ONE DIMENSIONAL BROWNIAN MOTION WITH HOLDING AND JUMPING BOUNDARY

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

On a Subclass of Analytic Functions That Are Starlike with Respect to a Boundary Point Involving Exponential Function

In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc D ≔ ς ∈ ℂ : ς < 1 , by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.

Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher.

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.

TOOTH LEG EXCESSIVE UNDERCUT ELIMINATION IN HELICAL CYLINDRICAL GEARS WITH PROTUBERANCE OF HOBBING CUTTER BASED ON GRAPHIC RUN-IN

In the paper there are considered procedures for designing a transition curved tooth leg of helical cylindrical gears. A significant parameter of a transition curve is a diameter of boundary points. The boundary point diameter belongs to a bottom point of the involute profile of the teeth side surface of a gear ring. The boundary point position must be lower of the design end point of the involute profile defined by the designer of gearing. A diameter value depends upon a great number of production factors: a profile and wear of a grinding disk, setting up parameters, teeth machining modes of a gear ring, but it is impossible to ensure the specified values of the diameter of boundary points without a correct design solution in the course of the form choice of milling cutter protuberance. The solution on protuberance acceptable parameters of a gear-cutting tool is made by the designer of a cutter during graphic run-in fulfillment. In the paper there are revealed conditions under which arise mistakes in the course of graphic run-in fulfillment within the limits of one teeth pitch of a milling cutter. There are shown recommendations for the fulfillment ensuring the diameter dimension of boundary points of the transition curve specified by the designer of gearing. The data on the design parameter impact of the hob protuberance upon the continuance of cutting edge interaction are shown. There are recommendations given to prevent undercut arising caused by the fulfillment of graphic two-dimensional run-ins of cylindrical helical gears. The work purpose: the elimination of tooth leg excessive undercut in helical cylindrical gears with the protuberance of a worm milling cutter at the expense of the fulfillment of graphic run-in conditions. The investigation methods: the graphical modeling of a run-in process. The investigation results and novelty: there are defined conditions of arising an excessive undercut in the tooth leg of helical cylindrical gears during the fulfillment of graphic run-ins of a tool rack. The conclusions: for mistake prevention in the calculations of the protuberance geometrical parameters of the helical milling cutter the graphic run-in must be carried out not less than on the 1.5 pitch of the milling cutter.

Effective Selection of Variable Point Neighbourhood for Feature Point Extraction from Aerial Building Point Cloud Data

Existing approaches that extract buildings from point cloud data do not select the appropriate neighbourhood for estimation of normals on individual points. However, the success of these approaches depends on correct estimation of the normal vector. In most cases, a fixed neighbourhood is selected without considering the geometric structure of the object and the distribution of the input point cloud. Thus, considering the object structure and the heterogeneous distribution of the point cloud, this paper proposes a new effective approach for selecting a minimal neighbourhood, which can vary for each input point. For each point, a minimal number of neighbouring points are iteratively selected. At each iteration, based on the calculated standard deviation from a fitted 3D line to the selected points, a decision is made adaptively about the neighbourhood. The selected minimal neighbouring points make the calculation of the normal vector accurate. The direction of the normal vector is then used to calculate the inside fold feature points. In addition, the Euclidean distance from a point to the calculated mean of its neighbouring points is used to make a decision about the boundary point. In the context of the accuracy evaluation, the experimental results confirm the competitive performance of the proposed approach of neighbourhood selection over the state-of-the-art methods. Based on our generated ground truth data, the proposed fold and boundary point extraction techniques show more than 90% F1-scores.