ESTIMATES OF DERIVATIVES OF (LOG) DENSITIES AND RELATED OBJECTS

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

Using Freshwater Heads to Analyze Flow Directions in Saline Aquifers of the Pingtung Plain, Taiwan

The hydraulic head is the most important parameter for the study of groundwater. However, a head measured from observation wells containing groundwater of variable density should be corrected to a reference density (e.g., a freshwater head). Some previous case studies have used unknown density hydraulic heads for calibrating flow models. Errors arising from the use of observed hydraulic head data of unknown density are, therefore, likely one of the most overlooked issues in flow simulations of seawater intrusion. Here, we present a case study that uses the freshwater head, instead of the observed hydraulic head, to analyze the flow paths of saline groundwater in the coastal region of the Pingtung Plain, Taiwan. Out of a total of 134 observation wells within the Pingtung Plain, 19 wells have been determined to be saline, with Electric Conductivity (EC) values higher than 1500 μS/cm during 2012. The misuse of observed hydraulic heads causes misinterpretation of the flow direction of saline groundwater. For such saline aquifers, the determination of a freshwater head requires density information obtained from an observation well. Instead of the purging and sampling method, we recommend EC logging using a month interval. Our research indicates that EC values within an observation well within saline aquifers vary not only vertically but also by season.

Density Fluctuations Yield Distinct Growth and Fitness Effects in Single Bacteria

Single-cells grow by increasing their biomass and size. Here, we report that while mass and size accumulation rates of single Escherichia coli cells are exponential, their density fluctuates during growth. As such, the rates of mass and size accumulation of a single-cell are generally not the same, but rather cells differentiate into increasing one rate with respect to the other. This differentiation yields a previously unknown density homeostasis mechanism, which we support mathematically. Further, growth differentiation challenges ongoing efforts to predict single-cell reproduction rates (or fitness-levels), through the accumulation rates of size or mass. In contrast, we observe that density fluctuations can predict fitness, with only high fitness individuals existing in the high density fluctuation regime. We detail our imaging approach and the invisible microfluidic arrays that critically enabled increased precision and throughput. Biochemical production, infections, and natural communities start from few, growing, cells, thus, underscoring the significance of density-fluctuations when considering non-genetic variability.

First Principles Density Functional Theory Prediction of the Crystal Structure and the Elastic Properties of Mo2ZrB2 and Mo2HfB2

The Molybdenum rich ternary alloys Mo-M-B (M = Zr, Hf) contain, next to the Mo solid solution (bcc Mo with small amounts of Zr or Hf as substitutional atoms), the binary borides Mo2B, MB and MB2. Recently, it was found that there is also ternary Mo2MB2, but the crystal structure and further properties are currently unknown. Density functional theory (DFT) calculations were used not only to predict the crystal structure of the Mo2MB2 phases, but also to estimate the isotropic and anisotropic elastic properties like bulk, shear and Young’s modulus, as well as the Vickers hardness of these new borides. Several known crystal structures that fulfill the criterion of the chemical composition were investigated, and the AlMn2B2 type structure seems to be the most stable crystal structure for Mo2HfB2 and Mo2ZrB2 as there are no signs of electronic or dynamic instability. Regarding the elastic properties, it was found that Mo2HfB2 shows higher elastic moduli and is less elastically anisotropic than Mo2ZrB2.

Minimum distance histograms with universal performance guarantees

We present a data-adaptive multivariate histogram estimator of an unknown density f based on n independent samples from it. Such histograms are based on binary trees called regular pavings (RPs). RPs represent a computationally convenient class of simple functions that remain closed under addition and scalar multiplication. Unlike other density estimation methods, including various regularization and Bayesian methods based on the likelihood, the minimum distance estimate (MDE) is guaranteed to be within an $$L_1$$ L 1 distance bound from f for a given n, no matter what the underlying f happens to be, and is thus said to have universal performance guarantees (Devroye and Lugosi, Combinatorial methods in density estimation. Springer, New York, 2001). Using a form of tree matrix arithmetic with RPs, we obtain the first generic constructions of an MDE, prove that it has universal performance guarantees and demonstrate its performance with simulated and real-world data. Our main contribution is a constructive implementation of an MDE histogram that can handle large multivariate data bursts using a tree-based partition that is computationally conducive to subsequent statistical operations.

Oblique Derivative Problem Solution for the Lavrentyev-Bitsadze Equation in a Half-Plane

The paper solves the boundary value problem of an oblique derivative for the Lavrent'ev – Bitsadze equation in a half-plane. The Lavrent'ev – Bitsadze equation is an equation of mixed (elliptic-hyperbolic) type. Mixed-type equations arise when solving many applied problems (for example, when simulating transonic flows of a compressible medium).In the paper, the domain of ellipticity is a half-plane, and that of hyperbolicity is its adjacent strip. On one of the straight lines bounding the strip, an oblique derivative is specified (in the direction that forms an acute angle with this straight line), and on the other straight line, which is the interface between the strip and the half-plane, the solutions are matched by boundary conditions of the fourth kind. In the hyperbolicity strip, the solution is represented by the d'Alembert formula, and in the half-plane, where the equation is elliptic, the bounded solution is represented by the Poisson integral with unknown density. For this unknown density of the Poisson integral, a singular integral equation is obtained, which is reduced to the Riemann boundary value problem with a shift for holomorphic functions. The solution of the Riemann problem is reduced to the solution of two functional equations. Solutions of these functional equations and the Sokhotsky formula for an integral of Cauchy type allowed us to find the unknown density of the Poisson integral. This allowed us to find a solution to the oblique derivative problem as the sum of a functional series (up to an arbitrary constant term).

Solution of Linearized Flat Problem of Hydrodynamics (IVF)

In the present paper a method of the generalized potential to planes is applied for the solution of the linearized according to Oseen of flat problem of hydrodynamics incompressible viscous fluid (IVF). Generalized potential simple layer containing McDonald function serves kernel for generalized potential to planes. For finding of an unknown density of the potential simple layer is received linear integral equation, containing double integral from curvilinear integral along border of the streamlined area. Sharing the pressure is in turn defined by potential simple layer with density of the potential, determined by linear integral equation, hanging from solution specified above integral equation. The offered method of the successive iterations, allowing elaborate the solution of the problem before achievement given to accuracy. As example of exhibit to theories is considered solution of the problem theory of hydrodynamic greasing.

Nonparametric kernel methods for curve estimation and measurement errors

We consider the problem of estimating an unknown density or regression curve from data. In the parametric setting, the curve to estimate is modelled by a function which is known up to the value of a finite number of parameters. We consider the nonparametric setting, where the curve is not modelled a priori. We focus on kernel methods, which are popular nonparametric techniques that can be used for both density and regression estimation. While these methods are appropriate when the data are observed accurately, they cannot be directly applied to astronomical data, which are often measured with a certain degree of error. It is well known in the statistics literature that when the observations are measured with errors, nonparametric procedures become biased, and need to be adjusted for the errors. Correction techniques have been developed, and are often referred to as deconvolution methods. We introduce those methods, in both the homoscedastic and heteroscedastic error cases, and discuss their practical implementation.