Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.

Nonconforming P1 Finite Element for the Biharmonic Problem and Its Application to Stokes Problem

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming [Formula: see text] finite element. For the solutions which admit a regularity greater than [Formula: see text], in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in [Formula: see text] an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.

A theorem of Roe and Strichartz on homogeneous trees

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

Enhancement of digital chest images using a modified Sobel edge detection algorithm

Computed tomography (CT) images are an essential factor in the diagnosing procedure for various diseases affecting the internal organs. Edge detection can be used for the appropriate enhancement of the lung CT scan images for the diagnosis of the various interstitial lung diseases (ILD). In order to solve the issues of edge detection provided by the traditional Sobel operator, the paper proposes a Sobel 12D edge detection algorithm which uses the additional direction templates for the better identification of the edge details. First, the vertical and horizontal directions available in the traditional Sobel operator are extended to few more directions (a total of 12 directions) which enhances the edge extraction ability. Next part, compute the edge detected image using the Sobel 12D, Laplace, Prewitt, Robert’s Cross and Scharr operators for edge detection separately. It is followed by image fusion method which optimizes the edge detection by combining the edge detected images obtained using the Sobel 12D approach and the Laplace operator. The experimental results shows that the proposed algorithms generates a better detection of the edges than the other edge detection operators.

Small Object Signal Formation Model for Optical System with Matrix Photodetector When Processed by the Laplace Operator

The authors review the formation of a small object signal by an optical system with a matrix photodetector. They develop a mathematical model of signal formation and test it. The authors calculate the signal/noise proportion for various small object signal shapes and amplitudes processed by the Laplace operator.

The Distribution of the Thermal Field in an Elliptical Electric Conductor Coated with Insulation

The paper determines the stationary thermal field in an elliptical cross-section electric conductor coated with insulation. Heat is generated by the flow of alternating current (AC) through the conducting core, and then dissipated from the insulation surface via convection and radiation. The authors have developed an original method for hybrid (analytical–numerical) modeling of a field. This method has been used to solve the relevant boundary problem of Poisson’s equation. While the eigenfunctions of the Laplace operator were determined analytically, the coefficients of the eigenfunctions were calculated by iteratively solving an appropriate system of algebraic equations. The proposed method enables the analysis of systems with an elliptical geometry and a heterogeneous layered structure (e.g., air, aluminum alloy, PCV), and does not require area discretization (grid). The developed analytical–numerical (AN) method has been positively verified using finite elements (FEs). The determined thermal field is illustrated graphically. The obtained solution has a good physical interpretation.

Characterization of Lagrangian Submanifolds by Geometric Inequalities in Complex Space Forms

In this paper, we give an estimate of the first eigenvalue of the Laplace operator on a Lagrangian submanifold M n minimally immersed in a complex space form. We provide sufficient conditions for a Lagrangian minimal submanifold in a complex space form with Ricci curvature bound to be isometric to a standard sphere S n . We also obtain Simons-type inequality for same ambient space form.

A three solutions theorem for Pucci's extremal operator and its application

In this article we prove a three solution type theorem for the following boundary value problem: \begin{equation*} \label{abs} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u) =f(u)& \text{in }\Omega,\\ u =0& \text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ and $f\colon [0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function. This is motivated by the work of Amann \cite{aman} and Shivaji \cite{shivaji1987remark}, where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when $f$ has a sublinear growth and $f(0)=0.$

Sufficient Turing instability conditions for the Schnakenberg system

A classical reaction-diffusion system, the Schnakenberg system, is under consideration in a bounded domain $\Omega\subset\mathbb{R}^m$ with Neumann boundary conditions. We study diffusion-driven instability of a stationary spatially homogeneous solution of this system, also called the Turing instability, which arises when the diffusion coefficient $d$ changes. An analytical description of the region of necessary and sufficient conditions for the Turing instability in the parameter plane is obtained by analyzing the linearized system in diffusionless and diffusion approximations. It is shown that one of the boundaries of the region of necessary conditions is an envelope of the family of curves that bound the region of sufficient conditions. Moreover, the intersection points of two consecutive curves of this family lie on a straight line whose slope depends on the eigenvalues of the Laplace operator and does not depend on the diffusion coefficient. We find an analytical expression for the critical diffusion coefficient at which the stability of the equilibrium position of the system is lost. We derive conditions under which the set of wavenumbers corresponding to neutral stability modes is countable, finite, or empty. It is shown that the semiaxis $d>1$ can be represented as a countable union of half-intervals with split points expressed in terms of the eigenvalues of the Laplace operator; each half-interval is characterized by the minimum wavenumber of loss of stability.

Dactylogram Processing System

Recognition of fingerprints (dactyloscopic images) is one of the practical application of signal processing. System of person identification by fingerprints is commonly-used by law enforcement bodies and Border services. This is also important in the field of access control systems and commercial devices where data security is not less important as reliability and data rate of processing algorithms. Existing systems of fingerprints processing are not fully ready for automatic recognition. Also, full modernization of existing equipment is not possible. The paper is devoted to the method of image processing. In particular, the preliminary processing of dactyloscopic images is considered as well as development of theoretical approach and practical realization of first stage of patterns forming – pre-processing of image for decreasing of its size and contrast increasing. The criteria for selecting ranges for sampling and quantization of images are given. Tasks of reducing the fingerprint image while increasing the contrast of the image were considered, analyzed and solved. Image reduction is based on the use of interpolation. It is shown that among the considered interpolation methods - linear, bilinear and bicubic - the latter one could provide the highest accuracy although it needs more hardware resources. However, when the dpi parameter (dots-per-inch) falls below 150, a rapid increase in the number of artifacts in the image is observed. Increasing of image sharpness is necessary for highlighting of colour transitions and consequently – for increasing the percentage of correct recognitions. Such increasing of image sharpness is proposed to achieve by using the Laplace operator (Laplasian calculation) and adding the result to the original image. The value of derivative at each pixel of the image depends linearly on sharpness level. Thus, it allows separating the areas with abrupt colour changes and gaps from the areas where the brightness is constant or changes slowly. The result of second derivative is much more for the areas with sharp changes than for the areas without them. The areas with constant or slowly-changing brightness after the second derivative calculation become almost the same dark colour. These areas could be restored to original image with retention of sharpness increasing effect. For this, transformed by Laplasian image should be added to the original one. Use of Laplasian allows to get an acceptable balance between the speed and computational complexity of the fingerprint recognition algorithm. The technical implementation of the device and illustration of its operation are given. Fingerprints image processing system is executed on the base of STM32f407 microcontroller with CortexM core. The system includes capasitive scanner, TFT LCD display and lab power source. The microcontroller software realizes, in particular, interpolation and contrast increasing. The system is module-compatible and able for scaling.