Convex Hulls of Random Walks

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in Rd. We prove convergence of the convex hull in the space of all convex and compact subsets of Rd, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk.

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

A machine learning model for screening thermodynamic stable lead-free halide double perovskites

We have investigated the problem of gender classification using a library of four hundred standard frontal facial images employing five classifiers, namely k-means, k-nearest neighbors, linear discriminant analysis (LDA), Mahalanobis distance based (MDB) classifiers and our modified KNN classifier. The image data independent discrete cosine transformation (DCT) basis is used for facial feature extraction. Areas under the convex hull (AUCH) of the classifiers are measured by varying the values of threshold for each feature subset in the receiver operating characteristics (ROC) curve. The scalar values of AUCH of the ROC curve increases with increasing number of features. More features yield a better representation of the gender facial image. The overall performance of classifiers is compared with different values of AUCH versus features under different conditions. It has been observed that when the number of features is increased beyond 5, AUCH starts to saturate. Our experimental results demonstrate that modified-KNN performs better than the rest of the conventional classifiers under all conditions. The LDA classifier did not perform well in the DCT domain; however, it gradually improved its performance with increasing number of features

We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general.Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise.The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Lévy processes with finite Laplace transform.

The isoperimetric problem for convex hulls and large deviations rate functionals of random walks

We prove a general theorem for the precise rate at which the convex hull of Brownian motion gets created. The latter result relates large deviation theory to P. Levy's geometric proof of Strassen's law of the iterated logarithm. This also answers a question of S. Evans. Moreover, we give a partial solution to a question of J. Hammersley and P. Levy regarding the slowness of the growth of the hull process. Several examples, some classical and some new, are given to illustrate the theorems. Finally, we present applications to the convex hull of random walks ind dimensions.

When the wing is treated as a composite sandwich plate and it is assumed that the wing chordwise section is rigid, mathematical formulations for the stiffened composite multicell wing structures are provided, and associated governing equations for the aeroelastic divergence are also derived by a direct approach. When the matrix notation is used, the system of equations is written into an explicit and simple mathematical expression that can then be solved explicitly by using the technique of Laplace integral transform. In this modeling, the composite wing skins and stringers (including the spar flanges) are simulated as the faces, whereas the spar webs and ribs are simulated as the core of the sandwiches. Because the wing cross section must have a streamline shape, unlike the usual uniform thickness modeling, it is more appropriate to simulate the wings as variable thickness sandwiches where the thickness is a function of the airfoil. Moreover, as is usual for assumptions for the sandwich plates, the effects of the transverse shear deformation are also considered. Because several special conditions have been studied in the literature, we first compare our solutions with some existing solutions. To show the generality, several illustrative examples are given to consider the effects of spars, skins (including the ply orientation and stacking sequence), stringers, swept angles, aspect ratio, shape of airfoil, and the warping restraints on the divergence dynamic pressures and the lift loads redistribution.

An alternative conclusion to the work of W. R. Hoegy [“Thermal electron heating rate: A derivation,” J. Geophys. Res. 89(A2), 977–985 (1984)] is presented, and rigorous analytical expressions for heating thermal Maxwellian electrons by the fast part of the energy distribution function (EDF) with an arbitrary dependence on energy are obtained. This makes the calculation of thermal electron heating under specific conditions possible, as well as scaling the obtained results. For the most convenient bi-Maxwell approximation of the EDF, comparatively simple analytical expressions for heating, convenient for practical calculations, are obtained. It has been shown that these expressions are in good agreement with the results of the fundamental Hoegy's work. Analytical expressions for heating thermal electrons by fast electrons, with a power-law distribution dependence on energy, are also presented.

An equilibrium model for the radial intensity distribution of analyte lines in the ICP discharge

