On Some Varieties of Soluble Lie Algebras

In this paper,we study a class of soluble Lie algebras with variety relations that the commutator of m and n is zero.The aim of the paper is to consider the relationship between the Lie algebra L with ...     

Conjugacy classes in finite permutation groups via homomorphic images

The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of non-solvable normal subgroups to compute the conjugacy classes of a finite group.

     

The L-sharp permutation groups

An answer is given to a problem proposed by Bannai and Ito for {l,l+s,l+s+t}-sharp permutation group, and the result is used to determine L-sharp groups for L={l,l+1,l+3}and{l,l+2,l+3}.     

Grassmannian Fixed Point Ratios

We provide estimates for the fixed point ratios in the permutation representations of a finite classical group over a field of order q on k-subspaces of its natural n-dimensional module. For sufficiently large n, each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q ^–dk where d is the codimension of the large eigenspace. The results here are tailored for our forthcoming proof of a conjecture of Guralnick and Thompson on composition factors of monodromy groups.     

Explicitly correlated MRCI-F12 potential energy surfaces for methane fit with several permutation invariant schemes and full-dimensional vibrational calculations

A data-set of nearly 100,000 symmetry unique multi-configurational ab initio points for methane were generated at the (AE)-MRCI-F12(Q)/CVQZ-F12 level, including energies beyond 30,000 cm^?1 above the minimum and fit into potential energy surfaces (PESs) by several permutation invariant schemes. A multi-expansion interpolative fit combining interpolating moving least squares (IMLS) fitting and permutation invariant polynomials (PIP) was able to fit the complete data-set to a root-mean-square deviation of 1.0 cm^?1 and thus was used to benchmark the other fitting methods. The other fitting methods include a single PIP expansion and two neural network (NN) based approaches, one of which combines NN with PIP. Full-dimensional variational vibrational calculations using a contracted-iterative method (and a Lanczos eigensolver) were used to assess the spectroscopic accuracy of the electronic structure method. The results show that the NN-based fitting approaches are able to fit the data-set remarkably accurately with the PIP–NN method producing levels in remarkably close agreement with the PIP–IMLS benchmark. The (AE)-MRCI-F12(Q)/CVQZ-F12 electronic structure method produces vibrational levels of near spectroscopic accuracy and a superb equilibrium geometry. The levels are systematically slightly too high, beginning at ～ 1–2 cm^?1 above the fundamentals and becoming correspondingly higher for overtones. The PES is therefore suitable for small ab initio or empirical corrections and since it is based on a multi-reference method, can be extended to represent dynamically relevant dissociation channels.     

零度为$n-7$的双圈符号图

Let Γ be a signed graph and A(Γ) be the adjacency matrix of Γ. The nullity ofΓ is the multiplicity of eigenvalue zero in the spectrum of A(Γ). In this paper, the connected bicyclic signed graphs(including simple bicyclic graphs) of order n with nullity n-7 are completely characterized.     

Probabilistic and combinatorial aspects of the card‐cyclic to random insertion shuffle

Consider a permutation σ ∈ S_n as a deck of cards numbered from 1 to n and laid out in a row, where σ_j denotes the number of the card that is in the j‐th position from the left. We study some probabilistic and combinatorial aspects of the shuffle on S_n defined by removing and then randomly reinserting each of the n cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 362–390, 2015     

Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.