2,459 research outputs found
Orthogonal nets and Clifford algebras
A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.Comment: Plain TeX, 16 pages, 4 picture
Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow
It is well known that the Poiseuille flow of a visco-elastic polymer fluid
between plates or through a tube is linearly stable in the zero Reynolds number
limit, although the stability is weak for large Weissenberg numbers. In this
paper we argue that recent experimental and theoretical work on the instability
of visco-elastic fluids in Taylor-Couette cells and numerical work on channel
flows suggest a scenario in which Poiseuille flow of visco-elastic polymer
fluids exhibits a nonlinear "subcritical" instability due to normal stress
effects, with a threshold which decreases for increasing Weissenberg number.
This proposal is confirmed by an explicit weakly nonlinear stability analysis
for Poiseuille flow of an UCM fluid. Our analysis yields explicit predictions
for the critical amplitude of velocity perturbations beyond which the flow is
nonlinearly unstable, and for the wavelength of the mode whose critical
amplitude is smallest. The nonlinear instability sets in quite abruptly at
Weissenberg numbers around 4 in the planar case and about 5.2 in the
cylindrical case, so that for Weissenberg numbers somewhat larger than these
values perturbations of the order of a few percent in the wall shear stress
suffice to make the flow unstable. We have suggested elsewhere that this
nonlinear instability could be an important intrinsic route to melt fracture
and that preliminary experiments are both qualitatively and quantitatively in
good agreement with these predictions.Comment: 20 pages, 16 figures. Accepted for publication in J. of Non-Newtonian
Fluid Mechanic
Game interpretation of Kolmogorov complexity
The Kolmogorov complexity function K can be relativized using any oracle A,
and most properties of K remain true for relativized versions. In section 1 we
provide an explanation for this observation by giving a game-theoretic
interpretation and showing that all "natural" properties are either true for
all sufficiently powerful oracles or false for all sufficiently powerful
oracles. This result is a simple consequence of Martin's determinacy theorem,
but its proof is instructive: it shows how one can prove statements about
Kolmogorov complexity by constructing a special game and a winning strategy in
this game. This technique is illustrated by several examples (total conditional
complexity, bijection complexity, randomness extraction, contrasting plain and
prefix complexities).Comment: 11 pages. Presented in 2009 at the conference on randomness in
Madison
Towards a Flexible Intra-Trustcenter Management Protocol
This paper proposes the Intra Trustcenter Protocol (ITP), a flexible and
secure management protocol for communication between arbitrary trustcenter
components. Unlike other existing protocols (like PKCS#7, CMP or XKMS) ITP
focuses on the communication within a trustcenter. It is powerful enough for
transferring complex messages which are machine and human readable and easy to
understand. In addition it includes an extension mechanism to be prepared for
future developments.Comment: 12 pages, 0 figures; in The Third International Workshop for Applied
PKI (IWAP2004
An analytic solution to the Busemann-Petty problem on sections of convex bodies
We derive a formula connecting the derivatives of parallel section functions
of an origin-symmetric star body in R^n with the Fourier transform of powers of
the radial function of the body. A parallel section function (or
(n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all
hyperplane sections of the body orthogonal to a given direction. This formula
provides a new characterization of intersection bodies in R^n and leads to a
unified analytic solution to the Busemann-Petty problem: Suppose that K and L
are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional)
volume of each central hyperplane section of K is smaller than the volume of
the corresponding section of L; is the (n-dimensional) volume of K smaller than
the volume of L? In conjunction with earlier established connections between
the Busemann-Petty problem, intersection bodies, and positive definite
distributions, our formula shows that the answer to the problem depends on the
behavior of the (n-2)-nd derivative of the parallel section functions. The
affirmative answer to the Busemann-Petty problem for n\le 4 and the negative
answer for n\ge 5 now follow from the fact that convexity controls the second
derivatives, but does not control the derivatives of higher orders.Comment: 13 pages, published versio
Analytic Equivalence Relations and Ulm-Type Classifications
Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations
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