A Monster Lie algebra?

This paper defines a remarkable Lie algebra of infinite dimension and rank, and conjectures that it may be related to the Fischer-Griess Monster group.     

Anti-Hadamard matrices

An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λ_i, singular values σ_i, and inverse B = (b_ij). We are interested in the four closely related problems of finding λ(n) = min_{A, i}|λ_i|, σ(n) = min_{A, i}σ_i, χ(n) = max_{A, i, j} |b_ij|, and μ(n) = max_AΣ_ijb²_ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between $\text{(2n)}{}^{\mathrm{?1}}\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ and c√n (2.274)^?n, where c is a constant, $\text{c}\text{(2.274)}{}^{\mathrm{n}}\text{?χ(n)?2(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, and $\text{c}\text{(5.172)}{}^{\mathrm{n}}\text{?μ(n)?4n}{}^2\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{n}}$. We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry.     

Most primitive groups have messy invariants

Suppose G is a finite group of complex n × n matrices, and let R^G be the ring of invariants of G: i.e., those polynomials fixed by G. Many authors, from Klein to the present day, have described R^G by writing it as a direct sum Σ^δ_j=1 η_j$\text{C}$[θ₁ ,…, θ_n]. For example, if G is a unitary group generated by reflections, δ = 1. In this note we show that in general this approach is hopeless by proving that, for any ? > 0, the smallest possible δ is greater than | G |^n-1-? for almost all primitive groups. Since for any group we can choose δ ? | G |^n-1, this means that most primitive groups are about as bad as they can be. The upper bound on δ follows from Dade's theorem that the θ_i can be chosen to have degrees dividing | G |.     

On sublattices of the hexagonal lattice

How many sublattices of index N are there in the planar hexagonal lattice? Which of them are the best from the point of view of packing density, signal-to-noise ratio, or energy? We answer the first question completely and give partial answers to the other questions.     

Dinitrogen Activation Upon Reduction of a Triiron(II) Complex

Reaction of a trinuclear iron(II) complex, Fe₃Br₃ L ( 1 ), with KC₈ under N₂ leads to dinitrogen activation products ( 2 ) from which Fe₃(NH)₃ L ( 2‐1 ; L is a cyclophane bridged by three β‐diketiminate arms) was characterized by X‐ray crystallography. ¹H NMR spectra of the protonolysis product of 2 synthesized under ¹⁴N₂ and ¹⁵N₂ confirm atmospheric N₂ reduction, and ammonia is detected by the indophenol assay (yield ～30 %). IR and Mössbauer spectroscopy, and elemental analysis on 2 and 2‐1 as well as the tri(amido)triiron(II) 3 and tri(methoxo)triiron 4 congeners support our assignment of the reduction product as containing protonated N‐atom bridges.     

McLaren’s improved snub cube and other new spherical designs in three dimensions

Evidence is presented to suggest that, in three dimensions, spherical 6-designs withN points exist forN=24, 26,≥28; 7-designs forN=24, 30, 32, 34,≥36; 8-designs forN=36, 40, 42,≥44; 9-designs forN=48, 50, 52,≥54; 10-designs forN=60, 62, ≥64; 11-designs forN=70, 72,≥74; and 12-designs forN=84,≥86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and—although not identified as such by McLaren—consists of the vertices of an “improved” snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5 designs exist forN=12, 16, 18, 20,≥22. It is conjectured, albeit with decreasing confidence fort≥9, that these lists oft-designs are complete and that no other exist. One of the constructions gives a sequence of putative sphericalt-designs withN=12m points (m≥2) whereN=1/2t ²(1+o(1)) ast→∞.     

Study of structural and electronic transport properties of Ce-doped LaMnO3

The structural and electronic transport properties of La_1−x Ce _x MnO₃ (x=0.0–1.0) have been studied. All the samples exhibit orthorhombic crystal symmetry and the unit cell volume decreases with Ce doping. They also make a metal-insulator transition (MIT) and transition temperature increases with increase in Ce concentration up to 50% doping. The system La_0.5Ce_0.5MnO₃ also exhibits MIT instead of charge-ordered state as observed in the hole doped systems of the same composition.