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Sep 21, 2013
09/13

by
Almut Burchard

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After n random polarizations of Borel set on a sphere, its expected symmetric difference from a polar cap is bounded by C/n, where the constant depends on the dimension [arXiv:1104.4103]. We show here that this power law is best possible, and that the constant grows at least linearly with the dimension.

Source: http://arxiv.org/abs/1108.5500v1

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54

Sep 19, 2013
09/13

by
Michael Aizenman; Almut Burchard

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Random systems of curves exhibiting fluctuating features on arbitrarily small scales ($\delta$) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit H\"older continuous parametrizations with a common exponent and a common random prefactor, which in the scaling limit ($\delta\to 0$) remains stochastically bounded. The regularity is used for the...

Source: http://arxiv.org/abs/math/9801027v4

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54

Sep 21, 2013
09/13

by
Almut Burchard; Marina Chugunova

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We study the problem of finding the instability index of certain non-selfadjoint fourth order differential operators that appear as linearizations of coating and rimming flows, where a thin layer of fluid coats a horizontal rotating cylinder. The main result reduces the computation of the instability index to a finite-dimensional space of trigonometric polynomials. The proof uses Lyapunov's method to associate the differential operator with a quadratic form, whose maximal positive subspace has...

Source: http://arxiv.org/abs/0905.1834v1

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23

Sep 21, 2013
09/13

by
Almut Burchard; Marc Fortier

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We derive conditions under which random sequences of polarizations (two-point symmetrizations) converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose distributions need not be uniform. The proof of convergence hinges on an estimate for the expected distance from the limit that also yields a bound on the rate of convergence. In the special case of i.i.d. sequences, we obtain almost sure convergence even for...

Source: http://arxiv.org/abs/1104.4103v4

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Jun 30, 2018
06/18

by
Almut Burchard; Adele Ferone

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The distance of an extremal of the P\'olya-Szeg\H{o} inequality from a translate of its symmetric decreasing rearrangement is controlled by the measure of the set of critical points.

Topics: Functional Analysis, Mathematics

Source: http://arxiv.org/abs/1407.6567

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37

Sep 20, 2013
09/13

by
Almut Burchard; Hichem Hajaiej

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The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u_1,..., u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and...

Source: http://arxiv.org/abs/math/0506336v2

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0.0

Jun 30, 2018
06/18

by
Almut Burchard; Gregory R. Chambers

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The Coulomb energy of a charge that is uniformly distributed on some set is maximized (among sets of given volume) by balls. It is shown here that near-maximizers are close to balls.

Topics: Functional Analysis, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1407.1918

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37

Sep 18, 2013
09/13

by
Almut Burchard; Lawrence E. Thomas

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Let C be a smooth closed curve of length 2 Pi in R^3, and let k(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrodinger operator H_C = -d^2/ds^2 + k^2 acting on the space of square integrable 2 Pi - periodic functions. A natural conjecture is that the lowest spectral value e(C) is bounded below by 1 for any C (this value is assumed when C is a circle). We study a family of curves {C} that includes the circle and for which e(C)=1 as...

Source: http://arxiv.org/abs/math/0505123v1

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58

Sep 18, 2013
09/13

by
Almut Burchard; Lawrence E. Thomas

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The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame...

Source: http://arxiv.org/abs/math/0202278v2

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68

Sep 18, 2013
09/13

by
Almut Burchard; Gregory R. Chambers

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Steiner symmetrization along n linearly independent directions transforms every compact subset of R^n into a set of finite perimeter.

Source: http://arxiv.org/abs/1209.4521v1

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Jun 28, 2018
06/18

by
Almut Burchard; Rustum Choksi; Ihsan Topaloglu

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We consider a class of nonlocal shape optimization problems for sets of fixed mass where the energy functional is given by an attractive/repulsive interaction potential in power-law form. We find that the existence of minimizers of this shape optimization problem depends crucially on the value of the mass. Our results include existence theorems for large mass and nonexistence theorems for small mass in the class where the attractive part of the potential is quadratic. In particular, for the...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1512.07282

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2.0

Jun 28, 2018
06/18

by
Alireza Shekaramiz; Jorg Liebeherr; Almut Burchard

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We present an extension of the window flow control analysis by R. Agrawal et.al. (Reference [1]), C.-S. Chang (Reference [6]), and C.-S. Chang et. al. (Reference [8]) to a system with random service time and fixed feedback delay. We consider two network service models. In the first model, the network service process itself has no time correlations. The second model addresses a two-state Markov-modulated service.

Topics: Performance, Computing Research Repository, Networking and Internet Architecture

Source: http://arxiv.org/abs/1507.04631

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4.0

Jun 28, 2018
06/18

by
Almut Burchard; Gregory R. Chambers; Anne Dranovski

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We consider the trajectories of points on $\mathbb{S}^{d - 1}$ under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is $d+1$.

Topics: Dynamical Systems, Mathematics

Source: http://arxiv.org/abs/1509.02454

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Sep 22, 2013
09/13

by
Yashar Ghiassi-Farrokhfal; Jorg Liebeherr; Almut Burchard

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We study how the choice of packet scheduling algorithms influences end-to-end performance on long network paths. Taking a network calculus approach, we consider both deterministic and statistical performance metrics. A key enabling contribution for our analysis is a significantly sharpened method for computing a statistical bound for the service given to a flow by the network as a whole. For a suitably parsimonious traffic model we develop closed-form expressions for end-to-end delays, backlog,...

Source: http://arxiv.org/abs/1101.1237v1

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Sep 21, 2013
09/13

by
Almut Burchard; Marina Chugunova; Benjamin K. Stephens

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The degenerate parabolic equation u_t + [u^3(u_xxx + u_x - sin x)]_x=0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each given mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region at the top with zero contact angle. The droplet minimizes the energy and attracts all strong solutions that satisfy certain energy and entropy inequalities. The distance of any solution...

Source: http://arxiv.org/abs/1011.2108v2

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Sep 20, 2013
09/13

by
Hussein Al-Zubaidy; Jorg Liebeherr; Almut Burchard

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A fundamental problem in the delay and backlog analysis across multi-hop paths in wireless networks is how to account for the random properties of the wireless channel. Since the usual statistical models for radio signals in a propagation environment do not lend themselves easily to a description of the available service rate on a wireless link, the performance analysis of wireless networks has resorted to higher-layer abstractions, e.g., using Markov chain models. In this work, we propose a...

Source: http://arxiv.org/abs/1207.6630v1

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28

Sep 19, 2013
09/13

by
Almut Burchard; Marina Chugunova; Benjamin K. Stephens

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Steady states of the thin film equation $u_t+[u^3 (u_xxx + \alpha^2 u_x -\sin(x) )]_x=0$ are considered on the periodic domain $\Omega = (-\pi,\pi)$. The equation defines a generalized gradient flow for an energy functional that controls the $H^1$-norm. The main result establishes that there exists for each given mass a unique nonnegative function of minimal energy. This minimizer is symmetric decreasing about $x=0$. For $\alpha

Source: http://arxiv.org/abs/1009.4092v1

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Sep 18, 2013
09/13

by
Michael Aizenman; Almut Burchard; Charles M. Newman; David B. Wilson

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A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean...

Source: http://arxiv.org/abs/math/9809145v3