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Journal articles on the topic 'Hyperplane projection method'
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1
Zheng, Lian. "New Double Projection Algorithm for Solving Variational Inequalities." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/714397.
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We propose a class of new double projection algorithms for solving variational inequality problem, which can be viewed as a framework of the method of Solodov and Svaiter by adopting a class of new hyperplanes. By the separation property of hyperplane, our method is proved to be globally convergent under very mild assumptions. In addition, we propose a modified version of our algorithm that finds a solution of variational inequality which is also a fixed point of a given nonexpansive mapping. If, in addition, a certain local error bound holds, we analyze the convergence rate of the iterative sequence. Numerical experiments prove that our algorithms are efficient.
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Ye, Ming-Lu. "A Cutting Hyperplane Projection Method for Solving Generalized Quasi-Variational Inequalities." Journal of the Operations Research Society of China 4, no. 4 (2016): 483–501. http://dx.doi.org/10.1007/s40305-016-0123-5.
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Chen, Lu, Bin Xin, and Jie Chen. "A Tradeoff-Based Interactive Multi-Objective Optimization Method Driven by Evolutionary Algorithms." Journal of Advanced Computational Intelligence and Intelligent Informatics 21, no. 2 (2017): 284–92. http://dx.doi.org/10.20965/jaciii.2017.p0284.
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Multi-objective optimization problems involve two or more conflicting objectives, and they have a set of Pareto optimal solutions instead of a single optimal solution. In order to support the decision maker (DM) to find his/her most preferred solution, we propose an interactive multi-objective optimization method based on the DM’s preferences in the form of indifference tradeoffs. The method combines evolutionary algorithms with the gradient-based interactive step tradeoff (GRIST) method. An evolutionary algorithm is used to generate an approximate Pareto optimal solution at each iteration. The DM is asked to provide indifference tradeoffs whose projection onto the tangent hyperplane of the Pareto front provides a tradeoff direction. An approach for approximating the normal vector of the tangent hyperplane is proposed which is used to calculate the projection. A water quality management problem is used to demonstrate the interaction process of the interactive method. In addition, three benchmark problems are used to test the accuracy of the normal vector approximation approach and compare the proposed method with GRIST.
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Liu, San-Yang, Yuan-Yuan Huang, and Hong-Wei Jiao. "Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/305643.
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Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.
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Ihm, Sun-Young, Aziz Nasridinov, and Young-Ho Park. "Grid-PPPS: A Skyline Method for Efficiently Handling Top-kQueries in Internet of Things." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/401618.
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A rapid development in wireless communication and radio frequency technology has enabled the Internet of Things (IoT) to enter every aspect of our life. However, as more and more sensors get connected to the Internet, they generate huge amounts of data. Thus, widespread deployment of IoT requires development of solutions for analyzing the potentially huge amounts of data they generate. A top-kquery processing can be applied to facilitate this task. The top-kqueries retrievektuples with the lowest or the highest scores among all of the tuples in the database. There are many methods to answer top-kqueries, where skyline methods are efficient when considering all attribute values of tuples. The representative skyline methods are soft-filter-skyline (SFS) algorithm, angle-based space partitioning (ABSP), and plane-project-parallel-skyline (PPPS). Among them, PPPS improves ABSP by partitioning data space into a number of spaces using hyperplane projection. However, PPPS has a high index building time in high-dimensional databases. In this paper, we propose a new skyline method (called Grid-PPPS) for efficiently handling top-kqueries in IoT applications. The proposed method first performs grid-based partitioning on data space and then partitions it once again using hyperplane projection. Experimental results show that our method improves the index building time compared to the existing state-of-the-art methods.
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Zhou, Yingjie, Yulun Wu, and Xiangrong Li. "A New Hybrid PRPFR Conjugate Gradient Method for Solving Nonlinear Monotone Equations and Image Restoration Problems." Mathematical Problems in Engineering 2020 (September 25, 2020): 1–13. http://dx.doi.org/10.1155/2020/6391321.
