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A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce index-row (and index-column) operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace [ Mem. Acad. Sciences Paris 1772] and Karlin (1968). In particular, for any square TN matrix A , these derived inequalities generalize – to every i th row of A and j th column of adj A – the classical Gantmacher–Krein fluctuating inequalities (1941) for i = j = 1. Furthermore, our index-row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices. The introduced index-row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying index-row/column operations add that these inequalities are offshoots of certain "complementary/higher" ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat–Gekhtman–Johnson [ Adv. Appl. Math. 2003] and later Skandera [ J. Algebraic Comb. 2004] (and revisited by Rhoades–Skandera [ Ann. Comb. 2005, J. Algebra 2006]). [ABSTRACT FROM AUTHOR]

Sensors commonly used to record sub-bandage pressures may be less reliable than is commonly believed. The techniques employed for the calibration or validation of these devices can sometimes produce a highly inaccurate prediction of their performance in vivo. In this review, the various calibration methods referenced in the literature are briefly described and the limitations of each discussed. The significance of these limitations is illustrated by reference to the results of published studies cited in a systematic review which report widely different values for the accuracy of one particular device, the Kikuhime pressure measuring system (ZiboCare, Denmark). The use of inappropriate calibration techniques therefore casts serious doubt upon the pressure values quoted historically in many published scientific and clinical studies in which these particular instruments have been employed. It is proposed that an international standard method be developed to ensure that all sensors intended for the measurement of sub-bandage pressure are characterised using a validated, clinically relevant technique. A programme of education is also required to correct some of the widely held misconceptions often expressed in the literature concerning the use of the Laplace equation to calculate sub-bandage pressure. [ABSTRACT FROM AUTHOR]

This research paper investigates the thermal stratification and chemical reaction effects on MHD Flow through oscillatory vertical plate in a porous medium with temperature variation and exponential mass diffusion. Through the application of the Laplace transform method, the paper derives analytical solutions that precisely depict the physical dynamics of the flow. The investigation utilizes sophisticated mathematical models to scrutinize the complex dynamics between Magnetohydrodynamics (MHD) and convective movements, considering a range of conditions involving temperature fluctuations and exponential rates of mass diffusion. A pivotal finding from this research is the detailed comparison between the outcomes of thermal stratification and those observed in environments lacking such stratification. It is observed that the implementation of stratification within the flow leads to a more rapid achievement of equilibrium or steady-state conditions. [ABSTRACT FROM AUTHOR]

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This paper re-analyzes the nonlinear Susceptible–Infected–Recovered (SIR) model using a hybrid approach based on the Laplace–Padé technique. The proposed approach is successfully applied to extract several analytic approximations for the infected and recovered individuals. The domains of applicability of such analytic approximations are addressed. In addition, the present results are validated through various comparisons with the Runge–Kutta numerical method. The obtained analytical results agree with the numerical ones for a wide range of numbers of contacts featured in the studied model. The efficiency of the present analysis reveals that it can be implemented to deal with other systems describing real-life phenomena. [ABSTRACT FROM AUTHOR]

The study explores the velocity of Casson fluid that is time-independent over an exponentially infinite isotherm vertical permeable sheet. The impact of magnetohydrodynamic (MHD) with Casson flow over the permeability sheet is examined. In contrast, thermal radiation and heat sink parameters have been incorporated. This study's primary goal is to determine the significance of thermal radiation on Casson flow with MHD using an analytical solution over a permeable sheet. The flow of the fluid occurs above the sheet when y is greater than zero, and the sheet extends far away in the x -direction. The model of governing equations is reduced by applying a suitable set of dimensionless parameters. These dimensionless systems of equations are solved through the Laplace transformation method. The impacts of various variables over velocity, temperature, concentration, skin friction, and Nuseelt number are scrutinized. These variables contain magnetic field M , Casson fluid parameter β , Ghroshof number Gr , modified Grashof number Gc , Prandtl number Pr , thermal radiation Rd and Scimdth number Sc. These plots are sketched for the considerable magnitude of these variables through the Mathematica Software, and these plots are discussed in detail. Results show that the increasing value of M reduces the fluid velocity but velocity of fluid is enhanced with larger values of Gr , Gc and β. The impact of Skin friction and Nusselt number is elaborated by tabular outlined. The study explores the velocity of Casson fluid that is time-independent over an exponentially infinite isotherm vertical permeable sheet. The impact of magnetohydrodynamic (MHD) with Casson flow over the permeability sheet is examined. We used Laplace transformation. Rising value casson parameters enhanced the fluid velocity. [ABSTRACT FROM AUTHOR]

