-based tape approach, which is the first-of-its-kind implementation in Fortran programming language, that can significantly reduce the memory footprint while improving the computational efficiency of the adjoint-based automatic differentiation (AD). The proposed expression-template-based approach is incorporated in our in-house AD toolbox, which currently is the only Fortran-based tool in the literature that uses a fixed-point-type operator-overloading adjoint sensitivity analysis. The improved version of the fast automatic differentiation using operator-overloading technique toolbox is then coupled with the in-house unstructured parallel compressible (UNPAC) flow solver for a robust design optimization framework (DOF), called UNPAC-DOF. The efficiency and robustness of the proposed technique and the resulting framework are tested for aerodynamic shape optimization problems applied to airfoil and wing geometries.

Snell’s law for the Schwarzschild black hole

The Wheeler equation, for electromagnetic disturbances in a gravitational field, was found by Fiziev to have exact solutions both above and below the event horizon, in the form of waves propagating both inwardly and outwardly. This observation can be interpreted and verified from the optical point of view, entirely on the basis of the Schwarzschild metric for length contraction and time dilation, in order to derive a differential version of Snell's law for the Schwarzschild black hole. It reveals interesting physics, including the correct amount of light deflection by the Sun, internal and external Oppenheimer-Snyder cones of the black hole, properties of its phonon sphere and the conclusion that light-rays are kept below the horizon by length contraction and time dilation rather than deflection.

On the solution calculation of nonlinear ordinary differential equations via exact quadratization

We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic σπ-ODEs (and, more in general, the real analytic σπ-reducible ODEs) in many indeterminates, characterized by an ODE-function given by generalized polynomials of the indeterminates and their derivatives, i.e. functions formally polynomial with exponents, though the exponent can be any real number, and whose coefficients are analytic time functions. The solution method consists in reducing the ODE to a certain canonical homogeneous quadratic ODE, named driver-type Riccati equation with a larger number of indeterminates, whose solutions include, as sub-solutions, the original solutions, and for which a recursive formula is shown, giving all coefficients of the solution power series directly from the ODE parameters. The reduction method is named exact quadratization and was formerly introduced in another our article, where we considered explicit ODEs only. In the present paper, which is self-contained to a large extent, we review and complete the theory of exact quadratization by solving issues, such as for instance the existence of a piecewise-quadratization, that had remained open, and also extend it to the more general case of an implicitly defined ODE. Finally, we argue that the result can be seen as a partial solution of a differential version of the 22nd Hilbert's problem.