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Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems. [ABSTRACT FROM AUTHOR]

In this paper, the scaling-law vector calculus, which is connected between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The scaling-law Gauss–Ostrogradsky-like, Stokes-like and Green-like theorems, and Green-like identities are considered in sense of the scaling-law vector calculus. The strong and weak conjectures for the scaling-law flows are obtained in detail. The obtained result is a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows. [ABSTRACT FROM AUTHOR]

Data normalization is a crucial step in multi-criteria decision-making (MCDM) processes. The choice of data normalization method significantly influences the ranking of alternatives. The available data normalization methods in the MARA (Magnitude of the Area for the Ranking of Alternatives) method may not be applicable in certain cases. This study aims to broaden the application scope of the MARA method. Therefore, an investigation into the compatibility of data normalization methods when combined with the MARA method has been conducted. Ten data normalization methods were utilized, including Linear normalization, Weitendorf normalization, Sum linear normalization, Vector normalization, Logarithmic normalization, Max linear normalization, Min linear normalization, Jüttler-Körth normalization, Peldschus normalization, and Stop normalization. The compatibility between the MARA method and these ten data normalization methods was tested in five different scenarios. In the initial four scenarios, variations in the number of alternatives, criteria, and criterion types were introduced. Eight out of the ten data normalization methods were proven to be suitable for integration with the MARA method. In the fifth scenario, a hypothetical situation was presented where the data normalization methods available in the MARA method could not be used. Alternative data normalization methods were employed, and their combination with the MARA method was compared to using other multi-criteria decision-making methods. The results affirmed the accuracy of these combinations. This exploration has expanded the application scope of the MARA method compared to its original version. [ABSTRACT FROM AUTHOR]

In this paper, we use differential forms to prove a number of theorems of integral vector calculus that are rarely found in textbooks. Two of them, as far as the author knows, have not been published before. Some possible applications to problems in physics are shared including a general approach for computing net forces and torques on current-carrying loops that yields insights that are not evident from the standard approach. Readers who are familiar with differential forms will enjoy seeing how they can be employed to prove several new vector calculus identities. But even readers who do not follow those derivations will benefit from seeing how these identities can be employed to find forces and torques on current-carrying loops. [ABSTRACT FROM AUTHOR]

A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new look at Finsler surfaces. Landsberg surfaces are Finsler surfaces that are solutions of a system of non-linear partial differential equations. Considering the unicorn's Landsberg problem, we reduce this system to a single non-linear PDE which we call the Landsberg's PDE. By making use of the new look of Finsler surfaces, we solve the Landsberg's PDE and get a class of solutions. Moreover, we show that these solutions and their conformal transformations are Berwaldain. [ABSTRACT FROM AUTHOR]

In this study, we investigate the relations between a multiple partitioned linear model and its correctly-reduced models. We consider the comparison problem of covariance matrices of the best linear unbiased predictors (BLUPs) of all unknown vectors including partial parameters under these models by using the block matrix rank and inertia formulas. We derive various inequalities and equalities for covariance matrices of BLUPs under some general assumptions. Also, results for special cases are given. [ABSTRACT FROM AUTHOR]

We prove that the realization space of the d-dimensional cube is contractible. For this we first show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. As an application we use this fact to define an analog of the connected sum construction for cubical d-polytopes, and apply this construction to certain cubical d-polytopes to conclude that the rays spanned by f -vectors of cubical d-polytopes are dense in Adin’s cone. The connectivity result on cubes extends to any product of simplices, and further it shows that the respective realization spaces are contractible. [ABSTRACT FROM AUTHOR]

The calculation of the view factor is an important consideration in radiative heat transfer analysis. Thermal engineering requires the view factor calculations between complex geometries containing parallel and inclined surfaces. There is currently no analytic formula available for this, so the summation rule (SR) must be used. However, when complex geometries are analyzed, processing SR becomes a very intensive task. Therefore, the present study aims to exploit vector calculus to develop a formula for calculating the view factors for radiative heat transfer from any two rectangular oblique or parallel surfaces. The exact solution obtained by means of the contour integration was compared with the numerical integration of the quadruple integral using the multiple Simpson's rule 1/3 with five intervals (MSR) and the summation rule via the Hamilton–Morgan solution (HMS). The lack of similar precedents in the literature reinforces the scientific and practical nature of the proposed method. [ABSTRACT FROM AUTHOR]

Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties curl α grad α = 0 and div α curl α = 0 hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size. [ABSTRACT FROM AUTHOR]

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. [ABSTRACT FROM AUTHOR]

We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and collate useful vector calculus identities for these coordinates. These results and provided code enable consistent accounting of geometric effects to arbitrary order for boundary layer asymptotics in a wide range of physical systems. [ABSTRACT FROM AUTHOR]

A novel nonlocal thermo‐hydro‐mechanical coupling model is proposed to investigate cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics (GPD). The nonlocal thermal conduction equation and the nonlocal flow diffusion equation are derived based on nonlocal vector calculus and the microscopic constitutive equation. Moreover, the coupling influences are considered based on the nonlocal equation and the coupling principle. The governing equation of the coupling system is established in the framework of GPD. The correctness of the proposed model is verified through comparing with the corresponding experimental data, and the numerical results are in good agreement with experimental results. Finally, the proposed model is applied to analyze the deformation and cracking behaviors of rock mass around tunnels, and the stress redistribution is revealed under thermo‐hydro‐mechanical coupling condition. Highlights: A novel nonlocal thermo‐hydro‐mechanical coupling model is proposed to investigate the cracking behaviors of rocks.The nonlocal thermal conduction equation and the nonlocal flow diffusion equation are derived.The governing equation of the coupling system is established in the framework of advanced general particle dynamics. [ABSTRACT FROM AUTHOR]

In this article we introduce a type of teaching scholarship called Lesson Analysis (LA) that helps build instructional knowledge, and an associated genre of writing called Lesson Analysis Manuscript (LAM) that shares this instructional knowledge. We briefly describe the problem LA is intended to solve, as well as key features of a LAM. In LA, instructors develop lessons that overcome specific instructional challenges. LAMs are lesson-specific manuscripts that help others understand details of the lesson that addressed the instructional challenge, and include such information as the instructional context, the rationale behind the instructional decisions, typical student thinking, and guidance on carrying out discussions and implementing tasks. We have illustrated this genre by including an entire example LAM as supplemental material located online. The example LAM focuses on helping students understand the connection between dot products and the connection of dot products to measures of similarity between vectors. [ABSTRACT FROM AUTHOR]

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study the Zariski equisingular families of surface, not necessarily isolated, singularities in C³. We show that a natural stratification of such a family, given by the singular set and the generic family of polar curves, provides a Lipschitz stratification in the sense of Mostowski. In particular such families are biLipschitz trivial, with trivializations obtained by integrating Lipschitz vector fields. [ABSTRACT FROM AUTHOR]

In this work, some set order relations are compared with each other. In addition, it is shown that every weak minimal solution of a set valued optimization problem with respect to vector optimization criterion, is also a weak minimal solution with respect to set optimization criterion considering some special set orders. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of L 2 -products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme. [ABSTRACT FROM AUTHOR]

Using Maxwell's mental imagery of a tube of fluid motion of an imaginary fluid, we derive his equations curl E = -∂B/∂t, curl H = ∂D/∂t + j, div D = ϱ, div B = 0, which together with the constituting relations D = "0ε, B = µ0H, form what we call today Maxwell's equations. Main tools are the divergence, curl and gradient integration theorems and a version of Poincare's lemma formulated in vector calculus notation. Remarks on the history of the development of electrodynamic theory, quotations and references to original and secondary literature complement the paper. [ABSTRACT FROM AUTHOR]

A model order reduction methodology for reducing the order of the peridynamic transient heat model is proposed. This methodology is based on the static condensation procedure. To set the development of the model reduction procedure on a sound mathematical setting, a nonlocal vector calculus was employed in the formulation of the heat transport problem. The model order reduction framework proposed in this study provides a technique to reduce the dimensionality of a peridynamic transport model while still maintaining accurate prediction of the model response. Moreover, the methodology can be adaptively applied to accommodate different resolution requirements for different sections of the model. Using numerical experiments, the proposed methodology is shown to be capable of accurately reproducing results of the full peridynamic transient heat transport problem. [ABSTRACT FROM AUTHOR]

Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions. [ABSTRACT FROM AUTHOR]

