Cross-Sections

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

A (differentiable) vector bundleof rank nconsists of a total space E,a base M,and a projection π :E → M,where Eand Mare differentiable manifolds, πis differentiable, each “fiber” Ex:= π -1(x) for x Є M,carries the structure of an n-dimensional (real) vector space, and the following local triviality requirement is satisfied: For each x Є M,there exist a neighborhood Uand a diffeomorphism.KeywordsRiemannian ManifoldVector BundleTangent BundleSpin StructureIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The combination of algebraic structures and graphs was carried out by investigating thoroughly the relation among the algebraic structure and the graph theoretic properties. Moreover, it needs to explore algebraic structure. The results of the combination of algebraic structures and graphs have many applications in the fields of Internet modeling, coding, etc. For example, the famous Cayley graph was constructed from groups and widely used in network models. Das introduced the subspace inclusion graph on finite-dimensional vector space over a finite filed, and studied that the graph is bipartite and some special properties if the dimension of the vector space is 3. In this paper, this bipartite inclusion graph in the case of 3-dimensional is extended to more general dimensional bipartite inclusion graph. The diameter, girth, clique number, covering number, independence number and matching number are studied and the properties are shown, such as regular, planar and Eulerian. Moreover, the authors also introduce a new results about the structure and the number of 1-dimensional and n-1-dimensional subspaces on n-dimensional vector space.

1. Let R be the additive group of all real numbers and t → Tt a representation of this group on an n-dimensional complex vector space X. This means that to each real number t a linear operator Tt : X → X is assigned in such a way that $${\text{T}}_{{\text{t + s}}} = {\text{T}}_{\text{t}} {\text{T}}_{\text{s}} , {\text{T}}_0 = {\text{I}}$$ (1) Since the set {Tt : t ∈ R} consists of operators which commute one to the other the irreducible representations are one dimensional. Hence we are lead to find all functions f: R → C (C is the set of all complex numbers) such that $${\text{f}}\left( {{\text{t}}\,{\text{ + }}\,{\text{s}}} \right) = {\text{f}}\left( {\text{t}} \right)\,\,{\text{f}}\,\,{\text{(s),}}\,\,\,\,\,{\text{f}}\,\,{\text{(0)}}\,{\text{ = }}\,\,{\text{1}}{\text{.}}$$ (2)

The KNN (The K nearest neighbor) is known as its simple efficient and widely used in classification problems or as a benchmark in classification problems. For different data types especially complex structure and high-dimensional data in real-life, the choice of distance metrics between sample points is a relatively complexity problem. The KNN's feature space is generally n-dimensional real vector space. This article converts the samples in the vector space to be the elements in line with the Lie group nature and then proposes a Li-KNN algorithm to solve the classification problem based on the theory of Lie groups. It shows good results by the experimental on handwritten numeral.

Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis

r-cross t-intersecting families for vector spaces

The Erdös-Ko-Rado theorem for vector spaces

Rocchio’s similarity-based Relevance feedback algorithm, one of the most important query reformation methods in information retrieval, is essentially an adaptive supervised learning algorithm from examples. In spite of its popularity in various applications there is little rigorous analysis of its learning complexity in literature. In this paper we show that in the Boolean vector space model, if the initial query vector is 0, then for any of the four typical similarities (inner product, dice coefficient, cosine coeffcient, and Jaccard coeffcient), Rocchio’s similarity- based relevance feedback algorithm makes at least n mistakes when used to search for a collection of documents represented by a monotone disjunction of at most k relevant features (or terms) over the n-dimensional Boolean vector space{0,1}n. When an arbitrary initial query vector in {0,1}n is used, it makes at least (n+k −3)/2 mistakes to search for the same collection of documents. The linear lower bounds are independent of the choices of the threshold and coefficients that the algorithm may use in updating its query vector and making its classification.

An integral formula for multiple summing norms of operators

On maximal compact subsemigroups of the endomorphism semigroup of an n-dimensional complex vector space

In this paper, we study the existence of extremal solutions for impulsive delay fuzzy differential equations in n-dimensional fuzzy vector space. This is an extension of the result of Kwun et al. [2] to impulsive fuzzy differential equations with delay condition.

It is shown in [4] that Rocchio's similarity-based relevance feedback algorithm makes Ωn mistakes in searching for a collection of documents represented by a monotone disjunction of at most k relevant features or terms over the n-dimensional binary vector space {0, 1}n. In practice, Rocchio's algorithm often uses a fixed query updating factor and a fixed classification threshold. When this is the case, we strengthen the work in [4] in this paper and prove that Rocchio's algorithm makes Ωkn---k mistakes in searching for the same collection of documents over the binary vector space {0, 1}n. A quadratic lower bound is obtained when k is proportional to n. An Okn---k2 upper bound is also obtained.

The family \(\mathcal {N}\) of n-variate normal distributions is parameterized by the cone of positive definite symmetric \(n\times n\)-matrices and the n-dimensional real vector space. Equipped with the Fisher information metric, \(\mathcal {N}\) becomes a Riemannian manifold. As such, it is diffeomorphic, but not isometric, to the Riemannian symmetric space \(\mathrm {Pos}_1(n+1,\mathbb {R})\) of unimodular positive definite symmetric \((n+1)\times (n+1)\)-matrices. As the computation of distances in the Fisher metric for \(n>1\) presents some difficulties, Lovrič et al. (J Multivar Anal 74:36–48, 2000) proposed to use the Killing metric on \(\mathrm {Pos}_1(n+1,\mathbb {R})\) as an alternative metric in which distances are easier to compute. In this work, we survey the geometric properties of the space \(\mathcal {N}\) and provide a quantitative analysis of the defect of certain geodesics for the Killing metric to be geodesics for the Fisher metric. We find that for these geodesics the use of the Killing metric as an approximation for the Fisher metric is indeed justified for long distances.

A. Convex Sets. We begin here with some basic definitions and elementary facts. Throughout this book we will work in an n-dimensional real vector space \(\mathcal{X}\). The choice of a basis in \(\mathcal{X}\) amounts to a linear isomorphism \(\mathcal{X}\mathop{\cong}\mathbb{R}^{n}\), but we will make this identification only in examples and some explicit computations.