Multidimensional Case Research Articles

Discontinuous Galerkin finite element method for the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes

Context. The study of the continuum radiative transfer problem inside circumstellar envelopes is both a theoretical and numerical challenge, especially in the frequency-dependent and multi-dimensional case. While approximate methods are easier to handle numerically, they often fail to accurately describe the radiation field inside complex geometries. For these cases, it is necessary to directly solve the radiative transfer equation numerically. Aims. We investigate the accuracy of the discontinuous Galerkin finite element method (DGFEM hereafter) applied to the frequency-dependent two-dimensional radiative transfer problem, and coupled with the radiative equilibrium equation. We next used this method in the context of axis-symmetric circumstellar envelopes. Methods. The DGFEM is a variant of finite element methods. It employs discontinuous elements and flux integrals along their boundaries, ensuring local flux conservation. However, as opposed to the classical finite element methods, the solution is discontinuous across element edges. We implemented this approach in a code and tested its accuracy by comparing our results with the benchmarks from the literature. Results. For all the tested cases, the temperatures profiles agree within one percent. Additionally, the emerging spectral energy distributions (SEDs) and images, obtained by ray-tracing techniques from the DGFEM emissivity, agree on average within 5% and 10%, respectively. Conclusions. We show that the DGFEM can accurately describe the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes. Consecutively the emerging SEDs and images are also well reproduced. The DGFEM provides an alternative method (other than Monte-Carlo methods for instance) for solving the radiative transfer equation, and it could be used in cases that are more difficult to handle with the other methods.

A Mathematical Aspect of Bloch’s Theorem

In this paper, by studying a class of 1-D Sturm-Liouville problems with periodic coefficients, we show and classify the solutions of periodic Schrödinger equations in a multidimensional case, which tells that not all the solutions are Bloch solutions. In addition, we also provide several properties of the solutions and quasimomenta and illustrate the relationship between bounded solutions and Bloch solutions.

Efficient finite element strategy using enhanced high-order and second-derivative-free variants of Newton's method

In this work, we propose a stable finite element approximation by extending higher-order Newton's method to the multidimensional case for solving nonlinear systems of partial differential equations. This approach relies solely on the evaluation of Jacobian matrices and residuals, eliminating the need for computing higher-order derivatives. Achieving third and fifth-order convergence, it ensures stability and allows for significantly larger time steps compared to explicit methods. We thoroughly address accuracy and convergence, focusing on the singular p-Laplacian problem and the time-dependent lid-driven cavity benchmark. A globalized variant incorporating a continuation technique is employed to effectively handle high Reynolds number regimes. Through two-dimensional and three-dimensional numerical experiments, we demonstrate that the improved cubically convergent variant outperforms others, leading to substantial computational savings, notably halving the computational cost for the lid-driven cavity test at large Reynolds numbers.

Existence and uniqueness of positive solution to a new class of nonlocal elliptic problem with parameter dependency

In this paper, we prove that under weak assumptions on the reaction terms and diffusion coefficients, a positive solution exists for a one-dimensional case and a positive radial solution to a multidimensional case of a nonlocal elliptic problem. Additionally, we establish the uniqueness of the solution, with the fixed point theorem being the primary tool employed. Our results are new and generalize several existing results.

The resultant method in higher dimensions

Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner [12], [11], became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for Rédei polynomials. We can prove a high dimensional analogue of the result of Szőnyi-Weiner [9], concerning the number of hyperplanes being skew to a point set of the space. We prove general results on “partial blocking sets”, using the tools we have developed.

A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds: II. Multi-dimensional case

