Search Results (14)

This paper considers some innovative theoretical features and practical applications of the normal system of equations used for estimating parameters in multiple linear regression. The Laplace expansion of a determinant by cofactors and double Laplace expansion are employed for resolving the normal system. Additional features are described, including the ridge regularization applied directly to the normal system, geometric interpretation as a unique hyperplane through the points of special weighted means, Mahalanobis distances from observations to these means for the linear link functions, and multidimensional interpolation. The found properties are useful for a better understanding and interpretation of multiple regression, and the numerical examples demonstrate convenience and applicability of these tools in data modeling. Full article

This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface $L{H}_{\mathit{\beta }}$, a critical lightlike surface $L{S}_{\mathit{\beta }}$, and a degenerate curve $L{C}_{\mathit{\beta }}$, with dimensions decreasing from 3D to 1D. Using singularity theory, we identify a novel geometric invariant $\sigma \left(t\right)$ that governs the emergence of specific singularity types, including $C(2,3)\times {\mathbb{R}}^2$, $SW\times \mathbb{R}$, $BF$, $C\left(BF\right)$, $C(2,3,4)\times \mathbb{R}$, and $(2,3,4,5)$-cusp. These singularities exhibit increasing degeneracy as the hierarchy progresses, with contact orders between the lightlike hyperplane $H{S}_{t_0}^L$ and the curve $\mathit{\beta }$ systematically intensifying. An explicit example demonstrates the construction of these objects and validates the theoretical results. This work establishes a systematic connection between null Cartan curves, stratified singularities, and contact geometry. Full article

This study explores a unique Finsler space with a Rander’s-type exponential metric, $\mathcal{G}(\alpha ,\beta )=(\alpha +\beta )e^{{\displaystyle \frac{\beta }{(\alpha +\beta )}}}$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form. We analyze the conditions under which its hypersurfaces behave like hyperplanes of the first, second, and third kinds. Additionally, we examine the reducibility of the Cartan tensor $\mathcal{C}$ for these hypersurfaces, providing insights into their geometric structure. Full article

This article considers a class of entire functions of several complex variables that are bounded in the Cartesian product of some half-planes. Each such hyperplane is defined on the condition that the real part of the corresponding variable is less than some r. For this class of functions, there are established analogs of the Wiman theorems. The first result describes the behavior of an entire function from the given class at the neighborhood of the point of the supremum of its modulus. The second result shows asymptotic equality for supremums of the modulus of the function and its real part outside some exceptional set. In addition, the analogs of Wiman’s theorem are obtained for entire multiple Dirichlet series with arbitrary non-negative exponents. These results are obtained as consequences of a new statement describing the behavior of an entire function $F\left(z\right)$ of several complex variables $z=(z_1,\dots ,z_p)$ at the neighborhood of a point w, where the value $F\left(w\right)$ is close to the supremum of its modulus on the boundary of polylinear domains. The paper has two moments of novelty: the results use a more general geometric exhaustion of p-dimensional complex space by polylinear domains than previously known; another aspect of novelty concerns the results obtained for entire multiple Dirichlet series. There is no restriction that every component of exponents is strictly increasing. These statements are valid for any non-negative exponents. Full article

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space ${\mathbb{R}}^n$ with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and spheres. The natural space for encoding such geometric objects is the real projective quadric with signature $(n+1,2)$. In this article, we first provide a general formula for calculating the angles formed by the geometric objects encoded by the points of the quadric. The main result is the determination of a very simple parametrization of a Laguerre-type subgroup that acts transitively on the quadric while preserving the geometric nature of its points. That is, points of the quadric representing oriented spheres, points, and oriented hyperplanes in ${\mathbb{R}}^n$ are mapped by the action to points of the same geometric type. We also provide simple parametrizations of the isotropies of the action. The action described in this article provides the foundation for an effective solution to geometric constraint problems. Full article

Loop closure detection plays an important role in the construction of reliable maps for intelligent agricultural machinery equipment. With the combination of convolutional neural networks (CNN), its accuracy and real-time performance are better than those based on traditional manual features. However, due to the use of small embedded devices in agricultural machinery and the need to handle multiple tasks simultaneously, achieving optimal response speeds becomes challenging, especially when operating on large networks. This emphasizes the need to study in depth the kind of lightweight CNN loop closure detection algorithm more suitable for intelligent agricultural machinery. This paper compares a variety of loop closure detection based on lightweight CNN features. Specifically, we prove that GhostNet with feature reuse can extract image features with both high-dimensional semantic information and low-dimensional geometric information, which can significantly improve the loop closure detection accuracy and real-time performance. To further enhance the speed of detection, we implement Multi-Probe Random Hyperplane Local Sensitive Hashing (LSH) algorithms. We evaluate our approach using both a public dataset and a proprietary greenhouse dataset, employing an incremental data processing method. The results demonstrate that GhostNet and the Linear Scanning Multi-Probe LSH algorithm synergize to meet the precision and real-time requirements of agricultural closed-loop detection. Full article