This paper concerns the problem of identifying the diffusion parameter in the nonlocal diffusion model. The parameter identification problem is formulated as an optimal control problem, and the objective of the control is a cost functional formulated by the energy functional. By using the nonlocal vector calculus, in a manner analogous to the local partial differential equations counterpart, we proved that the cost functional is a strictly convex functional with a unique global minimizer. Moreover, the existence and uniqueness of parameter identification are further demonstrated. Finally, one‐dimensional numerical experiments are given to illustrate our theoretical results and show that continuous and discontinuous parameters can be estimated. [ABSTRACT FROM AUTHOR]

The general particle dynamics (GPD), which is based on the kernel approximation method and Navier–Stokes equation, was developed from smoothed‐particle hydrodynamics to simulate the fracture behaviors by using the collective of damage variable to describe the damage evolution behaviors. In this paper, an advanced GPD with a nonlocal foundation is proposed for better description of the solid mechanics. The nonlocal vector calculus and microscopic constitutive model are employed to derive the governing equations in the proposed theory. The novel GPD with a nonlocal core is a generalization, which permits the inclusion of the microscopic constitutive model and the multi‐scale modeling compared with the previous GPD. To accomplish this generalization, the relationship between GPD and the previous kernel‐based theory is established. The novel GPD is superior in the fracture problems and multi‐scale modeling. Some numerical experiments are launched to verify the ability of the novel GPD. Highlights: Advanced GPD with nonlocal foundation for is proposed for better description of the solid mechanics.The governing equations are re‐derived by nonlocal vector calculus and micro‐bond model.The advanced GPD is good at handling with multi‐scale problem and fracture problems. [ABSTRACT FROM AUTHOR]

The mouth opening is an important indication of the functionality of the temporomandibular joint (TMJ). Mouth opening is usually evaluated by asking the patient to open their mouth as wide as possible and measuring the distance between the edges of the frontal incisors with a ruler or caliper. With the advancement of technology, new techniques have been proposed to record mandibular movement. The aim of this work is to present a novel technique based on 3D electromagnetic articulography and data postprocessing to analyze the mouth opening considering distances, trajectories, and angles. A maxilla-mandible phantom was used to simulate the mouth opening movement and fixed position mouth opening. This was recorded using the AG501 3D EMA (Carstens Medizinelektronik GmbH, Bovenden, Germany). The collected data was processed using Matlab (Mathworks, Natick, MA, USA). Fix and mobile mouth opening of 1, 2, 3 and 4 cm were simulated. It was possible to evaluate the mandibular opening through the vertical distance, the Euclidean distance, the trajectory, and the opening angle. All these values were calculated and the results were consistent with expectations. The trajectory was the highest value obtained while the vertical distance was the lowest. The angle increased as the mouth opening increased. This new technique opens up new possibilities in future research since oral opening can be analyzed using multiple variables without the need to use different devices or depending on the researcher's experience. This will make it possible to establish which parameter presents significant differences between groups of patients or between patients who have undergone some treatment. [ABSTRACT FROM AUTHOR]

Molecular dynamics and transmission electron microscopy are employed to study the formation and evolution of interstitial dislocation loops due to nearby growing helium bubbles. The study reveals a novel mechanism that suppresses further formation of loops. Mass transport of self-interstitial atoms between the bubbles is observed, followed by the loop formation, resulting in a loop-bubble complex in which the Burgers vector may rotate between <100> and 1/2 <111>. Such a complex can absorb free loops, effectively influencing the evolution of loop number density. These results provide a new understanding of loop-bubble complex formation and saturation of the loop density under continuous helium implantation in materials. [ABSTRACT FROM AUTHOR]

