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Linear Algebra for Machine Learning
Published in P. Kaliraj, T. Devi, Artificial Intelligence Theory, Models, and Applications, 2021
For the vector space W=(V,+,⋅), consider A={x1,…,xk} be the set of vectors. Then every vector νϵV is manifesting terms of the linear combination of the set of vectors in A, then such set A is called a generating set of W. The smallest set of A which contains the linear combinations form a span of A. If A spans W, we can write W=span[A] or W=span[x1,…,xk]. The generating sets generate the vector (sub)spaces, i.e., the vectors of the generating set can be used to represent every vector of W in the linear combination.
Conditional strong matching preclusion of the pancake graph
Published in International Journal of Parallel, Emergent and Distributed Systems, 2023
The n-dimensional pancake graph, denoted by , is a Cayley graph on the symmetric group , with the generating set , where . . It follows directly from the definition that is -regular graph of order . Although is vertex transitive, it is not edge transitive except for n = 3. The graph is the 6-cycle, and the graph is shown in Figure 1.
Diagnosability of arrangement graphs with missing edges under the MM* model
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
Let , and let be the symmetric group on containing all permutations of . The alternating group is the subgroup of containing all even permutations. It is well known that is a generating set for . The n-dimensional alternating group graph is the graph with vertex set = in which two vertices u, v are adjacent if and only if or , .
Weighted upper metric mean dimension for amenable group actions
Published in Dynamical Systems, 2020
Dingxuan Tang, Haiyan Wu, Zhiming Li
In reality, it is difficult to find a real orbit in the system. Pseudo orbits are good candidates to approximate the real orbits. There has been a lot of interests in the connection between entropies (resp. pressures) and pseudo-orbits. We will extend these good relations to weighted upper metric mean dimensions. In this section, we assume that G is a finitely generated countable discrete amenable group with a symmetric generating set , . Recall that a generating set is called symmetric if together with any element it contains .