Differentiable Research Articles

The altered functional status in vestibular migraine: A meta‐analysis

AbstractPurposeVestibular migraine (VM) is a disorder with prominent vestibular symptoms that are causally correlated with migraine and is the most prevalent neurological cause of episodic vertigo. Nevertheless, the functional underpinnings of VM remain largely unclear. This study aimed to reveal concordant alteration patterns of functional connectivity (FC) in VM patients.MethodsWe searched literature measuring resting‐state FC abnormalities of VM patients in PubMed, Embase, Cochrane, and Scopus databases before May 2023. Furthermore, we applied the anisotropic effect size‐signed differential mapping (AES‐SDM) to conduct a whole‐brain voxel‐wise meta‐analysis to identify the convergence of FC alterations in VM patients.ResultsNine studies containing 251 VM patients and 257 healthy controls (HCs) were included. Relative to HCs, VM patients showed reduced activity in the left superior temporal gyrus and left midcingulate/paracingulate gyri, and increased activity in the precuneus, right superior parietal gyrus, and right middle frontal gyrus. Jackknife's analysis and subgroup analysis further supported the generalization and robustness of the main results. Furthermore, meta‐regression analyses indicated that the Dizziness Handicap Inventory (DHI) ratings were positively correlated with the activity in the precuneus, while higher Headache Impact Test‐6 and DHI scores were associated with lower activity within the left midcingulate/paracingulate gyri.ConclusionsThe study indicates that VM is associated with specific functional deficits of VM patients in crucial regions involved in the vestibular and pain networks and provides further information on the pathophysiological mechanisms of VM.

Kernel density estimation of Tsalli’s entropy with applications in adaptive system training

Information theoretic learning plays a very important role in adaption learning systems. Many non-parametric entropy estimators have been proposed by the researchers. This work explores kernel density estimation based on Tsallis entropy. Firstly, it has been proved that for linearly independent samples and for equal samples, Tsallis-estimator is consistent for the PDF and minimum respectively. Also, it is investigated that Tsallis-estimator is smooth for differentiable, symmetric, and unimodal kernel function. Further, important properties of Tsallis-estimator such as scaling and invariance for both single and joint entropy estimation have been proved. The objective of the work is to understand the mathematics behind the underlying concept.

Convex optimization via inertial algorithms with vanishing Tikhonov regularization: fast convergence to the minimum norm solution

In a Hilbertian framework, for the minimization of a general convex differentiable function f, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of f with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time t, is associated with the strongly convex function obtained by adding to f a Tikhonov regularization term with vanishing coefficient ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document}. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regularization parameter ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document}. By adjusting the speed of convergence of ε(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon (t)$$\\end{document} towards zero, we will obtain both rapid convergence towards the infimal value of f, and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of f. In particular, we obtain an improved version of the dynamic of Su-Boyd-Candès for the accelerated gradient method of Nesterov. This study naturally leads to corresponding first-order algorithms obtained by temporal discretization. In the case of a proper lower semicontinuous and convex function f, we study the proximal algorithms in detail, and show that they benefit from similar properties.

Some Simpson- and Ostrowski-Type Integral Inequalities for Generalized Convex Functions in Multiplicative Calculus with Their Computational Analysis

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus.

Transparent, flexible, and eco-friendly starch-based films for reversible optoelectronic noses for food spoilage monitoring in smart packaging.

Areversible optoelectronic nose is presentedconsisting of ten acid-base indicators incorporated into a starch-based film, covering a wide pHrange. The starch substrate is odorless, biocompatible, flexible, and exhibits high tensile resistance. This optical artificial olfaction system was used to detect the early stages of food decomposition by exposing it to the volatile compounds produced during the spoialge process of three food products (beef, chicken, and pork).A smartphone was used to capture the color changes caused by intermolecular interactions between each dye and the emitted volatiles over time. Digital images were processed to generate a differential color map, which uses the observed color shifts to create a unique signature for each food product. To effectively discriminate among different samples and exposure times, we employed chemometric tools, including hierarchical cluster analysis (HCA) and principal component analysis (PCA). This approach detects food deterioration in a practical, cost-effective, and user-friendly manner, making it suitable for smart packaging. Additionally, the use of starch-based films in the food industry is preferable due to their biocompatibility and biodegradability characteristics.

Holomorphy over Finite-dimensional Commutative Associative C-algebras: Local Theory

To a morphism $$\mathcal{A} \xrightarrow{\varphi} \mathcal{B}$$ of finite-dimensional commutative associative unital Banach $$\mathbb{C}$$-algebras one can associate a sheaf $$\mathcal{O}_\varphi$$ on the underlying topological space $$|\mathcal{A}|$$ of $$\mathcal{A}$$ consisting of $$\mathcal{B}$$-valued differentiable functions $$f$$ with $$\mathcal{A}$$-linear differential $$Df$$. It turns out that this class of functions exhibits a theory very similar to the classical complex analysis of one variable. In this article we give only an overview of some new results concerning the fundamentals of the corresponding local theory while at the same time also strengthen various already existing results scattered throughout the literature.