A new method for the solution of Cauchy problems involving linear elliptic operators is introduced. Starting with an initial guess for the missing boundary condition, the algorithm obtains corrections to the assumed value at every iteration. For the updating part, it uses a sampling function to relate the error field to the unknown boundary condition. It uses the data evaluated at points on the boundary. A filtering technique is also introduced which is tailored to the specific problem at hand. The proposed method requires the Green's function for the linear operator. A number of systems including Laplace, Helmholtz and biharmonic equations are studied. A number of numerical examples are used to show the applicability of the method in the presence of noise. [ABSTRACT FROM AUTHOR]

Owing to the assembly state changes during aircraft assembly processes, assembly force-deformation problem occurs. To obtain the structure shape in the product assembly process efficiently and accurately, a three-dimensional (3D) mapping technology for the structural deformation during the aircraft assembly process is proposed combined with a fiber Bragg grating (FBG) optical fiber sensor and binocular vision measurement system. First, this study established a curvature transformation model using optical fiber monitoring data, obtained the 3D spatial deformation of the product, and completed the unification of the optical fiber wavelength change and spatial 3D point coordinate heterogeneous data. Second, a mesh deformation optimization algorithm based on point-cloud optimization was established. Subsequently, the deformation effects of four mesh deformation models were compared to verify the feasibility and accuracy of HEC-Laplace, and the 3D mapping of the product structure shape in the assembly process was realized. Finally, a cantilever wing model was used to verify the deformation of different loading modes. The results show that the product structure changes can be accurately obtained through the proposed technology, thereby improving the accuracy control and overall assembly quality in the aircraft assembly process and providing a theoretical basis for intelligent aircraft assembly. [ABSTRACT FROM AUTHOR]

Edge detection of an image is needed to obtain information related to the size and shape of an image. There are numerous methods for detecting edges, including the Prewitt, Laplace, and Kirsch operators. Each edge detection method has different performance and results. Therefore, this study aims to analyze the performance comparison of the Prewitt, Laplace, and Kirsch operators. The analysis process is carried out using MSE, PSNR and Image Contrast values. Based on the experiments that have been carried out, the best edge detection is produced by the Prewitt operator. The average MSE and PSNR values obtained were 4.63 and 41.79 dB. The Laplace operator is good in the contrast value of 17.77. However, the contrast value only serves as a supporting parameter to clarify the differences in the results of each edge detection operator. So it can be concluded that the Prewitt edge detection method is the best method among the other two methods. [ABSTRACT FROM AUTHOR]

Apresenta-se neste trabalho uma tradução do artigo “Mémorie sur la théorie de la Lune” de Pierre-Simon Laplace. Publicado originalmente em 1800, em francês, perante o Institut national des Sciences et Arts e anexado posteriormente em 1802 ao volume III, livro VII, do Traité de mécanique céleste e republicado em edições posteriores das Oeuvres de Laplace a pedido do rei francês Louis Philippe I em 1842. O artigo de Laplace é reconhecido por utilizar o cálculo diferencial com elegância e possui inúmeras implicações na Astronomia, além de grande importância nos subsequentes trabalhos dele, tais como o estudo da forma do planeta Terra e o movimento da Lua.

In this work, the solution of Poisson's equation in a semi-disc under a Dirichlet boundary condition at the base and a Neumann boundary condition on the circumference is calculated. The solution is determined in terms of Green's function, which is calculated in two ways, by the method of images and by solving its equation. In the particular case of Laplace's equation, it is presented a second way to solve it, which uses separation of variables and a Fourier transform.

