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We introduce two new measures for the dependence of n ≥ 2 random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted L²-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to n-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of n random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.

In this paper, we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically, deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper, we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and nonvanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.

In spontaneous parametric down conversion (SPDC) based quantum information processing (QIP) experiments, there is a tradeoff between the coincidence count rates (i.e. the pumping power of the SPDC), which limits the rate of the protocol, and the visibility of the quantum interference, which limits the quality of the protocol. This tradeoff is mainly caused by the multi-photon pair emissions from the SPDCs. In theory, the problem is how to model the experiments without truncating these multi-photon emissions while including practical imperfections. In this paper, we establish a method to theoretically simulate SPDC-based QIPs which fully incorporates the effect of multi-photon emissions and various practical imperfections. The key ingredient in our method is the application of the characteristic function formalism which has been used in continuous variable QIPs. We apply our method to three examples, the Hong-Ou-Mandel interference and the Einstein-Podolsky-Rosen interference experiments, and the concatenated entanglement swapping protocol. For the first two examples, we show that our theoretical results quantitatively agree with the recent experimental results. Also we provide the closed expressions for these interference visibilities with the full multi-photon components and various imperfections. For the last example, we provide the general theoretical form of the concatenated entanglement swapping protocol in our method and show the numerical results up to five concatenations. Our method requires only a small computational resource (a few minutes by a commercially available computer), which was not possible in the previous theoretical approach. Our method will have applications in a wide range of SPDC-based QIP protocols with high accuracy and a reasonable computational resource.

We consider the symmetric Markov random flight, also called the persistent random walk, performed by a particle that moves at constant finite speed in the Euclidean space R m , m ≥ 2 , and changes its direction at Poisson-distributed time instants by taking it at random according to the uniform distribution on the surface of the unit ( m - 1 ) -dimensional sphere. Such stochastic motion has become a very popular object of modern statistical physics because it can serve as an appropriate model for describing the isotropic finite-velocity transport in multidimensional Euclidean spaces. In recent decade this approach was also developed in the framework of the run-and-tumble theory. In this article we study one of the most important characteristics of the multidimensional symmetric Markov random flight, namely, its characteristic function. We derive two series representations of the characteristic function of the process with respect to Bessel functions with variable indices and with respect to the powers of time variable. The coefficients of these series are given by recurrent relations, as well as in the form of special determinants. As an application of these results, an asymptotic formula for the second moment function μ ( 2 , 2 , 2 ) ( t ) , t > 0 , of the three-dimensional Markov random flight, is presented. The moment function μ ( 2 , 0 , 0 ) ( t ) , t > 0 , is obtained in an explicit form.

Continuum topology optimization is a powerful structural optimization method, but its application often leads to thin-walled structures with complex and irregular stiffening patterns, making the manufacturing process challenging. To address this issue, a novel optimization method for thin-walled structures with directional stiffeners is proposed. By introducing the single-variable characteristic function into the Discrete Material Optimization (DMO) model, a diagonal elements scheme is developed. This approach significantly reduces the number of design variables and enables the rapid optimal design of thin-walled structures with directional stiffeners. Firstly, a set of single-variable are introduced to describe the stiffeners in a given direction. The original variables are transformed into a set of density functions using the single-variable characteristic function. These density functions are incorporated into the DMO model to construct a new interpolation model. The update of design variables is driven by the genetic algorithm. An adaptive smoothing strategy is employed to adjust the projection slope and penalty parameters, which improves optimization efficiency. A numerical example verifies that the proposed method not only successfully achieves optimized results but also ensures good manufacturability.