As the main way to reproduce flow response in subsurface reservoirs, the reservoir simulation could drastically assist in reducing the uncertainties in the geological characterization and in optimizing the field development strategies. However, the challenges in providing efficient and accurate solutions for complex field cases constrain further utilization of this technology. In this work, we develop a new reservoir simulation framework based on advanced spatial discretization and linearization schemes, the mimetic finite difference (MFD) and operator-based linearization (OBL), for fully implicit temporal discretization. The MFD has gained some popularity lately since it was developed to solve for unstructured grids and full tensor properties while mimicking the fundamental properties of the system (i.e. conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus). On the other hand, in the OBL the mass-based formulations are written in an operator form where the parametric space of the unknowns is treated in a piece-wise manner for the linearization process. Moreover, the values of these operators are usually precomputed into a nodal tabulation and with the implementation of multi-linear interpolation, the values of these operators and their derivatives during a simulation run can be determined in an efficient way for the Jacobian assembly at any time-step. This saves computational time during complex phase behavior computations. By first coupling these two schemes within a parallel framework, we can solve large and complex reservoir simulation problems in an efficient manner. Finally, we present a benchmark case that compares the numerical solutions to a Buckley-Leverett analytical solution to assure their accuracy and convergence. Moreover, we test three challenging field cases to demonstrate the performance of the advanced parallel framework for complex reservoir simulation. [ABSTRACT FROM AUTHOR]

The goal of this research to realize the control humanoid type prosthetic movement used for speech recognition principles. Also, the basic principles of speaker-dependent speech recognition algorithms are introduced, a speaker-dependent speech recognition algorithm in the implementation process of an upper machine is described, and the feature template library of voice command is established. Ultimately, this information was used to test the performance of a speech recognition system. Experimental results show that this method could detect blinks at high accuracy signals and obtain an eigenvector, which achieved the goal of accurate prosthetic limb movement control. [ABSTRACT FROM AUTHOR]

The basins of attraction of periodic orbits or focus equilibria of a given vector field are foliated by forward-time isochrons, defined as all initial conditions that synchronize under the flow with a given phase. Similarly, backward-time isochrons of repelling periodic orbits or focus equilibria foliate their respective basins of repulsion. We present a case study of a planar system that features a sequence of bifurcations, including a saddle-node bifurcation of periodic orbits, a homoclinic bifurcation and Hopf bifurcations, that change the nature and existence of periodic orbits. We explain how the basins and isochron foliations change throughout the sequence of bifurcations. In particular, we identify structurally stable tangencies between the foliations by forward-time and backward-time isochrons, which are curves in the plane, in regions of phase space where they exist simultaneously. Such tangencies are generically quadratic and associated with sharp turns of isochrons and phase sensitivity of the system. In contrast to the earlier reported cubic isochron foliation tangency (CIFT) mechanism, which generates a pair of tangency orbits, we find isochron foliation tangencies that occur along single specific orbits in the respective basin of attraction or repulsion. Moreover, the foliation tangencies we report arise from actual bifurcations of the system, while a CIFT is not associated with a topological change of the underlying phase portrait. The properties and interactions of isochron foliations are determined and illustrated by computing a representative number of forward-time and backward-time isochrons as arclength-parametrized curves with a boundary value problem set-up. Our algorithm for computing isochrons has been further refined and implemented in the Matlab package CoCo; it is made available as Matlab code in the supplementary material of this paper, together with a guide that walks the user through the computation of two specific isochron foliations. [ABSTRACT FROM AUTHOR]

In the paper, we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss–Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of the Hausdorff vector calculus. The formula is proposed as a mathematical tool to describe the real-world problems for the fractal power-law flow equations with the anomalous diffusion equation. A conjecture for the fractal power-law flow equations analogous to the Smale's 15th problem and one of Millennium Prize Problems for the Navier–Stokes equations is also addressed. [ABSTRACT FROM AUTHOR]

In this paper, mechanics of continuum with general form of nonlocality in space and time is considered. Some basic concepts of nonlocal continuum mechanics are discussed. General fractional calculus (GFC) and general fractional vector calculus (GFVC) are used as mathematical tools for constructing mechanics of media with general form of nonlocality in space and time. Balance equations for mass, momentum, and energy, which describe conservation laws for nonlocal continuum, are derived by using the fundamental theorems of the GFC. The general balance equation in the integral form are derived by using the second fundamental theorems of the GFC. The first fundamental theorems of GFC and the proposed fractional analogue of the Titchmarsh theorem are used to derive the differential form of general balance equations from the integral form of balance equations. Using the general fractional vector calculus, the equations of conservation of mass, momentum, and energy are also suggested for a wide class of regions and surfaces. [ABSTRACT FROM AUTHOR]

