Explore the vast range of titles available.

An integer-valued multiplicative function f is said to be polynomially-defined if there is a nonconstant separable polynomial F ( T ) ∈ Z [ T ] with f ( p ) = F ( p ) for all primes p . We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus q . For example, we show that the values ϕ ( n ) , sampled over integers n ≤ x with ϕ ( n ) coprime to q , are asymptotically equidistributed among the coprime classes modulo q , uniformly for moduli q coprime to 6 that are bounded by a fixed power of log x .

In this note, we study the convergence of functional sequences. A criterion for uniform distribution mod 1 is derived. Then we study the partitions, block sequences and the uniform distribution preserving mappings. In the last part, we prove that to each one to one sequence dense in [0, 1] a regular matrix summation method such that this sequence is uniformly distributed mod 1 with respect to this method exists.

Glasses-free three-dimensional (3D) display has attracted wide interest for providing stereoscopic virtual contents with depth cues. However, how to achieve high spatial and angular resolution while keeping ultrawide field of view (FOV) remains a significant challenge in 3D display. Here, we propose a light field 3D display with space-variant resolution for non-uniform distribution of information and energy. The spatial resolution of each view is modulated according to watching habit. A large-scale combination of pixelated 1D and 2D metagratings is used to manipulate dot and horizontal line views. With the joint modulation of pixel density and view arrangement, the information density and illuminance of high-demand views are at most 5.6 times and 16 times that of low-demand views, respectively. Furthermore, a full-color and video rate light field 3D display with non-uniform information distribution is demonstrated. The prototype provides 3D images with a high spatial resolution of 119.6 pixels per inch and a high angular resolution of 0.25 views per degree in the high-demand views. An ultrawide viewing angle of 140° is also provided. The proposed light field 3D display does not require ultrahigh-resolution display panels and has form factors of thin and light. Thus, it has the potential to be used in portable electronics, window display, exhibition display, as well as tabletop display.

This article serves as a continuation of our previously published work and focuses on loose material transport via sandwich belt conveyors. Experimental, analytical, stochastic, and numerical approaches are used to obtain and utilize the moduli of a bilateral Winkler elastic foundation that represent a loose material, which is wheat (Triticum aestivum) that is free of bran in this case. The solutions were obtained for a uniformly and nonuniformly distributed loose material. The task of the conveyor with loose material is simplified into a symmetric task, i.e., a beam on an elastic bilateral Winkler foundation, for an analytical solution and stochastic solution (Anthill and Matlab sw). In a numerical approach, this is considered a plane strain problem within the finite element method (Ansys and MSC.Marc sw). The experimental data are evaluated and used to obtain the functions of Winkler elastic foundation moduli, which are further considered in the numerical solution. The finite element method mainly serves as a verification tool. The acquired histograms of the elastic foundation moduli can be further applied in various scientific and research fields.

Discrete uniform distribution (DUD) is one of the simplest probability models, but it is now introduced as the main tool for the evaluation of resampling techniques which are rapidly entering data analysis and discovering useful information for the researchers.In this paper we evaluate whether the sample coefficient of variation(CV) is a good estimator for the population CV, when the random variable (r.v.) follows the DUD. A method is proposed to obtain the percentage of the number of samples where the CV lies within the bounds of the corresponding population CV and this value is used as a measure of goodness. Samples both with replacement and without replacement are examined, indicating that the goodness of the sample CV estimator increases with the sample size. The overall study gives a good idea of whether the sample CV is generally a good estimator. A real-life data set is analyzed to demonstrate the applicability of the proposed method in practice and the results are interpreted.

The assessment of the way distributive shocks, such as increased polarization or higher inequality, affect vertically differentiated markets has been severely hampered by the standard reference to uniform distributions. In this paper we offer the first proof of existence of a subgame perfect Nash equilibrium in a vertically differentiated duopoly with uncovered market, for a large set of symmetric and asymmetric distributions of consumers, including, among others, all logconcave distributions. The proof relies on the 'income share elasticity' representation of the consumers' density function. Some illustrative examples are also provided to assess the impact of distributive shocks on market equilibrium.

There are two main interrelated goals of this paper. Firstly we investigate the sums

Based upon a new general vector-valued vector product, generalized complex numbers with respect to certain positively homogeneous functionals including norms and antinorms are introduced and a vector-valued Euler type formula for them is derived using a vector valued exponential function. Furthermore, generalized Cauchy–Riemann differential equations for generalized complex differentiable functions are derived. For random versions of the considered new type of generalized complex numbers, moments are introduced and uniform distributions on discs with respect to functionals of the considered type are analyzed. Moreover, generalized uniform distributions on corresponding circles are studied and a connection with generalized circle numbers, which are natural relatives of π, is established. Finally, random generalized complex numbers are considered which are star-shaped distributed.

We propose a general new method, the conditional permutation test, for testing the conditional independence of variables X and Y given a potentially high dimensional random vector Z that may contain confounding factors. The test permutes entries of X non-uniformly, to respect the existing dependence between X and Z and thus to account for the presence of these confounders. Like the conditional randomization test of Candès and co-workers in 2018, our test relies on the availability of an approximation to the distribution of X|Z—whereas their test uses this estimate to draw new X-values, for our test we use this approximation to design an appropriate non-uniform distribution on permutations of the X-values already seen in the true data. We provide an efficient Markov chain Monte Carlo sampler for the implementation of our method and establish bounds on the type I error in terms of the error in the approximation of the conditional distribution of X|Z, finding that, for the worst-case test statistic, the inflation in type I error of the conditional permutation test is no larger than that of the conditional randomization test. We validate these theoretical results with experiments on simulated data and on the Capital Bikeshare data set.

Our study suggested a new distribution through the use of the cum. dis. function for the exp. dis., as well as the cum. dis. function for the uni. dis. in addition to the quantitative function of the unified generalization using the quantitative function. Therefore it is covered different mathematical and statistical properties .of this distribution.