A Grey System Approach for Estimating the Hölderian Regularity with an Application to Algerian Well Log Data

The Hölderian regularity is an important mathematical feature of a signal, connected with the physical nature of the measured parameter. Many algorithms have been proposed in literature for estimating the local Hölder exponent value, but all of them lead to biased estimates. This paper attempts to apply the grey system theory (GST) on the raw signal for improving the accuracy of Hölderian regularity estimation. First, synthetic logs data are generated by the successive random additions (SRA) method with different types of Hölder functions. The application on these simulated signals shows that the Hölder functions estimated by the GST are more precise than those derived from the raw data. Additionally, noisy signals are considered for the same experiment, and more accurate regularity is obtained using signals processed using GST. Second, the proposed technique is implemented on well log data measured at an Algerian exploration borehole. It is demonstrated that the regularity determined from the well logs analyzed by the GST is more reliable than that inferred from the raw data. In addition, the obtained Hölder functions almost reflect the lithological discontinuities encountered by the well. To conclude, the GST is a powerful tool for enhancing the estimation of the Hölderian regularity of signals.

GEOMETRICAL AND MEASURE-THEORETIC STRUCTURES OF MAPS WITH A MOSTLY EXPANDING CENTER

In this article we study physical measures for $\operatorname {C}^{1+\alpha }$ partially hyperbolic diffeomorphisms with a mostly expanding center. We show that every diffeomorphism with a mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology. Moreover, for each diffeomorphism with a mostly expanding center, there exists a $C^1$ neighbourhood, such that diffeomorphism among a $C^1$ residual subset of this neighbourhood admits finitely many physical measures, whose basins have full volume. We also show that the physical measures for diffeomorphisms with a mostly expanding center satisfy exponential decay of correlation for any Hölder observes. In particular, we prove that every $C^2$ , partially hyperbolic, accessible diffeomorphism with 1-dimensional center and nonvanishing center exponent has exponential decay of correlations for Hölder functions.

On approximately monotone and approximately Hölder functions

A real valued function f defined on a real open interval I is called $$\Phi $$ Φ -monotone if, for all $$x,y\in I$$ x , y ∈ I with $$x\le y$$ x ≤ y it satisfies $$\begin{aligned} f(x)\le f(y)+\Phi (y-x), \end{aligned}$$ f ( x ) ≤ f ( y ) + Φ ( y - x ) , where $$ \Phi :[0,\ell (I) [ \rightarrow \mathbb {R}_+$$ Φ : [ 0 , ℓ ( I ) [ → R + is a given nonnegative error function, where $$\ell (I)$$ ℓ ( I ) denotes the length of the interval I. If f and $$-f$$ - f are simultaneously $$\Phi $$ Φ -monotone, then f is said to be a $$\Phi $$ Φ -Hölder function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for $$\Phi $$ Φ -monotonicity and $$\Phi $$ Φ -Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper $$\Phi $$ Φ -monotone and $$\Phi $$ Φ -Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.

Summability on non-rectifiable Jordan curves

For a given parameterization of a Jordan curve, we define the notion of summability or classes of measurable functions on a contour where a new integral is introduced. It is shown that natural functional spaces defining summability for non-rectifiable Jordan curves are the Lebesgue spaces with the weighted norm. For non-rectifiable Jordan curves where an integral was previously defined for continuous (Hölder) functions [Y. Guseynov, Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 103], a weight function is constructed which, in general, is not summable by parameter, and a weighted functional space (summability) is defined where the new integral exists.