To better understand degradation in electrochemical converters and helping to correlate certain phenomena with specific operating conditions, machine learning (ML) methods are increasingly being applied. Success has already been achieved in the field of degradation analysis and prediction of capacity of lithium ion batteries1, for instance. In terms of Solid Oxide Cell (SOC) stacks ML methods have been applied mainly with the aim of identification of faulty operation modes and degradation related fault diagnosis2. ML approaches usually require a considerable amount of real training data, when used for forecasting models. A data consolidation and curation strategy was developed with the aim of processing the historic long-term test bench data of SOCs collected by Forschungszentrum Jülich over the past years. In comparison to other datasets developed in this field3, the one presented in this work contains SOC stack tests in fuel cell operation with significantly longer operating times under load. A compilation of the sample experiments and the consolidation into a hierarchical data format are presented. Further, an essential part of the strategy is the automatic curation and analysis of electrochemical impedance spectroscopy (EIS) measurements, using a specifically developed procedure in Python. The varying quality of measurements from past years, as well as recurring artefacts such as parasitic inductances, can be addressed in this way. Additional distribution of relaxation times (DRT) deconvolutions and equivalent circuit modelling (ECM) are performed, as part of the procedure to automatically retrieve feature values from measurements (cf. Fig. 1). The novel dataset, which to the authors’ knowledge includes some of the longest SOC stack tests available, serves as the basis for several evaluations. In addition to classification and clustering work to derive degradation patterns, in particular based on the EIS data, another focus is on the development of forecasting models. The current work is primarily concerned with long short-term memory (LSTM), as well as regression models that make use of both the time series data and the characterisation measurements, such as EIS. Acknowledgement The authors would like to thank their colleagues at Forschungszentrum Jülich GmbH for their great support and the Helmholtz Society as well as the German Federal Ministry of Education and Research for financing these activities as part of the WirLebenSOFC project (03SF0622B). References 1: Jones, P.K., Stimming, U. & Lee, A.A. Impedance-based forecasting of lithium-ion battery performance amid uneven usage. Nature Communications 13, 4806 (2022).2: B. Yang et al. Solid oxide fuel cell systems fault diagnosis: Critical summarization, classification, and perspectives. Journal of Energy Storage 34, 102153 (2021).3: A.K. Padinjarethil, S. Pollok & A. Hagen. Degradation studies using machine learning on novel solid oxide cell database. Fuel Cells 21, 566–576 (2021). Figure caption: Fig.1: Flow diagram of EIS data curation pipeline and curation results for example EIS measurement. Figure 1

In this work, three variant Time Domain Random Walk (TDRW) algorithms were developed for the problem of solute transport in a single fracture-matrix system, where an arbitrary inlet boundary condition can be applied. One approach performs an additional evaluation of integral in terms of injection boundary and the solution to a Dirac delta case. One method makes use of two functions, dependent on specific boundary conditions, to estimate the particle arrival time. The other additionally introduces the concept of solute injection time, resulting from the injection boundary, into the calculation of particle arrival time. To validate the developed variant algorithms, two benchmark cases are considered with respect to a general Dirichlet injection mode and a Robin injection boundary, respectively. The results from three approaches all make a good agreement with those of inverse Laplace transform method. However, the Monte Carlo nature of the TDRW algorithm implies that the accuracy of the computational result is highly dependent on the number of particles applied in the simulation.

Equivalences in strong limit theorems for renewal counting processes

Let \(\xi _1,\xi _2,\ldots \) be a sequence of independent copies of a random vector in \(\mathbb {R}^d\) having an absolutely continuous distribution. Consider a random walk \(S_i:=\xi _1+\cdots +\xi _i\), and let \(C_{n,d}:={{\,\mathrm{conv}\,}}(0,S_1,S_2,\ldots ,S_n)\) be the convex hull of the first \(n+1\) points it has visited. The polytope \(C_{n,d}\) is called k-neighborly if for any indices \(0\le i_1<\cdots < i_k\le n\) the convex hull of the k points \(S_{i_1},\ldots , S_{i_k}\) is a \((k-1)\)-dimensional face of \(C_{n,d}\). We study the probability that \(C_{n,d}\) is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to \(\infty \). There is an explicit formula for the expected number of \((k-1)\)-dimensional faces of \(C_{n,d}\) which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of \(C_{n,d}\). This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.

This article studies discrete height functions on the discrete hypertorus. These are functions on the vertices of this hypertorus graph for which the derivative satisfies a specific condition on each edge. We then perform a random walk on the set of such height functions, in the spirit of Diffusivity of a random walk on random walks, a work of Boissard, Cohen, Espinasse, and Norris. The goal is to estimate the diffusivity of this random walk in the mesh limit. It turns out that each height functions is characterised by a number of so-called fractures of the hypertorus. These fractures are then studied in isolation; we are able to understand their asymptotic behaviour in the mesh limit due to the recent understanding of the associated random surfaces. This allows for an asymptotic reduction to a one-dimensional continuous system consisting of $\operatorname{gcd}\mathbf n$ parts where $\mathbf n\in\mathbb N^d$ is the fundamental parameter of the original model. We then prove that the diffusivity of the random walk tends to $1/(1+2\operatorname{gcd}\mathbf n)$ in this mesh limit.