Let G be a compact connected Lie group of dimension m. Once a bi-invariant metric on G is fixed, left-invariant metrics on G are in correspondence with m × m positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on G in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets S of the space of left-invariant metrics M on G such that there exists a positive real number C depending on G and S such that λ1(G,g)diam(G,g)2 ≤ C for all g ∈ S . The existence of the constant C for S = M is the original conjecture. [ABSTRACT FROM AUTHOR]

The econometric framework of the contemporaneous asset pricing model used by Theodossiou and Savva and Savva and Theodossiou to investigate the relationship between risk and expected returns in financial markets is generalized to a class of two-sided, asymmetry separable distributions. The latter class of distributions includes as special cases the skewed forms for the normal, Student's t, Laplace, generalized error, generalized t, logistic and generalized type III logistic. All distributions document a positive and statistically significant relationship between risk and expected returns. A comparison of their data fitting ability shows that the generalized t distribution provides the best overall results. [ABSTRACT FROM AUTHOR]

Our problem is to express intertwined existential, epistemological and ethical aspects of a scientific method for informing action under conditions of high stakes and urgency. Addressing situations where, taking Descartes at his word, "...the pressure of things to be done does not allow us to stop and make a meticulous check," we set out a framework for multi-actor multi-criteria deliberative evaluation for building knowledge partnerships within wicked problems of action. With the metaphor of Borges' Library of Babel, we suggest a cataloguing process whose effect is to signal, or assign, the Qualities of a knowledge proposition along several axes of the context of their mobilisation. The protagonists in a controversy are actors de facto in a KQA deliberation process, contributors (willy-nilly) to a collective resource of putative knowledge claims, participants in Borges' eternally-unfinished Congress of the World. Finally, with urgency and high stakes is associated the passage - repeated, traumatic, ineluctable - through dilemmas. These passages, irreducible to rationalisation, are often painful and always transformative. Adopting a third Borges metaphor, The Garden of Forking Paths, we argue for a reflexivity accepting the ethical complexity of our status as vulnerable actors engaged hastily, in necessarily value-laden terms, in matters of life-and-death. • A structured deliberative approach to the mobilisation and evaluation of knowledge propositions, for Post-Normal conditions. • History of science context (from Descartes to Maxwell), in synthesis of Natural History, deterministic chaos and complexity. • Knowledge quality as fitness for purpose, triangulation [What, Why, for Whom?] integrating scientific and societal KQA. • A review of different published formulations since the 1990s of "matrixial" approaches to knowledge-action problems. • Sustained use of metaphors from Jorge Luis BORGES, to evoke dilemmas and impossibilities of working within wickedness. [ABSTRACT FROM AUTHOR]

Predicting the fate and transport of contaminants in soil or groundwater systems using analytical or numerical models is crucial for environmental researchers. While the analytical models are a flexible approach to quantifying the subsurface contamination and remediation because they are non-susceptible to numerical dispersions, economical and handier; two-dimensional analytical models that describe a bilateral flow coupled with both sink and decay factors are rarely reported. Motivated by the case of a non-bare soil ridge with constant point-solute source lying internally but parallel to the longitudinal flow direction, a (2 + 1) dimensional Advection-Diffusion-Reaction Equation (ADRE) of bilateral flow coupled with the linear sorption, decay, and sink is formulated to model the transport of dissolved solute in a homogenous and isotropic non-fractured porous medium. Then, a brief review of exact and analytical methods for solving the ADRE is conducted to establish the right solution methods. The Repeated Integral Transformation Method (RITM) is employed to derive the approximate analytical solutions for the formulated model, maintaining the model's original terms for the efficient sensitivity analysis. The RITM uses Laplace and Fourier transforms with a wide range of computed results. We compare the approximate solutions with numerical simulations in COMSOL to verify the accuracy of the approximate analytical models. Then, the application of the solutions is demonstrated through a systematic analysis of the effect on solute transport of advection-diffusion, reaction, sorption, retardation, sink, and pore water velocity. Results show that the presence of sink, mimicked by plant-root uptake activity, and decay tend to reduce the solute concentration in the medium. While both the retardation and sorption factors affect the movement of the dissolved solutes, water content and pore-water velocity promote the spreading of dissolved solutes. Solute concentration in the medium increases at low Peclet numbers, signifying the influence of diffusive coefficients. The current proposed RITM-based solutions can characterize contamination in the soil, and should be useful to environmental researchers.