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A new hybrid PRPFR conjugate gradient method is presented in this paper, which is designed such that it owns sufficient descent property and trust region property. This method can be considered as a convex combination of the PRP method and the FR method while using the hyperplane projection technique. Under accelerated step length, the global convergence property is gained with some appropriate assumptions. Comparing with other methods, the numerical experiments show that the PRPFR method is more competitive for solving nonlinear equations and image restoration problems.
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Zhou, Hao, and Jin Ping Sun. "A New SVM Considering the Sample Distribution." Advanced Materials Research 403-408 (November 2011): 1302–5. http://dx.doi.org/10.4028/www.scientific.net/amr.403-408.1302.
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As the continuation of former work, a new SVM is discussed. It is similar to the Fisher Discriminant Analysis (FDA) that the normal vector of SVM separating hyperplane is a one-dimension vector in the high-dimension hidden space, which can be used as a projection orientation to classify data. So, after the SVM training, the projection of samples can be calculated by kernel function. Finally, the threshold of the classifier is ascertained according to the distribution of projection. With the sample distribution considered, SVM performance is improved. Simulation manifests the validity of the method in this paper.
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Feng, Haishan, and Tingting Li. "An Accelerated Conjugate Gradient Algorithm for Solving Nonlinear Monotone Equations and Image Restoration Problems." Mathematical Problems in Engineering 2020 (October 5, 2020): 1–12. http://dx.doi.org/10.1155/2020/7945467.
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Combining the three-term conjugate gradient method of Yuan and Zhang and the acceleration step length of Andrei with the hyperplane projection method of Solodov and Svaiter, we propose an accelerated conjugate gradient algorithm for solving nonlinear monotone equations in this paper. The presented algorithm has the following properties: (i) All search directions generated by the algorithm satisfy the sufficient descent and trust region properties independent of the line search technique. (ii) A derivative-free search technique is proposed along the direction to obtain the step length αk. (iii) If ϕk=−αkhk−hwkTdk>0, then an acceleration scheme is used to modify the step length in a multiplicative manner and create a point. (iv) If the point satisfies the given condition, then it is the next point; otherwise, the hyperplane projection technique is used to obtain the next point. (v) The global convergence of the proposed algorithm is established under some suitable conditions. Numerical comparisons with other conjugate gradient algorithms show that the accelerated computing scheme is more competitive. In addition, the presented algorithm can also be applied to image restoration.
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Morkun, V. S., N. V. Morkun, and A. V. Pikilnyak. "AUTOMATIC CONTROL FORMATION OF THE ULTRASONIC OSCILLATIONS INTENSITY IN THE FLOTATION PROCESS." Metallurgicheskaya i gornorudnaya promyshlennost, №5, 2018, no. 5 (October 2018): 58–66. http://dx.doi.org/10.33101/s005-4789456.
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Purpose. The method development of adaptive control of the high-energy ultrasound intensity and frequency, which are formed by means of an ultrasonic phased array. Methodology. The problem of automatic control automation for the ultrasonic oscillations intensity in flotation process using the PID controller, Lyapunov's and gradient methods based on the adaptive control system with a reference model, self-tuning regulator based on the Kaczmarz projection iteration method is considered. The method of adaptive control of the high-energy ultrasound intensity at a given point in space, which is generated by an ultrasonic phased array based on the successive orthogonal projection of the approximation on a hyperplane is developed. Findings. The computer simulation results of the ultrasonic exposure on the pulp flow using adaptive and PID regulators with a given intensity and time of the transient process with a linearly increasing pulp flow rate are shown. The developed method is implemented in software and allows to maintain the desired intensity of ultrasound exposure at a given point of flotation machine space with high accuracy. Originality of the work lies in the fact that a method of adaptive control of the intensity of high-energy ultrasound at a given point in space, formed by using an ultrasonic phased array based on a sequential orthogonal design of the approximation on the hyperplane, has been developed, with error not exceeding 1.2%. Рractical value consists in the development of algorithms and programs for adaptive control of the parameters of the gas phase of the pulp in the process of flotation.
