Cylinder | EPFL Graph Search

A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the right circular cylinder. Types The definitions and results in this section are taken from the 1913 text Plane and Solid Geometry by George Wentworth and David Eugene Smith . A is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parall

Aspects of two dimensional magnetic Schrödinger operatorsquantum hall systems and magnetic Stark resonances

Christian Ferrari

In this PhD thesis we deal with two mathematical problems arising from quantum mechanics. We consider a spinless non relativistic quantum particle whose configuration space is a two dimensional surface S. We also suppose that the particle feels the effect of an homogeneous magnetic field perpendicular to the surface S. In the first case S = R × SL1, the infinite cylinder of circumference L, corresponding to periodic boundary conditions, while in the second one S = R2. In both cases the particle feels the effect of an additional suitable potential. We are thus left with the study of two specific classes of Schrödinger operators. The operator of the first class generates the dynamics of the particle when it is submitted to an Anderson-type random potential, as well as to a non random potential confining the particle along the cylinder axis in an interval of length L. In this case we describe the spectrum and classify it by the quantum mechanical current carried by the corresponding eigenfunctions. We prove that there are spectral regions in which all the eigenvalues have an order one current with respect to L, and spectral regions where eigenvalues with order one current and eigenvalues with infinitesimal current with respect to L are intermixed. These results are relevant for the theory of the integer quantum Hall effect. The second Schrödinger operator class corresponds to the physical situation where the potential is the sum of a "local" potential and of a potential due to a weak constant electric field F. In this case we show that the resonant states, induced by the electric field, decay exponentially at a rate given by the imaginary part of the eigenvalues of some non self-adjoint operator. Moreover we prove an upper bound on this imaginary part that turns out to be of order exp(-1/F2) as F goes to zero. Therefore the lifetime of the resonant states is at least of order exp(-1/F2).

EPFL2003

Propriétés hors-équilibre d'une paroi adiabatique

Séverine Pache

We study the evolution of a system composed of N non-interacting particles of mass m distributed in a cylinder of length L. The cylinder is separated into two parts by an adiabatic piston of a mass M ≫ m. The length of the cylinder is a fix parameter and can be finite or infinite (in this case N is infinite). For the infinite case we carry out a perturbative analysis using Boltzmann's equation based on a development of the velocity distribution of the piston in function of a small dimensionless parameter ε = √(m/M). The non-stationary case is solved up to the order ε ;; our analysis shows that the system tends exponentially fast towards a stationary state where the piston has an average velocity V. The characteristic time scale for this relaxation is proportional to the mass of the piston (τ0 = M/A where A is the cross-section of the piston). We show that for equal pressures the collisions of the particles induce asymmetric fluctuations of the velocity of the piston which leads to a macroscopic movement of the piston in the direction of the higher temperature. In the case of the finite model a perturbative approach based on Liouville's equation (using the parameter α = 2m/(M + m)) shows that the evolution towards thermal equilibrium happens on two well separated time scales. The first relaxation step is a fast, deterministic and adiabatic evolution towards a state of mechanical equilibrium with approximately equal pressures but different temperatures. The movement of the piston is more or less damped. This damping qualitatively depends on whether the ratio R = Mgas/M between the total mass of the gas and the mass of the piston is small (R < 2) or large (R > 4). The second part of the evolution is much slower ; the typical time scales are proportional to the mass of the piston. There is a stochastic evolution including heat transfer leading to thermal equilibrium. A microscopic analysis yields the relation XM(t) = L(1/2 - ξ(at)) where the function ξ is independent of M. Using the hypothesis of homogeneity (i.e. the values of the densities, pressures and temperatures at the surface of the piston can be replaced by their respective average values) introduced in the previous analysis the observed damping does not show up. This can be explained by shock waves propagating between the piston and the walls at the extremities of the cylinder. In order to study the behaviour of the system there is hence a need to adequately describe the non-equilibrium fluids around the piston. We carry out an analysis of the infinite case, based on the perturbative approach introduced earlier. In this case the initial conditions are chosen in such a manner that the piston on average stays at the origin. It is shown that it is possible to describe the evolution of the fluids in such a way that it is coherent with the two laws of thermodynamics and the phenomenological relationships. Finally we study the case of a constant velocity of the piston in a finite cylinder. Such a condition and elastic collisions allow us to derive an explicit expression for the distribution of the fluids and hence for the hydrodynamics fields. This expression reveals the presence of shock waves between the piston and the extremities of the cylinder.