Sabe-se que muitas vezes a Física não é uma das matérias favoritas das pessoas durante o período escolar. Tratando de assuntos que muitas vezes parecem distantes da realidade dos estudantes essa disciplina muitas vezes acaba pecando no que tange a sua relação com o mundo e com a cultura humana de forma geral. Este ensaio busca discutir como a Física é mostrada na literatura e para tal, usa a obra “Memórias do Subsolo” de Fiódor Dostoiévski para demonstrar que a literatura de certa forma condiciona a visão do leitor ao usar da Física como metáfora em seus escritos. Na obra em questão, o determinismo de Laplace aparece como visão de mundo para o personagem anônimo de Dostoiévski, fazendo com que se estabeleça determinada relação entre o saber científico e o homem.

Since uncertainties in many physical systems have impulsive properties poorly modeled by Gaussian distributions, heavier-tailed distributions, such as Laplace, may be used to improve model characteristics. From insights obtained through development of the scalar Laplace estimator, an algorithm is determined for the vector-state case. For a discrete-time vector linear system with scalar additive Laplace-distributed process and measurement noises, the a priori and a posteriori conditional probability density functions (pdfs) of the system states given the measurement history are propagated recursively and in closed form. The conditional pdfs are composed of signs and absolute values of affine functions, and a basis composed of signs of affine functions is constructed to simplify their representation. The a posteriori conditional mean and variance are derived analytically from the conditional pdf using characteristic functions. Generalization to independent vector measurement and process noises is straightforward. From the general method for deriving Laplace estimators in n-dimensions, a two-dimensional minimum variance estimator is explicitly developed, and a simulation is presented. [ABSTRACT FROM AUTHOR]

Using magnetic resonance (MR) images to evaluate changes in the shape of the hippocampus has been an active research topic. This paper presents a new shape analysis approach to quantify and visualize deformations of the hippocampus in epilepsy. The proposed method is based on Laplace–Beltrami (LB) eigenvalues and eigenfunctions as isometric invariant shape features, and thus, the procedure does not require any image registration. In addition to the LB-based shape features, total hippocampal volume and surface area are calculated using manually segmented images. Theses shape and volumetric descriptors are used to distinguish the patients with temporal lobe epilepsy (TLE) (N = 55) from healthy control subjects (N = 12, age = 32.2 ± 9.1, sex (M/F) = 6/6) and patients with right TLE (N = 26, age = 45.1 ± 11.0, sex (M/F) = 9/17) from left TLE (N = 29, age = 45.4 ± 11.9, sex (M/F) = 10/19). Experimental results illustrate the usefulness of the proposed approach for the diagnosis and lateralization of TLE with 93.0% and 86.4% of the cases, respectively. Moreover, the proposed method outperforms the volumetric analysis in terms of both sensitivity (94.9% vs. 88.1%) and specificity (83.3% vs. 50.0%) of the lateralization. The analysis of local hippocampal thickness variations suggests significant deformation in both ipsilateral and contralateral hippocampi of epileptic patients, while there were no differences between right and left hippocampi in controls. It is anticipated that the proposed method could be advantageous in the presurgical evaluation of patients with drug-resistant epilepsy; however, further validation of the method using a larger dataset is required. [ABSTRACT FROM AUTHOR]