The family of Generalized Gaussian (GG) distributions has received considerable attention from the engineering community, due to the flexible parametric form of its probability density function, in modeling many physical phenomena. However, very little is known about the analytical properties of this family of distributions, and the aim of this work is to fill this gap. Roughly, this work consists of four parts. The first part of the paper analyzes properties of moments, absolute moments, the Mellin transform, and the cumulative distribution function. For example, it is shown that the family of GG distributions has a natural order with respect to second-order stochastic dominance. The second part of the paper studies product decompositions of GG random variables. In particular, it is shown that a GG random variable can be decomposed into a product of a GG random variable (of a different order) and an independent positive random variable. The properties of this decomposition are carefully examined. The third part of the paper examines properties of the characteristic function of the GG distribution. For example, the distribution of the zeros of the characteristic function is analyzed. Moreover, asymptotically tight bounds on the characteristic function are derived that give an exact tail behavior of the characteristic function. Finally, a complete characterization of conditions under which GG random variables are infinitely divisible and self-decomposable is given. The fourth part of the paper concludes this work by summarizing a number of important open questions.

The present research demonstrates the utility of the linear canonical transform (LCT) in constructing the characteristic function of real random variables. We refer to this construction as the linear canonical characteristic function (LCCF). The proposed LCCF aims to address the limitations of the classical characteristic function in both theoretical and applied aspects. Using this approach, we investigate its properties, such as Hermitian symmetry, continuity, convolution, and derivatives, which are generalized forms of the classical characteristic function in the literature. Finally, we implement the obtained results by calculating several probability density functions in the LCCF domains.

Herein, we propose a novel strategy for implementing a direct readout of the symmetric characteristic function of the quantum states of quantum fields without the involvement of idealized measurements, an aspect that has always been deemed ill-defined in quantum field theory. This proposed scheme relies on the quantum control and measurements of an auxiliary qubit locally coupled to the quantum fields. By mapping the expectation values of both the real and imaginary parts of the field displacement operator to the qubit states, the qubit’s readout provides complete information regarding the symmetric characteristic function. We characterize our technique by applying it to the Kubo-Martin-Schwinger (thermal) and squeezed states of a quantum scalar field. In addition, we have discussed general applications of this approach to analogue-gravity systems, such as Bose-Einstein condensates, within the scope of state-of-the-art experimental capabilities. This proposed strategy may serve as an essential in understanding and optimizing the control of quantum fields for relativistic quantum information applications, particularly in exploring the interplay between gravity and quantum, for example, the relation to locality, causality, and information.

The main purpose of this work is solving a generalized (2 + 1)-dimensional nonlinear wave equation via ∂ ¯ -dressing method. The key to this process is to establish connection between characteristic functions and ∂ ¯ -problem. With use of Fourier transformation and Fourier inverse transformation, we obtain explicit expressions of Green’s function and give two characteristic functions corresponding to general potential. Further, the ∂ ¯ -problem is constructed by calculating ∂ ¯ derivative of characteristic function. The solution of ∂ ¯ -problem can be shown by Cauchy–Green formula, and after determining time evolution of scatter data, we can give solutions of the (2 + 1)-dimensional equation.

In this paper, a new kind of multivariate global sensitivity index based on energy distance is proposed. The covariance decomposition based index has been widely used for multivariate global sensitivity analysis. However, it just considers the variance of multivariate model output and ignores the correlation between different outputs. The proposed index considers the whole probability distribution of dynamic output based on characteristic function and contains more information of uncertainty than the covariance decomposition based index. The multivariate probability integral transformation based index is an extension of the popularly used moment-independent sensitivity analysis index. Although it considers the whole probability distribution of dynamic output, it is difficult to estimate the joint cumulative distribution function of dynamic output. The proposed sensitivity index can be easily estimated, especially for models with high dimensional outputs. Compared to the classic sensitivity indices, the proposed sensitivity index can be easily used for dynamic systems and obtain reasonable results. An efficient method based on the idea of the given-data method is used to estimate the proposed sensitivity index with only one set of input-output samples. The numerical and engineering examples are employed to compare the proposed index and the covariance decomposition based index. The results show that the input variables may have different effect on the whole probability distribution and variance of dynamic model output since the proposed index and the covariance decomposition based index measure the effects of input variables on the whole distribution and variance of model output separately.