We extend to multi-dimensions the work of [1], where new fully explicit kinetic methods were built for the approximation of linear and non-linear convection-diffusion problems. The fundamental principles from the earlier work are retained: (1) rather than aiming for the desired equations in the strict limit of a vanishing relaxation parameter, as is commonly done in the diffusion limit of kinetic methods, diffusion terms are sought as a first-order correction of this limit in a Chapman-Enskog expansion, (2) introducing a coupling between the conserved variables within the relaxation process by a specifically designed collision matrix makes it possible to systematically match a desired diffusion. Extending this strategy to multi-dimensions cannot, however, be achieved through simple directional splitting, as diffusion is likely to couple space directions with each other, such as with shear viscosity in the Navier-Stokes equations. In this work, we show how rewriting the collision matrix in terms of moments can address this issue, regardless of the number of kinetic waves, while ensuring conservation systematically. This rewriting allows for introducing a new class of kinetic models called regularized models, simplifying the numerical methods and establishing connections with Jin-Xin models. Subsequently, new explicit arbitrary high-order kinetic schemes are formulated and validated on standard two-dimensional cases from the literature. Excellent results are obtained in the simulation of a shock-boundary layer interaction, validating their ability to approximate the Navier-Stokes equations with kinetic speeds obeying nothing but a subcharacteristic condition along with a hyperbolic constraint on the time step.

Propagation phenomena for a nonlocal reaction-diffusion model with bounded phenotypic traits

In this paper we study the stability of cylinder front waves and the propagation of solutions of a nonlocal Fisher-type model describing the propagation of a population with nonlocal competition among bounded and continuous phenotypic traits. By applying spectral analysis and separation of variables we prove the spectral and local exponential stability of the cylinder waves with the noncritical speeds in some exponentially weighted spaces. By combining the detailed analysis with the spectral expansion and the special construction of sub-supersolutions, we further prove the uniform boundedness of the solutions and the global asymptotic stability of the cylinder waves for more general nonnegative bounded initial data, and prove that the spreading speeds and the asymptotic behavior of the solutions are determined by the decay rates of the initial data. Our results also extend some classical results on the stability of planar waves for Fisher-KPP equation to the nonlocal Fisher model in multi-dimensional cylinder case.

Construction method of fault function test case for Primary-Secondary fusion switch in distribution network

Test case design is the key to effective quality inspection of electrical equipment. To tackle the challenges of inadequate scenario coverage and disparate difficulty levels in the detection of single-phase ground faults in power distribution primary and secondary fusion switch, we propose a case construction method utilizing Latin hypercube sampling (LHS) based on NSGA-II double-objective optimization. Initially, the method investigates the primary factors influencing waveform detection difficulty and determines their respective weights through measurement and calculation, thus establishing a comprehensive model for calculating difficulty coefficients. Then, based on the dual objectives of uniformity and difficulty balance, we introduce a double-objective optimization LHS generation algorithm to construct a multi-dimensional detection cases of single-phase ground fault in power distribution primary and secondary fusion switch. Finally, the difficulty coefficient calculation model and the constructed test case undergo validation using actual test results, demonstrating a 43.59% efficiency improvement on the premise of ensuring the detection efficacy.

Multi-Dimensional Integral Transform with Fox Function in Kernel in Lebesgue-Type Spaces

This paper is devoted to the study of the multi-dimensional integral transform with the Fox H-function in the kernel in weighted spaces with integrable functions in the domain R+n with positive coordinates. Due to the generality of the Fox H-function, many special integral transforms have the form studied in this paper, including operators with such kernels as generalized hypergeometric functions, classical hypergeometric functions, Bessel and modified Bessel functions and so on. Moreover, most important fractional integral operators, such as the Riemann–Liouville type, are covered by the class under consideration. The mapping properties in Lebesgue-weighted spaces, such as the boundedness, the range and the representations of the considered transformation, are established. In special cases, it is applied to the specific integral transforms mentioned above. We use a modern technique based on the extensive use of the Mellin transform and its properties. Moreover, we generalize our own previous results from the one-dimensional case to the multi-dimensional one. The multi-dimensional case is more complex and needs more delicate techniques.

Size and Shape Selective Classification of Nanoparticles

As nanoparticle syntheses on a large scale usually yield products with broad size and shape distributions, the properties of nanoparticle-based products need to be tuned after synthesis by narrowing the size and shape distributions or via the removal of undesired fractions. The development of property-selective classification processes requires a universal framework for the quantitative evaluation of multi-dimensional particle fractionation processes. This framework must be applicable to any property and any particle classification process. We extended the well-known one-dimensional methodology commonly used for describing particle size distributions and fractionation processes to the multi-dimensional case to account for the higher complexity of the property distribution and separation functions. In particular, multi-dimensional lognormal distributions are introduced and applied to diameter and length distributions of gold nanorods. The fractionation of nanorods via centrifugation and by orthogonal centrifugal and electric forces is modeled. Moreover, we demonstrate that analytical ultracentrifugation with a multi-wavelength detector (MWL-AUC) is a fast and very accurate method for the measurement of two-dimensional particle size distributions in suspension. The MWL-AUC method is widely applicable to any class of nanoparticles with size-, shape- or composition-dependent optical properties. In addition, we obtained distributions of the lateral diameter and the number of layers of molybdenum disulfide nanosheets via stepwise centrifugation and spectroscopic evaluation of the size fractions.

