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This paper focuses on the directional derivative of a set optimization problem with a set order relation defined by the Minkowski difference. Firstly, we investigate a nonlinear scalar function related to the Minkowski difference and propose a Hausdorff-type distance of sets. Some properties of Hausdorff-type distance are given. Secondly, a directional derivative of set-valued mappings is introduced, and some properties are examined. Finally, the obtained results are applied to establish optimality conditions for set optimization problems. [ABSTRACT FROM AUTHOR]

The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov Ḃ ∞ , ∞ − 1 space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato Ṁ 2 , 3 r space. More precisely, we show that the smooth solution (u , d) can be extended beyond T , provided that ∫ 0 T (∥ ∇ h u ̃ (⋅ , t) ∥ B ̇ ∞ , ∞ − 1 2 + ∥ ∂ 3 d (⋅ , t) ∥ Ṁ 2 , 3 r 2) d t < ∞. [ABSTRACT FROM AUTHOR]

This paper proposes a method for visualizing three-dimensional non-uniformly sampled data from multibeam echosounder systems (MBESs), aimed at addressing the requirements of monitoring complex and dynamic underwater flow fields. To tackle the challenges associated with spatially non-uniform sampling, the proposed method employs linear interpolation along the radial direction and arc length weighted interpolation in the beam direction. This approach ensures consistent resolution of three-dimensional data across the same dimension. Additionally, an opacity transfer function is generated to enhance the visualization performance of the ray casting algorithm. This function leverages data values and gradient information, including the first and second directional derivatives, to suppress the rendering of background and non-interest regions while emphasizing target areas and boundary features. The simulation and experimental results demonstrate that, compared to conventional two-dimensional beam images and three-dimensional images, the proposed algorithm provides a more intuitive and accurate representation of three-dimensional data, offering significant support for the observation and analysis of spatial flow field characteristics. [ABSTRACT FROM AUTHOR]

Biometallic ions play a crucial role in regulating the immune system. In recent years, cancer immunotherapy has become a breakthrough in cancer treatment, achieving good efficacy in a wide range of cancers with its specificity and durability advantages. However, existing therapies still face challenges, such as immune tolerance and immune escape. Biometallic ions (e.g. zinc, copper, magnesium, manganese, etc.) can assist in enhancing the efficacy of immunotherapy through the activation of immune cells, enhancement of tumor antigen presentation, and improvement of the tumor microenvironment. In addition, biometallic ions and derivatives can directly inhibit tumor cell progression and offer the possibility of effectively overcoming the limitations of current cancer immunotherapy by promoting immune responses and reducing immunosuppressive signals. This review explores the role and potential application prospects of biometallic ions in cancer immunotherapy, providing new ideas for future clinical application of metal ions as part of cancer immunotherapy and helping to guide the development of more effective and safe therapeutic regimens. [ABSTRACT FROM AUTHOR]

New fullerene derivatives keep attracting attention of chemists due to their possible applications. Fullerene adducts with saturated polycyclic hydrocarbons represent one of the promising classes of fullerene compounds but still have rare examples. To enhance molecular diversity of fullerene derivatives in this direction, we have synthesized the representatives of C60–bicyclo[2.2.1]heptanes. For this purpose, we first produced the unsubstituted organic hydrazones from various bicyclo[2.2.1]heptane derivatives and then used the organic precursors in the reactions with C60. New homofullerenes and methanofullerenes with one bicyclo[2.2.1]heptane fragment have been obtained under metal-complex catalysis conditions at the fullerene C60:hydrazone molar ratio equal to 1:1.5. If fullerene C60:hydrazone ratio was 1:10, homofullerenes with six bicyclo[2.2.1]heptane fragments were synthesized, which can be isomerized into methanofullerenes when boiling in o-dichlorobenzene. [ABSTRACT FROM AUTHOR]

