Related papers: Densities of 4-ranks of $K_2(\mathcal{O})$

There has been recent progress on computing the 4-rank of the tame kernel $K_2(\mathcal{O}_F)$ for $F$ a quadratic number field. For certain quadratic number fields, this progress has led to "density results'' concerning the 4-rank of tame…

For certain real quadratic number fields, we prove density results concerning 4-ranks of tame kernels. We also discuss a relationship between 4-ranks of tame kernels and 4-class ranks of narrow ideal class groups. Additionally, we give a…

Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set…

By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for…

Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and $k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$ is a prime integer and $d$ an odd square-free integer coprime to $p$. In this work,…

We study the kernel estimator of the transition density of bifurcating Markov chains. Under some ergodic and regularity properties, we prove that this estimator is consistent and asymptotically normal. Next, in the numerical studies, we…

Let $K$ be a number field of degree $n$ over ${\mathbb Q}$. Then the 4-rank of the strict class group of $K$ is at least ${\text{rank}_2 \, } ({ E_{K}^{+} } / E_K^2) - \lfloor n /2 \rfloor$ where $E_K$ and ${ E_{K}^{+} }$ denote the units…

Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\cal K}^{ijkl}$. This tensor density is, by definition,…

The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for…

Given a set of points $P\subset \mathbb{R}^{d}$ and a kernel $k$, the Kernel Density Estimate at a point $x\in\mathbb{R}^{d}$ is defined as $\mathrm{KDE}_{P}(x)=\frac{1}{|P|}\sum_{y\in P} k(x,y)$. We study the problem of designing a data…

Kernel density estimation is a convenient way to estimate the probability density of a distribution given the sample of data points. However, it has certain drawbacks: proper description of the density using narrow kernels needs large data…

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of…

For an important class of arithmetic Dedekind domains O including the ring of integers of not totally complex number fields, we describe explicitly the group of linear characters of SL_2(O). For this, we determine, for arbitrary Dedekind…

The determination of the density functions for products of random elements from specified classes of matrices is a basic problem in random matrix theory and is also of interest in theoretical physics. For connected simple Lie groups of…

We determine some properties of the narrow 2-class field tower of those real quadratic number fields whose discriminants are not a sum of two squares and for which their 2-class groups are elementary of order $4$. Here in Part I, we…

We introduce a nonparametric way to estimate the global probability density function for a random persistence diagram. Precisely, a kernel density function centered at a given persistence diagram and a given bandwidth is constructed. Our…

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2x2-matrix…

This work explores new classes of nonstationary stochastic sequences associated with polynomial hypergroups. Their covariance structures are analyzed through positive definite kernels and corresponding Hilbert spaces. Novel consistent…

Based on [1], we study the complexity of horizontality in each twistor space $\hat{E}_{\varepsilon}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over the $2$-torus $T^2$, and obtain…

In this paper we examine curves defined over a field of characteristic 2 which are $(\ZZ/2\ZZ)^2$-covers of the projective line. In particular, we prove which 2-ranks occur for such curves of a given genus and where possible we give…