Related papers: $B'$

A number $\alpha$ has a representation with respect to the numbers $\alpha_1,...,\alpha_n$, if there exist the non-negative integers $\lambda_1,... ,\lambda_n$ such that $\alpha=\lambda_1\alpha_1+...+\lambda_n \alpha_n$. The largest natural…

Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of…

Feigin and Shoikhet conjectured in math/0610410 that successive quotients $B_m(A_n)$ of the lower central series filtration of a free associative algebra $A_n$ have polynomial growth. In this paper we give a proof of this conjecture, using…

For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erd\H{o}s problem 684 asks for bounds on $f(n)$. We resolve the…

We derive a lower bound for the top possibly-non-zero Betti number of the Milnor fiber of an analytic function in terms of the zero-dimensional L\^e number and the internal monodromy of the vanishing cycles restricted to the complex link of…

Let $\mathcal{U(\alpha, \lambda)}$, $0<\alpha <1$, $0 < \lambda <1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda$$ in the unit disc ${\mathbb…

In this paper, we generalize several Berezin number inequalities involving product of operators. For instance, we show that if $A, B$ are positive operators and $X$ is any operator, then \begin{align*}…

Let $P(n)$ be the number of polyominoes of $n$ cells and $\lambda$ be Klarner's constant, that is, $\lambda=\lim_{n\to\infty} \sqrt[n]{P(n)}$. We show that there exist some positive numbers $A,T$, so that for every $n$ \[ P(n) \ge…

We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds…

Let $\mathbb{R}^m$ be endowed with the Euclidean metric. The covering radius of a lattice $\Lambda \subset \mathbb{R}^m$ is the least distance $r$ such that, given any point of $\mathbb{R}^m$, the distance from that point to $\Lambda$ is…

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of…

Let $k$ be a number field, let $\theta$ be a nonzero algebraic number, and let $H(\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of…

We show that any Algebraic Branching Program (ABP) computing the polynomial $\sum_{i = 1}^n x_i^n$ has at least $\Omega(n^2)$ vertices. This improves upon the lower bound of $\Omega(n\log n)$, which follows from the classical result of Baur…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational…

The present paper deals with the problem of computing (or at least estimating) the LW-number $\lambda(n)$, i.e., the supremum of all $\gamma$ such that for each convex body $K$ in $\mathbb{R}^n$ there exists an orthonormal basis…

Let $A$ be a positive operator on a complex Hilbert space $\mathcal{H}.$ We present inequalities concerning upper and lower bounds for $A$-numerical radius of operators, which improve on and generalize the existing ones, studied recently in…

Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given a positive operator $A\in\B(\h)$, and a number $\lambda\in [0,1]$, a…

we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of…

For every positive integer $n$, consider the linear operator $\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1 - t)^{d + 1}} = \sum_{m \geq 0} g(m) t^{m}$, for some…