Set Of Planes Research Articles

On the impact of nanometric γ’ precipitates on the tensile deformation of superelastic Co49Ni21Ga30

Results are presented reporting on the martensite domain variant selection and stress-induced martensite morphology in [001]-oriented superelastic Co49Ni21Ga30 shape memory alloy (SMA) single crystals under tensile load. In situ neutron diffraction, as well as in situ optical- and confocal laser scanning microscopy were conducted focusing on three differently treated samples, i.e. in the as-grown, solution-annealed and aged condition. An aging treatment performed at 350 °C promotes the precipitation of nanoprecipitates. These second phase precipitates contribute to an increase of the number of habit plane interfaces, while reducing lamellar martensite plate thickness compared to the as-grown and solution-annealed (precipitate free) samples. During tensile loading, all samples show a stress-induced formation of martensite, characterized by one single domain variant (“detwinned”) and one set of parallel habit planes in a shear band. The results clearly show that γ’ nanoprecipitates do not necessarily promote multi-variant interaction during tensile loading. Thus, reduced recoverability in Co-Ni-Ga SMAs upon aging cannot be solely attributed to this kind of interaction as has been proposed in literature so far.

Billiard Ordered Games and Books

The aim of this work is to put together two novel concepts from the theory of integrable billiards: billiard ordered games and confocal billiard books. Billiard books appeared recently in the work of Fomenko's school, in particular of V. Vedyushkina. These more complex billiard domains are obtained by gluing planar sets bounded by arcs of confocal conics along common edges. Such domains are used in this paper to construct the configuration space for billiard ordered games. We analyse dynamical and topological properties of the systems obtained in that way.

Dynamic clustering of software defined network switches and controller placement using deep reinforcement learning

Software defined networking (SDN) has emerged as a promising alternative to the traditional networks, offering many advantages, including flexibility in network management, network programmability and guaranteeing application Quality-of-Service (QoS) requirements. In SDN, the control plane is separated from the data plane, and deployed as a logically centralized controller. However, due to the large scale of networks as well as latency and reliability requirements, it is necessary to deploy multiple controllers to satisfy these requirements. The distributed deployment of SDN controllers unveiled new challenges in terms of determining the number of controllers needed, their locations and the assignment of switches to controllers that minimizes flow set delay. In this context, we propose, in this paper, a new method that dynamically computes the optimal number of controllers, determines their optimal locations, and at the same time partitions the set of data plane switches into clusters and assigns them to these controllers. First, we mathematically formulate the controller placement as an optimization problem, whose objectives are to minimize the controller response time, that is the delay between the SDN controller and assigned switches, the Control Load (CL), the Intra-Cluster Delay (ICD) and the Intra-Cluster Throughput (ICT). Second, we propose a simple yet computationally efficient heuristic, called Deep Q-Network based Dynamic Clustering and Placement (DDCP), that leverages the potential of reinforcement and deep learning techniques to solve the aforementioned optimization problem. Experimental results using ONOS controller show that the proposed approach can significantly improve the network performances in terms of response time and resource utilization.

On the Hausdorff dimension of the residual set of a packing by smooth curves

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $\sigma$-finite Hausdorff 1-measure.

Identification of Enantiomers Using Low-Frequency Raman Spectroscopy.

Distinguishing between d and l enantiomers is of important scientific interest, especially for the pharmaceutical industry. Enantiomeric differentiation in the solid form is repeatedly presented as a challenge in the research community. Raman spectroscopy is a nondestructive tool, widely used for the characterization of different materials by probing their vibrational modes. The low-frequency region of the Raman spectrum reveals lattice-level interactions and global fluctuations in the molecule. Lower frequencies correspond to vibrations arising from weaker bonds and long-range interactions and hence are very susceptible to polarization changes. This work presents low-frequency Raman (LFR) spectroscopy as a facile technique to identify enantiomers. The optical setup of conventional Raman spectroscopy is engineered such that the excitation and collection geometries use an asymmetrical focal cone. In addition, a half-wave retarder is added to the excitation path and a Glan-Taylor polarizer is added to the collection path, and these modifications allow us to select the polarization plane for both excitation and collection geometries. The asymmetry in the foci when using a polarized beam for excitation provides different intensities of the collected signal for each polarization plane. In a calibrated system, one can define the chirality of an analyte by comparing the intensity of the LFR signal along orthogonal sets of polarization planes. For nonchiral molecules, the spectral intensity is always higher in the co-polarized plane when compared to the orthogonally depolarized plane, as expected. This contrast in the intensity of Raman spectra serves as a distinct tool for identifying enantiomers.

The Branch Set of Minimal Disks in Metric Spaces

AbstractWe study the structure of the branch set of solutions to Plateau’s problem in metric spaces satisfying a quadratic isoperimetric inequality. In our 1st result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi )^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak–Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau’s problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau’s problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.