It has been thirty years since Perona and Malik (PM) introduced the nonlinear diffusion equation in image processing and analysis. The problem's complexity was to find a suitable and adaptive diffusion function that smooths away noise or textures while preserving sharp edges of a sufficiently smooth intensity. This paper provides a new two-dimensional analysis of the PM diffusion equation to examine its behavior during scales and an explicit formula to select the right diffusion function adequately. In this context, we study the PM equation at the zero crossings of the first and second directional derivatives of a sufficiently smooth function in the gradient direction. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the hyperbolic Keller-Segel equations in H s (T d) on torus with d ≥ 1. Firstly, developing the dissipative mechanism through translation, we establish the global well-posedness in H s (T d) (s > 1 + d 2) with initial data near some equilibrium state. Secondly, by capturing the feature of the preservation of zero directional derivative, we give a class of initial date that lead to finite time blow-up. It's worth noting that our method of proving blow-up phenomenon does not require any conservation law. Finally, the characterization of this blow-up motivates us to show the ill-posedness of this system in H 3 2 (T d) in the sense of "norm inflation", which implies that our ill-posedness result for this system is sharp on one dimensional torus. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study stability of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems with respect to the perturbations of feasible sets and objective mappings. We introduce a new metric between two set-valued mappings by utilizing a Hausdorff-type distance proposed by Han [A Hausdorff-type distance, the Clarke generalized directional derivative and applications in set optimization problems. Appl Anal. 2022;101:1243–1260]. The new metric between two set-valued mappings allows us to discuss set optimization problems with respect to the perturbation of objective mappings. Then, we establish semicontinuity and Lipschitz continuity of l-minimal approximate solution mapping and weak l-minimal approximate solution mapping to parametric set optimization problems by using the scalarization method and a density result. Finally, our main results are applied to stability of the approximate solution sets for vector optimization problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish new results on a class of generalized mixed quasi-variational-hemivariational inequalities (GMQVHVI, for short) via regularized gap functions. First, we introduce the new regularized gap function of GMQVHVI. Then we establish the criterion of the Levitin-Polyak well-posedness for GMQVHVI under suitable conditions. Further, we provide the equivalence between the Levitin-Polyak well-posedness in the generalized sense for GMQVHVI and that for a quasi-optimization problem using the new regularized gap function. Finally, the global error bound for GMQVHVI in term of the regularized gap function is derived by employing some the properties of the Clarke generalized directional derivative and strong monotonicity conditions. [ABSTRACT FROM AUTHOR]

This article focuses on the stability of error bounds, both local and global, for semi-infinite convex constraint systems in Banach spaces. We present primal characterizations of the stability of these error bounds under small perturbations. These characterizations are expressed through the directional derivatives of the functions that define the systems. It is shown that ensuring stability of the error bounds is closely tied to verifying that the optimal values of several minimax problems, formulated using the directional derivatives, remain outside a certain neighbourhood of zero. Furthermore, this stability condition only requires that all component functions of the system share the same linear perturbation. When applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, these stability results yield primal criteria that guarantee the uniform boundedness of Hoffman's constants under perturbations in the problem data. [ABSTRACT FROM AUTHOR]

Qing-Chengzi (QCZ) is an important silver-gold mining area in the eastern part of the Northeast China Craton. The shallow minerals in this area are almost completely depleted, leading to a demand for exploration to find deeper, concealed deposits. However, due to the rugged terrain, few high-precision ground surveys have been executed in this area, resulting in an insufficient understanding of the unexposed ores. To address this issue, this study implemented a high-precision ground magnetic survey to identify faults and potential rocks in this area. To achieve these goals, remanence was analyzed to reduce its adverse effect on processing. Then, lineament enhancement with directional derivatives was conducted on the pre-processed magnetic anomalies to highlight structural features. Based on the results, eight major and twenty-one minor faults were identified, among which three major faults correspond well to the known faults. Most of the major faults run N–S, and the others run NW/NE. Furthermore, 3D inversion was conducted to locate potential rocks. Our inversion results indicate that there are six hidden rocks in the underground, extending from a depth of a few hundred meters to no more than three km. Two of the rocks correspond well to the already mined areas. This study provides support for subsequent exploration in the QCZ area. [ABSTRACT FROM AUTHOR]

