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Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

The main focus of this paper is to study multi-valued linear monotone operators in the context of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-)representability are studied and criteria are provided.

During the processing and planting of soybeans, it is greatly significant that a reliable, rapid, and accurate technique is used to detect soybean varieties. Traditional chemical analysis methods of soybean variety sampling (e.g., mass spectrometry and high-performance liquid chromatography) are destructive and time-consuming. In this paper, a robust and accurate method for nondestructive soybean classification is developed through hyperspectral imaging and ensemble machine learning algorithms. Image acquisition, preprocessing, and feature selection are used to obtain different types of soybean hyperspectral features. Based on these features, one of ensemble classifiers-random subspace linear discriminant (RSLD) algorithm is used to classify soybean seeds. Compared with the linear discrimination (LD) and linear support vector machine (LSVM) methods, the results show that the RSLD algorithm in this paper is more stable and reliable. In classifying soybeans in 10, 15, 20, and 25 categories, the RSLD method achieves the highest classification accuracy. When 155 features are used to classify 15 types of soybeans, the classification accuracy of the RSLD method reaches 99.2%, while the classification accuracies of the LD and LSVM methods are only 98.6% and 69.7%, respectively. Therefore, the ensemble classification algorithm RSLD can maintain high classification accuracy when different types and different classification features are used.

We provide a recursive formula for the dimension of the algebraic sum of finitely many subspaces in a finite-dimensional vector space over an arbitrary field.

We prove that if M is an infinite complete metric space, then the set of strongly norm-attaining Lipschitz functions SNA(M) contains a linear subspace isomorphic to c_0 . This solves an open question posed by V. Kadets and Ó. Roldán.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Let d and k be integers with \\[1 k d-1\\]. Let \\[ \\] be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in \\[ K\\]. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional \\[n n\\] grid is at least \\[ (n^d(d-k)/(d-1)- )\\] and at most \\[O (n^d(d-k)/(d-1) )\\], where \\[ >0\\] is an arbitrarily small constant. This nearly settles a problem mentioned in the book by Brass et al. (Research problems in discrete geometry, Springer, New York, 2005). We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover \\[ K\\]. We use these new results to improve the best known lower bound for the maximum number of point–hyperplane incidences by Brass and Knauer (Comput Geom 25(1–2):13–20, 2003). For \\[d 3\\] and \\[ ın (0,1)\\], we show that there is an integer \\[r=r(d, )\\] such that for all positive integers n, m the following statement is true. There is a set of n points in \\[ R^d\\] and an arrangement of m hyperplanes in \\[ R^d\\] with no \\[K_r,r\\] in their incidence graph and with at least \\[ ((mn)^1-(2d+3)/((d+2)(d+3)) - )\\] incidences if d is odd and \\[ ((mn)^1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - )\\] incidences if d is even.

We first introduce a new notion called statistical convergence of order α and primarily show that it gives rise to a decreasing chain of closed linear subspaces of the space of all bounded real sequences with sup norm which never coincides with the class of convergent sequences and in fact their intersection properly contains the class of convergent sequences. We then show that the same method can be applied for double sequences also and introduce the notion of statistical convergence of order (α,β).

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals. Our work ties together and generalizes results in Bauer et al. (Int Math Res Not IMRN 24:7459–7514, 2019), Dumnicki et al. (J Algebra 393:24–29, 2013), Harbourne and Seceleanu (J Pure Appl Algebra 219(4):1062–1072, 2015) and Malara and Szpond (J Pure Appl Algebra 222(8):2323–2329, 2018) under a unified approach.

Consider a random walk Si =ξ₁ + … +ξi, i ∊ N, whose increments ξ₁, ξ₂, … are independent identically distributed random vectors in ℝ d such that ξ₁ has the same law as -ξ₁ and ℙ[ξ₁ ∊ H] = 0 for every affine hyperplane H ⊂ ℝd. Our main result is the distribution-free formula E ∑ 1 ≤ i 1 < … < i k ≤ n 1 0 ∉ Conv ( S i , … , S i k ) = 2 n k B ( k , d − 1 ) + B ( k , d − 3 ) + … 2 k k ! where the B(k, j)’s are defined by their generating function (t + 1)(t + 3) … (t + 2k-1) = ∑ j = 0 k B ( k , j ) t j . The expected number of k-tuples above admits the following geometric interpretation: it is the expected number of k-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type Bn that are not intersected by a generic linear subspace L ⊂ ℝ n of codimension d. The case d = 1 turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required.