ABELIAN DIFFERENCE SETS AS LATTICE COVERINGS AND LATTICE TILINGS

AbstractWe demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $ . This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

Peano dimension of fundamental groups

We define the Peano dimension for groups arising as fundamental groups, which generalizes the classical definition of geometric dimension of finitely presented groups. We conjecture that the Peano dimension of the fundamental group of a aspherical Peano continuum X is equal to the homotopy dimension of X. We prove the conjecture for one-dimensional or planar Peano continua. This answers a question posed by Cannon and Conner in 2007 concerning the homotopy dimension of planar sets.

Rigorous accounting diffraction on non-plane gratings irradiated by non-planar waves

The modified boundary integral equation method (MIM) is considered a rigorous theoretical application for the diffraction of cylindrical waves by arbitrary profiled plane gratings, as well as for the diffraction of plane/non-planar waves by concave/convex gratings. This study investigates 2D diffraction problems of the filiform source electromagnetic field scattered by a plane lamellar grating and of plane waves scattered by a similar cylindrical-shaped grating. Unlike the problem of plane wave diffraction by a plane grating, the field of a localised source does not satisfy the quasi-periodicity requirement. Fourier transform is used to reduce the solution of the problem of localised source diffraction by the grating in the whole region to the solution of the problem of diffraction inside one Floquet channel. By considering the periodicity of the geometry structure, the problem of Floquet terms for the image can be formulated so that it enables the application of the MIM developed for plane wave diffraction problems. Accounting of the local structure of an incident field enables both the prediction of the corresponding efficiencies and the specification of the bounds within which the approximation of the incident field with plane waves is correct. For 2D diffraction problems of the high-conductive plane grating irradiated by cylindrical waves and the cylindrical high-conductive grating irradiated by plane waves, decompositions in sets of plane waves/sections are investigated. The application of such decomposition, including the dependence on the number of plane waves/sections and radii of the grating and wave front shape, was demonstrated for lamellar, sinusoidal and saw-tooth grating examples in the 0th and −1st orders as well as in the transverse electric and transverse magnetic polarisations. The primary effects of plane wave/section partitions of non-planar wave fronts and curved grating shapes on the exact solutions for 2D and 3D (conical) diffraction problems are discussed.

Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When ∂C is C2 regardless of curvature, we prove that for every set PN of N points in T2 we have the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN1/2. When ∂C is only piecewise C2 and is not a polygon we prove the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN2/5. We also give a whole range of intermediate sharp results between N2/5 and N1/2. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.

Forced vibrations of a viscoelastic three-layer plate

In this article, forced vibrations of a viscoelastic three-layer plate of a particular type are considered and some solution of numerical computation by Maple is given. On the basis of the obtained refined equations of vibrations, the problem of harmonic vibrations of a trisyllabic plate is solved. Various methods and approaches are used to reduce the three-dimensional in spatial coordinates of the problem of the theory of plates to the two-dimensional one. The article is devoted to the development of the theory of symmetric vibrations of a three-layer elastic plate in a plane setting with respect to two unknown functions, which are the main parts of the displacements of some “intermediate” surface of the plate.

The Effect of Pre-Thermal and -Load Conditions on IN-718 High Temperature Fatigue Life

Ni-based superalloys are largely used in the aerospace industry as critical components for turbine engines due to their excellent mechanical properties and fatigue resistance at high temperatures. A hypothesis to explain this atypical characteristic among metals is the presence of a cross-slip mechanism. Previous work on the role of thermal activation on cubic slip has shown strain accommodation in two sets of slip planes, which resembled the activation of {100} cubic slip systems along of the octahedral slip planes {111} in Ni-based superalloys under high strain and temperature, exhibiting a more homogeneous strain distribution and less strain localization. Following those previous literature evaluations of initial conditions that can potentially activate cubic-slip planes and provide the level of accommodation and strain homogenization within the grain, this paper presents some experimental procedures and results of Ni-based superalloy (IN-718) tested at 500°C under operational loading condition, without and after being submitted to an overload and overtemperature. The experiments have shown that a pre-condition of 1% strain at 700°C would increase the fatigue life of the IN-718 at 500°C by four times when compared to pristine tested samples. The present results bring up the potential of improving this material fatigue performance, opening the need to further investigate the microstructure as the precondition is applied.

Higher dimensional spiral Delone sets

A Delone set in $\mathbb{R}^n$ is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point in $\mathbb{R}^n$ to the set is uniformly bounded above.Delone sets are thus sets of points enjoying nice spacing properties, and they therefore appear naturally in mathematical models for quasicrystals. Define a spiral set in $\mathbb{R}^n$ as a set of points of the form $\{\sqrt[n]{k}\cdot\bm{u}_k\}_{k\ge 1}$, where $(\bm{u}_k)_{k\ge 1}$ is a sequence in the unit sphere $\mathbb{S}^{n-1}$. In the planar case $n=2$, spiral sets serve as natural theoretical models in phyllotaxis (the study of configurations of leaves on a plant stem), and an important example in this class includes the sunflower spiral.Recent works by Akiyama, Marklof and Yudin provide a reasonable complete characterisation of planar spiral sets which are also Delone. A related problem that has emerged in several places in the literature over the past few years is to determine whether this theory can be extended to higher dimensions, and in particular to show the existence of spiral Delone sets in any dimension. This paper addresses this question by characterising the Delone property of a spiral set in terms of packing and covering conditions satisfied by the spherical sequence $(\mathbf{u}_k)_{k\ge 1}$. This allows for the construction of explicit examples of spiral Delone sets in $\mathbb{R}^n$ for all $n\ge 2$, which boils down to finding a sequence of points in $\mathbb{S}^{n-1}$ enjoying some optimal distribution properties.

