Differentiable Research Articles

An Inverse Inequality for Fractional Sobolev Norms in Unbounded Domains

The nonlocal operators have found applications in various areas of contemporary science. The anomalous diffusion phenomena have been modeled by the fractional Poisson boundary-value problem. Electromagnetic fluids have been described by fractional differential equations. The fractional differential operators have found applications in material sciences, planar and space elasticity, probabilistic theory, harmonic analysis, and even in finance. The inverse inequality plays an important role in Numerical Analysis. The well-known results on inverse inequalities have been obtained in bounded domains and finite-dimensional spaces. Naturally, a new challenge arises to obtain inverse inequalities in the fractional Sobolev spaces. This paper is devoted to differential inequalities between fractional Sobolev norms. We expand the notion of a monotone function into a new notion supermonotone function and rigorously prove an inverse inequality for a class of differentiable functions in unbounded domains. Examples that demonstrate the theory are presented.

Strong stationarity for non-smooth control problems with fractional semi-linear elliptic equations in dimension $$N\le 3$$

AbstractIn this paper, we investigate the optimal control of a semi-linear fractional PDEs involving the spectral diffusion operator, or the realization of the integral fractional Laplace operator with the zero Dirichlet exterior condition, both of order s with $$s\in (0,1)$$ s ∈ ( 0 , 1 ) . The state equation contains a non-smooth nonlinearity, and the objective functional is convex in the control variable but contains non-smooth terms. As the mappings involved may not be Gâteaux differentiable, we use a regularization technique to regularize these nonlinear terms, aiming to obtain Gâteaux differentiable mappings. By employing this regularization technique, we are able to derive the first-order optimality condition for the regularized control problem by using the associated adjoint system. Furthermore, we conduct a limit analysis on the regularized term resulting in an optimality system for the non-smooth problem of C-stationary type. Subsequently, we establish a primal optimality condition, specifically B-stationarity. Under the assumption of “constraint qualification”, we derive the strong stationarity conditions for the non-smooth optimization problem with control constraints and establish the equivalence between B-stationarity and strong stationarity conditions.

Optimization schemes on manifolds for structured matrices with fixed eigenvalues

AbstractSeveral manifold optimization schemes are presented and analyzed for solving a specialized inverse structured symmetric matrix problem with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered. The rest of the entries are free, some of them preferably away from zero. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem is related to the problem of determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our optimization schemes are based on considering the eigenvector matrix as the only unknown and iteratively moving on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose Riemannian gradient-type methods combined with two different well-known retractions, and with two well-known constrained optimization strategies: penalization and augmented Lagrangian. We also present a block alternating technique that takes advantage of a proper separation of variables. Convergence properties of the penalty alternating approach are established. Finally, we present initial numerical results to demonstrate the effectiveness of our proposals.

LTM-NeRF: Embedding 3D Local Tone Mapping in HDR Neural Radiance Field.

Recent advances in Neural Radiance Fields (NeRF) have provided a new geometric primitive for novel view synthesis. High Dynamic Range NeRF (HDR NeRF) can render novel views with a higher dynamic range. However, effectively displaying the scene contents of HDR NeRF on diverse devices with limited dynamic range poses a significant challenge. To address this, we present LTM-NeRF, a method designed to recover HDR NeRF and support 3D local tone mapping. LTM-NeRF allows for the synthesis of HDR views, tone-mapped views, and LDR views under different exposure settings, using only the multi-view multi-exposure LDR inputs for supervision. Specifically, we propose a differentiable Camera Response Function (CRF) module for HDR NeRF reconstruction, globally mapping the scene's HDR radiance to LDR pixels. Moreover, we introduce a Neural Exposure Field (NeEF) to represent the spatially varying exposure time of an HDR NeRF to achieve 3D local tone mapping, for compatibility with various displays. Comprehensive experiments demonstrate that our method can not only synthesize HDR views and exposure-varying LDR views accurately but also render locally tone-mapped views naturally.

A New Inclusion on Inequalities of the Hermite–Hadamard–Mercer Type for Three-Times Differentiable Functions

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities.

Procedural Material Generation with Reinforcement Learning

Modern 3D content creation heavily relies on procedural assets. In particular, procedural materials are ubiquitous in the industry, but their manipulation remains challenging. Previous work [Hu et al. 2023] conditionally generates procedural graphs that match a given input image. However, the parameter generation step limits how accurately the generated graph matches the input image, due to a reliance on supervision with scarcely available procedural data. We propose to improve parameter prediction accuracy for image-conditioned procedural material generation by leveraging reinforcement learning (RL) and present the first RL approach for procedural materials. RL circumvents the limited availability of procedural data, the domain gap between real and synthetic materials, and the need for end-to-end differentiable loss functions. Given a target image, we retrieve a procedural material and use an RL-trained transformer model to predict a set of parameters that reconstruct the target image as closely as possible. We show that using RL significantly improves parameter prediction to match a given target image compared to supervised methods on both synthetic and real target images.