Modern video surveillance systems mainly rely on human operators to monitor and interpret the behavior of individuals in real time, which may lead to severe delays in responding to an emergency. Therefore, there is a need for continued research into the designing of interpretable and more transparent emotion recognition models that can effectively detect emotions in safety video surveillance systems. This study proposes a novel technique incorporating a straightforward model for detecting sudden changes in a person’s emotional state using low-resolution photos and video frames from surveillance cameras. The proposed technique includes a method of the geometric interpretation of facial areas to extract features of facial expression, the method of hyperplane classification for identifying emotional states in the feature vector space, and the principles of visual analytics and “human in the loop” to obtain transparent and interpretable classifiers. The experimental testing using the developed software prototype validates the scientific claims of the proposed technique. Its implementation improves the reliability of abnormal behavior detection via facial expressions by 0.91–2.20%, depending on different emotions and environmental conditions. Moreover, it decreases the error probability in identifying sudden emotional shifts by 0.23–2.21% compared to existing counterparts. Future research will aim to improve the approach quantitatively and address the limitations discussed in this paper. Full article

We study certain physically-relevant subgeometries of binary symplectic polar spaces $W(2N-1,2)$ of small rank N, when the points of these spaces canonically encode N-qubit observables. Key characteristics of a subspace of such a space $W(2N-1,2)$ are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of $W(2N-1,2)$ and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of $W(2N-1,2)$ whose rank is $N-1$. $W(3,2)$ features three negative lines of the same type and its $W(1,2)$’s are of five different types. $W(5,2)$ is endowed with 90 negative lines of two types and its $W(3,2)$’s split into 13 types. A total of 279 out of 480 $W(3,2)$’s with three negative lines are composite, i.e., they all originate from the two-qubit $W(3,2)$. Given a three-qubit $W(3,2)$ and any of its geometric hyperplanes, there are three other $W(3,2)$’s possessing the same hyperplane. The same holds if a geometric hyperplane is replaced by a ‘planar’ tricentric triad. A hyperbolic quadric of $W(5,2)$ is found to host particular sets of seven $W(3,2)$’s, each of them being uniquely tied to a Conwell heptad with respect to the quadric. There is also a particular type of $W(3,2)$’s, a representative of which features a point each line through which is negative. Finally, $W(7,2)$ is found to possess 1908 negative lines of five types and its $W(5,2)$’s fall into as many as 29 types. A total of 1524 out of 1560 $W(5,2)$’s with 90 negative lines originate from the three-qubit $W(5,2)$. Remarkably, the difference in the number of negative lines for any two distinct types of four-qubit $W(5,2)$’s is a multiple of four. Full article

In this paper, we propose a novel binary classification method called the kernel-free quadratic surface minimax probability machine (QSMPM), that makes use of the kernel-free techniques of the quadratic surface support vector machine (QSSVM) and inherits the advantage of the minimax probability machine (MPM) without any parameters. Specifically, it attempts to find a quadratic hypersurface that separates two classes of samples with maximum probability. However, the optimization problem derived directly was too difficult to solve. Therefore, a nonlinear transformation was introduced to change the quadratic function involved into a linear function. Through such processing, our optimization problem finally became a second-order cone programming problem, which was solved efficiently by an alternate iteration method. It should be pointed out that our method is both kernel-free and parameter-free, making it easy to use. In addition, the quadratic hypersurface obtained by our method was allowed to be any general form of quadratic hypersurface. It has better interpretability than the methods with the kernel function. Finally, in order to demonstrate the geometric interpretation of our QSMPM, five artificial datasets were implemented, including showing the ability to obtain a linear separating hyperplane. Furthermore, numerical experiments on benchmark datasets confirmed that the proposed method had better accuracy and less CPU time than corresponding methods. Full article

The Cayley–Klein (CK) formalism is applied to the real algebra $\mathfrak{so}(5)$ by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for $\mathfrak{so}(5)$ together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter $\mathfrak{so}(4,1)$ and anti-de Sitter $\mathfrak{so}(3,2)$ as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature $\mathsf{\Lambda }$ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all $\kappa$-deformations as particular cases. Full article

The geometric tolerance of notching machines used in the fabrication of components for induction motor stators and rotators is less than 50 µm. The blunt edges of worn molds can cause the edge of the sheet metal to form a burr, which can seriously impede assembly and reduce the efficiency of the resulting motor. The overuse of molds without sufficient maintenance leads to wasted sheet material, whereas excessive maintenance shortens the life of the punch/die plate. Diagnosing the mechanical performance of die molds requires extensive experience and fine-grained sensor data. In this study, we embedded polyvinylidene fluoride (PVDF) films within the mechanical mold of a notching machine to obtain direct measurements of the reaction forces imposed by the punch. We also developed an automated diagnosis program based on a support vector machine (SVM) to characterize the performance of the mechanical mold. The proposed cyber-physical system (CPS) facilitated the real-time monitoring of machinery for preventative maintenance as well as the implementation of early warning alarms. The cloud server used to gather mold-related data also generated data logs for managers. The hyperplane of the CPS-PVDF was calibrated using a variety of parameters pertaining to the edge characteristics of punches. Stereo-microscopy analysis of the punched workpiece verified that the accuracy of the fault classification was 97.6%. Full article

In the paper, the authors derive an integral representation, present a double inequality, supply an asymptotic formula, find an inequality, and verify complete monotonicity of a function involving the gamma function and originating from geometric probability for pairs of hyperplanes intersecting with a convex body. Full article

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups. Full article

Based on geometric invariance properties, we derive an explicit prior distribution for the parameters of multivariate linear regression problems in the absence of further prior information. The problem is formulated as a rotationally-invariant distribution of \(L\)-dimensional hyperplanes in \(N\) dimensions, and the associated system of partial differential equations is solved. The derived prior distribution generalizes the already known special cases, e.g., 2D plane in three dimensions. Full article