Related papers: Mild parametrizations of power-subanalytic sets

We show that if $X$ is an $m$-dimensional definable set in $\mathbb{R}^\text{pow}_\text{an}$, the structure of real subanalytic sets with real power maps added, then for any positive integer r there exists a $C^r$-parameterization of X…

We prove that the function $P_\alpha(x) = \exp(1-x^{-\alpha})$ with $\alpha > 0$, is $1/\alpha$-mild. We apply this result to obtain a uniform $1/\alpha$-mild parametrization of the family of curves $\{xy = \epsilon^2 \mid (x,y) \in…

We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.

We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable…

Note that the family of closed curves C_N={(x,y)\in R^2;x^(2N)+y^(2N)=1} for N=1,2,3,... approaches the boundary of [-1,1]^2 as N \to \infty. In this paper we exhibit a natural parameterization of these curves and generalize to a larger…

We prove that for an o-minimal expansion of the real additive group $\cal R$ and a set $P\subseteq \mathbb{R}$ of dimension $0$ such that $\langle\mathcal{R},P\rangle$ is sparse, has definable choice and every definable set has interior or…

For a given discrete decomposable graphical model, we identify several alternative parametrizations, and construct the corresponding reference priors for suitable groupings of the parameters. Specifically, assuming that the cliques of the…

We prove Schlichting's theorem for approximate subgroups: if $\mathcal{X}$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with…

We study ideals $\mathcal{I}$ on $\mathbb{N}$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_{A} \colon A \in \mathcal{I} \}$ is a family of nowhere dense subsets of $X$ with $X_{A} \subset X_{B}$…

In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this…

We prove that a general planar NURBS curve parametrization $\phi: [u_0,u_m] \xrightarrow{} C \subset \mathbb{R}^2$ admits an inverse map $\phi^{-1}: C \xrightarrow{} [u_0,u_m]$ defined by rational splines. More specifically, we construct a…

We study families of positive and completely positive maps acting on a bipartite system $\mathbb{C}^M\otimes \mathbb{C}^N$ (with $M\leq N$). The maps have a property that when applied to any state (of a given entanglement class) they result…

We look at topological equisingularity of a holomorphic family of reduced mapping germs f_t:(C^3,O)->C over a contractible base T having non-isolated singularities, by means of their normalisations. We introduce the notion of…

We describe the norming sets for the space of global holomorphic sections to a $k$-power of a positive holomorphic line bundle on a compact complex manifold $X$. We characterize in metric terms the sequence of measurable subsets…

Recall that in a laminar family, any two sets are either disjoint or contained one in the other. Here, a parametrized weakening of this condition is introduced. Let us say that a set system $\mathcal{F} \subseteq 2^X$ is $t$-laminar if $A,B…

A set partition is $c$-uniform if every block has size $c$. Two families of $c$-uniform partitions of a finite set are said to be cross $t$-intersecting if two partitions from different families share at least $t$ blocks. In this paper, we…

We provide results demonstrating the smoothness of some marginal log-linear parameterizations for distributions on multi-way contingency tables. First we give an analytical relationship between log-linear parameters defined within different…

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $\mu_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to…