This paper studies the global existence of bounded smooth solutions for the compressible ideal magnetohydrodynamic (MHD) system. We assume that the solutions have planar symmetry. Then, the MHD system can be reduced to a 7 × 7 first-order quasilinear hyperbolic system. We find a sufficient condition on the initial data to ensure the global existence of bounded smooth solutions. The main difficulty for the global existence is to establish a priori estimates for the derivatives of the solution. To this end, we derive a group of delicate characteristic decompositions for the MHD system. These characteristic decompositions can be seen as a system of "nonlinear ordinary differential equations" for C + and C - characteristic directional derivatives of the unknown functions. Owing to the good structures of the characteristic decompositions, we use "maximum principle" to obtain some uniform a priori estimates for the derivatives of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized differentiability of the metric projection operator in Hilbert spaces. We find exact expressions for Mordukhovich derivatives (which are also called Mordukhovich coderivatives) for the metric projection operator onto closed balls in Hilbert spaces and positive cones in Euclidean spaces and in real Hilbert space l2. We investigate the connection between Frèchet derivatives, Gâteaux directional derivatives and the Mordukhovich derivatives of the metric projection in Hilbert spaces. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on the numerical study of fourth-order nonlinear dissipative symmetric regular long wave equation. We propose two new methods: the Multiple Varying Bounds Integral (MVBI) method and Taylor Function Fitted (TFF) method. With the multiple varying bounds integral method, all the derivatives in the space direction of the differential equation can be eliminated and we can get different numerical formats by adjusting the integral bound parameters. According to the physical properties of the original differential equation, we can choose an appropriate format from them. Meanwhile, with the Taylor function fitted method, the derivatives of the function at one point, such as first-order and second-order, can be approximated by the original function value at the points around it. Hence, with the MVBI method and TFF method, we can establish two compact and high-precision numerical schemes. In addition, we prove that these numerical schemes are consistent with the original equation on the energy property. Next, the convergence and stability of numerical solution U and P ̃ are both proved. Finally, numerical experiments are carried out to verify the effectiveness of numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the higher-order lower and upper radial Hadamard directional derivatives for set-valued maps, which can capture the global information of concerning maps, and discuss the relationships with other existed higher-order derivatives. Second, based on the introduced derivatives, we establish optimality conditions for Henig efficient solutions of a set-valued equilibrium problem with constraints. Particularly, the optimality conditions hold in the case where the derivatives of objective and constraint functions are separated. Finally, we give some duality theorems for a mixed type of primal-dual set-valued equilibrium problem. The main results of this paper are illustrated by several concrete examples. [ABSTRACT FROM AUTHOR]

We introduce a novel algorithm that converges to level set convex viscosity solutions of high-dimensional Hamilton–Jacobi equations. The algorithm is applicable to a broad class of curvature motion PDEs, as well as a recently developed Hamilton–Jacobi equation for the Tukey depth, which is a statistical depth measure of data points. A main contribution of our work is a new monotone scheme for approximating the direction of the gradient, which allows for monotone discretizations of pure partial derivatives in the direction of, and orthogonal to, the gradient. We provide a convergence analysis of the algorithm on both regular Cartesian grids and unstructured point clouds in any dimension, and present numerical experiments that demonstrate the effectiveness of the algorithm in approximating solutions of the affine flow in two dimensions and the Tukey depth measure of high-dimensional datasets such as MNIST and FashionMNIST. [ABSTRACT FROM AUTHOR]

We study second-order differential sensitivity in parametrized vector optimization problems with inclusion constraints. First, we consider a set-valued unconstrained problem and establish a sufficient condition for the second-order directional Dini derivative of the marginal map to be equal to the minimum of that of the objective map. We then extend our research to vector optimization problems with general inclusion constraints and demonstrate that the first- and second-order directional Dini derivatives of the objective image map are equal to the union of those of the objective map. Using advanced proof techniques, we derive a formula for the second-order directional Dini derivative of the marginal map and prove the second-order semi-derivability of the feasible objective and marginal/efficient-value maps. Examples are provided to illustrate the novelty and depth of our results. [ABSTRACT FROM AUTHOR]

First of all, we study the Hilbert boundary value problem for the Beltrami equations with sources in Jordan domains of the complex plane. Assuming that the coefficient of the problem is a function of countable bounded variation and the boundary date is measurable with respect to the logarithmic capacity, we prove the existence of nonclassical solutions of the problem in the sense of limits along all non-tangential paths quasi-everywhere on the boundary. In another case, when the notion of the boundary limits is weakened and is understood in the sense of limits along arbitrary systems of arcs by Bagemihl–Seidel, the Hilbert problem is already solved by us with arbitrary measurable coefficients. In this case, we also prove here similar solvability theorems for the Riemann boundary value problem on the conjugation, including also nonlinear boundary conditions. Our final theorems on mixed boundary conditions allow us, in particular, to obtain a solution to the Poincare boundary value problem on directional derivatives, including the nonlinear case. Note that all found solutions are represented through generalized analytic functions with sources. [ABSTRACT FROM AUTHOR]

The well-known MacCullagh formula that considers deviations of a body's shape from a spherical one for the gravitational potential U of any body at a large distance r from its center of mass to an external test point is presented via Maxwell's representation for homogeneous harmonic functions in the form of a superposition of directional derivatives of the fundamental solution r−1 of the Laplace equation ∆U = 0. In the case of the general mass distribution, this representation is determined by one scalar value and two unit vectors, h1 and h2, located in a plane orthogonal to the middle principal axis of inertia of the body. At the same time, the axis of inertia of the body corresponding to its smallest moment of inertia is the bisector of the angle formed by these vectors. The geometric meaning of the vectors is established: they are orthogonal to the circular sections of the ellipsoid of inertia of the body constructed at its center of mass. This research allows one to propose an approach to finding the central principle axes of inertia of a body according to Maxwell's representation of its gravitational potential. [ABSTRACT FROM AUTHOR]

