Loop laws and collineations of planes

In translation planes, collineations are determined whose existence is necessary and sufficient for the validity of certain laws in the multiplicative loop of the coordinatizing quasifield. A general method is developed for loop laws derived from loop isostrophisms. Dedicated to the memory of Professor Eri Jabotinsky     

Non-existence of infinitesimally invariant measures on loop groups

Let G be a compact Lie group, L(G) the associated loop group, ω the canonical symplectic form on L(G). Set H the Hamiltonian function for which the associated ω-Hamiltonian vector field is the infinitesimal rotation. Then H generates a canonical semi-definite Riemannian structure on L(G), which induces a Riemannian structure on the free loop groupL(G)/G=L₀(G). This metric corresponds to the Sobolev norm H¹. Using orthonormal frame methodology the positivity and finiteness of the Ricci curvature of L₀(G) is proved. By studying the dissipation towards high modes of a unitary group valued SDE it is proved that the loop group does not have any infinitesimally invariant measure.     

Non-existence of positive solutions of some elliptic equations in positive-type domains

In this work we prove a non-existence result for a positive solution of a Dirichlet boundary value problem in a class of domains that are more general than star-shaped ones. Our result extends some earlier non-existence results for the star-shaped case. We also present an example of a domain which satisfies our conditions but not star-shaped essentially.     

A finite loop space not rationally equivalent to a compact Lie group

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5. Mathematics Subject Classification (2000) Primary: 55P35; Secondary: 55P15, 55R35     

Products of central collineations

An n by n matrix M over a (commutative) field F is said to be central if M ? I has rank 1. We say that M is an involution if M²=I; if M is also central we call M a simple involution. We will prove that any n-by-n matrix M satisfying detM=±1 is the product of n+2 or fewer simple involutions. This can be reduced to n+1 if F contains no roots of the equation xⁿ=(?1)ⁿ other than ±1. Any ordered field is of this kind. Our main result is that if M is any n-by-n nonsingular nonscalar matrix and if x_i ∈ F such that x₁?x_n=detM, then there exist central matrices M_i such that M=M₁?M_n and x_i=detM_i for i=1,…,n. We will apply this result to the projective group PGL(n,F) and to the little projective group PSL(n,F).     

Conservation laws of some lattice equations

We derive infinitely many conservation laws for some multidimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schrödinger equation, modified lattice Boussinesq equation, Hietarinta’s Boussinesq-type equations, Schwarzian lattice Boussinesq equation, and Toda-modified lattice Boussinesq equation.     

On some determinantal identities formation laws

We show that Muir’s law of extensible minors, Cayley’s law of complementaries and Jacobi’s identity for minors of the adjugate [Determinantal identities Linear Algebra and its Applications 52/53 (1983) pp. 769–791] are equivalent. We also show our generalization of Mühlbach/Muir’s extension principle [A generalization of Mühlbach’s extension principle for determinantal identities. Linear and Multilinear Algebra 61 (10) (2013) pp. 1363–1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach–Gasca–(Lopez-Carmona)–Ramirez identity [A generalization of Sylvester’s identity on determinants and some applications. Linear Algebra and its Applications 66 (1985) pp. 221–234/On extending determinantal identities. Linear Algebra and its Applications 132 (1990) pp. 145–162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications. Linear Algebra and its Applications 197,198 (1994) pp. 93–112].     

拟调和球的不存在性

证明了当N的万有覆盖上存在一个多项式增长的非负严格凸函数时,不存在从R3到N的非平凡拟调和球.于是在dim M=3时推广了Fells-Sampson的定理.     

Laplace's method for the laws of heat processes on loop spaces

In this paper, we will consider Laplace's method for a class of heat processes on loop spaces. We will obtain the first term of the asymptotics under assumptions that the function under consideration attains its minimum at a unique point and that the Hessian at the point is non-degenerate. This kind of process was first introduced by P. Malliavin in 1990 for the loop group case and then gradually generalized by various authors. Our tool is the rough path theory of T. Lyons. This technique was pioneered by S. Aida for finite-dimensional processes in his unpublished paper.     

Polarities and planar collineations of Moufang planes

We show that conjugacy classes of Baer involutions and non-elliptic polarities, respectively, of proper (i.e., non-desarguesian) Moufang planes are interrelated. Restriction of the conjugating group to the stabilizer of a triangle or a quadrangle does not refine the classes. These results are applied to prove transitivity properties for the centralizers of these polarities. Along the way, a new proof is obtained for the fact that the automorphism group of a Moufang plane acts transitively on quadrangles.     

Non-existence of KAM torus

Given an integrable Hamiltonian h ₀ with n-degrees of freedom and a Diophantine frequency ω, then, arbitrarily close to h ₀ in the C ^r topology with r < 2n, there exists an analytical Hamiltonian h _ε with no KAM torus of rotation vector ω. In contrast with it, KAM tori exist if perturbations are small in C ^r topology with r > 2n.     

拟调和球的不存在性

证明了当N的万有覆盖上存在一个多项式增长的非负严格凸函数时,不存在从R~3到N的非平凡拟调和球.于是在dim M=3时推广了Eells-Sampson的定理.     

A tail empirical process approach to some nonstandard laws of the iterated logarithm

We show that a number of nonstandard laws of the iterated logarithm have limiting constants which may be expressed as the extreme values of functionals off, wheref varies over suitable compact sets of functions. By solving the corresponding extremal problems, we show how these constants are generated. In addition, several new laws are presented.     

Non-existence of a minimizer to the magnetic Hartree-Fock functional

In the presence of an external magnetic field, we prove absence of a ground state within the Hartree-Fock theory of atoms and molecules. The result is established for a wide class of magnetic fields when the number of electrons is greater than or equal to 2Z + K, where Z is the total charge of K nuclei. Positivity properties are instrumental in the proof of this bound for the maximal ionization.      

A note on some laws of the iterated logarithm

Some function space laws of the iterated logarithm for Brownian motion with values in finite and infinite dimensional vector spaces are shown to follow from Hincin's classical law of the iterated logarithm and some martingale techniques. A law of the iterated logarithm for Brownian motion in a differentible manifold is also stated.     

Non-existence of certain topological expansions

Summary This note proves the following theorem: Theorem. — For For every cardinal τ that τ>2^c there exists a completely regular space of dispersion character τ which cannot be expanded to a normal space without changing the dispersion character. This result answers a question raised by Edwin Hewitt. Dedicated to the sixtieth birthday of Frof. Edgar R. Lorch This research was supported in part by the National Science Foundation, U.S.A.