Cauchy distribution

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.

A flexible and effective algorithm GRPF (Global complex Roots and Poles Finding) for complex roots and poles finding is presented. A wide class of analytic functions can be analyzed, and any arbitrarily shaped search region can be considered. The method is very simple and intuitive. It is based on sampling a function at the nodes of a regular mesh, and on the analysis of the function phase. As a result, a set of candidate regions is created and then the roots/poles are verified using a discretized Cauchy's argument principle. The accuracy of the results can be improved by the application of a self-adaptive mesh. The effectiveness of the presented technique is supported by numerical tests involving different types of functions (microwave and optical applications). The results are verified, and the computational efficiency of the method is examined.

In this work, we discuss various types of $\mathcal{I}_2$-uniform convergence and equi-continuous for double sequences of functions. Also, we introduce the concepts of $\mathcal{I}_2$-uniform convergence, $\mathcal{I}_2^*$-uniform convergence, $\mathcal{I}_2$-uniformly Cauchy sequences and $\mathcal{I}_2^*$-uniformly Cauchy sequences for double sequences of functions in $2$-normed spaces. Then, we show the relationships between these new concepts.

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.~e. continuous and differentiable in the sense of Gateaux) functions of the variable $\sum_{j=1}^k x_j\,e_j$, where $x_1,x_2,\ldots,x_k$ are real, and obtain a constructive description of all mentioned functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux derivatives of all orders. The present article is generalized of the author's paper [1], where mentioned results are obtained for $k=3$.

Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let \theta be a contact form for (V,HT(V)), \xi_0 the Reeb vector field determined by \theta and assume that \xi_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of \xi_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Amp\`ere equation (dd^c u)^(n+1)=0 on N which is a defining equation for V. Moreover, the Monge-Amp\`ere foliation associated to u induces on V that one associated to the Reeb vector field. The converse is also true. The result is obtained solving a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one which is of independent interest (cf. Theorem 3.1). Comment: 15 pages

Let (S, •) be a semigroup, (H, +) an abelian group and f : S → H. The first and second order Cauchy differences of f are Higher order Cauchy differences C k f are defined recursively. In the case of H = R, a ring where multiplication is distributive over addition, we show that functions f : S → R with vanishing finite Cauchy differences are closed under multiplication. The equation C k f = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups.

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

We present a general construction of initial data for Einstein's equations containing an arbitrary number of black holes, each of which is instantaneously in equilibrium. Each black hole is taken to be a marginally trapped surface and plays the role of the inner boundary of the Cauchy surface. The black hole is taken to be instantaneously isolated if its outgoing null rays are shear-free. Starting from the choice of a conformal metric and the freely specifiable part of the extrinsic curvature in the bulk, we give a prescription for choosing the shape of the inner boundaries and the boundary conditions that must be imposed there. We show rigorously that with these choices, the resulting non-linear elliptic system always admits solutions.

When the production function of a productive phenomenon is estimated by statistical methods, the economic optimum, the point where net income is maximized, is usually a non-linear function of the estimated parameters and thus a random variable. In the univariate case, with a quadratic response function the optimum point estimator follows a Cauchy distribution with infinite mean and variance, hence

Euler's formulae for zeta(2n) are recovered from the computation in two dierent manners of the even moments of log(|C1C2|), for C1 and C2 two independent standard Cauchy variables. The method employed is generalized first to the L function associated to the character chi(4) and then to other trigonometric series.

The main aim of this paper is to introduce and analyze the notions of subspace almost periodicity and subspace weak almost periodicity for $C$-distribution semigroups and $C$-distribution cosine functions in Banach spaces. We continue our previous research study of almost periodicity of abstract Volterra integro-differential equations \cite{aot}, focusing our attention on the abstract ill-posed Cauchy problems of first order.

who initiated me to the theory of large continuum deformations. Special thanks to N. Ramaniraka who has, with me, arrived at some important results in the last chapter devoted to the two models of microcracked solids. Last but not least, I am greatly indebted to my wife Oly and my children Rindra, Herinarivo, and Haga, to whom this book is dedicated, for their understanding and moral support.

