arXiv subjects
Publications and source records attributed to Ziyang Zhu.
We determine all orthogonal transformations of a quadratic space over reals such that any orthogonal transformation which is conjugate to one of them in the linear group is conjugate in the orthogonal group.
We study the equivalence of canonical and microcanonical ensembles for quadratic Hamiltonians on Euclidean lattices. The answer depends on the arithmetic structure of the energy spectrum: exact-energy microcanonical ensembles converge locally to the canonical Gibbs law in the lattice case, but may retain additional arithmetic information in the nonlattice case. We show that this obstruction disappears after replacing an exact energy shell by any fixed-width energy window. Using uniform local central limit theorems, we also obtain sharp asymptotics for the corresponding state counts and recover Bost's thermodynamic entropy.
It is well-known that if $n\times n$ integral matrices $A$ and $B$ of a number field $K$ are similar over all completions of the ring of integers of $K$, then $A$ and $B$ are similar over the ring of integers of a finite extension of $K$. We prove that there is no uniform bound of the degree of extension of $K$ valid for all $n\times n$ matrices. On the other hand, we provide a upper bound of the degree of extension of $K$ for a given separable characteristic polynomial.
We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent by a finite covering and verify this conjecture for $2\times2$ matrices and separable characteristic polynomials.
In this paper, we presents a method for factoring morphisms between arithmetic surfaces based on the regularity of arithmetic surfaces. Using this factorization, we derive a Riemann-Hurwitz formula satisfied by the ramification divisor and the canonical divisor on arithmetic surfaces. We also extend this formula to Arakelov theory.