Time-frequency analysis of nonlinear structures under nonstationary excitation by directly solving structural dynamic equation through absolute displacement

The time-frequency evolution of seismic ground motion energy under strong earthquakes, which can be described by the evolutionary power spectral density (EPSD) matrix, will amplify the most unfavorable responses of members in nonlinear structural systems. However, the existing general finite element software fails to reduce the dimension of the EPSD matrix and then decouple time from frequency in frequency domain analysis. This failure greatly limits the simulation and calculation of multi-dimensional nonlinear structures with multiple degrees of freedom by the stochastic vibration theory. In this study, the pseudo excitation method (PEM), which can direct solve structural dynamic equations using absolute displacement, is used to derive the self-spectral and cross-spectral densities of seismic ground motions in one-dimensional and multi-dimensional cases. In MATLAB, the EPSD matrix is transformed into a four-dimensional uniform modulation excitation matrix independent of the time variable. Nonlinear iteration is conducted and the precise integration method (PIM) is employed through a transient analysis module. Furthermore, a simple one-dimensional example is presented to verify the correctness of the proposed algorithm herein. Finally, as an example, we select important parameters of the coherence model at different supports of a half-through three-span concrete-filled steel tubular (CFST) tied-arch bridge in view of real engineering conditions, and calculate the time-varying power spectrum and variance of the responses of arch ribs, arch feet and the bridge deck system. The results show that the theoretical derivation and the algorithm constructed in this paper can simulate the time-delay effect of the nonlinear structural system. This proves the feasibility of general finite element software in solving and analyzing random responses to multi-dimensional and multi-support non-stationary excitation of nonlinear structures.

On the distributional fractional derivative: From unidimensional to multidimensional

In this paper, we use the generalized notions of Riemann–Liouville (fractional calculus with respect to a regular function) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, and the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.

Generalizations of the Jacobi identity to the case of the Lauricella function F D ( N )

ABSTRACT The paper considers the issue of constructing differential relations for the Lauricella hypergeometric function F D ( N ) . The formulas found give explicit expressions for the derivative of the product of the function F D ( N ) and some binomials through the product of other binomials and a combination of adjacent Lauricella functions whose parameters differ by 1 or − 1 . The derived differential relations generalize the Jacobi identity and other differential formulas known in the theory of the Gauss hypergeometric function on the multidimensional case. With special parameter values F D ( N ) , the right-hand side of such Jacobi-type formulas are simplified and have the form of a product of binomials and a polynomial. Such formulas can be used to transform a Cauchy-type integral to the form of the Schwartz–Christoffel integral and have an application to the Riemann–Hilbert problem with piecewise constant coefficients.

The meaning landscape of the concept of the total derivative in multivariable real analysis textbooks: an analysis based on a new model of meaning

AbstractIn this paper, we propose a novel conceptual framework tailored for modeling the meaning of mathematical concepts in university-level mathematics, addressing their rigorous nature and their relationships with related concepts as well as interpretations in various contexts. Within this framework, we present a model of meaning for the concepts of total differentiability and total derivative that provides a variety of possible interpretations and aspects. We then use the proposed model of meaning as a tool for analyzing three different textbooks for mathematics majors on the topic of multidimensional differentiability. Our paper is an example of a subject matter analysis of a topic in university mathematics carried out in a structured way. The model of meaning for total differentiability presented in this paper could inspire course design and analysis including the design of assignments and assessments for students. Moreover, it could serve as a valuable research tool for further analyses. For example, it could be used as a framework for analyzing courses taught or as a basis for developing a test instrument to assess students’ understanding. With our textbook analysis, we begin to examine the landscape of textbooks regarding differentiability concepts in the multidimensional case, shedding light on the diversity of meaning facets that are covered in the textbooks. These results could be useful for guiding instructors and learners in selecting and using textbooks for teaching and learning based on their respective needs.

