Absence of Critical Mass Densities for a Vibrating Membrane

We study the dependence of the eigenvalues of a N-dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass.     

Low and high frequency approximations to eigenvibrations of string with double contrasts

We study eigenvibrations for inhomogeneous string consisting of two parts with strongly contrasting stiffness and mass density. In this work we treat a critical case for the high frequency approximations, namely the case when the order of mass density inhomogeneity is the same as the order of stiffness inhomogeneity, with heavier part being softer. The limit problem for high frequency approximations depends nonlinearly on the spectral parameter. The quantization of the spectral semiaxis is applied in order to get a close approximations of eigenvalues as well as eigenfunctions for the prime problem under perturbation.     

Refinement of the gravitational energy definition

We analyze the error arising in the well-known “energy” method for calculating the central density of a white dwarf with maximum possible mass in the framework of the theory of tetrad energy-momentum complexes of the gravitational field. We choose the preferable complex by comparing the central density calculated using each of the three complexes under study with its value obtained by numerically integrating the Einstein equations.     

A Generalized Single Linkage Method for Estimating the Cluster Tree of a Density

The goal of clustering is to detect the presence of distinct groups in a dataset and assign group labels to the observations. Nonparametric clustering is based on the premise that the observations may be regarded as a sample from some underlying density in feature space and that groups correspond to modes of this density. The goal then is to find the modes and assign each observation to the domain of attraction of a mode. The modal structure of a density is summarized by its cluster tree; modes of the density correspond to leaves of the cluster tree. Estimating the cluster tree is the primary goal of nonparametric cluster analysis. We adopt a plug-in approach to cluster tree estimation: estimate the cluster tree of the feature density by the cluster tree of a density estimate. For some density estimates the cluster tree can be computed exactly; for others we have to be content with an approximation. We present a graph-based method that can approximate the cluster tree of any density estimate. Density estimates tend to have spurious modes caused by sampling variability, leading to spurious branches in the graph cluster tree. We propose excess mass as a measure for the size of a branch, reflecting the height of the corresponding peak of the density above the surrounding valley floor as well as its spatial extent. Excess mass can be used as a guide for pruning the graph cluster tree. We point out mathematical and algorithmic connections to single linkage clustering and illustrate our approach on several examples. Supplemental materials for the article, including an R package implementing generalized single linkage clustering, all datasets used in the examples, and R code producing the figures and numerical results, are available online.     

Weak solutions for a hyperbolic system with unilateral constraint and mass loss

We consider isentropic gas dynamics equations with unilateral constraint on the density and mass loss. The γ and pressureless pressure laws are considered. We propose an entropy weak formulation of the system that incorporates the constraint and Lagrange multiplier, for which we prove weak stability and existence of solutions. The nonzero pressure model is approximated by a kinetic BGK relaxation model, while the pressureless model is approximated by a sticky-blocks dynamics with mass loss.     

Stability and convergence for a complete model of mass diffusion

We propose a fully discrete scheme for approximating a three-dimensional, strongly nonlinear model of mass diffusion, also called the complete Kazhikhov–Smagulov model. The scheme uses a C⁰ finite-element approximation for all unknowns (density, velocity and pressure), even though the density limit, solution of the continuous problem, belongs to H². A first-order time discretization is used such that, at each time step, one only needs to solve two decoupled linear problems for the discrete density and the velocity–pressure, separately.We extend to the complete model, some stability and convergence results already obtained by the last two authors for a simplified model where λ²-terms are not considered, λ being the mass diffusion coefficient. Now, different arguments must be introduced, based mainly on an induction process with respect to the time step, obtaining at the same time the three main properties of the scheme: an approximate discrete maximum principle for the density, weak estimates for the velocity and strong ones for the density. Furthermore, the convergence towards a weak solution of the density-dependent Navier–Stokes problem is also obtained as λ→0 (jointly with the space and time parameters).Finally, some numerical computations prove the practical usefulness of the scheme.     

Potential of a simple layer for a torus with density depending on the distance from a tangent plane to the center

Using a bulk potential, we construct a one-dimensional toroidal layer with corresponding density for the mass distribution. We determine the density of a simple layer, and also find the potential of a simple layer distributed along the surface of an elliptical torus of revolution with density depending on distance as measured along the normal between a tangent plane and the center of the torus.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 85–87, 1989.     

Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks

Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity conditions. We extend these arguments to discrete mass functions and their associated measures using the concept that the mass function can be embedded in a joint density function with continuous arguments. We give two different conditions, monotonicity and convergence on the unit sphere, both of which can make the discrete function embeddable. Our results are then applied to the preferential attachment network model, and we conclude that the joint mass function of in- and out-degree is embeddable and thus regularly varying.     

Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A₁, A₁⊕A₂, 3A₁, or A3,3⊕A₁. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A₁ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250].     

Cell polarisation model: The 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a decay rate for 1D-viscous compressible barotropic fluid equations

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H ¹ as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L ²-norm and H ¹-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions). Received December 20, 2000; accepted February 27, 2001.     

Long-time behavior of one-dimensional compressible Bingham flows

Barotropic flows of one-dimensional compressible Bingham fluids are considered. Long-time behavior of the corresponding initial-boundary problem is investigated. The uniform upper and lower bounds for the density and a decay of the kinetic energy are established. We admit a class of mass forces not considered for similar problems to Newtonian fluids. Under additional assumptions on the mass force, we achieve strong estimates for the solution (uniformly in time) and decays of the velocity and its derivatives. Received: April 14, 2004; revised: November 22, 2004     

Complexity penalized support estimation

We consider the estimation of the support of a probability density function with iid observations. The estimator to be considered is a minimizer of a complexity penalized excess mass criterion. We present a fast algorithm for the construction of the estimator. The estimator is able to estimate supports which consists of disconnected regions. We will prove that the estimator achieves minimax rates of convergence up to a logarithmic factor simultaneously over a scale of Hölder smoothness classes for the boundary of the support. The proof assumes a sharp boundary for the support.     

Numerical modeling of processes of heat and mass transfer in a capillary-porous material

We propose a method of solving a nonlinear heat and mass transfer boundary-value problem for a conductive drying process in a capillary-porous material. We study the effect of the characteristics of the material and the magnitude of the heat flux on the time change in temperature, bulk moisture saturation, and air density.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 149–154.     

Random creation and dispersion of mass

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge “in law” to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1–52). We show that solutions exist and that non-deterministic solutions are “smeared” continuous-state branching diffusions.     

Inhomogeneous beams that may possess a prescribed polynomial second mode

The following problem is posed and solved in this study: Reconstruct the stiffness variation of an inhomogeneous Bernoulli–Euler beam, whose second shape expressed as a polynomial function possesses a node at any pre-selected location. It is shown that for polynomial mode shapes there are regions of allowable position of the node for three specific cases studied in detail: (1) a beam with constant mass density; (2) linear mass density and (3) parabolically varying mass density.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks

We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation of mass condition is satisfied exactly. The mathematical model is formulated in terms of density, pressure, and mass flux variables, and as a result permits the use of a general equation of state to define the relation between the gas density and pressure for a given temperature. In a single pipe, the model represents the dynamics of the density by propagation of a non-linear wave according to a variable wave speed. We derive compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors. We compare our staggered grid method with an explicit operator splitting method and a lumped element scheme, and perform numerical experiments to validate the convergence order of the new discretization approach. In addition, we perform several computations to investigate the influence of non-ideal equation of state models and temperature effects on pipeline simulations with boundary conditions on various time and space scales.