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Sun, Min, and Jing Liu. "A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/386030.
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A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factorγis attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.
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Awwal, Aliyu Muhammed, Lin Wang, Poom Kumam, and Hassan Mohammad. "A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems." Symmetry 12, no. 6 (2020): 874. http://dx.doi.org/10.3390/sym12060874.
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In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The BB spectral parameters can be viewed as the approximations of Jacobians with scalar multiple of identity matrices. If the Jacobians are close to symmetric matrices with clustered eigenvalues then the BB parameters are expected to behave nicely. We present a new line search technique for generating the separating hyperplane projection step of Solodov and Svaiter (1998) that generalizes the one used in most of the existing literature. We establish the convergence result of the algorithm under some suitable assumptions. Preliminary numerical experiments demonstrate the efficiency and computational advantage of the algorithm over some existing algorithms designed for solving similar problems. Finally, we apply the proposed algorithm to solve image deblurring problem.
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Dai, Zhifeng, and Huan Zhu. "A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations." Mathematics 8, no. 2 (2020): 168. http://dx.doi.org/10.3390/math8020168.
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The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.
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Яганов, П. О., та І. В. Редько. "РЕГРЕСІЙНА МОДЕЛЬ ПРОСТОРУ СТАНІВ ТЕПЛОВОГО КОМФОРТУ". Bulletin of the Kyiv National University of Technologies and Design. Technical Science Series 138, № 5 (2020): 26–34. http://dx.doi.org/10.30857/1813-6796.2019.5.3.
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Modeling the hyperspace of system states by the method of regression analysis, mathematical analysis of regression equations that model the thermal comfort index, optimization of thermal comfort by determining the coordinates of the projection of the vector image of thermal comfort onto a state hyperplane and solving an equation with many variables. Development of methods and models for a formal description of the space of states of human thermal comfort. The use of the method of regression analysis as a tool for modeling the state of thermal comfort in multidimensional hyperspace of system states is investigated. The procedure for calculating the PMV thermal comfort index and the formation of control commands for ACS of the thermal environment have been simplified. It is proved that a linear and quasilinear model of the state surface of the TC system within the selected TC intervals with sufficient accuracy for practical use calculates the PMV thermal comfort index. The regression analysis method is developed and extended to the class of thermal comfort systems. For the first time, a formal model of the space of states of thermal comfort is represented by linear and quasilinear polynomials. The state model of a complex multifactor system for providing thermal comfort for a person is represented by regression polynomials in a linear and quasilinear form, which greatly simplifies the calculation of the PMV index, classification of the state of thermal comfort, the search for optimal operating modes and command formation for actuators that are part of the ACS of the TC.
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Ortiz, Andres, F. Lozano, Juan M. Górriz, Javier Ramírez, Francisco J. Martínez Murcia, and for the Alzheimer’s Disease Neuroimaging Initiative. "Discriminative Sparse Features for Alzheimer's Disease Diagnosis Using Multimodal Image Data." Current Alzheimer Research 15, no. 1 (2017): 67–79. http://dx.doi.org/10.2174/1567205014666170922101135.
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Background: Feature extraction in medical image processing still remains a challenge, especially in high-dimensionality datasets, where the expected number of available samples is considerably lower than the dimension of the feature space. This is a common problem in real-world data, and, specifically, in medical image pro- cessing as, while images are composed of hundreds of thousands voxels, only a reduced number of patients are available. Objective: Extracting descriptive and discriminative features to represent each sample (image) by a small number of features, which is particularly important in classification task, due to the curse of dimensionality problem. Methods: In this paper we solve this recognition problem by means of sparse representations of the data, which also provides an arena to multimodal image (PET and MRI) data classification by combining specialized classifiers. Thus, a novel method to effectively combine SVC classifiers is presented here, which uses the distance to the hyperplane computed for each class in each classifier allowing to select the most discriminative image modality in each case. The discriminative power of each modality also provides information about the illness evolution; while functional changes are clearly found in Alzheimer’s diagnosed patients (AD) when compared to control subjects (CN), structural changes seem to be more relevant at the early stages of the illness, affecting Mild Cognitive Impairment (MCI) patients. Results: Classification experiments using 68 CN, 70 AD and 111 MCI images from the Alzheimer's Disease Neuroimaging Initiative database have been performed and assessed by cross-validation to show the effectiveness of the proposed method. Accuracy values of up to 92% and 84% for CN/AD and CN/MCI classification are achieved. Conclusions: The method presented in this work shows that sparse representations of brain images are of importance for codifying and transferring relevant image features, as they may capture the salient features while maintaining lightweight data transactions. In fact, the method proposed in this work outperforms the classification results obtained using projection methods such as Principal Component Analysis for extracting representative features of the images.