HBMIRT: A SAS macro for estimating uni- and multidimensional 1- and 2-parameter item response models in small (and large!) samples

Item response theory (IRT) has evolved as a standard psychometric approach in recent years, in particular for test construction based on dichotomous (i.e., true/false) items. Unfortunately, large samples are typically needed for item refinement in unidimensional models and even more so in the multidimensional case. However, Bayesian IRT approaches with hierarchical priors have recently been shown to be promising for estimating even complex models in small samples. Still, it may be challenging for applied researchers to set up such IRT models in general purpose or specialized statistical computer programs. Therefore, we developed a user-friendly tool – a SAS macro called HBMIRT – that allows to estimate uni- and multidimensional IRT models with dichotomous items. We explain the capabilities and features of the macro and demonstrate the particular advantages of the implemented hierarchical priors in rather small samples over weakly informative priors and traditional maximum likelihood estimation with the help of a simulation study. The macro can also be used with the online version of SAS OnDemand for Academics that is freely accessible for academic researchers.

Dissipative structures for the system of Moore–Gibson–Thompson thermoelasticity in the whole space

We investigate the dissipative structure for the system of Moore–Gibson–Thompson (MGT) thermoelasticity in the whole space. To analyze the dissipative structure, it is very useful to rewrite the equations into a symmetric hyperbolic system and apply the so‐called stability condition. When we rewrite our system into the symmetric hyperbolic form in the multidimensional case, it is important to take the constraint conditions into account. Indeed, the stability condition with the constraint conditions guarantees the dissipative property for our system in some cases. In this paper, we introduce the stability condition with constraints for the general problem and apply this argument to the system of MGT thermoelasticity. Furthermore, we discuss the optimality of the decay estimates we obtain, together with the no regularity‐loss phenomenon.

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part II: The multidimensional case

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved.

On the hierarchy and fine structure of blowups and gradient catastrophes for multidimensional homogeneous Euler equation *

Blowups of derivatives and gradient catastrophes for the n-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blowups exhibit a fine structure in accordance with the admissible ranks of certain matrix generated by the initial data. Blowups form a hierarchy composed by n + 1 levels with the singularity of derivatives given by ∂ui/∂xk∼|δx|−(m+1)/(m+2) , m=1,…,n along certain critical directions. It is demonstrated that in the multi-dimensional case there is certain bounded linear superposition of blowup derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.

Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates

A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method φS=1.618… to super-quadratic φT=φS+1=2.618… and the quadratic convergence of the Newton method φN=2 to cubic φT=φN+1=3 in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases.

Aceptabilidad de un modelo de formulación diagnóstica de casos clínicos como estrategia educativa en una residencia de psiquiatría del niño y del adolescente

IntroductionThe aim of the study is to analyze the acceptability of an Integrative and Multidimensional Case Formulation Model implemented as an educational strategy in the first year of a child and adolescent psychiatry residency by a clinical supervisor within a one-way mirror room. Materials and methodsA qualitative methodology is used through content analysis of interviews conducted with residents of the program who had participated consecutively in clinical supervision between 2018 and 2021. For the analysis, 5 theoretical constructs derived from the Theory of Planned Behavior and Fazio's MODE Model are used: attitudes, subjective norm, perceived behavioral control, motivation and opportunity. ResultsTwenty-six thematic codes that influence acceptability emerged from the analysis: 8 attitudinal codes (e.g. acceptability is favored by the usefulness of the Model to gain theoretical insight of the cases but with a high amount of time required); 4 normative codes (e.g. the approval or disapproval of the model by scholars outside the clinical supervision); 7 perceived behavioral control codes (e.g. the perceived self-efficacy increases over time); 4 motivational codes (e.g. the desire to learn) and 3 opportunity codes (e.g. joint deliberation with peers). ConclusionThe results show valuable information for the development of educational strategies in the specialty residency program by approaching significant cognitive and affective factors which influence learning during the training periods. Aspects of the strategy that stand out as favoring the educational process are identified, along with other aspects that can be improved for better acceptability of the educational strategy.