Differentiable Photon Mapping using Generalized Path Gradients

Photon mapping is a fundamental and practical Monte Carlo rendering technique for efficiently simulating global illumination effects, especially for caustics and specular-diffuse-specular (SDS) paths. In this paper, we present the first differentiable rendering method for photon mapping. The core of our method is a newly introduced concept named generalized path gradients. Based on the extended path space manifolds (EPSMs) [Xing et al. 2023], the generalized path gradients define the derivatives of the vertex positions and color contributions of a path with respect to scene parameters under given geometric constraints. By formalizing photon mapping as a path sampling technique through vertex merging [Georgiev et al. 2012] and incorporating a smooth differentiable density estimation kernel, we enable the differentiation of the photon mapping algorithms based on the theoretical results of generalized path gradients. Experiments demonstrate that our method is more effective than state-of-the-art physics-based differentiable rendering methods in inverse rendering applications involving difficult illumination paths, especially SDS paths.

All you need is rotation: Construction of developable strips

We present a novel approach to generate developable strips along a space curve. The key idea of the new method is to use the rotation angle between the Frenet frame of the input space curve, and its Darboux frame of the curve on the resulting developable strip as a free design parameter, thereby revolving the strip around the tangential axis of the input space curve. This angle is not restricted to be constant but it can be any differentiable function defined on the curve, thereby creating a large design space of developable strips that share a common directrix curve. The range of possibilities for choosing the rotation angle is diverse, encompassing constant angles, linearly varying angles, sinusoidal patterns, and even solutions derived from initial value problems involving ordinary differential equations. This enables the potential of the proposed method to be used for a wide range of practical applications, spanning fields such as architectural design, industrial design, and papercraft modeling. In our computational and physical examples, we demonstrate the flexibility of the method by constructing, among others, toroidal and helical windmill blades for papercraft models, curved foldings, triply orthogonal structures, and developable strips featuring a log-aesthetic directrix curve.

Fractional Newton‐type integral inequalities for the Caputo fractional operator

In this paper, we present a set of Newton‐type inequalities for n‐times differentiable convex functions using the Caputo fractional operator, extending classical results into the fractional calculus domain. Our exploration also includes the derivation of Newton‐type inequalities for various classes of functions by employing the Caputo fractional operator, thereby broadening the scope of these inequalities beyond convexity. In addition, we establish several fractional Newton‐type inequalities by using bounded functions in conjunction with fractional integrals. Furthermore, we develop specific fractional Newton‐type inequalities tailored to Lipschitzian functions. Moreover, the paper emphasizes the significance of fractional calculus in refining classical inequalities and demonstrates how the Caputo fractional operator provides a more generalized framework for addressing problems involving non‐integer order differentiation. The inclusion of bounded and Lipschitzian functions introduces additional layers of complexity, allowing for a more comprehensive analysis of function behaviors under fractional operations.

Semiclassical instanton theory for reaction rates at any temperature: How a rigorous real-time derivation solves the crossover temperature problem.

Instanton theory relates the rate constant for tunneling through a barrier to the periodic classical trajectory on the upturned potential energy surface, whose period is τ = ℏ/(kBT). Unfortunately, the standard theory is only applicable below the "crossover temperature," where the periodic orbit first appears. This paper presents a rigorous semiclassical (ℏ → 0) theory for the rate that is valid at any temperature. The theory is derived by combining Bleistein's method for generating uniform asymptotic expansions with a real-time modification of Richardson's flux-correlation function derivation of instanton theory. The resulting theory smoothly connects the instanton result at low temperature to the parabolic correction to Eyring transition state theory at high-temperature. Although the derivation involves real time, the final theory only involves imaginary-time (thermal) properties, consistent with the standard version of instanton theory. Therefore, it is no more difficult to compute than the standard theory. The theory is illustrated with application to model systems, where it is shown to give excellent numerical results. Finally, the first-principles approach taken here results in a number of advantages over previous attempts to extend the imaginary free-energy formulation of instanton theory. In addition to producing a theory that is a smooth (continuously differentiable) function of temperature, the derivation also naturally incorporates hyperasymptotic (i.e., multi-orbit) terms and provides a framework for further extensions of the theory.