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Wang, Xiaoguang, Yunbo Hu, and Qi Lin. "Workspace analysis and verification of cable-driven parallel mechanism for wind tunnel test." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 231, no. 6 (2016): 1012–21. http://dx.doi.org/10.1177/0954410016646601.
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Cable-driven parallel mechanism is a special kind of parallel robot in which traditional rigid links are replaced by actuated cables. This provides a new suspension method for wind tunnel test, in which an aircraft model is driven by a number of parallel cables to fulfil 6-DOF motion. The workspace of such a cable robot is limited due to the geometrical and unilateral force constraints, the investigation of which is important for applications requiring large flight space. This paper focuses on the workspace analysis and verification of a redundant constraint 6-DOF cable-driven parallel suspension system. Based on the system motion and dynamic equations, the geometrical interference (either intersection between two cables or between a cable and the aircraft) and cable tension restraint conditions are constructed and analyzed. The hyperplane vector projection strategy is used to solve the aircraft’s orientation and position workspace. Moreover, software ADAMS is used to check the workspace, and experiments are done on the prototype, which adopts a camera to monitor the actual motion space. In addition, the system construction is designed by using a built-in six-component balance to measure the aerodynamic force. The results of simulation and tests show a good consistency, which means that the restraint conditions and workspace solution strategy are valid and can be used to provide guidance for the cable-driven parallel suspension system’s application in wind tunnel tests.
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Amini, Keyvan, and Ahmad Kamandi. "A new line search strategy for finding separating hyperplane in projection-based methods." Numerical Algorithms 70, no. 3 (2015): 559–70. http://dx.doi.org/10.1007/s11075-015-9961-1.
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Lauder, Alan G. B. "A Recursive Method for Computing Zeta Functions of Varieties." LMS Journal of Computation and Mathematics 9 (2006): 222–69. http://dx.doi.org/10.1112/s1461157000001261.
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AbstractWe present an algorithm that reduces the problem of calculating a numerical approximation to the action of absolute Frobenius on the middle-dimensional rigid cohomology of a smooth projective variety over a finite held, to that of performing the same calculation for a smooth hyperplane section. When combined with standard geometric techniques, this yields a method for computing zeta functions which proceeds ‘by induction on the dimension’. The ‘inductive step’ combines previous work of the author on the deformation of Frobenius with a higher rank generalisation of Kedlaya's algorithm. The analysis of the loss of precision during the algorithm uses a deep theorem of Christol and Dwork on p-adic solutions to differential systems at regular singular points. We apply our algorithm to compute the zeta functions of compactifications of certain surfaces which are double covers of the affine plane.
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Filisbino, Tiene A., Gilson A. Giraldi, and Carlos E. Thomaz. "Comparing Ranking Methods for Tensor Components in Multilinear and Concurrent Subspace Analysis with Applications in Face Images." International Journal of Image and Graphics 15, no. 01 (2015): 1550006. http://dx.doi.org/10.1142/s0219467815500060.