On Grüss, Ostrowski and trapezoid-type inequalities via nabla integral on time scales

AbstractOstrowski inequality gives the absolute deviation of the function from its integral mean. Delta and nabla calculi are first two approaches to study time scales calculus. This article presents the Ostrowski inequality for univariate first order nabla differentiable function by using Montgomery identity established for nabla integrals. Some extensions of dynamic Ostrowski-type inequality are investigated with the help of integration by parts for nabla integrals, properties of the modulus and polynomials on time scales. Furthermore, dynamic Grüss and trapezoid-type inequalities are established in their generalized form for twice nabla differentiable functions by utilizing the Montgomery identity. In addition, the obtained inequalities are discussed for continuous and discrete time scales.

Honeycomb effect elimination in differential phase fiber-bundle-based endoscopy

Fiber-bundle-based endoscopy, with its ultrathin probe and micrometer-level resolution, has become a widely adopted imaging modality for in vivo imaging. However, the fiber bundles introduce a significant honeycomb effect, primarily due to the multi-core structure and crosstalk of adjacent fiber cores, which superposes the honeycomb pattern image on the original image. To tackle this issue, we propose an iterative-free spatial pixel shifting (SPS) algorithm, designed to suppress the honeycomb effect and enhance real-time imaging performance. The process involves the creation of three additional sub-images by shifting the original image by one pixel at 0, 45, and 90 degree angles. These four sub-images are then used to compute differential maps in the x and y directions. By performing spiral integration on these differential maps, we reconstruct a honeycomb-free image with improved details. Our simulations and experimental results, conducted on a self-built fiber bundle-based endoscopy system, demonstrate the effectiveness of the SPS algorithm. SPS significantly improves the image quality of reflective objects and unlabeled transparent scattered objects, laying a solid foundation for biomedical endoscopic applications.

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

Regularization of Solutions of One Class of Systems of Linear Volterra Integral Equations of the First Kind in the Space of Differentiable Vector Functions on the Axis

Regularizing operators according to Lavrentiev are constructed and sufficient conditions for the uniqueness of solutions to systems of linear Volterra integral equations of the first kind on the axis are established.

New error bounds for Newton’s formula associated with tempered fractional integrals

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson’s second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature.

Harmonic conformable refinements of Hermite-Hadamard Mercer inequalities by support line and related applications

ABSTRACT We establish new conformable fractional Hermite-Hadamard (H–H) Mercer type inequalities for harmonically convex functions using the concept of support line. We introduce two new conformable fractional auxiliary equalities in the Mercer sense and apply them to differentiable functions with harmonic convexity. We also use Power-mean, Hölder’s and improved Hölder inequality to derive new Mercer type inequalities via conformable fractional integrals. The accuracy and superiority of the offered technique are clearly depicted through impactful visual illustrations. We also use our technique to derive new estimates for hypergeometric functions and special means of real numbers that are more precise than existing ones. Some applications are provided as well. Our results generalize and extend some existing ones in the literature.

Some considerations about l’H ˆopital-type rules for the monotonicity

The aim of this note is to present some results associated to the l'H\^{o}pital-type rules for the monotonicity (LMR). We will prove the monotonicity of the ratio $\frac{f\left( x\right) -f\left( c\right) }{g\left( x\right)-g\left( c\right) },$ were $c$ is an arbitrary point from the interval $% \left( a,b\right) $. Also, we extend LMR to the ratio of two higher-order differentiable functions. We complete with some applications of our results.

Exploring the geometry of the bifurcation sets in parameter space

By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p-1)$$\\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.

Safety-critical multi-tasks control for switched systems via several barrier functions

This paper investigates the problem of safety-critical multi-tasks control (SCMTC) for switched systems, where the multi-tasks control is achieved by ensuring the invariance and attraction of the task set. A new definition of safety is presented under the switched systems framework, which extends the existing safety definition for non-switched systems and reveals the essence of safety for switched systems. For two different types of task sets, a single barrier function (SBF) method and a multiple barrier functions method are proposed to solve the SCMTC problem, where the former requires each task set to be defined by a continuously differentiable function, while the latter requires that to be defined by multiple continuously differentiable functions. Both the two methods allow the safety and/or the invariance and attraction of the task sets to be unnecessary for individual subsystems. Also, two novel task-based switching signals are designed to guide the switching among subsystems that do not possess the invariance and attraction of task sets. Furthermore, a common barrier function method, as a special case of the SBF method, is proposed to solve the SCMTC problem for switched systems under arbitrary switchings. Finally, an illustrate example is given to verify the effectiveness of the proposed methods.