The diversity in organismal responses to environmental changes (i.e., response diversity) plays a crucial role in shaping community and ecosystem stability. However, existing measures of response diversity only consider a single environmental variable, whereas natural communities are commonly exposed to changes in multiple environmental variables simultaneously. Thus far, no approach exists to integrate multifarious environmental change and the measurement of response diversity. Here, we show how to consider and quantify response diversity in the context of multifarious environmental change, and in doing so introduce a distinction between response diversity to a defined or anticipated environmental change, and the response capacity to any possible set of (defined or undefined) future environmental changes. First, we describe and illustrate the concepts with empirical data. We reveal the role of the trajectory of environmental change in shaping response diversity when multiple environmental variables fluctuate over time. We show that, when the trajectory of the environmental change is undefined (i.e., there is no information or a priori expectation about how an environmental condition will change in future), we can quantify the response capacity of a community to any possible environmental change scenario. That is, we can estimate the capacity of a system to respond under a range of realistic or extreme environmental changes, with utility for predicting future responses to even multifarious environmental change. Finally, we investigate determinants of response diversity within a multifarious environmental change context. We identify factors such as the diversity of species responses to each environmental variable, the relative influence of different environmental variables and temporal means of environmental variable values as important determinants of response diversity. In doing so, we take an important step towards measuring and understanding the insurance capacity of ecological communities exposed to multifarious environmental change. [ABSTRACT FROM AUTHOR]

This paper is concerned with the directional derivative of the value function for a very general set-constrained optimization problem under perturbation. Under reasonable assumptions, we obtain upper and lower estimates for the upper and lower Dini directional derivative of the value function respectively, from which we obtain Hadamard directional differentiability of the value function when the set of multipliers is a singleton. Our results do not require convexity of the set involved. Even in the case of a parametric nonlinear program, our results improve the classical ones in that our regularity conditions are weaker and the directional solution set is used which is in general smaller than its nondirectional counterparts. [ABSTRACT FROM AUTHOR]

We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area. [ABSTRACT FROM AUTHOR]

This article investigates the design of time-accurate input signals in the angle-of-attack and pitch rate space to identify the aerodynamic characteristics of a generic triple-delta wing configuration at subsonic speeds. Regression models were created from the time history of signal simulations in DoD HPCMP CREATETM-AV/Kestrel software. The input signals included chirp, Schroeder, pseudorandom binary sequence (PRBS), random, and sinusoidal signals. Although similar in structure, the coefficients of these regression models were estimated based on the specific input signals. The signals covered a wide range of angle-of-attack and pitch rate space, resulting in varying regression coefficients for each signal. After creating and validating the models, they were used to predict static aerodynamic data at a wide range of angles of attack but with zero pitch rate. Next, slope coefficients and dynamic derivatives in the pitch direction were estimated from each signal. These predictions were compared with each other as well as with the ONERA wind tunnel data and some CFD calculations from the DLR TAU code provided by the NATO Science and Technology Organization research task group AVT-351. Subsequently, the models were used to predict different pitch oscillations at various mean angles of attack with given amplitudes and frequencies. Again, the model predictions were compared with wind tunnel data. Final predictions involved responses to new signals from different models. A feed-forward neural network was then used to model pressure coefficients on the upper surface of the vehicle at different spanwise sections for each signal and the validated models were used to predict pressure data at different angles of attack. Overall, the models predict similar integrated forces and moments, with the main discrepancies appearing at higher angles of attack. All models failed to predict the stall behavior observed in the measurements and CFD data. Regarding the pressure data, the PRBS signal provided the best accuracy among all the models. [ABSTRACT FROM AUTHOR]

We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ-stationary point of any directionally differentiable Lipschitz objective using O(ϵ−4) calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves O(ϵ−5) for any difference-of-convex objective and O(ϵ−4) for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense. Funding: This work was supported by the National Science Foundation [Grant DMS-2006990]. [ABSTRACT FROM AUTHOR]

The fractional derivative approximations offer many approaches to understanding real‐world problems. The conformable fractional derivative operator that is one of the fractional derivative operators has recently attracted a lot of interesting. In this paper, a new approach to the directional derivative obtained with the help of the conformal fractional derivative is presented. Considering this approach, the definition of the fractional partial derivative is reformulated. In addition, a new definition of fractional gradient, fractional curl, and fractional divergence is given, and the properties of these new concepts are examined. [ABSTRACT FROM AUTHOR]

