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Micha Wasem

Publications and source records attributed to Micha Wasem.

14 recordsLinked to original sources
arXiv2025

A sampling method based on highest density regions: Applications to surrogate models

This paper introduces a practical, one-shot design-of-experiments strategy for training surrogate models in the context of uncertainty quantification. Instead of drawing the experimental design according to the input distribution, which favours training points around its mode, we propose a heuristic method to sample uniformly within a highest density region (HDR) of the input random vector, aiming at a more homogeneous approximation accuracy over the whole domain. The approach is assessed through four error metrics: two task-agnostic accuracy measures (leave-one-out and relative mean square error) and two task-oriented measures quantifying the recovery of reference global sensitivity indices and of the probability of failure. Across the benchmark, the HDR-based designs are comparable to, or more accurate than, the distribution-based designs for most surrogate-task combinations, with the clearest gains in predictive accuracy and reliability and essentially no effect on the recovery of sensitivity indices. The method operates in a black-box context and is compatible with existing uncertainty quantification frameworks for low-dimensional and moderately correlated inputs, the curse of dimensionality affecting the sample generation rather than the design strategy itself. It is most useful whenever a single surrogate is reused across several downstream UQ tasks.

stat.ME↗
arXiv2025

Poncelet Curves

We examine pairs of closed plane curves that have the same closing property as two conic sections in Poncelet's porism. We show how the vertex curve can be computed for a given envelope and vice versa. Our formulas are universal in the sense that they produce all possible sufficiently regular pairs of such Poncelet curves. We arrive at similar results for sets of curves, analogous to the pencil of conic sections in the full Poncelet theorem. We also study the case of Poncelet curves that carry Poncelet polygons which are equiangular or even congruent.

math.DG↗
arXiv2024

Application of a Fourier-Type Series Approach based on Triangles of Constant Width to Letterforms

In this work, we present a novel approach to type design by using Fourier-type series to generate letterforms. We construct a Fourier-type series for functions in $L^2(S^1,\mathbb C)$ based on triangles of constant width instead of circles to model the curves and shapes that define individual characters. In order to compute the coefficients of the series, we construct an isomorphism $\mathcal R:L^2(S^1,\mathbb C)\to L^2(S^1,\mathbb C)$ and study its application to letterforms, thus presenting an alternative to the common use of B\'ezier curves. The proposed method demonstrates potential for creative experimentation in modern type design.

math.CA↗
arXiv2024

Counting Nodes in Smolyak Grids

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, M\"uller-Gronbach, Novak-Ritter and Ullrich.

math.CO↗
arXiv2021

Polyfunctions over Commutative Rings

A function $f:R\to R$, where $R$ is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative $p\in R[x]$. Based on this notion we introduce ring invariants which associate to $R$ the numbers $s(R)$ and $s(R';R)$, where $R'$ is the subring generated by $1$. For the ring $R=\mathbb Z/n\mathbb Z$ the invariant $s(R)$ coincides with the number theoretic \emph{Smarandache function} $s(n)$. If every function in a ring $R$ is a polyfunction, then $R$ is a finite field according to the R\'edei-Szele theorem, and it holds that $s(R)=|R|$. However, the condition $s(R)=|R|$ does not imply that every function $f:R\to R$ is a polyfunction. We classify all finite commutative rings $R$ with unit element which satisfy $s(R)=|R|$. For infinite rings $R$, we obtain a bound on the cardinality of the subring $R'$ and for $s(R';R)$ in terms of $s(R)$. In particular we show that $|R'|\leqslant s(R)!$. We also give two new proofs for the R\'edei-Szele theorem which are based on our results.

math.RA↗
arXiv2021

The Ring of Polyfunctions over $\mathbb Z/n\mathbb Z$

We study the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. The ring of polyfunctions over a commutative ring $R$ with unit element is the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense that $f(x)= p(x)$ for all $x\in R$. This allows to define a ring invariant $s$ which associates to a commutative ring $R$ with unit element a value in $\mathbb N\cup\{\infty\}$. The function $s$ generalizes the number theoretic Smarandache function. For the ring $R=\mathbb Z/n\mathbb Z$ we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number $\Psi(n)$ of polyfunctions over $\mathbb Z/n\mathbb Z$. We also investigate algebraic properties of the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over $\mathbb Z/n\mathbb Z$, and we compute the number of polyfunctions which are units of the ring.

math.CO↗
arXiv2020

On Absolute and Relative Change

Based on an axiomatic approach we propose two related novel one-parameter families of indicators of change which put in a relation classical indicators of change such as absolute change, relative change and the log-ratio.

econ.TH↗
arXiv2019

Equilibria of plane convex bodies

We obtain a formula for the number of horizontal equilibria of a planar convex body $K$ with respect to a center of mass $O$ in terms of the winding number of the evolute of $\partial K$ with respect to $O$. The formula extends to the case where $O$ lies on the evolute of $\partial K$ and a suitably modified version holds true for non-horizontal equilibria.

math.DG↗
arXiv2018

An integral that counts the zeros of a function

Given a real function $f$ on an interval $[a,b]$ satisfying mild regularity conditions, we determine the number of zeros of $f$ by evaluating a certain integral. The integrand depends on $f, f'$ and $f''$. In particular, by approximating the integral with the trapezoidal rule on a fine enough grid, we can compute the number of zeros of $f$ by evaluating finitely many values of $f,f'$ and $f''$. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.

math.CA↗
arXiv2018

Non-integer valued winding numbers and a generalized Residue Theorem

We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal value, but is also possible in a real version via an integral with bounded integrand. The new winding number allows to establish a generalized residue theorem which covers also the situation where singularities lie on the cycle. This residue theorem can be used to calculate the value of improper integrals for which the standard technique with the classical residue theorem does not apply.

math.CA↗
arXiv2015

$h$-Principle for Curves with Prescribed Curvature

We prove that every immersed $C^2$-curve $\gamma$ in $\mathbb R^n$, $n\geqslant 3$ with curvature $k_{\gamma}$ can be $C^1$-approximated by immersed $C^2$-curves having prescribed curvature $k>k_{\gamma}$. The approximating curves satisfy a $C^1$-dense $h$-principle. As an application we obtain the existence of $C^2$-knots of arbitrary positive curvature in each isotopy class, which generalizes a similar result by McAtee for $C^2$-knots of constant curvature.

math.DG↗
arXiv2015

Equidimensional Isometric Extensions

Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:\Sigma\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean standard metric. We prove a general curvature obstruction to the existence of merely differentiable extensions and an obstruction to the existence of Lipschitz extensions of $f$ using a length comparison argument. Using a weak form of convex integration, we then construct one-sided isometric Lipschitz extensions of which we compute the Hausdorff dimension of the singular set and obtain an accompanying density result. As an application we obtain the existence of infinitely many Lipschitz isometries collapsing the standard two-sphere to the closed standard unit $2$-disk mapping a great-circle to the boundary of the disk.

math.DG↗
arXiv2014

The One-Sided Isometric Extension Problem

Let $\Sigma$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $\Sigma$ into $\mathbb R^q$ equipped with the standard Euclidean metric, $q\geqslant n+1$, to be locally isometrically $C^1$-extendable to $M$. Even if this condition is not met, "one-sided" isometric $C^1$-extensions may exist and turn out to satisfy a $C^0$-dense parametric $h$-principle in the sense of Gromov.

math.DG↗