Related papers: On Universal Binary Hermitian Forms

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

We apply the algebraic consideration of holonomy systems to study Hermitian manifolds whose Chern connection is Ambrose-Singer and prove structure theorems for such manifolds. The main result (Theorem 1.2) asserts that the universal cover…

Let $\mathscr{C}$ be a small category. For every commutative ring $R$ with unity, we associate an $R\mathrm{-linear}$ abelian category with the universal homotopy category of $\mathscr{C}$, where we can do the corresponding homological…

We consider dynamically defined Hermitian matrices generated from orbits of the doubling map. We prove that their spectra fall into the GUE universality class from random matrix theory.

A positive definite quadratic form is called perfect, if it is uniquely determined by its arithmetical minimum and the integral vectors attaining it. In this self-contained survey we explain how to enumerate perfect forms in $d$ variables…

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume…

Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion $K$-algebra $Q$ to quadratic forms over the function field $K(Q)$…

We extend the theory of bilinear forms on the Green ring of a finite group developed by Benson and Parker to the context of the Grothendieck group of a triangulated category with Auslander-Reiten triangles, taking only relations given by…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms…

A (positive definite and integral) quadratic form $f$ is said to be $\textit{universal}$ if it represents all positive integers, and is said to be $\textit{primitively universal}$ if it represents all positive integers primitively. We also…

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…

Using the methods of classical invariant theory a general approach to finding of identities for Bernulli, Euler and Hermite polynomials is proposed.

Let $B$ be some invertible Hermitian or skew-Hermitian matrix. A matrix $A$ is called $B$-normal if $AA^\star = A^\star A$ holds for $A$ and its adjoint matrix $A^\star := B^{-1}A^HB$. In addition, a matrix $Q$ is called $B$-unitary, if…

Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of…

We use the notion of universal extension in a linear abelian category to study extensions of variations of mixed Hodge structure and convergent and overconvergent isocrystals. The results we obtain apply, for example, to prove the exactness…

We generalize cubic norm structures to cubic norm pairs and extend hermitian cubic norm structures to arbitrary commutative unital rings. For the associated ``skew dimension one structurable algebra" of these pairs, we construct a…

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a…

We introduce the projective Hermite constant for positive definite binary hermitian forms associated with an imaginary quadratic number field $K$. It is a lower bound for the classical Hermite constant, and these two constants coincide when…

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…