An improved Aleksandrov–Bakel’man–Pucci estimate for a second-order elliptic operator with unbounded drift

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue [Formula: see text]-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue [Formula: see text]-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.

Seamless Detection of Cutoff Low and Preexisting Trough

<p><strong>Cutoff lows are cyclones existing in the upper troposphere developing from precursory preexisting troughs. We introduce a new method to seamlessly detect cutoff lows and even preexisting troughs aiming to improve lead time of meso scale disturbances like tornadoes. The method is based on a geometric character; in this method, a slope defined as the tangential line from a minimum point of each height depression is measured on an isobaric surface. This slope evaluates an intensity and horizontal extension (radius) of each depression. Adopting a mathematical assumption, we successfully achieved to make an algorithm to separate the depression and the local background flow. To remove the background flow enables us to detect both cutoff lows and preexisting troughs seamlessly in reanalysis height fields. So, our method would allow the life cycle to be illustrated continuously from the birth of the cutoff low, that is, from the precursory preexisting trough, and is expected to contribute to the improvement of the lead time for predicting severe weathers. Some further application examples, including tornado accompanying cases, and even for blocking highs, would be shown.</strong></p>

On Optimal Interpolation by Linear Functions on an n-Dimensional Cube

Let \(n\in{\mathbb N}\), and let \(Q_n\) be the unit cube \([0,1]^n\). By \(C(Q_n)\) we denote the space of continuous functions \(f:Q_n\to{\mathbb R}\) with the norm \(\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,\) by \(\Pi_1\left({\mathbb R}^n\right)\) --- the set of polynomials of \(n\) variables of degree \(\leq 1\) (or linear functions). Let \(x^{(j)},\) \(1\leq j\leq n+1,\) be the vertices of \(n\)-dimnsional nondegenerate simplex \(S\subset Q_n\). An interpolation projector \(P:C(Q_n)\to \Pi_1({\mathbb R}^n)\) corresponding to the simplex \(S\) is defined by equalities \(Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).\) The norm of \(P\) as an operator from \(C(Q_n)\) to \(C(Q_n)\) may be calculated by the formula \(\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.\) Here \(\lambda_j\) are the basic Lagrange polynomials with respect to \(S,\) \(ver(Q_n)\) is the set of vertices of \(Q_n\). Let us denote by \(\theta_n\) the minimal possible value of \(\|P\|.\) Earlier, the first author proved various relations and estimates for values \(\|P\|\) and \(\theta_n\), in particular, having geometric character. The equivalence \(\theta_n\asymp \sqrt{n}\) takes place. For example, the appropriate, according to dimension \(n\), inequalities may be written in the form \linebreak \(\frac{1}{4}\sqrt{n}\) \(<\theta_n\) \(<3\sqrt{n}.\) If the nodes of the projector \(P^*\) coincide with vertices of an arbitrary simplex with maximum possible volume, we have \(\|P^*\|\asymp\theta_n.\)When an Hadamard matrix of order \(n+1\) exists, holds \(\theta_n\leq\sqrt{n+1}.\) In the paper, we give more precise upper bounds of numbers \(\theta_n\) for \(21\leq n \leq 26\). These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements \(\pm 1.\) Also, we systematize and comment the best nowaday upper and low estimates of numbers \(\theta_n\) for a concrete \(n.\)

Permeability and porosity models of bi-fractal porous media

In previous studies, it is found that the frame and pore in porous media both possess the fractal geometric character. So the permeability and porosity models of bi-fractal porous media are derived based on the assumption that a porous media consists of fractal solid clusters and capillary bundles. The expressions of presented models are constituted by the fractal parameters of solid cluster and those of capillary bundle. Good agreement between model predictions and experimental data is obtained. This verifies the validity of the permeability and porosity models for bi-fractal porous media. The sensitive parameters that influence the permeability and porosity are specified, and their effects on the relationship between permeability and porosity are discussed.

Concept System and Function of Dynamic Symmetry in Mechanical Product Structure

Dynamic symmetry is a common phenomenon in mechanical product structures, and is used for describing the regular motion process over time in mechanical systems. It plays an important role in implementing product functions, transfering and transforming movement motion, increasing the strength of product functions, expanding the scope of the functions. For the case that the lack of research about dynamic symmetry and imperfection of symmetric system, based on the analysis of instances, a new concept system of mechanical dynamic structure symmetry was established by taking the different properties of the geometric character and the time character of the motion process as standards. The concept system is made up of rotation dynamic symmetry, translation dynamic symmetry, scaling dynamic symmetry, combinatorial dynamic symmetry, time-translation dynamic symmetry and time-inversion dynamic symmetry. Among the concept systemthe author has a symmetry theory analysis of many instances to verify the validity and rationality of the established symmetry concept system. Finally, the functions of dynamic symmetry in realizing the function of transferring or transforming motion, matching the space-time characters and improving the performance of mechanical product functions was proposed. The concept system can completely describes the existences of dynamic structure symmetries in mechanical systems, and can offer an academic basis for further research on the functions and application laws of dynamic structure symmetry in mechanical systems.

Using origami design principles to fold reprogrammable mechanical metamaterials

Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami’s scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.

The Flexible Control System Development of the Sawing and Milling Machining Center for PVC Door and Window Profile

In the paper, with the secondary development of FANUC-0MD system, the flexible control system which meets the requirements of the sawing and milling machining center for PVC door and window profile is realized. Based on researching the structure, the function, the process of the technologies and the control requirements of the machining center, the CNC system has implemented, including the machine tool origin and work piece coordinate origin establishment and auxiliary control system M code development. In this flexible control system, through studying the geometric character, the technical character and the mutual relation between them, various windows shape on the structure and the machining figuration is possessed of. We gain the feature subclasses aggregate of the machining technology of windows shape. Consequently, CAM system is achieved.

Geometric algorithms for sensor networks

This paper surveys the use of geometric methods for wireless sensor networks. The close relationship of sensor nodes with their embedded physical space imposes a unique geometric character on such systems. The physical locations of the sensor nodes greatly impact on system design in all aspects, from low-level networking and organization to high-level information processing and applications. This paper reviews work in the past 10 years on topics such as network localization, geometric routing, information discovery, data-centric routing and topology discovery.

Geometric character of domain and the Harnack inequality of solution of a singular parabolic equation

The geometric character of domains of solutions of a singular parabolic equation with the Neumann boundary condition are discussed in this work. We prove a gradient estimate and a Harnack type inequality and then, show that the domains of the solutions are exactly columns in the space-time. Specially, the altitudes of the columns are calculated accurately.