Related papers: Computing the exponent of a Lebesgue space
In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show…
For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the…
We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel.…
Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any \epsilon>0, we show that X contains O(B^{D+\epsilon}) rational points of…
Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points…
The TTE computability notion in effective metric spaces is usually defined by using Cauchy representations. Under some weak assumptions, we characterize this notion in a way which avoids using the representations.
In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper…
The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small…
The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and…
The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs…
To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the…
Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$…
A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in…
A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number…
We investigate computability in the lattice of equivalence relations on the natural numbers. We mostly investigate whether the subsets of appropriately defined subrecursive equivalence relations -for example the set of all polynomial-time…
We solve the covering problem for Demuth randomness, showing that a computably enumerable set is computable from a Demuth random set if and only if it is strongly jump-traceable. We show that on the other hand, the class of sets which form…
An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Levy process is given in [3, Theorem 2] for almost all irrational $\alpha$. This result cannot be made stronger in the following sense:…
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on $(-1, +1)$ augmented by the point $-1$ and with the Radau quadrature points on either $(-1, +1]$ or $[-1, +1)$. It is shown that the Lebesgue…