Cauchy Difference Research Articles

First-principles analysis of physical properties of the novel calcium-based hydrides for hydrogen storage application

The current work aims to investigate the physical properties of the new calcium-based perovskite hydride CaXH3 (X=Cu, and Zn) for hydrogen storage applications using DFT studies. Both materials were found to be thermodynamically and mechanically stable owing to their negative formation enthalpies, cohesive energy, and elastic constants. The absence of negative frequencies in the phonon dispersion curve reveals the dynamic stability of the structure. The optimized structure lattice constants were found 3.62Å, and 3.78Å with a bulk modulus of 117.40, and 82.39 GPa, respectively. The Poisson and Pugh ratios were confirmed to have ionic bonding, while the Cauchy difference and Pugh ratio (B/G) results exhibited the brittle behavior of both materials. In addition, both materials have anisotropic characteristics, and CaCuH3 is stiffer than CaZnH3. The electronic properties confirmed that both compounds have a metallic nature owing to the zero band gap and high contribution of the Ca and X (Cu, Zn) d states near the Fermi level. These compounds were found non-magnetic by the investigation of magnetic properties. The optical results indicate that CaZnH3 plays a significant role in optoelectronics, solar cells, and photonics applications because of its higher conductivity, refractive index, dielectric function, and reflectivity, as well as its higher adsorption than CaCuH3.The hydrogen storage capacities of CaCuH3 and CaZnH3 hydrides were calculated to be 2.88 wt. % and 2.75 wt.%, respectively, and it was predicted that both compounds have the potential to store hydrogen, as evidenced by storage application.

Geometry Parameter Estimation for Sparse X-Ray Log Imaging

We consider geometry parameter estimation in industrial sawmill fan-beam X-ray tomography. In such industrial settings, scanners do not always allow identification of the location of the source–detector pair, which creates the issue of unknown geometry. This work considers an approach for geometry estimation based on the calibration object. We parametrise the geometry using a set of 5 parameters. To estimate the geometry parameters, we calculate the maximum cross-correlation between a known-sized calibration object image and its filtered backprojection reconstruction and use differential evolution as an optimiser. The approach allows estimating geometry parameters from full-angle measurements as well as from sparse measurements. We show numerically that different sets of parameters can be used for artefact-free reconstruction. We deploy Bayesian inversion with first-order isotropic Cauchy difference priors for reconstruction of synthetic and real sawmill data with a very low number of measurements.

Solution for 3rd order Cauchy Difference Equation in Free Monoid

— Let k: C ? D be a function, where (C,.) is a Group and (D,+) is an Abelian Group. In this article, the Third Order Cauchy difference of k:E(3) k (c1,c2,c3,c4)= k(c1c2c3c4) – k(c1c2c3) – k(c1c3c4) – k (c1c2c3c4) –k (c2c3c4) + k(c1c2) + k (c1c3) +k(c1c4)+k(c2c3)+ k(c2c4)+k(c3c4) –k(c1) – k(c2) –k(c3) –k(c4) ? c1,c2,c3,c4 C is studied. We give solutions of E(3)k=0 on Free Monoid.

Cauchy Markov random field priors for Bayesian inversion

The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed. In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo methods are usually required. In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison. We firstly propose a one-dimensional second-order Cauchy difference prior, and construct new first- and second-order two-dimensional isotropic Cauchy difference priors. Another new Cauchy prior is based on the stochastic partial differential equation approach, derived from Matérn type Gaussian presentation. The comparison also includes Cauchy sheets. Our numerical computations are based on both maximum a posteriori and conditional mean estimation. We exploit state-of-the-art MCMC methodologies such as Metropolis-within-Gibbs, Repelling-Attracting Metropolis, and No-U-Turn sampler variant of Hamiltonian Monte Carlo. We demonstrate the models and methods constructed for one-dimensional and two-dimensional deconvolution problems. Thorough MCMC statistics are provided for all test cases, including potential scale reduction factors.