Directional derivative-based edge detectors are commonly used to outline the abrupt lateral variations in densities or magnetizations of geological sources from potential-field datasets. Some subtle traces of faults, lithological contacts, and lineaments can be made more visible on anomaly maps using these algorithms. However, these algorithms generally suffer from common limitations such as blurry edges with low resolution or faulty boundaries. To minimize these challenges, we propose a new efficient edge detector that benefits from the hyperbolic tangent function. The performance of this detector was tested on the responses of a synthetic gravity model and Bishop's complex magnetic model. Gaussian noise content was also added to both synthetic cases to test the response of the proposed operator. In a real-data experiment, residual gravity anomalies of the Aegean Graben System from western Türkiye, obtained via finite element method, were used to delineate abrupt lateral variations in mass densities in the region. The traces of basin-bounding normal faults and the boundaries of major grabens were improved in the resultant image map. A modified nonlocal means filter was also applied in both synthetic and real-data cases to attenuate undesired high-frequency effects. The proposed edge detection operator produced satisfactory outputs in each case. Comparative studies performed with some traditional and recently introduced algorithms further showed the superiority of the operator; thus, this detector is useful in revealing geological features that cannot be easily seen in potential-field anomaly maps and it is a strong alternative to currently used algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems. [ABSTRACT FROM AUTHOR]

In this article, we study the quasi-translation-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron–Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron–Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators. [ABSTRACT FROM AUTHOR]

The synucleinopathies Parkinson disease (PD), multiple system atrophy (MSA), and the Lewy body form of pure autonomic failure (PAF) entail intra‐cytoplasmic deposition of the protein alpha‐synuclein and pathogenic catecholaminergic neurodegeneration. Cerebrospinal fluid (CSF) levels of catecholamines and their metabolites are thought to provide a "neurochemical window" on central catecholaminergic innervation and can identify specific intra‐neuronal dysfunctions in synucleinopathies. We asked whether there are CSF concentration gradients for catechols such as 3,4‐dihydroxyphenylacetic acid (DOPAC), the main neuronal metabolite of dopamine, and if so whether the gradients influence neurochemical differences among synucleinopathies. In a retrospective cohort study, we reviewed data about concentrations of catechols in the first, sixth, and twelfth 1‐mL aliquots from 33 PD, 28 MSA, and 15 PAF patients and 41 controls. There were concentration gradients for DOPAC, dopamine, norepinephrine, and 3,4‐dihydroxyphenylglycol (the main neuronal metabolite of norepinephrine) and gradients in the opposite direction for 5‐S‐cysteinyldopa and 5‐S‐cysteinyldopamine. In all 3 aliquots, CSF DOPAC was low in PD and MSA compared with controls (p < 0.0001 each) and normal in PAF. Synucleinopathies differ in CSF catechols regardless of concentration gradients. Concentration gradients for 5‐S‐cysteinyl derivatives in opposite directions from the parent catechols may provide biomarkers of spontaneous oxidation in the CSF space. [ABSTRACT FROM AUTHOR]

This paper investigates second-order optimality conditions for general constrained set-valued optimization problems in normed vector spaces under the set criterion. To this aim we introduce several new concepts of second-order directional derivatives for set-valued maps by means of excess from a set to another one, and discuss some of their properties. By virtue of these directional derivatives and by adopting the notion of set criterion intoduced by Kuroiwa, we obtain second-order necessary and sufficient optimality conditions in the primal form. Moreover, under some additional assumptions we obtain dual second-order necessary optimality conditions in terms of Lagrange–Fritz–John and in terms of Lagrange–Karush–Kuhn–Tucker multipliers. [ABSTRACT FROM AUTHOR]