COMPLEX PHYSICAL THERAPY OF PATIENTS WITH ADHESIVE CAPSULITIS

Adhesive capsulitis is predominantly an idiopathic condition and is more common in patients with metabolic disorders. PS contracture can be of varying degrees of severity, depending on the pathology of the PS with which we are dealing. This condition greatly aff ects work capacity and quality of life due to the presence of pain, discomfort and limitation of daily life activities. Prolonged pain syndrome can provoke sleep disorders and depression. Disruption of normal sleep, pain and depression form a pathological closed circle and contribute to the maintenance of the pathological process in the locus morbi.The development of a comprehensive program of physical therapy for patients with adhesive capsulitis with the use of modern means of recovery, which will reduce the risk of the disease re-emerging, and will also contribute to the faster recovery of the functional capacity of the musculoskeletal system of the patients.The objective of the study. The develop and practically substantiate a comprehensive program of therapy for patients with adhesive capsulitis using physical therapy.Materials and Research Methods. The study is based on a comprehensive therapeutic approach for 36 patients diagnosed with adhesive capsulitis in the acute phase of the condition. The assessed parameters included X-ray and computed tomography results, evaluation of range of motion in the shoulder joint, joint- muscle apparatus, and analysis of pain intensity indicators.Conclusions. To improve the condition of patients and optimize the process of treatment and rehabilitation of patients with adhesive capsulitis of the shoulder joint in the acute period, it is advisable to use with the diff erentiation of means of therapeutic intervention according to the diagnosis of the international classifi cation of functioning, optimization of physiotherapeutic means and special means of physical therapy allowed to shorten the terms of treatment and recovery of patients with adhesive capsulitis. It was determined that the best combination of means of therapeutic intervention for patients with adhesive capsulitis is the use of a complex of ideomotor exercises to create the patient’s idea of correct movement; correct positioning of the limb during sleep, everyday and professional activity; passive exercises to increase the volume of painless movements (bending, external rotation); restorationof the pattern of typical motor actions and optimization of physiotherapeutic means due to the use of shock wave and TENS therapy signifi cantly improved the eff ectiveness of their recovery in the short-term and long-term stages of the acute period of rehabilitation.

Multidimensional Quantum Generative Modeling by Quantum Hartley Transform

AbstractAn approach for building quantum models based on the exponentially growing orthonormal basis of Hartley kernel functions is developed. First, a differentiable Hartley feature map parameterized by real‐valued argument that enables quantum models suitable for solving stochastic differential equations and regression problems is designed. Unlike the naturally complex Fourier encoding, the proposed Hartley feature map circuit leads to quantum states with real‐valued amplitudes, introducing an inductive bias and natural regularization. Next, a quantum Hartley transform circuit is proposed as a map between computational and Hartley basis. The developed paradigm is applied to generative modeling from solutions of stochastic differential equations, and utilize the quantum Hartley transform for fine sampling from parameterized distributions through an extended register. Finally, the capability of multivariate quantum generative modeling is demonstrated for both correlated and uncorrelated distributions. As a result, the developed quantum Hartley‐based generative models (QHGMs) offer a distinct quantum approach to generative AI at increasing scale.

Wide area maps of building differential deformations

Abstract. The European Ground Motion Service (EGMS) uses Sentinel-1 data to monitor and measure land displacement across Europe, delivering information on ground motion phenomena. The EGMS service includes three types of products, enabling the analysis of annually updated ground motion data. The authors are working on a project aimed at computing the spatial gradient of land deformation and generating wide-area differential deformation maps. This type of maps can help identifying potentially vulnerable buildings and urban structures. The project involves developing a software pipeline to compute and convert original deformation data and creating a customized web viewer to display the results. Given the large volume of the processed datasets, with several millions of measurement points an accurate algorithm implementation is needed. The resulting information will aid in monitoring anthropogenic phenomena and their impact on building and infrastructures. This will be essential for urban management and risk mitigation planning.

Novel versions of Hölder's-Like and related inequalities with newly defined [formula omitted] space, and their applications over fuzzy domain

It is widely recognized that fuzzy number theory relies on the characteristic function. However, within the fuzzy realm, the characteristic function transforms into a membership function contingent upon the interval [0,1]. This implies that real numbers and intervals represent exceptional cases of fuzzy numbers. By considering this approach, this paper introduces a new LP space and novel refinements for integral variations of Hölder's inequality which is known as Hölder's-like inequality over fuzzy domain. Numerous prevailing inequalities associated with Hölder's-like inequality can be enhanced through the newly acquired inequalities, as demonstrated through an application. By using newly defined special means, some new versions of integral inequalities have obtained where differentiable mappings are real-valued convex-like (or convex fuzzy) mappings Lastly, nontrivial numerical examples are also included to validate the accuracy of the presented inequalities as they vary with the parameter ꙍ.