Nice operators and nice spaces

In this paper we obtain a necessary and sufficient condition for an operator on a uniform algebra to be nice. We characterize nice operators on an expansive class of Banach spaces. Then as examples, we give a complete description of nice operators on the space of differentiable functions on compact perfect plane sets, Bloch and Zygmund spaces and the space of Lipschitz functions on intervals. We also prove that every surjective nice operators are isometries on some of these Banach spaces.

On maximizing the likelihood function of general geostatistical models

AbstractGeneral geostatistical models are powerful tools for analyzing spatial datasets. A two‐step estimation based on the likelihood function is widely used by researchers, but several theoretical and computational challenges remain to be addressed. First, it is unclear whether there is a unique global maximizer of the log‐likelihood function, a seemingly simple but theoretically challenging question. The second challenge is the convexity of the log‐likelihood function. Besides these two challenges in maximizing the likelihood function, we also study the theoretical property of the two‐step estimation. Unlike many previous works, our results can apply to the non‐twice differentiable covariance functions. In the simulation studies, three optimization algorithms are evaluated in terms of maximizing the log‐likelihood functions.

Emission line predictions for mock galaxy catalogues: a new differentiable and empirical mapping from DESI

ABSTRACT We present a simple, differentiable method for predicting emission line strengths from rest-frame optical continua using an empirically determined mapping. Extensive work has been done to develop mock galaxy catalogues that include robust predictions for galaxy photometry, but reliably predicting the strengths of emission lines has remained challenging. Our new mapping is a simple neural network implemented using the JAX Python automatic differentiation library. It is trained on Dark Energy Spectroscopic Instrument Early Release data to predict the equivalent widths (EWs) of the eight brightest optical emission lines (including H α, H β, [O ii], and [O iii]) from a galaxy’s rest-frame optical continuum. The predicted EW distributions are consistent with the observed ones when noise is accounted for, and we find Spearman’s rank correlation coefficient ρs > 0.87 between predictions and observations for most lines. Using a non-linear dimensionality reduction technique, we show that this is true for galaxies across the full range of observed spectral energy distributions. In addition, we find that adding measurement uncertainties to the predicted line strengths is essential for reproducing the distribution of observed line-ratios in the BPT diagram. Our trained network can easily be incorporated into a differentiable stellar population synthesis pipeline without hindering differentiability or scalability with GPUs. A synthetic catalogue generated with such a pipeline can be used to characterize and account for biases in the spectroscopic training sets used for training and calibration of photo-z’s, improving the modelling of systematic incompleteness for the Rubin Observatory LSST and other surveys.

An accelerated hybrid Riemannian conjugate gradient method for unconstrained optimization

With the goal to deal with a series of optimization problems on general matrix manifolds with differentiable objective functions, we propose an accelerated hybrid Riemannian conjugate gradient technique. Specifically, the acceleration scheme of the proposed method using a modified stepsize which is multiplicatively determined by the Wolfe line search. The search direction of the proposed algorithm is determined by the hybrid conjugate parameter with computationally promising. We showed that the suggested approach converges globally to a stationary point. Our approach performs better than the state of art Riemannian conjugate gradient algorithms, as illustrated by computations on problems such as the orthogonal Procrustes problem and the Brockett-cost-function minimization problem.

Circular Differential Photocurrent Mapping of Hot Electron Response from Individual Plasmonic Nanohelicoids.

Chiral plasmonic nanocrystals have recently attracted increasing attention in circular polarization-dependent photocatalysis driven by hot carriers. While being concealed in traditional ensemble measurements, the individual chiral photocatalytic activity of nanocrystals can exclusively be revealed by directly correlating the circular differential photocurrent response to helical morphologies using single-particle techniques. Herein, we develop a method named circular differential photocurrent mapping (CDPM) and demonstrate that CDPM can be used to characterize the circular differential hot electron (CDHE) response from individual Au nanohelicoids (AuNHs) on a TiO2 photoanode in a photoelectrochemical cell. The single-particle circular differential scattering and CDHE measurements were interpreted with calculations performed on a model in direct correlation to the helical morphologies of the nanocrystal. While CDHE response was found inactive at a dipolar resonance of 750 nm, helicity-convoluted sites of HE generation were identified on the AuNH at a specific higher-order mode of 550 nm, resulting in a significant response of CDHE in association with the handedness of the AuNH. Details of circular differential contributions were further resolved by examining the efficiencies of individual AuNHs in terms of g-factors. Our study provides a powerful microscopic method at the single-particle level for the photocatalytic characterization of chiral nanocrystals, gaining fundamental insights into the photocatalysis of chirality, especially toward plasmon-induced asymmetrical photochemistry or photoelectrochemistry.