An important step in designing a fuzzy system is the elicitation of the membership functions for the fuzzy sets used. Often the membership functions are obtained from data in a traininglike manner. They are expected to match or be at least compatible with those obtained from experts knowledgeable of the domain and the problem being addressed. In cases when neither are possible, e.g., insufficient data or unavailability of experts, we are faced with the question of hypothesizing the membership function. We have previously argued in favor of Cauchy membership functions (thus named because their expression is similar to that of the Cauchy distributions) and supported this choice from the point of view of efficiency of training. This paper looks at the same family of membership functions from the point of view of reliability.

In this paper, we will introduce the Cauchy numbers of both kinds in type B and produce their corresponding exponential generating functions. Then we will provide some identities involving Cauchy, Lah, and Stirling numbers in type B through combinatorial methods.

Discrete particle models have been widely used as an alternative to classical continuum mechanics to describe the mechanical behavior of solids, fluids and granular matter. Lattice particle models (LPM) are a subset of discrete particle models which treat the body as a set of interacting point masses with fixed sets of neighbors. Heretofore, the types of interaction forces considered in LPMs have been very limited; consisting mainly of linear pairwise interactions or empirically motivated generalizations thereof. Thus LPMs have been able to describe a very limited range of macroscopic constitutive behavior and the properties have generally been strongly dependent on the discretization. We present a new paradigm for incorporating multibody interactions in LPMs using a mesoscale analog of the Cauchy-Born rule which relates the deformation of the continuum to that of the underlying atomic lattice. We consider a Delaunay triangulation of an irregular lattice of particles comprising the body in its reference configuration. We calculate linear transformations to map the particle positions of each triangle in the current configuration to its reference configuration. Invoking the analogy of the Cauchy-Born rule, we identify the linear transformations with the continuum deformation gradient. Using this we express the continuum constitutive free energy of the body, which is a function of the deformation gradient, in terms of the current positions of the particles. The neighbor interaction forces on each of the particles are taken to be the gradients of this energy. The equations of motion for each particle are then solved numerically to obtain the particle trajectories and the overall deformation of the body. This model is validated using comparisons with classical results of stress distribution in isotropic and anisotropic material plates with circular holes. We then show how this model is able to describe formation of fine microstructure in two different materials: crack branching in brittle materials and twinning in phase transforming materials. c

An implementation of the reduced multiplication scheme of the Rys-Gauss quadrature to compute the gradients of electron repulsion integrals is discussed. The study demonstrates that the Rys-Gauss quadrature is very suitable for efficient utilization of simplifications as offered by the direct computation of symmetry adapted gradients and the use of the translational invariance of the integrals. The introduction of the so-called intermediate products is also demonstrated to further reduce the floating point operation count. Two prescreening techniques based on the 2nd order density matrix in the basis of the uncontracted Gaussian functions is proposed and investigated in the paper. This investigation gives on hand that it is not necessary to employ the Cauchy-Schwarz inequality to achieve efficient prescreening. All the features mentioned above were demonstrated by their implementation into the gradient program ALASKA. The paper offers a theoretical and practical assessment of the modified Rys-Gauss quadrature in comparison with other methods and implementations and a detailed analysis of the behavior of the method as suggested above as a function of changes with respect to symmetry, basis set quality, molecular size, and prescreening threshold.

With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained.

In the present work, the authors determine coefficient bounds for functions in certain subclasses of starlike and convex functions of complex order, which are introduced here by means of a family of nonhomogeneous Cauchy-Euler differential equations. Several corollaries and consequences of the main results are also considered.

We consider a Levy flyer that starts from a point on an interval with absorbing boundaries. Using fractional differential equations, we find a simple analytical expression for the average number of flights the flyer takes before it is absorbed. When the parameter of the Levy distribution alpha is larger than 2, i.e. when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We also find a closed form expression for the total length of the flights before absorption, and show that it can be well approximated by the product of the average number of flights and the average length of a single flight. We show that when the starting point is close to the the absorbing boundary, the average total length has a minimum at alpha equals to 1. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

In this paper, the linear non homogeneous integral equation of H-functions is considered to find a new form of H-function as its solution .The Wiener-Hopf technique is used to express a known function into two functions with different zones of analyticity. The linear non homogeneous integral equation is thereafter expressed into two different sets of functions having the different zones of regularity.The modified form of Liouville's theorem is thereafter used , Cauchy's Integral formulae are used to determine functional representation over the cut region in a complex plane .The new form of H-function is derived both for conservative and non-conservative cases .The existence of solution of linear non homogeneous integral equations and its uniqueness are shown. For numerical calculation of this new H-function , a set of useful formulae are derived both for conservative and non-conservative cases.

