149 research outputs found
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
On the logarithmic probability that a random integral ideal is -free
This extends a theorem of Davenport and Erd\"os on sequences of rational
integers to sequences of integral ideals in arbitrary number fields . More
precisely, we introduce a logarithmic density for sets of integral ideals in
and provide a formula for the logarithmic density of the set of so-called
-free ideals, i.e. integral ideals that are not multiples of any
ideal from a fixed set .Comment: 9 pages, to appear in S. Ferenczi, J. Ku{\l}aga-Przymus and M.
Lema\'nczyk (eds.), Chowla's conjecture: from the Liouville function to the
M\"obius function, Lecture Notes in Math., Springe
An Analytical Construction of the SRB Measures for Baker-type Maps
For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle
and Bowen showed the existence of an invariant measure (SRB measure) weakly
attracting the temporal average of any initial distribution that is absolutely
continuous with respect to the Lebesgue measure. Recently, the SRB measures
were found to be related to the nonequilibrium stationary state distribution
functions for thermostated or open systems. Inspite of the importance of these
SRB measures, it is difficult to handle them analytically because they are
often singular functions. In this article, for three kinds of Baker-type maps,
the SRB measures are analytically constructed with the aid of a functional
equation, which was proposed by de Rham in order to deal with a class of
singular functions. We first briefly review the properties of singular
functions including those of de Rham. Then, the Baker-type maps are described,
one of which is non-conservative but time reversible, the second has a
Cantor-like invariant set, and the third is a model of a simple chemical
reaction . For the second example, the
cases with and without escape are considered. For the last example, we consider
the reaction processes in a closed system and in an open system under a flux
boundary condition. In all cases, we show that the evolution equation of the
distribution functions partially integrated over the unstable direction is very
similar to de Rham's functional equation and, employing this analogy, we
explicitly construct the SRB measures.Comment: 53 pages, 10 figures, to appear in CHAO
Birth and death processes and quantum spin chains
This papers underscores the intimate connection between the quantum walks
generated by certain spin chain Hamiltonians and classical birth and death
processes. It is observed that transition amplitudes between single excitation
states of the spin chains have an expression in terms of orthogonal polynomials
which is analogous to the Karlin-McGregor representation formula of the
transition probability functions for classes of birth and death processes. As
an application, we present a characterization of spin systems for which the
probability to return to the point of origin at some time is 1 or almost 1.Comment: 14 page
Almost perfect state transfer in quantum spin chains
The natural notion of almost perfect state transfer (APST) is examined. It is
applied to the modelling of efficient quantum wires with the help of spin
chains. It is shown that APST occurs in mirror-symmetric systems, when the
1-excitation energies of the chains are linearly independent over rational
numbers. This result is obtained as a corollary of the Kronecker theorem in
Diophantine approximation. APST happens under much less restrictive conditions
than perfect state transfer (PST) and moreover accommodates the unavoidable
imperfections. Some examples are discussed.Comment: 11 page
Characterization of n-rectifiability in terms of Jones' square function: Part II
We show that a Radon measure in which is absolutely
continuous with respect to the -dimensional Hausdorff measure is
-rectifiable if the so called Jones' square function is finite -almost
everywhere. The converse of this result is proven in a companion paper by the
second author, and hence these two results give a classification of all
-rectifiable measures which are absolutely continuous with respect to
. Further, in this paper we also investigate the relationship between
the Jones' square function and the so called Menger curvature of a measure with
linear growth.Comment: A corollary regarding analytic capacity and a few new references have
been adde
Asymptotic solvers for ordinary differential equations with multiple frequencies
We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focusing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question. Numerical examples illustrate the effectiveness of the method
Fractional differentiability of nowhere differentiable functions and dimensions
Weierstrass's everywhere continuous but nowhere differentiable function is
shown to be locally continuously fractionally differentiable everywhere for all
orders below the `critical order' 2-s and not so for orders between 2-s and 1,
where s, 1<s<2 is the box dimension of the graph of the function. This
observation is consolidated in the general result showing a direct connection
between local fractional differentiability and the box dimension/ local Holder
exponent. Levy index for one dimensional Levy flights is shown to be the
critical order of its characteristic function. Local fractional derivatives of
multifractal signals (non-random functions) are shown to provide the local
Holder exponent. It is argued that Local fractional derivatives provide a
powerful tool to analyze pointwise behavior of irregular signals.Comment: minor changes, 19 pages, Late
Holder exponents of irregular signals and local fractional derivatives
It has been recognized recently that fractional calculus is useful for
handling scaling structures and processes. We begin this survey by pointing out
the relevance of the subject to physical situations. Then the essential
definitions and formulae from fractional calculus are summarized and their
immediate use in the study of scaling in physical systems is given. This is
followed by a brief summary of classical results. The main theme of the review
rests on the notion of local fractional derivatives. There is a direct
connection between local fractional differentiability properties and the
dimensions/ local Holder exponents of nowhere differentiable functions. It is
argued that local fractional derivatives provide a powerful tool to analyse the
pointwise behaviour of irregular signals and functions.Comment: 20 pages, Late
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