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In the area of multi-dimensional image databases modeling, the multilinear principal component analysis (MPCA) and concurrent subspace analysis (CSA) approaches were independently proposed and applied for mining image databases. The former follows the classical principal component analysis (PCA) paradigm that centers the sample data before subspace learning. The CSA, on the other hand, performs the learning procedure using the raw data. Besides, the corresponding tensor components have been ranked in order to identify the principal tensor subspaces for separating sample groups for face image analysis and gait recognition. In this paper, we first demonstrate that if CSA receives centered input samples and we consider full projection matrices then the obtained solution is equal to the one generated by MPCA. Then, we consider the general problem of ranking tensor components. We examine the theoretical aspects of typical solutions in this field: (a) Estimating the covariance structure of the database; (b) Computing discriminant weights through separating hyperplanes; (c) Application of Fisher criterium. We discuss these solutions for tensor subspaces learned using centered data (MPCA) and raw data (CSA). In the experimental results we focus on tensor principal components selected by the mentioned techniques for face image analysis considering gender classification as well as reconstruction problems.
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Ляшков, А., A. Lyashkov, К. Панчук, K. Panchuk, Л. Варепо, and L. Varepo. "Four-Dimensional Space’s Hypersurface Mapping Singularity." Geometry & Graphics 5, no. 3 (2017): 3–10. http://dx.doi.org/10.12737/article_59bfa3078af4c1.45321238.
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Investigation of singularity related to a mapping by the orthogonal projection in a four-dimensional space of a hypersurface given by parametric equations is presented in this paper. On this investigation’s basis have been proposed in a united way three approaches to the hypersurface’s discriminant determination. Thus, have been defined conditions which the investigated hypersurface’s discriminant set and criminant are satisfied. Have been obtained dependencies settling the relationship between parameters of the hypersurface at its discriminant points. They are used to determine the singularity of the hypersurface mapping by analytical methods in general form. The complexity of this approach (the first one) is that an equation connecting the hypersurface’s parameters contains its differential characteristics, and often is the transcendental one in applications, that causes certain difficulties when solving it. Have been obtained dependences in carrying out of which the hypersurface’s discriminant has an edge of regression. A study of hypersurface sections by hyperplanes which are parallel to coordinate hyperplanes has been performed. The last ones contain a coordinate axis along which the hypersurface mapping is performed. It has been established that curves obtained in these sections have extreme points belonging to the hypersurface’s discriminant. Such property is used to calculate the points of the hypersurface’s discriminant by numerical methods without using the hypersurface’s differential characteristics, and it is a basis for the second approach to solving the problem posed. It has been also demonstrated the use of 3D modeling for study of hypersurface’s different sections, as well as its discriminant that represents the third approach to the study. All three approaches having a common basis can be used both independently and complement each other in determining the envelope for one-parameter family of surfaces. As an example has been considered a hypersurface formed by a family of spheres. Based on stated results have been obtained equations determining the hypersurface’s discriminant and this family’s corresponding envelope, as well as various sections. These equations have been used for creating of polygonal 3D models of the hypersurface’s discriminant, and some of the hypersurface’s sections.
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Rakdi, M. A., and N. Midoune. "Weights of the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$ over a finite field." Carpathian Mathematical Publications 11, no. 2 (2019): 422–30. http://dx.doi.org/10.15330/cmp.11.2.422-430.
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Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by Nogin for general $q.$ These codes are conveniently described using the correspondence between non-degenerate $[n,k]_{q}$ linear codes on one hand and non-degenerate $[n,k]$ projective systems on the other hand. A non-degenerate $[n,k]$ projective system is simply a collection of $n$ points in projective space $\mathbb{P}^{k-1}$ satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ contains all the $n$ points under consideration. In this paper we will determine the weight of linear codes $C(3,8)$ associated with Grassmann varieties $G(3,8)$ over an arbitrary finite field $\mathbb{F}_{q}$. We use a formula for the weight of a codeword of $C(3,8)$, in terms of the cardinalities certain varieties associated with alternating trilinear forms on $\mathbb{F}_{q}^{8}.$ For $m=6$ and $7,$ the weight spectrum of $C(3,m)$ associated with $G(3,m),$ have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension $n$ of the base space. For $n\leq 8 $ there exist only finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\mathbb{\bar{F}}$ to $\mathbb{F}.$ This program is partially determined by Noui L. and Midoune N. and the classification of trilinear alternating forms on a vector space of dimension $8$ over a finite field $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L. and Midoune N. We describe the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use to determine the weight of the $\mathbb{F}_{q}$-forms.