SUMMARY: Minimum-structure, or Occam's style of, inversion introduces a regularization function into the underdetermined geophysical inverse problems to stabilize the inverse problem and mitigate its non-uniqueness. The regularization function is typically designed such that it can incorporate a priori information into the inversion framework, thus constructing models that have more plausible representations of the true Earth's subsurface structure. One type of a priori information is geological orientation information such as strike, dip and tilt angles of the subsurface structure. This type of information can be incorporated into inverse problems through the roughness operators. Designing such roughness operators for inversion frameworks using unstructured tetrahedral meshes is not as straightforward as for inversion frameworks using structured meshes due to the arbitrary and complex geometry of unstructured meshes. Researchers have developed methods which allow us to incorporate geological orientation information into inversion frameworks with unstructured tetrahedral meshes. The majority of these methods consider each cell in a package with its neighbours, hence, the constructed models are not as sharp as desired if the regularization function is measured using an $\ell _1$ -type measure instead of the $\ell _2$ norm. To address this issue, we propose a method that calculates the directional derivatives of physical property differences between two adjacent cells normalized by the distance between the cell centroids. This approach is able to both incorporate geological orientation information into the inversion framework and construct models with sharp boundaries for the scenarios in which the regularization term is quantified by an $\ell _1$ -type measure. This method is an integral-based approach, therefore, the roughness operators are scaled appropriately by the cell volumes, which is an important characteristic for the inversions with unstructured meshes. To assess the performance and the capability of the proposed method, it was applied to 3-D synthetic gravity and magnetotelluric examples. The gravity example was also used to investigate the impact of applying the depth weighting function inside and outside the roughness operators for the scenarios that the model objective function is measured by an $\ell _1$ norm. The examples show that the proposed method is able to construct models with a reasonable representation of the strike and dip directions of the true subsurface model with sharper boundaries if the regularization function is quantified by an $\ell _1$ -type measure. The examples also demonstrate the proposed method behaves numerically well, and has a fast convergence rate. [ABSTRACT FROM AUTHOR]

We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations (PDEs). Such PDEs contain constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that direct smoothing of the ReLU network with the aim of using classical numerical solvers can have disadvantages, such as potentially introducing multiple solutions for the corresponding PDE. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown. [ABSTRACT FROM AUTHOR]

The majorization approximation procedure consists in replacing the resolution of a non-linear optimization problem by solving a sequence of simpler ones, whose objective and constraint functions upper estimate those of the original problem. For generalized fractional programming, i.e., constrained minimization programs whose objective functions are maximums of finite ratios of functions, we propose an adapted scheme that simultaneously upper approximates parametric functions formed by the objective and constraint functions. For directionally convex functions, that is, functions whose directional derivatives are convex with respect to directions, we will establish that every cluster point of the generated sequence satisfies Karush–Kuhn–Tucker type conditions expressed in terms of directional derivatives. The proposed procedure unifies several existing methods and gives rise to new ones. Numerical problems are solved to test the efficiency of our methods, and comparisons with different approaches are given. [ABSTRACT FROM AUTHOR]

Determining the edge and horizontal position of geologic structures is one of the fundamental steps in interpreting potential field data. Several filters have been introduced that use the concept of curvature to determine the edge of potential field data. However, these filters have advantages and disadvantages in detecting causative sources. Therefore, it seems necessary to introduce more efficient approaches. In this work, the most positive and most negative curvatures of gravity field data were analyzed, and a more efficient filter was introduced and applied that uses the concept of curvature and its combination to delineate the edges of geological structures and buried sources. The proposed method, called the hybrid positive and negative curvature (PNH) approach, combines the most positive and most negative curvatures into one curvature by fitting the formula and weighted summation. The proposed strategy takes advantage of both positive and negative curvatures to improve the edge detection of gravity field data. To this end, the performance of the PNH procedure was investigated considering different density assumptions (positive, negative, and positive-negative) for the relatively imposed synthetic gravity model resulting from buried prisms. The results obtained on synthetic models with and without noise show that the PNH procedure can detect the horizontal boundaries of buried structures relatively well. Of course, due to the use of directional derivatives in the filter of the hybrid positive and negative curvature approach, it seems very necessary to use noise-reducing filters before applying edge detection methods. Moreover, conventional filters such as the second vertical derivative (SVD) and the tilt angle (TDR) were used to compare the performance of the hybrid positive and negative curvature filter on the synthetic model. However, the obtained results show that the second vertical derivative and the tilt angle do not have the required capability to determine the edge of the synthetic model. In the following, the quality of the most positive and most negative curvatures filter and the hybrid positive and negative curvature were investigated using real data from a gold mine in the Witwatersrand area (South Africa) and also gravity data from the Aji-chai salt dome, East Azerbaijan province (Iran) and then using WGM-2012 derived gravity data belonging to the Marian trench area. Due to the sensitivity of the filters to noise, the upward continuation filter was applied before determining the edge of the buried structures. The edge maps from the Witwatersrand area and the data from the Aji-chai salt dome obtained using the hybrid positive and negative curvature determination method, demonstrate acceptable accuracy of this filter in determining the edge and representing the horizontal position of various geological structures. By using the PNH filter, the lateral boundaries of the main structures and other subsurface sources are well detected. Of course, due to the noise sensitivity of this filter, which is due to the use of secondorder gravity derivatives, good quality data without noise must be used. Therefore, it is suggested that noise attenuate filters, such as upward continuation method, must be used prior to creating the maps to determine the edge. Therefore, the PNH edge detection method can be reliably used for qualitative interpretation of gravity field data. [ABSTRACT FROM AUTHOR]

