Structural Heterogeneity of an Amorphous-Nanocrystalline Alloy Fe77Cu1Si16B6 in the Nanometer Range

In this article, an alloy of the Finemet type Fe77Cu1Si16B6 obtained by quenching from a liquid state (spinning method) in the initial state is investigated. The main research methods were scanning and transmission electron microscopy. Methods for describing multiscale structural heterogeneities in amorphous-nanocrystalline alloys have been developed, allowing the structural state to be described and its influence on the physicochemical and technical properties to be determined depending on the technological conditions for obtaining these alloys. Representation of electron microscopic images in the form of Fourier spectra made it possible to reveal the nature of the formation of short- and middle-order in amorphous-nanocrystalline alloys according to the principle of self-similar spatial structures. The analysis of electron microscopic images by integral Lebesgue measures revealed density fluctuations over the alloy volume, which corresponds to the hierarchical representation of structural inhomogeneities in amorphous metallic alloys.

Curve Based Approximation of Measures on Manifolds by Discrepancy Minimization

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve’s length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on $$\mathbb R^3$$ R 3 and the Grassmannian of all 2-dimensional linear subspaces of $${\mathbb {R}}^4$$ R 4 . Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.

Characterizations of the lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part

In this paper, we study a class of Borel measures on [Formula: see text] that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the [Formula: see text]-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.

On a Choquet-Stieltjes type integral on intervals

In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

Surface Morphology of Spinning Tapes Fe-(Cu, Nb)-(Si, B) with Different Content of Metalloid

Using the methods of spectral analysis with application of the formalism of the integral function of Lebesgue measures and Kullback's divergence, the structural features of the interfaces of spinning Fe- (Cu, Nb) - (Si, B) tapes with different metalloid contents are investigated. An increase in the total concentration of the metalloid ambiguously affects the nature of the morphology of rapidly quenched alloys. The minimum difference in the statistics of contact and free surfaces, formed in the spinning process, is demonstrated by fusionFe74Cu1Nb3Si16B6.

The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures

One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction.

Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line. We firstly give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive measure. Furthermore, the operational schemes of above several classes of integrals on a discrete set are investigated which enable us to calculate Choquet integrals in some applications. Secondly, we give a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. Finally, an example is given to illustrate the main results at the end of the paper.

ABSOLUTE CONTINUITY THEOREM FOR RANDOM DYNAMICAL SYSTEMS ON Rd

In this paper we provide a proof of the so-called absolute continuity theorem for random dynamical systems on Rd which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds, the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincaré map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.