COMPUTATIONAL INVESTIGATION OF A4-GRAPHS FOR CERTAIN LEECH LATTICE GROUPS

In the case of a finite simple group , and -conjugacy class of element of order 3, The A4-graph is define as simple graph denoted by A4 has as vertex set and are adjacent if and only if x≠y and xy-1 = yx-1.We aim to investigate computationally the structure of theA4 when Leech Lattice groups.

Investigation of Commuting Graphs for Elements of Order 3 in Certain Leech Lattice Groups

Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) and has a set of vertices X with two distinct vertices x, y Î X, being connected together on the condition of xy = yx. In this paper, we investigate the structure of Ϲ(G,X) when G is a particular type of Leech lattice groups, namely Higman–Sims group HS and Janko group J2, along with X as a G-conjugacy class of elements of order 3. We will pay particular attention to analyze the discs’ structure and determinate the diameters, girths, and clique number for these graphs.

Automorphisms and periods of cubic fourfolds

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.

Eigenfunctions of the Fourier transform with specified zeros

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

Root lattices in number fields

We explore whether a root lattice may be similar to the lattice [Formula: see text] of integers of a number field [Formula: see text] endowed with the inner product [Formula: see text], where [Formula: see text] is an involution of [Formula: see text]. We classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is similar to either an even root lattice or the root lattice [Formula: see text]. We also classify all pairs [Formula: see text], [Formula: see text] such that [Formula: see text] is a root lattice. In addition to this, we show that [Formula: see text] is never similar to a positive-definite even unimodular lattice of rank [Formula: see text], in particular, [Formula: see text] is not similar to the Leech lattice. In Appendix B, we give a general cyclicity criterion for the primary components of the discriminant group of [Formula: see text].

Erratum: Energy Minimization, Periodic Sets, and Spherical Designs

In [ 3] we considered energy minimization of pair potentials among periodic sets of a fixed-point density. For a large class of potentials we presented sufficient conditions for a point lattice to give a local optimum among periodic sets. We hereby, in particular, derived a local version of Cohn and Kumar’s conjecture [ 1, Conjecture 9.4] by which the hexagonal lattice $\textsf{A}_2$, the root lattice $\textsf{E}_8$, and the Leech lattice are globally universally optimal. Latter conjecture has recently been proved for $\textsf{E}_8$ and the Leech lattice by Cohn et al. [ 2].

Restriction enzymes use a 24 dimensional coding space to recognize 6 base long DNA sequences

Restriction enzymes recognize and bind to specific sequences on invading bacteriophage DNA. Like a key in a lock, these proteins require many contacts to specify the correct DNA sequence. Using information theory we develop an equation that defines the number of independent contacts, which is the dimensionality of the binding. We show that EcoRI, which binds to the sequence GAATTC, functions in 24 dimensions. Information theory represents messages as spheres in high dimensional spaces. Better sphere packing leads to better communications systems. The densest known packing of hyperspheres occurs on the Leech lattice in 24 dimensions. We suggest that the single protein EcoRI molecule employs a Leech lattice in its operation. Optimizing density of sphere packing explains why 6 base restriction enzymes are so common.