• After introducing the Laplacian strategy and Nelder-Mead simplex mechanism to the basic hunger games search, a novel LNMHGS algorithm is developed. • The proposed LNMHGS compares against the original HGS and other algorithms to optimize unknown parameters of static PV systems. • The LNMHGS method is used to extract SDM and DDM models' parameters on ST40, KC200GT and SM55 datasheet under different environmental conditions. • The results show the fast convergence and stable convergence of the proposed LNMHGS algorithm to identify parameters for PV models. Photovoltaic (PV) technology can convert solar energy to electric power, which is an essential tool for future years. Subsequently, several static solar PV models have been designed to simulate the current in a PV cell. However, the modeling process of PV systems requires extracting the unknown parameters of these cells, which can be modeled as an optimization problem. However, this is a very challenging task as it is multimodal and nonlinear. In this case, a Laplacian Nelder-Mead hunger games search (HGS) method, denoted as LNMHGS, is proposed for parameter extraction of static solar PV models. It realizes the equilibrium between exploitation and exploration by introducing the Laplacian strategy and Nelder-Mead simplex mechanism to hunger name search. LNMHGS compares against many recent methods and stands out by efficiently estimating static PV models' parameters. The simulation outcomes show that the root mean square error (RMSE) values and standard deviation offered by LNMHGS are smaller than other algorithms. Meanwhile, LNMHGS obtains the best performance both in different light conditions and at different temperature conditions. Therefore, LNMHGS is a promising, reliable, and feasible alternative for optimizing the parameters for PV systems. [ABSTRACT FROM AUTHOR]

This paper re-analyzes the nonlinear Susceptible–Infected–Recovered (SIR) model using a hybrid approach based on the Laplace–Padé technique. The proposed approach is successfully applied to extract several analytic approximations for the infected and recovered individuals. The domains of applicability of such analytic approximations are addressed. In addition, the present results are validated through various comparisons with the Runge–Kutta numerical method. The obtained analytical results agree with the numerical ones for a wide range of numbers of contacts featured in the studied model. The efficiency of the present analysis reveals that it can be implemented to deal with other systems describing real-life phenomena.

Generalized Linear Mixed Model (GLMM) is a framework that has a response variable, fixed effects, and random effects. The response variable comes from an exponential family, whereas random effects have a normal distribution. Estimating parameters can be calculated using the maximum likelihood method using the Laplace approach or the Gauss-Hermite Quadrature (GHQ) approach. The purpose of this study was to identify factors that trigger student's interest to continue studying at Universitas Syiah Kuala (USK) using both techniques. The GLMM is suitable for the data because the variable response has a Bernoulli distribution, and the random effects are assumed to be having a normal distribution. Also, the model helps identify the relationship between the dependent variable and the predictors. This study utilizes data from six high schools in Banda Aceh city drawn using a two-stage sampling technique. Stage 1, we randomly chose six out of sixteen public senior high schools in Banda Aceh. Stage 2, we selected students from each school from four different major classes. The GLMM model includes one binary response variable, five numerical fixed-effects, and two random effects. The response variable is the interest of high school students to continue study at USK (yes or no). The five fixed effects in the model including scores of collaboration (C), Action (A), Emotion (E), Purposes (P), and Hope (H). Finally, the random effects are schools (S) and majors (M). In this study, both Laplace and GHQ techniques produce identical results. The predictors that can explain student interest are A, E, and H. These predictors have a positive effect. The random effects of schools and majors are not significantly different from zero. The model with three significant predictors is better than the complete predictor model.

Owing to the assembly state changes during aircraft assembly processes, assembly force-deformation problem occurs. To obtain the structure shape in the product assembly process efficiently and accurately, a three-dimensional (3D) mapping technology for the structural deformation during the aircraft assembly process is proposed combined with a fiber Bragg grating (FBG) optical fiber sensor and binocular vision measurement system. First, this study established a curvature transformation model using optical fiber monitoring data, obtained the 3D spatial deformation of the product, and completed the unification of the optical fiber wavelength change and spatial 3D point coordinate heterogeneous data. Second, a mesh deformation optimization algorithm based on point-cloud optimization was established. Subsequently, the deformation effects of four mesh deformation models were compared to verify the feasibility and accuracy of HEC-Laplace, and the 3D mapping of the product structure shape in the assembly process was realized. Finally, a cantilever wing model was used to verify the deformation of different loading modes. The results show that the product structure changes can be accurately obtained through the proposed technology, thereby improving the accuracy control and overall assembly quality in the aircraft assembly process and providing a theoretical basis for intelligent aircraft assembly.