This comprehensive investigation explores the application and significance of q-differential calculus in the realm of vector functions of several variables, addressing critical aspects such as q-Rolle’s theorem, the q-Mean-value theorem, and q-chain rule for vector functions. Additionally, we investigate the q-gradient, q-Jacobian, and q-Hessian operators, elucidating their roles in quantifying rates of change, determining directional derivatives, and characterizing critical points of multivariate functions. Furthermore, this research provides a rigorous treatment of Multivariate and Bivariate Taylor theorems in the context of q-differential calculus, presenting analytical expansions of functions around specific points and showcasing their utility in approximating functions in higher dimensions. The q-Maximum and Minimum are demonstrated and discussed as well. [ABSTRACT FROM AUTHOR]

Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that many changes to the active set can be made at the cost of computing a single search direction. As the objective function is not differentiable along the search path, it is not possible to use a projected-search method with a step that satisfies the Wolfe conditions, which require the directional derivative of the objective function at a point on the path. For this reason, methods based in full or in part on a simple backtracking procedure must be used to give a step that satisfies an 'Armijo-like' sufficient decrease condition. As a consequence, conventional projected-search methods are unable to exploit sophisticated safeguarded polynomial interpolation techniques that have been shown to be effective for the unconstrained case. This paper describes a new framework for the development of a general class of projected-search methods for bound-constrained optimization. At each iteration, a descent direction is computed with respect to a certain extended active set. This direction is used to specify a search direction that is used in conjunction with a step length computed by a quasi-Wolfe search. The quasi-Wolfe search is designed specifically for use with a piecewise-linear search path and is similar to a conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the restriction of the objective function on the search path is not differentiable. Standard existence and convergence results associated with a conventional Wolfe line search are extended to the quasi-Wolfe case. In addition, it is shown that under a standard nondegeneracy assumption, any method within the framework will identify the optimal active set in a finite number of iterations. Computational results are given for a specific projected-search method that uses a limited-memory quasi-Newton approximation of the Hessian. The results show that, in this context, a quasi-Wolfe search is substantially more efficient and reliable than an Armijo-like search based on simple backtracking. Comparisons with a state-of-the-art bound-constrained optimization package are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate optimality conditions of nonsmooth sparsity multiobjective optimization problem (shortly, SMOP) by the advanced variational analysis. We present the variational analysis characterizations, such as tangent cones, normal cones, dual cones and second-order tangent set, of the sparse set, and give the relationships among the sparse set and its tangent cones and second-order tangent set. The first-order necessary conditions for local weakly Pareto efficient solution of SMOP are established under some suitable conditions. We also obtain the equivalence between basic feasible point and stationary point defined by the Fréchet normal cone of SMOP. The sufficient optimality conditions of SMOP are derived under the pseudoconvexity. Moreover, the second-order necessary and sufficient optimality conditions of SMOP are established by the Dini directional derivatives of the objective function and the Bouligand tangent cone and second-order tangent set of the sparse set. [ABSTRACT FROM AUTHOR]

Shallow, dominantly silt‐ and clay‐filled erosional troughs in Quaternary sediments under the Flathead Valley (northwestern Montana, USA) are very likely to be hydraulic barriers limiting the horizontal flow of groundwater. Accurately mapping them is important because of increasing demand for groundwater. We used a legacy Bouguer gravity map measured in 1968. The directional derivatives of the map are computed, and the map was enhanced by implementing edge detection tools. We produced generalized derivative, maximum horizontal gradient, total gradient and tilt gradient maps through two‐dimensional Fourier transform analysis. These maps were remarkably successful in locating buried troughs in the northern and northwestern parts of the study area, closely matching locations determined previously from compiled borehole data. Our results also identify hitherto unknown extensions of troughs and indicate that some of the buried troughs may be connected. [ABSTRACT FROM AUTHOR]

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives. [ABSTRACT FROM AUTHOR]

Motivated by the work, in which the author studied the convexity of nonlinear mappings defined between bounded linear operator spaces, our research extends this inquiry. In this work, we continue the study of the convexity of nonlinear mappings defined between bounded linear operator spaces and we establish a characterization in terms of the second order directional derivative. We apply the main result to prove the convexity and the nonconvexity of well-known nonlinear mappings. The case of nondifferentiable mappings is also treated in the last section. [ABSTRACT FROM AUTHOR]

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values − ∞ ˜ and + ∞ ˜ at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional g -derivative for the extended fuzzy-valued convex function. [ABSTRACT FROM AUTHOR]

