Related papers: Transfinite diameter and the resultant

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$…

Let $K$ be an inversive difference-differential field and $L$ a (not necessarily inversive) finitely generated difference-differential field extension of $K$. We consider the natural filtration of the extension $L/K$ associated with a…

We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is either a polytope or of dimension two. An example shows that the proof…

In this short note we prove a lemma about the dimension of certain algebraic sets of matrices. This result is needed in our paper arXiv:1201.1672. The result presented here has also applications in other situations and so it should appear…

Let $F\in W^{1,n}_{\text{loc}}(\Omega; \Bbb R^n)$ be a mapping with nonnegative Jacobian $J_F(x)=\det DF(x)\ge 0$ for a.e. $x$ in a domain $\Omega\subset\Bbb R^n$. The {\it dilatation} of $F$ is defined (almost everywhere in $\Omega$) by…

Let g be a basic classical Lie superalgebra over C. In the case of a typical weight whose every nonnegative integer multiple is also typical, we compute a closed form for the Hilbert series whose coefficients encode the dimensions of…

We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain…

Nanowires show a large potential for various electrooptical devices, such as light emitting diodes, solar cells and nanowire lasers. We present a direct method developed to calculate the modal reflection and transmission matrix at the end…

We show that a finite set of integers $A \subseteq \mathbb{Z}$ with $|A+A| \le K |A|$ contains a large piece $X \subseteq A$ with Fre\u{i}man dimension $O(\log K)$, where large means $|A|/|X| \ll \exp(O(\log^2 K))$. This can be thought of…

We give an extension of the Fekete's Subadditive Lemma for a set of submultiplicative functionals on countable product of compact spaces. Our method can be considered as an unfolding of the ideas [1]Theorem 3.1 and our main result is an…

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is…

Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Given a square matrix $A \in \mathbb{F}^{n \times n}$ and a polynomial $f \in \mathbb{F}[w]$, we determine the Jordan canonical form of the formal Fr\'{e}chet…

This paper studies the effects on Zernike coefficients of aperture scaling, translation and rotation, when a given aberrated wavefront is described on the Zernike polynomial basis. It proposes a new analytical method for computing the…

Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a…

We show that there is an absolute constant $c > 1/2$ such that the Mahler measure of the Fekete polynomials $f_p$ of the form $$f_p(z) := \sum_{k=1}^{p-1}{\left( \frac kp \right)z^k}\,,$$ (where the coefficients are the usual Legendre…

Let U be the open unit disc in C and let B be the open unit ball in C^2. We prove that every discrete subset of B is contained in the range f(U) of a complete, proper holomorphic embedding f:U-->B. Here the completeness of f means that for…

We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of…

The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which…

Consider any representation $\phi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $\Lambda(g)$. We…

It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if…