Data envelopment analysis (DEA) evaluates the cost efficiency of systems with multiple inputs and multiple outputs in two scenarios, one where the input prices are the same in all decision-making units (DMUs) (competitive space), and the other where the input prices differ from one DMU to another (non-competitive space). In many situations, the DMUs could have a multi-stage network structure with intermediate measures. Although a model has been presented for calculating the network cost efficiency in a competitive space in such cases, no method has been proposed so far to calculate the cost efficiency of such networks in a non-competitive space. The present article focuses on the concept of cost efficiency in DEA for DMUs with network structures and varying input prices. To this end, a cost-based production possibility set (PPS) is first introduced for the network systems and their sub-systems, and after calculating the network cost efficiency, it is decomposed into efficiencies such as technical, price, and allocative efficiencies, and the reasons behind the occurrence of extra costs due to various types of inefficiency are explained. [ABSTRACT FROM AUTHOR]

The article discusses the use of large language models (LLMs) like GPT-3 and GPT-4, specifically in the conversational format of ChatGPT, to solve problems in physics, math, and engineering. The authors tested ChatGPT-3.5, 4, and 4 with Code Interpreter (4/CI) on a set of 13 electromagnetism problems and found that ChatGPT-4/CI was able to solve all of the problems correctly most of the time. They recommend that instructors and students be aware of the increasing capabilities of AI tools and test them for teaching and learning purposes. They also suggest incorporating in-class assessments and creating assignments that allow the use of AI tools. [Extracted from the article]

This paper discusses the relationship between statistical mechanics and nonlocal mechanical theories. Specifically, the relationship among statistical mechanics, nonlocal vector calculus, smoothed particle hydrodynamics (SPH), and peridynamics is established. Furthermore, a general particle dynamics method is proposed to solve the solid mechanics problem based on the nonlocal vector calculus and peridynamic bond constitutive model. The proposed method is a successful nonlocal mechanics theory; the convergence of the nonlocal property is verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Data Envelopment Analysis is a non-parametric method for evaluating the efficiency of those Decision Making Units (DMUs) that have the same functionality and use multiple inputs to generate multiple outputs. DMUs may sometimes be divided into several groups according to a series of criteria, and it is intended to evaluate a group of similar DMUs. In this paper groups are fairly evaluated under a com- mon platform, each group was considered as a player in a cooperative game and a subset of groups was considered as a coalition. Assuming the Production Possibility Set(PPS) is made up of DMUs belonging to the groups that are a coalition's member, a characteristic function was defined in terms of the sum of the efficiency of all units to determine the marginal effect of each group in different coalitions. The groups were then evaluated using the Shapley Value as a unique solution of the cooperative game. Some Examples were provided to describe and apply the method. [ABSTRACT FROM AUTHOR]

In this article, we address a new model for the scaling law heat conduction problem by using the scaling law vector calculus associated with the Korcak scaling law. The scaling law heat conduction equations are discussed in detail. The scaling law vector calculus formulas are proposed as an efficiently mathematical tool to describe the Korcak scaling -law phenomena in heat transport system. [ABSTRACT FROM AUTHOR]

In this paper, all normal extensions of the minimal operator associated with the singular degenerate linear differential expressions for first order in the Hilbert space of vector-functions are investigated in terms of boundary values. As well, it analyses the spectrum parts of any normal extension. [ABSTRACT FROM AUTHOR]

We consider Galton-Watson branching processes with countable typeset X. We study the vectors q(A) = (qx(A))x∈X recording the conditional probabilities of extinction in subsets of types A ⊆ X, given that the type of the initial individual is x. We first investigate the location of the vectors q(A) in the set of fixed points of the progeny generating vector and prove that qx({x}) is larger than or equal to the xth entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for qx(A) < qx(B) for any initial type x and A, B ⊆ X. Finally, we develop a general framework to characterise all distinct extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions. [ABSTRACT FROM AUTHOR]

A novel Giant Magnetostrictive Actuator (GMA) experimental system with Fiber Bragg Grating (FBG) sensing technique and its modeling method based on data driven principle are proposed. The FBG sensors are adopted to gather the multi-physics fields' status data of GMA considering the strong nonlinearity of the Giant Magnetostrictive Material and GMA micro-actuated structure. The feedback features are obtained from the raw dynamic status data, which are preprocessed by data fill and abnormal value detection algorithms. Correspondingly the Least Squares Support Vector Machine method is utilized to realize GMA online nonlinear modeling with data driven principle. The model performance and its relative algorithms are experimentally evaluated. The model can regularly run in the frequency range from 10 to 1000 Hz and temperature range from 20 to 100°C with the minimum prediction error stable in the range from -1.2% to 1.1%. [ABSTRACT FROM AUTHOR]