From the physical, mathematical, and conceptual points of view, Roger Joseph Boscovich's original 1758 formulation of the principle of physical determinism and Pierre-Simon Laplace's later 1814 rendering of the principle appear identical. Both Boscovich and Laplace expressed epistemological and ontological commitments to the ideas that (a) the present state of the universe is a necessary effect of its preceding states and the immediate cause of the states that follow, and (b) that knowledge of the present state of the universe allows the full knowledge of its future and past states. However, their respective metaphysical readings of the principle of determinism radically differ. Contrary to the secularized reading of the principle given by Laplace, Boscovich's reading of the principle strongly affirms the natural theology he inherited from Newton. In particular, in addition to enhancing the traditional teleological and cosmological arguments for the existence of God based on probability theory and his specific views of the nature of infinity, Boscovich also offered an innovative version of the cosmological argument based on the very principle of physical determinism he was the first to formulate. The purpose of this article is to analyze the content and context of these metaphysical differences between Boscovich and Laplace in respect to the principle of physical determinism, which nevertheless leave intact the scientific content of the principle. [ABSTRACT FROM AUTHOR]

In this paper we shall introducd the Laplce transforms, develop its fundamental properties and illustrate its use to solve linear differential equations with bounday conditions. The paper gives theory, few examples worked on properties and application of Laplace transform. [ABSTRACT FROM AUTHOR]

Alternative specifications of univariate asymmetric Laplace models are described and investigated. Amore general mixture model is then introduced. Bivariate extensions of these models are discussed in some detail, with particular emphasis on associated parameter estimation strategies. Multivariate versions of the models are briefly introduced. [ABSTRACT FROM AUTHOR]

This contribution describes a Decision Tree intended to guide the selection of statistical models and data reduction procedures in key comparisons (KCs). The Decision Tree addresses a specific need of the Inorganic Analysis Working Group (IAWG) of the Consultative Committee (CC) for Amount of Substance, Metrology in Chemistry and Biology (CCQM), of the International Committee for Weights and Measures (CIPM), and it is likely to address similar needs of other working groups and consultative committees. Because the portfolio of KCs previously organized by the CCQM-IAWG affords a full range of opportunities to demonstrate the capabilities of the Decision Tree, the majority of the illustrative examples of application of the Decision Tree are from this working group. However, the Decision Tree is widely applicable in other areas of metrology, as illustrated in examples of application to measurements of radionuclides and of the efficiency of a thermistor power sensor. The Decision Tree is intended for use after choices will have been made about the measurement results that qualify for inclusion in the calculation of the key comparison reference value (KCRV), and about the measurement results for which degrees of equivalence should be produced. Both these choices should be based on substantive considerations, not on purely statistical criteria. However, the Decision Tree does not require that the measurement results selected for either purpose be mutually consistent. The Decision Tree should be used as a guide, not as the sole and autonomous determinant of the model that should be selected for the measurement results obtained in a KC, or of the procedure that should be employed to reduce these results. The scientists running the KCs ultimately have the freedom and responsibility to make the corresponding choices that they deem most appropriate and that best fit the purpose of each KC. The Decision Tree involves three statistical tests, and comprises five terminal leaves, which correspond to as many alternative ways in which the KCRV, its associated uncertainty, and the degrees of equivalence (DoEs) may be computed. This contribution does not purport to suggest that any of the KCRVs, associated uncertainties, or DoEs, presented in previously approved final reports issued by working groups of the CCs should be modified. Neither do the alternative results question existing, demonstrated calibration and measurement capabilities (CMCs), nor do they support any new CMCs. [ABSTRACT FROM AUTHOR]

We study the nonlinear eigenvalue problem for quasilinear operator Lu = −∆pu+ V |u| p−2u and we give some geometric estimates of the first eigenvalue of the operator L on closed Riemannian manifold with some geometric conditions as integral curvature condition. [ABSTRACT FROM AUTHOR]