Fault detection is an important issue in today’s distribution networks, the structure of which is becoming more complex. In this article, a data-based Cauchy distribution weighting M-estimate RVFLNs method is proposed for short-circuit fault detection in distribution networks. The proposed method detects short circuits based on current and voltage measurements. In addition, noises were added to the data to ensure the robustness of the method. The performance of the method was examined in the RTDS RTS simulator using the IEEE 33-bus-bar system model with the help of real-time simulations. The success rate of the proposed method is between 98% and 100% for low-impedance (0 ohm) short-circuit faults, depending on the fault type. The success rate of high-impedance (100 ohm) short-circuit faults, which are more difficult to detect, is between 80% and 92%, depending on the fault type.

This paper is the second of a two-part series that discusses the implementation issues and test results of a robust Unscented Kalman Filter (UKF) for power system dynamic state estimation with non-Gaussian synchrophasor measurement noise. The tuning of the parameters of our Generalized Maximum-Likelihood-type robust UKF (GM-UKF) is presented and discussed in a systematic way. Using simulations carried out on the IEEE 39-bus system, its performance is evaluated under different scenarios, including i) the occurrence of two different types of noises following thick-tailed distributions, namely the Laplace or Cauchy probability distributions for real and reactive power measurements; ii) the occurrence of observation and innovation outliers; iii) the occurrence of PMU measurement losses due to communication failures; iv) cyber attacks; and v) strong system nonlinearities. It is also compared to the UKF and the Generalized Maximum-Likelihood-type robust iterated EKF (GM-IEKF). Simulation results reveal that the GM-UKF outperforms the GM-IEKF and the UKF in all scenarios considered. In particular, when the system is operating under stressed conditions, inducing system nonlinearities, the GM-IEKF and the UKF diverge while our GM-UKF does converge. In addition, when the power measurement noises obey a Cauchy distribution, our GM-UKF converges to a state estimate vector that exhibits a much higher statistical efficiency than that of the GM-IEKF; by contrast, the UKF fails to converge. Finally, potential applications and future work of the proposed GM-UKF are discussed in concluding remarks section.

This paper develops the theoretical framework and the equations of a new robust Generalized Maximum-likelihoodtype Unscented Kalman Filter (GM-UKF) that is able to suppress observation and innovation outliers while filtering out non-Gaussian measurement noise. Because the errors of the real and reactive power measurements calculated using Phasor Measurement Units (PMUs) follow long-tailed probability distributions, the conventional UKF provides strongly biased state estimates since it relies on the weighted least squares estimator. By contrast, the state estimates and residuals of our GM-UKF are proved to be roughly Gaussian, allowing the sigma points to reliably approximate the mean and the covariance matrices of the predicted and corrected state vectors. To develop our GM-UKF, we first derive a batch-mode regression form by processing the predictions and observations simultaneously, where the statistical linearization approach is used. We show that the set of equations so derived are equivalent to those of the unscented transformation. Then, a robust GM-estimator that minimizes a convex Huber cost function while using weights calculated via Projection Statistics (PS's) is proposed. The PS's are applied to a two-dimensional matrix that consists of serially correlated predicted state and innovation vectors to detect observation and innovation outliers. These outliers are suppressed by the GMestimator using the iteratively reweighted least squares algorithm. Finally, the asymptotic error covariance matrix of the GM-UKF state estimates is derived from the total influence function. In the companion paper, extensive simulation results will be shown to verify the effectiveness and robustness of the proposed method.

