54 research outputs found
Lines Missing Every Random Point
We prove that there is, in every direction in Euclidean space, a line that
misses every computably random point. We also prove that there exist, in every
direction in Euclidean space, arbitrarily long line segments missing every
double exponential time random point.Comment: Added a section: "Betting in Doubly Exponential Time.
Time-domain harmonic balance method for aerodynamic and aeroelastic simulations of turbomachinery flows
A time-domain Harmonic Balance method is applied to simulate the blade row interactions and vibrations of state- of-the-art industrial turbomachinery configurations. The present harmonic balance approach is a time-integration scheme that turns a periodic or almost-periodic flow problem into the coupled resolution of several steady computations at different time samples of the period of interest. The coupling is performed by a spectral time-derivative operator that appears as a source term of all the steady problems. These are converged simultaneously making the method parallel in time. In this paper, a non-uniform time sampling is used to improve the robustness and accuracy regardless of the considered frequency set. Blade row interactions are studied within a 3.5-stage high-pressure axial compressor representative of the high-pressure core of modern turbofan engines. Comparisons with reference time-accurate computations show that four frequencies allow a fair match of the compressor performance, with a reduction of the computational time up to a factor 30. Finally, an aeroelastic study is performed for a counter-rotating fan stage, where the rear blade is submitted to a prescribed harmonic vibration along its first torsion mode. The aerodynamic damping is analysed, showing possible flutter
A Generalization of the Convex Kakeya Problem
Given a set of line segments in the plane, not necessarily finite, what is a
convex region of smallest area that contains a translate of each input segment?
This question can be seen as a generalization of Kakeya's problem of finding a
convex region of smallest area such that a needle can be rotated through 360
degrees within this region. We show that there is always an optimal region that
is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute
such a triangle for a given set of n segments. We also show that, if the goal
is to minimize the perimeter of the region instead of its area, then placing
the segments with their midpoint at the origin and taking their convex hull
results in an optimal solution. Finally, we show that for any compact convex
figure G, the smallest enclosing disk of G is a smallest-perimeter region
containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure
Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment
In this paper, we show that the concept of sigma-convergence associated to
stochastic processes can tackle the homogenization of stochastic partial
differential equations. In this regard, the homogenization problem for a
stochastic nonlinear partial differential equation is studied. Using some deep
compactness results such as the Prokhorov and Skorokhod theorems, we prove that
the sequence of solutions of this problem converges in probability towards the
solution of an equation of the same type. To proceed with, we use a suitable
version of sigma-convergence method, the sigma-convergence for stochastic
processes, which takes into account both the deterministic and random
behaviours of the solutions of the problem. We apply the homogenization result
to some concrete physical situations such as the periodicity, the almost
periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
Dynamical percolation on general trees
H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of
percolation on a graph . When is a tree they derived a necessary and
sufficient condition for percolation to exist at some time . In the case
that is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif
(1997) derived a necessary and sufficient condition for percolation to exist at
some time in a given target set . The main result of the present paper
is a necessary and sufficient condition for the existence of percolation, at
some time , in the case that the underlying tree is not necessary
spherically symmetric. This answers a question of Yuval Peres (personal
communication). We present also a formula for the Hausdorff dimension of the
set of exceptional times of percolation.Comment: 24 pages; to appear in Probability Theory and Related Field
- …