EPFL2004

Simple topological graphs

Andres Jacinto Ruiz Vargas

This thesis is devoted to the understanding of topological graphs. We consider the following four problems: 1. A \emph{simple topological graph} is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once, at endpoint or at a crossing. Let

G

be a complete simple topological graph on

n

vertices. The three edges induced by any triplet of vertices in

G

form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an \emph{empty triangle}. In 1998, Harborth proved that

G

has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least

2n/3

. We settle Harborth's conjecture in the affirmative. 2. A \emph{monotone cylindrical} graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called \emph{simple} if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are \emph{disjoint} if they do not intersect. We show that every simple complete monotone cylindrical graph on

n

vertices contains

\Omega(n^{1-\epsilon})

pairwise disjoint edges for any

\epsilon>0

. As a consequence, we show that every simple complete topological graph (drawn in the plane) with

n

vertices contains

\Omega(n^{\frac 12-\epsilon})

pairwise disjoint edges for any

\epsilon>0

. By extending some of the ideas here we are then able to get rid of the

\epsilon

term in the exponent, showing that in fact we can always guarantee a set with

\Omega(n^{\frac 12})

pairwise disjoint edges. This improves the previous lower bound of

\Omega(n^\frac 13)

by Suk and independently by Fulek. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges. 3. A {\em tangle} is a graph drawn in the plane such that its edges are represented by continuous arcs, and any two edges share precisely one point, which is either a common endpoint or an interior point at which the two edges are tangent to each other. These points of tangencies are assumed to be distinct. If we drop the last assumption, that is, more than two edges may touch one another at the same point, then the drawing is called a {\em degenerate tangle}. We settle a problem of Pach, Radoi\v ci'c, and T'oth \cite{TTpaper}, by showing that every degenerate tangle has at most as many edges as vertices. We also give a complete characterization of tangles. 4. We show that for a constant

t\in \NN

, every simple topological graph on

n

vertices has

O(n)

edges if the graph has no two sets of

t

edges each such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is

K_{t,t}

-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i'c, and T'oth: Every

n

-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most

O(n)

edges.

EPFL2016

Simple topological graphs

Andres Jacinto Ruiz Vargas

G

be a complete simple topological graph on

n

vertices. The three edges induced by any triplet of vertices in

G

form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an \emph{empty triangle}. In 1998, Harborth proved that

G

has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least

2n/3

n

vertices contains

\Omega(n^{1-\epsilon})

pairwise disjoint edges for any

\epsilon>0

. As a consequence, we show that every simple complete topological graph (drawn in the plane) with

n

vertices contains

\Omega(n^{\frac 12-\epsilon})

pairwise disjoint edges for any

\epsilon>0

. By extending some of the ideas here we are then able to get rid of the

\epsilon

term in the exponent, showing that in fact we can always guarantee a set with

\Omega(n^{\frac 12})

pairwise disjoint edges. This improves the previous lower bound of

\Omega(n^\frac 13)

t\in \NN

, every simple topological graph on

n

vertices has

O(n)

edges if the graph has no two sets of

t

edges each such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is

K_{t,t}

-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i'c, and T'oth: Every

n

-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most

O(n)

edges.

EPFL2016

Aspects of two dimensional magnetic Schrödinger operatorsquantum hall systems and magnetic Stark resonances

Christian Ferrari

EPFL2003

Propriétés hors-équilibre d'une paroi adiabatique

Séverine Pache

EPFL2004