Schwartz's solution to the Björling problem leads to an equivalence class of spatial strips S(t)=(c(t),n(t)) which produce equivalent minimal surfaces. For the particular case when the generating strip S(t) belongs to some plane E and c(t) is symmetric with respect to some straight line in E, the symmetries of the minimal surface permit us to identify another planar curve ~c(t) that we call the CPG curve to c(t). A simple symmetric argument shows that self-CPG curves produce minimal surfaces whose adjoint surface contains another self-CPG curve. We ask for minimal surfaces generated by self-CPG curves which are self-adjoints.

This paper presents a method for solving a class of inverse problems for elliptic equations known as the data completion problem. The goal is to recover missing data on the inaccessible part of the boundary using measurements from the accessible part. The inherent difficulty of this problem arises from its ill-posed nature, as it is susceptible to variations in the input data. To address this challenge, the proposed approach integrates Tikhonov regularization to enhance the stability of the problem. To solve this problem, we use a metaheuristic approach, specifically, the Bat Algorithm (BA) inspired by the echolocation behavior of bats. The performed numerical results show that the Bat Algorithm yields stable, convergent, and accurate solutions.

The Cauchy problem associated with the Helmholtz equation is an ill-posed inverse problem that is challenging to solve due to its instability and sensitivity to noise. In this paper, we propose a metaheuristic approach to solve this problem using Genetic Algorithms in conjunction with Tikhonov regularization. Our approach is able to produce stable, convergent, and accurate solutions for the Cauchy problem, even in the presence of noise. Numerical results on both regular and irregular domains show the effectiveness and accuracy of our approach.

In this paper we provide a probabilistic representation of Lagrange's identity which we use to obtain Papathanasiou-type variance expansions of arbitrary order. Our expansions lead to generalized sequences of weights which depend on an arbitrarily chosen sequence of (non-decreasing) test functions. The expansions hold for arbitrary univariate target distribution under weak assumptions, in particular they hold for continuous and discrete distributions alike. The weights are studied under different sets of assumptions either on the test functions or on the underlying distributions. Many concrete illustrations for standard probability distributions are provided (including Pearson, Ord, Laplace, Rayleigh, Cauchy, and Levy distributions).

In this paper we provide a probabilistic representation of Lagrange's identity which we use to obtain Papathanasiou-type variance expansions of arbitrary order. Our expansions lead to generalized sequences of weights which depend on an arbitrarily chosen sequence of (non-decreasing) test functions. The expansions hold for univariate target distribution under weak assumptions, in particular they hold for continuous and lattice distributions alike. The weights are studied under different sets of assumptions either on the test functions or on the underlying distributions. Many concrete illustrations for standard probability distributions are provided (including Pearson, Ord, Laplace, Rayleigh, Cauchy, and Levy distributions).

In this paper, following on from [49, 50, 63] we present a minimal formalism for Stein operators which leads to different probabilistic representations of solutions to Stein equations. These in turn provide a wide family of Stein-Covariance identities which we put to use for revisiting the very classical topic of bounding the variance of functionals of random variables. Applying the Cauchy-Schwarz inequality yields first order upper and lower Klaassen [45]-type variance bounds. A probabilistic representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder) leads to Papathanasiou [60]-type variance expansions of arbitrary order. A matrix Cauchy-Schwarz inequality leads to Olkin-Shepp [59] type covariance bounds. All results hold for univariate target distribution under very weak assumptions (in particular they hold for continuous and discrete distributions alike). Many concrete illustrations are provided.

In a previous paper we have presented a new method for solving a class of Cauchy integral equations. In this work we discuss in detail how to manage this method numerically, when only a finite and noisy data set is available: particular attention is focused on the question of the numerical stability.

We provide three large classes of control functions that ensure the hyperstability of the Cauchy equation on restricted domains included in various types of commutative semigroups. Among other consequences, we obtain significant improvements on similar results known from the literature for several Aoki-Rassias-type control functions.

We provide three large classes of control functions that ensure the hyperstability of the Cauchy equation on restricted domains included in various types of commutative semigroups. Among other consequences, we obtain significant improvements on similar results known from the literature for several Aoki-Rassias-type control functions. Mathematics Subject Classification (2010). Primary 39B82, 39B52; Secondary 39A70, 39B62, 39A10.