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HEERSINK, BYRON. "Equidistribution of Farey sequences on horospheres in covers of and applications." Ergodic Theory and Dynamical Systems 41, no. 2 (2019): 471–93. http://dx.doi.org/10.1017/etds.2019.71.
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We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.
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Koorapetse, Mompati, P. Kaelo, and S. Kooepile-Reikeletseng. "A modified Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations." RAIRO - Operations Research, August 5, 2021. http://dx.doi.org/10.1051/ro/2021117.
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In this paper, a new modified Perry-type derivative-free projection method for solving large-scale nonlinear monotone equations is presented. The method is developed by combining a modified Perry's conjugate gradient method with the hyperplane projection technique. Global convergence and numerical results of the proposed method are established. Preliminary numerical results show that the proposed method is promising and efficient compared to some existing methods in the literature.
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Jia, Xiaoxi, Zhiren Sun, and Lingling Xu. "An improved hyperplane projection method for generalized Nash equilibrium problems with extrapolation technique." Optimization, February 19, 2021, 1–21. http://dx.doi.org/10.1080/02331934.2021.1887182.
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DAYAN, YIFTACH. "Random fractals and their intersection with winning sets." Mathematical Proceedings of the Cambridge Philosophical Society, July 7, 2021, 1–30. http://dx.doi.org/10.1017/s0305004121000360.
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Abstract We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$ ), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$ , $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$ , where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.
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Chini, Francesco, and Niels Martin Møller. "Bi-Halfspace and Convex Hull Theorems for Translating Solitons." International Mathematics Research Notices, October 8, 2019. http://dx.doi.org/10.1093/imrn/rnz183.
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AbstractWhile it is well known from examples that no interesting “halfspace theorem” holds for properly immersed $n$-dimensional self-translating mean curvature flow solitons in Euclidean space $\mathbb{R}^{n+1}$, we show that they must all obey a general “bi-halfspace theorem” (aka “wedge theorem”): two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori–Yau maximum principle. As an application we classify the closed convex hulls of all properly immersed (possibly with compact boundary) $n$-dimensional mean curvature flow self-translating solitons $\Sigma ^n$ in ${\mathbb{R}}^{n+1}$ up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman–Meeks in 1989 for minimal submanifolds: all of ${\mathbb{R}}^{n}$, halfspaces, slabs, hyperplanes, and convex compacts in ${\mathbb{R}}^{n}$.
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Rana, Priyanka, Arcot Sowmya, Erik Meijering, and Yang Song. "Estimation of three-dimensional chromatin morphology for nuclear classification and characterisation." Scientific Reports 11, no. 1 (2021). http://dx.doi.org/10.1038/s41598-021-82985-9.
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AbstractClassification and characterisation of cellular morphological states are vital for understanding cell differentiation, development, proliferation and diverse pathological conditions. As the onset of morphological changes transpires following genetic alterations in the chromatin configuration inside the nucleus, the nuclear texture as one of the low-level properties if detected and quantified accurately has the potential to provide insights on nuclear organisation and enable early diagnosis and prognosis. This study presents a three dimensional (3D) nuclear texture description method for cell nucleus classification and variation measurement in chromatin patterns on the transition to another phenotypic state. The proposed approach includes third plane information using hyperplanes into the design of the Sorted Random Projections (SRP) texture feature and is evaluated on publicly available 3D image datasets of human fibroblast and human prostate cancer cell lines obtained from the Statistics Online Computational Resource. Results show that 3D SRP and 3D Local Binary Pattern provide better classification results than other feature descriptors. In addition, the proposed metrics based on 3D SRP validate the change in intensity and aggregation of heterochromatin on transition to another state and characterise the intermediate and ultimate phenotypic states.