Some remarks on the stability of the Cauchy equation and completeness

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

Alternative Cauchy equation in three unknown functions

In this paper we deal with the product of two or three Cauchy differences equaled to zero. We show that in the case of two Cauchy differences, the condition of absolute continuity and differentiability of the two functions involved implies that one of them must be linear, i.e., we have a trivial solution. In the case of the product of three Cauchy differences the situation changes drastically: there exists non trivial {mathcal {C}}^{infty } solutions, while in the case of real analytic functions we obtain that at least one of the functions involved must be linear. Some open problems are then presented.

Generalized Cauchy difference iterative forecasting model for wind speed based on fractal time series

The local irregularity and global correlation of wind speed can be described by the fractal dimension D and the Hurst parameter H. However, the existing mathematical models imply a linear relationship between the fractal dimension D and the Hurst parameter H, which is not adequate to describe the complete characteristics of the wind speed. To overcome the descriptive limitations of the $$D-H$$ linearity assumption, in this paper, we introduce novel model, based on the generalized Cauchy (GC) process, for simulation and forecasting of wind speed. In the model, the fractal dimension D and Hurst parameter H can be combined arbitrarily to describe the local irregularity and global correlation of the wind speed. Furthermore, the GC process is taken as the disturbance fluctuation term in the forecasting model, and a difference iteration form is obtained as difference equation and incremental distribution. In such model, the maximum forecasting range of the wind speed is determined by the maximum Lyapunov exponent. To illustrate the properties of the model and its performance, simulating and forecasting of the actual wind speeds in two regions are reported.

Rough variation on lacunary quasi Cauchy triple difference sequences

Abstract In the present paper, we extend the notion of rough ideal lacunary statistical quasi Cauchy triple difference sequence of order α using the concept of ideals, which automatically extends the earlier notions of rough convergence and rough statistical convergence. We prove several results associated with this notion.

Cauchy difference priors for edge-preserving Bayesian inversion

Abstract We consider inverse problems in which the unknown target includes sharp edges, for example interfaces between different materials. Such problems are typical in image reconstruction, tomography, and other inverse problems algorithms. A common solution for edge-preserving inversion is to use total variation (TV) priors. However, as shown by Lassas and Siltanen 2004, TV-prior is not discretization-invariant: the edge-preserving property is lost when the computational mesh is made denser and denser. In this paper we propose another class of priors for edge-preserving Bayesian inversion, the Cauchy difference priors. We construct Cauchy priors starting from continuous one-dimensional Cauchy motion, and show that its discretized version, Cauchy random walk, can be used as a non-Gaussian prior for edge-preserving Bayesian inversion. We generalize the methodology to two-dimensional Cauchy fields, and briefly consider a generalization of the Cauchy priors to Lévy α-stable random field priors. We develop a suitable posterior distribution sampling algorithm for conditional mean estimates with single-component Metropolis–Hastings. We apply the methodology to one-dimensional deconvolution and two-dimensional X-ray tomography problems.

Generalizations of an asymptotic stability theorem of Bahyrycz, Páles and Piszczek on Cauchy differences to generalized cocycles

We prove some straightforward analogues and generalizations of a recent asymptotic stability theorem of A. Bahyrrycz, Zs. Pales and M. Piszczek on Cauchy differences to semi-cocycles and speudo-cocycles introduced in a former paper by the present author.

Homogeneous Beta-type functions

All beta-type functions, which are p-homogeneous, are determined. Applying this result, we show that a beta-type function is a homogeneous mean iff it is the harmonic one. A reformulation of a result due to Heuvers in terms of a Cauchy difference and the harmonic mean is given.

The kernel of the second order Cauchy difference on semigroups

Let S be a semigroup, H a 2-torsion free, abelian group and $$C^2f$$ the second order Cauchy difference of a function $$f:S \rightarrow H$$ . Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of $$C^2f = 0$$ are the functions of the form $$f(x) = j(x) + B(x,x)$$ , where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of $$C^2f = 0$$ to Frechet’s functional equation and to polynomials of degree less than or equal to 2.