Multidimensional correlation spectroscopy is emerging as a novel MRI modality that is well suited for microstructure and microdynamic imaging studies, especially of biological specimens. Conventional MRI methods only provide voxel‐averaged and mostly macroscopically averaged information; these methods cannot disentangle intra‐voxel heterogeneity on the basis of both water mobility and local chemical interactions. By correlating multiple MR contrast mechanisms and processing the data in an integrated manner, correlation spectroscopy is able to resolve the distribution of water populations according to their chemical and physical interactions with the environment. The use of a non‐parametric, phenomenological representation of the multidimensional MR signal makes no assumptions about tissue structure, thereby allowing the study of microscopic structure and composition of complex heterogeneous biological systems. However, until recently, vast data requirements have confined these types of measurement to non‐localized NMR applications and prevented them from being widely and successfully used in conjunction with imaging. Recent groundbreaking advancements have allowed this powerful NMR methodology to be migrated to MRI, initiating its emergence as a promising imaging approach. This review is not intended to cover the entire field of multidimensional MR; instead, it focuses on pioneering imaging applications and the challenges involved. In addition, the background and motivation that have led to multidimensional correlation MR development are discussed, along with the basic underlying mathematical concepts. The goal of the present work is to provide the reader with a fundamental understanding of the techniques developed and their potential benefits, and to provide guidance to help refine future applications of this technology. [ABSTRACT FROM AUTHOR]

In this article, we present Bembel, the C++ library featuring higher order isogeometric Galerkin boundary element methods for Laplace, Helmholtz, and Maxwell problems. Bembel is compatible with geometries from the Octave NURBS package, and provides an interface to the Eigen template library for linear algebra operations. For computational efficiency, it applies an embedded fast multipole method tailored to the isogeometric analysis framework and a parallel matrix assembly based on OpenMP.

We developed Variational Laplace for Bayesian neural networks (BNNs), which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is the log-likelihood plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a posteriori inference and standard sampling-based variational inference, despite using the same variational approximate posterior. Finally, we emphasize the care needed in benchmarking standard VI, as there is a risk of stopping before the variance parameters have converged. We show that early-stopping can be avoided by increasing the learning rate for the variance parameters.

To improve the reliability and accuracy of dynamic machine in design process, high precision and efficiency of numerical computation is essential means to identify dynamic characteristics of mechanical system. In this paper, a new computation approach is introduced to improve accuracy and efficiency of computation for coupling vibrating system. The proposed method is a combination of piecewise constant method and Laplace transformation, which is simply called as Piecewise-Laplace method. In the solving process of the proposed method, the dynamic system is first sliced by a series of continuous segments to reserve physical attribute of the original system; Laplace transformation is employed to separate coupling variables in segment system, and solutions of system in complex domain can be determined; then, considering reverse Laplace transformation and residues theorem, solution in time domain can be obtained; finally, semi-analytical solution of system is given based on continuity condition. Through comparison of numerical computation, it can be found that precision and efficiency of numerical results with the Piecewise-Laplace method is better than Runge-Kutta method within same time step. If a high-accuracy solution is required, the Piecewise-Laplace method is more suitable than Runge-Kutta method. [ABSTRACT FROM AUTHOR]

In this article, we study the evolution, monotonicity for the first eigenvalue of the clamped plate on closed Riemannian manifold along the Ricci flow. We prove that the first nonzero eigenvalue is nondecreasing under the Ricci flow under certain geometric conditions and find some applications in 2-dimensional manifolds. [ABSTRACT FROM AUTHOR]

We study strongly harmonic functions in Carnot–Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that appropriate spherical harmonic polynomials in ℍ1 are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples. [ABSTRACT FROM AUTHOR]

Let M be an n-dimensional closed Riemannian manifold with metric g, dμ = e-ø(x)dv be the weighted measure and Δp, ø be the weighted p-Laplacian. In this article we will investigate monotonicity for the first eigenvalue problem of the weighted p-Laplace operator act- ing on the space of functions along the Ricci-Bourguignon ow on closed Riemannian manifolds. We find the first variation formula for the eigen-values of the weighted p-Laplacian on a closed Riemannian manifold evolving by the Ricci-Bourguignon ow and we obtain various monotonic quantities. At the end we find some applications in 2-dimensional and 3-dimensional manifolds and give an example. [ABSTRACT FROM AUTHOR]