Enhancing the detection accuracy of the edges of the geological features within the subsurface remains a significant objective in geophysical data interpretation. Despite numerous advancements, approaches stemming from the directional gradients of gravity and magnetic fields still grapple with challenges such as low-resolution outcomes and susceptibility to noise contamination. In this study, we introduce a novel filtering framework based on the total horizontal gradient and its derivatives, designed to yield more precise and coherent edges free from false boundaries or disruptive artifacts. Validation using synthetic Bishop complex magnetic and gravity datasets, alongside Tuangiao aeromagnetic data from Vietnam, substantiates the robustness and applicability of our modified approach. Furthermore, recognizing the inherent susceptibility of edge detection filters to noise contamination resulting from directional derivatives, we employ the recently developed modified non-local means (MNLM) algorithm to alleviate noise effects prior to the analysis of noisy synthetic and real datasets. Our findings confirm the efficacy of the proposed method in reducing false artifacts and identifying edges with heightened precision, positioning MHGA as a valuable alternative for processing potential field data. [ABSTRACT FROM AUTHOR]

In this study, the thermo-mechanical behavior of rectangular plates functionally graded (FG) in two directions along the plane is investigated with three-dimensional (3D) elasticity equations. In numerical analysis, the finite difference method (FDM) is used and the directional variation of the derivatives of the material properties is taken into account. The thermo-mechanical behavior of the plates is investigated for different compositional gradient exponents and it is emphasized that the variation of the compositional gradient exponent in both directions significantly changed the thermal stress distribution and the results are confirmed. [ABSTRACT FROM AUTHOR]

For a given direction b ∈ C n \ {0} we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of L-index in the direction with a positive continuous function L satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions as the coefficients of the equation. Our considerations use some estimates involving a directional logarithmic derivative and distribution of zeros on all directional slices in the unit ball. [ABSTRACT FROM AUTHOR]

Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let PC: X → C denote the (standard) metric projection operator. In this paper, we define the G a ^ teaux directional differentiability of PC. We investigate some properties of the G a ^ teaux directional differentiability of PC. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of PC. [ABSTRACT FROM AUTHOR]

We present a concise review of selected parts of axion electrodynamics and their application to Casimir physics. We present the general formalism including the boundary conditions at a dielectric surface, derive the dispersion relation in the case where the axion parameter has a constant spatial derivative in the direction normal to the conducting plates, and calculate the Casimir energy for the simple case of scalar electrodynamics using dimensional regularization. [ABSTRACT FROM AUTHOR]

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as the lower limit (i.e., limit inferior) of the classical derivative of the mapwhere it exists. The new representations lead to significantly shorter proofs for the basic properties of the subgradient and the generalised Jacobian including the chain rule. We establish that a sequence of locally Lipschitz maps between finite dimensional Euclidean spaces converges to a given locally Lipschitz map in the L-topology--that is, theweakest refinement of the sup norm topology on the space of locally Lipschitz maps that makes the generalised Jacobian a continuous functional--if and only if the limit superior of the sequence of directional derivatives of the maps in a given vector direction coincides with the generalised directional derivative of the given map in that direction, with the convergence to the limit superior being uniform for all unit vectors. We then prove our main result that the subspace of Lipschitz C8 maps between finite dimensional Euclidean spaces is dense in the space of Lipschitz maps equipped with the L-topology, and, for a given Lipschitz map, we explicitly construct a sequence of Lipschitz C8 maps converging to it in the L-topology, allowing global smooth approximation of a Lipschitz map and its differential properties. As an application, we obtain a short proof of the extension of Green's theorem to interval-valued vector fields. For infinite dimensions, we show that the subgradient of a Lipschitz map on a Banach space is upper continuous, and, for a given real-valued Lipschitz map on a separable Banach space, we construct a sequence of Gateaux differentiable functions that converges to the map in the sup norm topology such that the limit superior of the directional derivatives in any direction coincides with the generalised directional derivative of the Lipschitz map in that direction. [ABSTRACT FROM AUTHOR]

This work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as p$$ p $$‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear. [ABSTRACT FROM AUTHOR]

In this expository review paper, we show that co-kriging, a widely used geostatistical multivariate optimal linear estimator, has a diverse range of extensions that we have collected and illustrated to show the potential of this spatial interpolator. In the context of spatial stochastic processes, this paper covers scenarios including increasing the spatial resolution of a spatial variable (downscaling), solving inverse problems, estimating directional derivatives, and spatial interpolation taking boundary conditions into account. All these spatial interpolators are optimal linear estimators in the sense of being unbiased and minimising the variance of the estimation error. [ABSTRACT FROM AUTHOR]