Multifrequency-STD NMR unveils the first Michaelis complex of an intramolecular trans-sialidase from Ruminococcus gnavus

RgNanH is an intramolecular trans-sialidase expressed by the human gut symbiont Ruminococcus gnavus, to utilise intestinal sialylated mucin glycan epitopes. Its catalytic domain, belonging to glycoside hydrolase GH33 family, cleaves off terminal sialic acid residues from mucins, releasing 2,7-anhydro-Neu5Ac which is then used as metabolic substrate by R. gnavus to proliferate in the mucosal environment. RgNanH is one of the three intramolecular trans-sialidases (IT-sialidases) characterised to date, and the first from a gut commensal organism. Here, saturation transfer difference NMR (STD NMR) in combination with computational techniques (molecular docking and CORCEMA-ST) were used to elucidate the specificity, kinetics and relative affinity of RgNanH for sialoglycans and 2,7-anhydro-Neu5Ac. We propose the first 3D model for the Michaelis complex of an IT-sialidase. This confirms the sialic acid to be the main recognition element for the interaction in the enzymatic cleft and highlights the crucial role of Trp698 to make CH-π stacking with the galactose residue of the substrate 3′-sialyllactose. The same contact is shown not to be possible for 6′-sialyllactose, due to geometrical constrains of the α-2,6 linkage. Indeed 6′-sialyllactose is not a substrate, even though it is shown to bind to RgNanH by STD NMR. These findings corroborate the role of Trp698 for the α-2,3 specificity of IT-sialidases. In this structural study, the use of Differential Epitope Mapping STD NMR (DEEP-STD NMR) approach allowed the validation of the proposed 3D models in solution. These structural approaches are shown to be instrumental in shedding light on the molecular mechanisms underpinning enzymatic reactions in the absence of enzyme-substrate X-ray structures.

On multiobjective fractional programs with vanishing constraints

The aim of this article is to combine the study of fractional programming and mathematical programs with vanishing constraints for the first time in literature. This paper deals with multiobjective fractional programs with vanishing constraints (MFPVC) involving continuously differentiable functions. Necessary and sufficient optimality conditions are derived for a feasible point to be an efficient (or local efficient) solution of the (MFPVC). A parametric dual model has been formulated and duality results are established with the primal (MFPVC).

The first derivative test and the classification of stationary points

Given a real differentiable function f we say that a point x0 is a stationary point of f if f′ (x0) = 0.In any standard single-variable calculus class, students learn how to determine the nature of a stationary point by checking the sign of f(x) in intervals to the left and to the right of the stationary point. In doing so, they are performing the first derivative test.

Generalization of some integral inequalities in multiplicative calculus with their computational analysis

UDC 517.9 We focus on generalizing some multiplicative integral inequalities for twice differentiable functions. First, we derive a multiplicative integral identity for multiplicatively twice differentiable functions. Then, with the help of the integral identity, we prove a family of integral inequalities, such as Simpson, Hermite–Hadamard, midpoint, trapezoid, and Bullen types inequalities for multiplicatively convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities to prove the validity of the results for multiplicatively convex functions. The generalized forms obtained in our research offer valuable tools for researchers in various fields.

TopoSinGAN: Learning a Topology-Aware Generative Model from a Single Image

Generative adversarial networks (GANs) have significantly advanced synthetic image generation, yet ensuring topological coherence remains a challenge. This paper introduces TopoSinGAN, a topology-aware extension of the SinGAN framework, designed to enhance the topological accuracy of generated images. TopoSinGAN incorporates a novel, differentiable topology loss function that minimizes terminal node counts along predicted segmentation boundaries, thereby addressing topological anomalies not captured by traditional losses. We evaluate TopoSinGAN using agricultural and dendrological case studies, demonstrating its capability to maintain boundary continuity and reduce undesired loop openness. A novel evaluation metric, Node Topology Clustering (NTC), is proposed to assess topological attributes independently of geometric variations. TopoSinGAN significantly improves topological accuracy, reducing NTC index values from 15.15 to 3.94 for agriculture and 14.55 to 2.44 for dendrology, compared to the baseline SinGAN. Modified FID evaluations also show improved realism, with lower FID scores: 0.1914 for agricultural fields compared to 0.2485 for SinGAN, and 0.0013 versus 0.0014 for dendrology. The topology loss enables end-to-end training with direct topological feedback. This new framework advances the generation of topologically accurate synthetic images, with applications in fields requiring precise structural representations, such as geographic information systems (GIS) and medical imaging.

All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

AbstractWe construct a differentiable locally Lipschitz function in with the property that for every convex body there exists such that coincides with the set of limits of derivatives of sequences converging to . The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite‐dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.