Functions on semigroups with vanishing finite Cauchy differences

Let \({(S,\cdot)}\) be a semigroup, (H, +) an abelian group and \({f: S \to H}\). The first and second order Cauchy differences of f are $$\begin{aligned} C^{1}f(x, y)= & {} f(xy) - f(x) - f(y),\\ C^{2}f(x, y, z)= & {} f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$ Higher order Cauchy differences Ckf are defined recursively. In the case of H = R, a ring where multiplication is distributive over addition, we show that functions \({f: S\to R}\) with vanishing finite Cauchy differences are closed under multiplication. The equation Ckf = 0 is considered for cyclic groups, free abelian groups and other selected quotients of the free groups.

Generalized Pexider difference equations

We extend previous results on generalized Cauchy difference equations (in which f = g = h) to functional equations of the form $$f(x)+g(y)-h(x+y)=k(L(x,y))$$ where L is a given function and f, g, h, k are unknown. We prove results assuming either that L is sufficiently smooth or that \({L(x,y)=\phi(x)+\phi(y)-\phi(x+y)}\) for some function ϕ. We illustrate these results with applications to several examples, including the Olkin–Baker equation.

Solutions of the nth-order Cauchy difference equation on groups

Let be a function, where is a group and is an abelian group. In this paper, the following third order Cauchy difference of = − − − − + + + + + + − − − − ( ), is studied. We first give some special solutions of on free groups. Then sufficient and necessary conditions on finite cyclic groups and symmetric groups are also obtained. MSC:39B52, 39A70.

P-means and the Solution of a Functional Equation Involving Cauchy Differences

Solutions to the functional equation $$f(x + y) - f(x) - f(y) = 2f(\Phi (x, y)), x, y > 0, \qquad\qquad (1)$$ are sought for the admissible pairs \({(f, \Phi)}\) constituted by a strictly monotonic function f and a strictly increasing in both variables mean \({\Phi}\). A related class of means, P-means, is introduced, studied and then employed in solving (1) under additional hypotheses on \({\Phi}\). For instance, Ger has proved that the unique P-mean which is also quasiarithmetic is the geometric mean \({G(x,y)=\sqrt{xy}}\). An elementary proof to this result is given in this paper. Moreover, as a consequence of a fundamental result on the uniqueness of representation of P-means it is proved that the geometric mean G is the unique homogeneous P-mean.

Kernels of higher order Cauchy differences on free groups

Let (G, .) be a group, (H, +) an abelian group and f : G → H. The first and the second Cauchy differences of f are $$\begin{aligned} & \quad C^{1}f(x,y) = f(xy) - f(x) - f(y), \\ & C^{2}f(x,y,z) = f(xyz) - f(xy) - f(yz) - f(xz) + f(x) + f(y) + f(z).\end{aligned}$$ Higher order Cauchy differences \({C^{m}f}\) are defined recursively. The functional equation $$C^{m}f = 0$$ is studied. Some earlier results on the equation \({C^{2}f = 0}\) are extended to higher m. For m = 3 we present its general solution on free groups G. When the free group has just one generator the solution is obtained for all m.

Solutions of the third order Cauchy difference equation on groups

Let be a function, where is a group and is an abelian group. In this paper, the following third order Cauchy difference of = − − − − + + + + + + − − − − ( ), is studied. We first give some special solutions of on free groups. Then sufficient and necessary conditions on finite cyclic groups and symmetric groups are also obtained. MSC:39B52, 39A70.

Kernel of the second order Cauchy difference on groups

Let (G, ·) be a group, (H, +) be an abelian group, and \({f:G\rightarrow H}\). The second order Cauchy difference of f is $$C^{(2)}f(x,y,z)=f(xyz)-f(xy)-f(yz)-f(xz)+f(x)+f(y)+f(z).$$ The functional equation $$C^{(2)}f(x,y,z)=0$$ is studied. We present its general solution on free groups. Solutions on other selected groups are also given.

Hyers–Ulam stability of derivations and linear functions

In this paper the following implication is verified for certain basic algebraic curves: if the additive real function $f$ approximately (i.e., with a bounded error) satisfies the derivation rule along the graph of the algebraic curve in consideration, then $f$ can be represented as the sum of a derivation and a linear function. When, instead of the additivity of $f$, it is assumed that, in addition, the Cauchy difference of $f$ is bounded, a stability theorem is obtained for such characterizations of derivations.