scholarly journals Unit Subduced Cycle Indices with and without Chirality Fittingness for IhGroup. An Application to Systematic Enumeration of Dodecahedrane Derivatives

1990 ◽  
Vol 63 (10) ◽  
pp. 2759-2769 ◽  
Author(s):  
Shinsaku Fujita
Keyword(s):  
Systematic Enumeration ◽  
Chirality Fittingness ◽  
Cycle Indices

scholarly journals Standard mark table and standard USCI-CF table of the point group D6h. Unification of candidates derived from different D6h-skeletons

2021 ◽  
Vol 87 (3) ◽  
pp. 481-525
Author(s):  
Shinsaku Fujita ◽  
Keyword(s):  
Point Group ◽  
Gap Function ◽  
Combinatorial Enumeration ◽  
Cycle Index ◽  
Single Standard ◽  
Partial Cycle ◽  
Unit Subduced Cycle Index ◽  
Chirality Fittingness ◽  
Usci Approach ◽  
Cycle Indices

Combined-permutation representations (CPRs) for characterizing -skeletons (a benzene skeleton, a Haworth-projected skeleton, a superphane skeleton, and a coronene skeleton) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to treat both achiral and chiral substituents under the GAP system. Thereby, the CPR of degree 8 (= 6 + 2) for the benzene skeleton, the CPR of degree 14 (= 12 + 2) for the Haworth-projected skeleton, the CPR of degree 14 (= 12 + 2) for the superphane skeleton, the CPR of degree 14 (= 12 + 2) for the coronene skeleton are generated to give primary mark tables (tables of marks) based on these CPRs. These primary mark tables generated by the GAP system are different in the sequence of subgroups from each other, although they stem from the same point group . They are unified into a single standard mark table by means of a newly-devised GAP function MarkTableforUSCI. Moreover, another newly-devised GAP function constructUSCITable is employed to construct a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table concordantly. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

scholarly journals Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

2017 ◽  
Vol 5 (1) ◽  
pp. 158-201
Author(s):  
Jan Brandts ◽  
Apo Cihangir
Keyword(s):  
Convex Hull ◽  
Computer Program ◽  
Explicit Form ◽  
Linear Algebra ◽  
Dihedral Angles ◽  
Cycle Index ◽  
Hyperoctahedral Group ◽  
Diagonally Dominant ◽  
Group B ◽  
Cycle Indices

Abstract The convex hull of n + 1 affinely independent vertices of the unit n-cube In is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in In can be described by nonsingular 0/1-matrices P of size n × n whose Gramians G = PTP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries [6, 7]. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn [17] of symmetries of In. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ 6. Using the computed cycle indices for B3, . . . ,B11 in combination with Pólya’s theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler’s Tree of Fractions [1, 24] that enumerates ℚ ⋂ (0, 1). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14, 26] matrices.

scholarly journals Recurrence quantification analysis of two solar cycle indices

2017 ◽  
Vol 7 ◽  
pp. A5 ◽  
Author(s):  
Marco Stangalini ◽  
Ilaria Ermolli ◽  
Giuseppe Consolini ◽  
Fabrizio Giorgi
Keyword(s):  
Solar Cycle ◽  
Recurrence Quantification Analysis ◽  
Recurrence Quantification ◽  
Quantification Analysis ◽  
Cycle Indices

Local Environmental Groups: A Systematic Enumeration in Two Geographical Areas*

2009 ◽  
Vol 66 (4) ◽  
pp. 557-578 ◽  
Author(s):  
Willett Kempton ◽  
Dorothy C. Holland ◽  
Katherine Bunting-Howarth ◽  
Erin Hannan ◽  
Christopher Payne
Keyword(s):  
Environmental Groups ◽  
Systematic Enumeration
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