221 research outputs found

    Curves of Finite Total Curvature

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    We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.Comment: 25 pages, 4 figures; final version, to appear in "Discrete Differential Geometry", Oberwolfach Seminars 38, Birkhauser, 200

    The opaque square

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    The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length 2+62=2.6389\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots. The current best lower bound for the length of a (not necessarily connected) barrier is 22, as established by Jones about 50 years ago. No better lower bound is known even if the barrier is restricted to lie in the square or in its close vicinity. Under a suitable locality assumption, we replace this lower bound by 2+10122+10^{-12}, which represents the first, albeit small, step in a long time toward finding the length of the shortest barrier. A sharper bound is obtained for interior barriers: the length of any interior barrier for the unit square is at least 2+1052 + 10^{-5}. Two of the key elements in our proofs are: (i) formulas established by Sylvester for the measure of all lines that meet two disjoint planar convex bodies, and (ii) a procedure for detecting lines that are witness to the invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure

    Fractal analysis of Hopf bifurcation for a class of completely integrable nonlinear Schrödinger Cauchy problems

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    We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schrödinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schrödinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in L2(Ω)L^2(\Omega) is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schrödinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in Rn\mathbb{R}^n

    Exceptional sets in projection and slicing theorems

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    The dissertation Exceptional sets in projection and slicing theorems contains a treatment of two classical topics in fractal geometry: projections and slicing. The thesis consists of an introductory chapter and two scientific articles; the new results extend a long line of research originated by J. M. Marstrand in 1954. The first paper deals with projecting a planar set K onto lines. The fractal geometer is interested in the following question: what is the relation between the dimensions of K and its projections? In 1954, Marstrand proved that if the dimension of K lies between zero and one, then the projections tend to preserve dimension; for almost every line the dimension of the projection equals dim K. During its nearly 60 years of existence, this theme has spawned countless variations. In the thesis, special attention is given to scrutinizing the words almost every line in Marstrand s theorem. The words cannot be entirely omitted (an illustrative example is given by projecting the y-axis onto the x-axis), but they can be sharpened in many cases. The definition used by Marstrand allows for a fairly large set of exceptional lines , the projection onto which fails to preserve the dimension of K. It turns out that better bounds for the size of this exceptional set can be obtained through a more intricate analysis. The second paper is thematically close akin to the first; it takes up another 1954 result by Marstrand, the slicing theorem , and examines the exceptional set estimates therein. To explain the slicing theorem, fix a planar set K with dimension greater than one. This time, the set K is intersected with various planar lines. What is the dimension of these slices of K? In general, one cannot expect to find a single constant answering the question: if K is bounded, many lines evade K altogether, and the corresponding slices have dimension zero. However, not all slices of K can be so small. Marstrand showed that in almost every direction many lines meet K in a set of dimension dim K 1. In Marstrand s original formulation, the same definition for the words almost every was used as in the projection theorem, and, again, bounds for the size of the exceptional set can be improved with new techniques. In the thesis, similar estimates are also derived in a variant of the theorem where planar lines are replaced by more complicated curves.Matematiikan väitöskirjani Exceptional sets in projection and slicing theorems (suom. Poikkeusjoukot projektio- ja viipalointilauseissa) käsittelee kahta klassista ongelmaa fraktaaligeometriassa: joukkojen projisointia ja viipalointia. Väitöskirja koostuu johdannosta sekä kahdesta tieteellisestä artikkelista; näiden sisältämät uudet tulokset ovat jatkoa kehityskululle, joka sai alkunsa J. M. Marstrandin työstä vuonna 1954. Väitöskirjan ensimmäinen artikkeli käsittelee tason osajoukon K projisointia suorille. Fraktaaligeometrian kannalta olennainen kysymys on seuraava: millainen yhteys on joukon K ja sen projektioiden dimensioilla? Marstrand osoitti vuonna 1954, että jos K:n dimensio dim K on nollan ja yhden välillä, projektioiden dimensio on dim K melkein kaikilla suorilla. Tämä teema on lähes 60 vuodessa poikinut lukemattomia muunnelmia, ja näihin lukeutuvat myös artikkelini tulokset. Erityisroolissa on sanojen melkein jokaisella täsmentäminen. Sanoista ei voida tyystin luopua Marstrandin lauseessa (havainnollinen esimerkki on y-akselin projisointi x-akselille) mutta ne voidaan korvata tarkemmalla väitteellä. Marstrandin käyttämä määritelmä sallii melko suuren joukon poikkeuksellisia suoria, joilla projektion dimensio poikkeaa luvusta dim K; tämän poikkeusjoukon koolle saadaan parempia arvioita analyysiä tarkentamalla. Väitöskirjan toinen artikkeli on aiheeltaan lähellä ensimmäistä: taas tarkennetaan poikkeusjoukkoarvioita Marstrandin lauseessa vuodelta 1954, tosin kyse on eri lauseesta kuin yllä. Kiinnitetään tason osajoukko K, jonka dimensio on yli yhden. Tällä kertaa joukkoa K ei projisoida vaan sitä leikellään suorilla. Mikä on näiden K:n viipaleiden dimensio? Yleisesti ei ole toivoa löytää vain yhtä vakiota, joka olisi sama melkein kaikilla suorilla: jos K on rajoitettu, leikkaus on usein tyhjä, ja silloin viipaleiden dimensio on nolla. Osoittautuu kuitenkin, etteivät kaikki viipaleet voi olla näin pieniä; Marstrand todisti, että melkein jokaisessa suunnassa monet suorat leikkaavat joukkoa K dimensiossa dim K 1. Lauseen alkuperäisessä muotoilussa käytetään samaa melkein jokaisessa sanojen määritelmää kuin projektioiden yhteydessä, ja poikkeusjoukkoarvioita voidaan jälleen tarkentaa uusilla tekniikoilla. Väitöskirjassani johdetaan vastaavia arvioita myös tapauksessa, jossa tason suorat korvataan monimutkaisemmilla käyräperheillä

    Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences

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    A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformizing coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichm\"uller mapping on the Riemann sphere.Comment: 75 pages, 18 figure

    Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination

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    This work is devoted so show the appearance of different cracking modes in linearly elastic thin film systems by means of an asymptotic analysis as the thickness tends to zero. By superposing two thin plates, and upon suitable scaling law assumptions on the elasticity and fracture parameters, it is proven that either debonding or transverse cracks can emerge in the limit. A model coupling debonding, transverse cracks and delamination is also discussed
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