Семейства изолиний в пространстве с прямоугольной метрикой

The paper is devoted to planar linear sets generated by linear conditions which are realized for rectangular or Manhattan metrics. The aim of this paper is to give an approach to researching the properties of linear conditions defined on Manhattan plane. Linear conditions are written as a finite sum of products Manhattan distances and real numerical factors. The paper presents a constructive method to solve the next general geometric problem. Let finite sets of linear figures (segments of line, polygons) are given on Manhattan plane at general positions. Find a planar set corresponding to given linear condition based on the given sets. The set of broken lines corresponding to given conditions generates a family of isolines for given sets. Constructive algorithm to build the family is suggested. The algorithm is based on the rectangular Hanan lattice and calculation numerical values in each cell. Some theorems applied to these cases are formulated in the paper.

Influence of Pitch Angle Errors in 3D Scene Reconstruction Based on U-V Disparity: A Sensitivity Study.

U-V disparity is a technique that is commonly used to detect obstacles in 3D scenes, modeling them as a set of vertical planes. In this paper, the authors describe the general lines of a method based on this technique for fully reconstructing 3D scenes, and conduct an analytical study of its performance and sensitivity to errors in the pitch angle of the stereoscopic vision system. The equations of the planes calculated for a given error in this angle yield the deviation with respect to the ideal planes (with a zero error in the angle) for a large test set consisting of planes with different orientations, which is represented graphically to analyze the method’s qualitative and quantitative performance. The relationship between the deviation of the planes and the error in the pitch angle is observed to be linear. Two major conclusions are drawn from this study: first, that the deviation between the calculated and ideal planes is always less than or equal to the error considered in the pitch angle; and second, that even though in some cases the deviation of the plane is zero or very small, the probability that a plane of the scene deviates from the ideal by the greatest amount possible, which matches the error in the pitch angle, is very high.

Expected size of random Tukey layers and convex layers

We study the Tukey layers and convex layers of a planar point set, which consists of n points independently and uniformly sampled from a convex polygon with k vertices. We show that the expected number of vertices on the first t Tukey layers is O(ktlog⁡(n/k)) and the expected number of vertices on the first t convex layers is O(kt3log⁡(n/(kt2))). We also show a lower bound of Ω(tlog⁡n) for both quantities in the special cases where k=3,4. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.

Minimizing capacity among linear images of rotationally invariant conductors

Logarithmic capacity is shown to be minimal for a planar set having N-fold rotational symmetry (\(N \ge 3\)), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is Pólya and Schiffer’s lower bound on capacity in terms of moment of inertia.

Refining and evaluating a Horvitz–Thompson-like stand density estimator in individual tree detection based on airborne laser scanning

Horvitz–Thompson-like stand density estimation is a method for estimating the stand density from tree crown objects extracted from airborne laser scanning data through individual tree detection. The estimator is based on stochastic geometry and mathematical morphology of the (planar) set formed by the detected tree crowns. This set is used to approximate the detection probabilities of trees. These probabilities are then used to calculate the estimate. The method includes a tuning parameter, which needs to be known to apply the method. We present a refinement of the method to allow more general detection conditions than those of previous papers. We also present and discuss the methods for estimating the tuning parameter of the estimator using a functional k-nearest neighbors method. We test the model fitting and prediction in two spatially separate data sets and examine the plot-level accuracy of estimation. The estimator produced a 13% lower RMSE (root-mean-square error) than the benchmark method in an external validation data set. We also analyze the effects of similarity and dissimilarity of training and validation data on the results.

Faster counting empty convex polygons in a planar point set

In this paper, we study the problem of counting the number of convex empty polygons in a given set of n points in general position in the plane. We present two algorithms: one counts the number of all convex empty polygons in O(T) time and the other counts the number of convex empty m-gons for all 3≤m≤k in O(kT) time, where T denotes the number of empty triangles in S. Note that T varies from Ω(n2) to O(n3), while its expected value is known to be Θ(n2) when n input points are chosen uniformly and independently at random from a convex and bounded body.

Stability of configurations in 3D

Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbours or angles formed by pairs of first neighbours. Such configurations are called angle-rigid. We tackle this question in the class of finite configurations in , seen as planar three-dimensional point sets. A sufficient condition preventing angle-rigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.