The floating occupation molecular orbital complete active space configuration interaction (FOMOCASCI) method is quite promising for the study of nonadiabatic processes. Use of this method directly in nonadiabatic dynamics simulations has been limited by the lack of available first-order nonadiabatic coupling vectors. Here, an analytic formulation of these derivative coupling vectors is presented for FOMO-CASCI wavefunctions using a simple Lagrangian-based approach. The derivative coupling vectors are applied in the optimization of minimum energy conical intersections of an aqueously solvated model compound for the chromophore of the green fluorescent protein (including 100 water molecules). The computational cost of the FOMO-CASCI derivative coupling vector is shown to scale quadratically, O(N²), with system size and is applied to systems with up to 1000 atoms. [ABSTRACT FROM AUTHOR]

Can convergence of random variables, random vectors or, more generally, random elements of a metric space be studied using fuzzy integrals, in particular the seminormed integral? Among the positive results we present, it is shown that for large families of t-seminorms convergence in distribution and convergence in probability can be characterized via seminormed integrals. This suggests that, while fuzzy integrals were devised for working with non-additive fuzzy measures, some of their theoretical properties within the realm of probability measures can be comparable to those of the Lebesgue integral. [ABSTRACT FROM AUTHOR]

This paper addresses the exploitation of vector based contraction theory to synchronize the motion of agents in multiagent systems. The convergence analysis of the agents using this theory provides a flexible frame as the components of a vector distance norm follow less stringent conditions than classical contraction analysis. The key feature is that it encompasses the formulation of some comparison system, as a result, the problem of convergence analysis of a complex multiagent system reduces to the convergence analysis of a much simpler comparison system. Hence, despite the nonlinearities, the methods of designing consensus controls remain simple as compared to the existing literature. We address four different cases: 1) consensus of third‐order dynamics multiagent systems subjected to acceleration and input constraints; 2) consensus of multiagent systems with heterogeneous nodes under communication imperfections; 3) synchronization of multiagent systems with connected and disconnected switching topologies; 4) synchronization in networked systems with time‐varying couplings. Moreover, an effective synchronization control scheme for a class of multiagent systems with underactuated agent dynamics is developed using the method of aggregation of comparison systems. Finally, simulation results are illustrated that show the validity of the mathematical results. [ABSTRACT FROM AUTHOR]

Summary: This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non‐sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses. The analysis and understanding of how AC circuits operate in steady state are of a paramount importance for all the electrical engineers and practitioners around the world. Typically, lecturers of most undergraduate courses teach circuit theory using complex phasors, vector calculus, or linear algebra. However, these approaches have some important limitations in practice, which requires the development of strategies to improve teaching‐learning process related to AC circuit analysis. By formulating a new mathematical framework, the paper presents and discusses how GA can be of help in this approach. It is also described how the results obtained by using GA are validated using computer‐based simulations. It is highlighted how the proposed teaching methodology based on GA theory can be effective in helping students learn AC circuit analysis since it has several potential benefits derived from its simplicity, compactness, and use of easily identifiable geometrical concepts. [ABSTRACT FROM AUTHOR]

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green's identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green's identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus. [ABSTRACT FROM AUTHOR]

We consider four- and five-body problems with symmetrical masses (Caledonian problems). Families of periodic orbits originate from the collinear Schubart orbits. We present and discuss some of these periodic orbits. [ABSTRACT FROM AUTHOR]

Conics are relevant in many undergraduate course on classical physics, from free fall to Rutherford scattering. A pedagogical derivation of the intrinsic expressions of the curvature radius of different types of conic sections is presented. Our proof is carried out without resorting to any coordinate systems, but rather on using only elementary kinematic concepts together with basics of vector calculus and the very definition of conics. As a byproduct application of the present analysis, a simple and compact deduction of the Newton 'inverse square law' for gravitation from the three Kepler laws is also presented. [ABSTRACT FROM AUTHOR]

