Related papers: Quadratic Lagrange spectrum: I

This paper deals with the quadratic integers of small norms and asserts that in some sense R >> (log D)^2 is true for almost all real quadratic number fields. (A few errata is corrected.)

We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group $\mathbf H_4$. They are identical to the Markoff and Lagrange spectra of the unit circle. The Markoff spectrum on $\mathbf H_4$ is also known as the Markoff…

The Lagrange and Markov spectra have been studied since late 19th century, concerning badly approximable real numbers. The Mordell-Gruber spectrum has been studied since 1936, concerning the supremum of the area of a rectangle centered at…

In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2--spectral set for the matrix A, we propose a study of various constants. We review some partial results, many problems are still open. We describe our…

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

We study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth maps in terms of scalar Lagrange functions.

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

I discuss the spectrum of hadrons containing heavy quarks ($b$ or $c$), and how well the experimental results are matched by theoretical ideas. Useful insights come from potential models and applications of Heavy Quark Symmetry and these…

This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur…

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with…

For the quadratic Lagrange interpolation function, an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation. The upper bound for the…

We study the real counterpart of double Hurwitz numbers, called real double Hurwitz numbers here. We establish a lower bound for these numbers with respect to their dependence on the distribution of branch points. We use it to prove, under…

The field of formal Laurent series is a natural analogue of the real numbers, and mathematicians have been translating well-known results about rational approximations to that setting. In the framework of power series over the rational…

Well-known results of Lagrange and Jacobi prove that the every $m \in \mathbb N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we…

The presence of chiral modes on the edges of quantum Hall samples is essential to our understanding of the quantum Hall effect. In particular, these edge modes should support ballistic transport and therefore, in a single particle picture,…

We obtain a sufficient condition for the convexity of quaternionic numerical range for complex matrices in terms of its complex numerical range. It is also shown that the Bild coincides with complex numerical range for real matrices. From…

We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and may assume…