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# Wasem Micha

### Assoziierter Professor FH

#### Main skills

### Assoziierter Professor FH

Phone: +41 26 429 67 22

Desktop: HEIA_D30.03

Boulevard de Pérolles 80, 1700 Fribourg, CH

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Ongoing

**Role: **Collaborator

**Description du projet : **

Modellierung und statistrische Analyse eines Spiels

**Research team within HES-SO:**
Wasem Micha

**Statut: ** Ongoing

**Role: **Collaborator

**Description du projet : **

Statistische Analyse und Modellierung eines Spielautomaten

**Research team within HES-SO:**
Wasem Micha

**Statut: ** Ongoing

**Role: **Co-applicant

**Description du projet : **

In 2017, Shalev-Schwartz, Shammah and Shashua published an article on the preprint server arXiv entitled «On a Formal Model of Safe and Scalable Self-driving Cars». In this paper, a deterministic safety model called RSS is introduced and the article claims that if every road user adheres to the rules given by RSS, then no accidents can happen. The goal of this project was to inspect the article and to either validate the model or to find counterexamples to the claims made therein. Although the approach presented in the article seems promising, we were able to find shortcomings of the work by constructing scenarios in which RSS does not provide a guarantee for no accidents. Furthermore, we detected lines of thought along which the model could be refined.

Joint work with Corinne Hager Jörin and Florence Yerly.

**Research team within HES-SO:**
Wasem Micha

**Statut: ** Ongoing

2022

**Summary:**

A function f:R→R, where R is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative p∈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=Z/nZ the invariant s(R) coincides with the number theoretic Smarandache or Kempner functions(n). If every function in a ring R is a polyfunction, then R is a finite field according to the Rédei–Szele theorem, and it holds that s(R)=|R|. However, the condition s(R)=|R| does not imply that every function f:R→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′|≤s(R)!. We also give two new proofs for the Rédei–Szele theorem which are based on our results.

**Summary:**

We study the ring of polyfunctions over Z/nZ. The ring of polyfunctions over a commutative ring R with unit element is the ring of functions f:R→R which admit a polynomial representative p∈R[x] in the sense that f(x)=p(x) for all x∈R. This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in N∪{∞}. The function s generalizes the number theoretic Smarandache function. For the ring R=Z/nZ we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number Ψ(n) of polyfunctions over Z/nZ. We also investigate algebraic properties of the ring of polyfunctions over Z/nZ. 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 Z/nZ, and we compute the number of polyfunctions which are units of the ring.

2021

**Summary:**

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 ∂K with respect to O. The formula extends to the case where O lies on the evolute of ∂K and a suitably modified version holds true for non-horizontal equilibria.

**Summary:**

Let Σ be a hypersurface in an n-dimensional Riemannian manifold M, n≥2. We study the isometric extension problem for isometric immersions f:Σ→Rn, where Rn 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.

2020

**Summary:**

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

2019

**Summary:**

We define a generalization of the winding number of a piecewise C1 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.

2018

**Summary:**

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.

2017

2016

Achievements