In this article, we suggest a new model for the heat-conduction problem by using the scaling-law vector calculus with Mandelbrot scaling law. The linear and non-linear scaling-law heat conduction equations are considered as analogues to the work of Fourier, Laplace, and Burgers. The obtained results are considered as typical examples to deal with the Mandelbrots scaling-law phenomena in heat transport system. [ABSTRACT FROM AUTHOR]

This paper addresses a non-traditional approach for the scaling-law fluid-flows described by fractal scaling-law vector calculus associated with the Mandelbrot scaling law. Their quantum equations were proposed to control the fluid-flows associated with the Mandelbrot scaling law. This gives a new insight into the descriptions for the scaling-law behaviors of the fluid-flows in the Mandelbrot scaling-law phenomena. [ABSTRACT FROM AUTHOR]

Measuring similarities/dissimilarities between atomic structures is important for the exploration of potential energy landscapes. However, the cell vectors together with the coordinates of the atoms, which are generally used to describe periodic systems, are quantities not directly suitable as fingerprints to distinguish structures. Based on a characterization of the local environment of all atoms in a cell, we introduce crystal fingerprints that can be calculated easily and define configurational distances between crystalline structures that satisfy the mathematical properties of a metric. This distance between two configurations is a measure of their similarity/dissimilarity and it allows in particular to distinguish structures. The new method can be a useful tool within various energy landscape exploration schemes, such as minima hopping, random search, swarm intelligence algorithms, and high-throughput screenings. [ABSTRACT FROM AUTHOR]

Summary: The future of nuclear energy in the energy mix faces a permanent scrutiny of safety aspects in conciliation with bridled costs, either fission‐ or fusion‐based. This affects to all the exciting milestones pursued in XXIst. To name few in the field of fission, the deployment of the IVth generation reactors is expected or the definitive solution to the radioactive wastes is sought. A mention apart is made to fusion technology, with ITER as the flagship project. It seeks a virtually infinite energy source, intrinsically safe and with reduced radioactive waste production with respect to fission the first commercial reactor. All these, and many other endeavors, share the operation of sophisticated devices in the presence of intense ionizing radiation fields. Humans and electronics must be protected to ensure safe and reliable performance, while shielding normally represents a large fraction of the budget. This involves nuclear analysis in the design phase to forecast the radiation conditions. The complexity of the simulation of 3D radiation fields that is computationally affordable nowadays is unprecedented. While sophistication in geometries and source definitions has become routine, the resulting complexity of these scalar fields makes their analysis increasingly difficult. The need of enhancement of the analysis techniques is evident today. Vector calculus is proposed following a physical interpretation of the field lines that boosts the analysis capabilities. It identifies the trajectories around which shielding is weakest in an automated errorless and effortless approach. Its power is illustrated with an example relevant to the ITER reactor. [ABSTRACT FROM AUTHOR]

This study deals with the problem of outlier censoring from the secondary data in a radar scenario, where the number of outliers is unknown. To this end, a procedure consisting of joint likelihood estimation and statistical model order selection (MOS) is proposed. Since the maximum likelihood (ML) estimation of the outlier subset requires to solve a combinatorial problem, an approximate ML (AML) method is employed to reduce the complexity. Therefore, to determine the number of outliers, different MOS criteria based on likelihood function are applied. At the analysis stage, the performance of the proposed methods is assessed based on simulated data. The results highlight that the devised algorithms exhibit satisfactory performance with efficient complexity at the same time. [ABSTRACT FROM AUTHOR]

Here, a novel off‐grid direction‐of‐arrival (DOA) estimation algorithm for nested arrays in the context of sparse recovery is proposed. The framework of existing off‐grid approaches is to correct the on‐grid parameter with a grid mismatch variable. From another perspective, we treat the direction‐of‐arrival deviation as the steering vector deviation equivalently, and the concept of steering vector uncertainty set is introduced to approximate the desired steering vector. Based on this idea, an optimization problem with respect to the uncertainty set for the purpose of steering vector reconstruction is formulated. After that, phase‐ambiguity resolving and phase‐difference approximation are orderly performed on the optimized uncertainty set to extract the off‐grid direction of arrival. Even though a coarse grid is given, simulation results are presented to demonstrate the performance advantage of the proposed direction‐of‐arrival estimation algorithm compared with several existing approaches. [ABSTRACT FROM AUTHOR]