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Customizing A* Heuristics for Network Routing
Published in Takushi Tanaka, Setsuo Ohsuga, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2022
In the following, Sn≡(Γ,•) with Γ={π1,π2,.πn!} denotes the permutation group of order n!. As usual, a specific permutation π∈Γ is depicted by a string of ndigits, the identical permutation 123. nby I.For example, S3corresponds to the group ({ 123, 132, 213, 231, 312, 321 }, •), 321•231 is an abbreviation for the composition of the two bijective functions πl π2 to be resolved from right to left yielding 213 as the result.
Mathematical Morphology with Noncommutative Symmetry Groups
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
Let X be a non-empty set. A bijection X → X is called a permutation of X. By SymX we denote the group of all permutations of X. If X is a finite set of n elements, we write Sn instead of SymX. A subgroup Γ of SymX is called a permutation group or transformation group on X. We also say that Γ is a group action on X or that Γ acts onX. Each element g ∈ Γ is a mapping X → X : x → g (x), satisfying (i)gh(x)=g(h(x)),(ii)e(x)=x
Group of L-homeomorphisms and L f -representability of Permutation Groups
Published in Fuzzy Information and Engineering, 2020
Let be a non-trivial permutation group on a set . Let . Define , which is a permutation group on . Note that moves all the elements of and By Theorem 3.3, it follows that, if is -representable on , then is -representable on X. So if is an -topological space which is not rigid and then without loss of generality, we can assume that moves all the elements of .
Symmetry properties of the electron density and following from it limits on the KS-DFT applications
Published in Molecular Physics, 2018
This means that the diagonal element of the full density matrix (and all reduced density matrices as well) transforms according to the totally symmetric one-dimensional representation A1 of G regardless of the dimension of representation Γ(α). In Ref. [19], this was proved for the arbitrary point group, but it is correct for any finite group. For the permutation group, this result was used in Refs. [54,55] in analysis of the foundations of the Pauli exclusion principle. To the best of our knowledge, it was not discussed in literature. Even in the specialised monograph by Davidson [56], the symmetry of the reduced density matrices is discussed only for non-degenerate states, but the latter is evident.
Universal approximation with neural networks on function spaces
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Wataru Kumagai, Akiyoshi Sannai, Makoto Kawano
Gordon et al. (2019) and Kawano et al. (2021) showed that two kinds of symmetries for data are important to enhance the effectiveness of learning algorithms. The first symmetry is represented by a permutation of the dataset. Let for and let be the permutation group on . The action of on is defined as