42 research outputs found
Homotopy type of the complement of an immersion and classification of embeddings of tori
This paper is devoted to the classification of embeddings of higher
dimensional manifolds. We study the case of embeddings ,
which we call knotted tori. The set of knotted tori in the the space of
sufficiently high dimension, namely in the metastable range ,
, which is a natural limit for the classical methods of embedding
theory, has been explicitely described earlier. The aim of this note is to
present an approach which allows for results in lower dimension
The boundary value problem for discrete analytic functions
This paper is on further development of discrete complex analysis introduced
by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying
in the complex plane and having quadrilateral faces. A function on the vertices
is called discrete analytic, if for each face the difference quotients along
the two diagonals are equal.
We prove that the Dirichlet boundary value problem for the real part of a
discrete analytic function has a unique solution. In the case when each face
has orthogonal diagonals we prove that this solution uniformly converges to a
harmonic function in the scaling limit. This solves a problem of S. Smirnov
from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L.
Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G.
Ciarlet-P.-A. Raviart for rhombic lattices.
In particular, our result implies uniform convergence of the finite element
method on Delaunay triangulations. This solves a problem of A. Bobenko from
2011. The methodology is based on energy estimates inspired by
alternating-current network theory.Comment: 22 pages, 6 figures. Several changes: Theorem 1.2 generalized,
several assertions added, minor correction in the proofs of Lemma 2.5, 3.3,
Example 3.6, Corollary 5.
Feynman checkers: towards algorithmic quantum theory
We survey and develop the most elementary model of electron motion introduced
by R.Feynman. It is a game, in which a checker moves on a checkerboard by
simple rules, and we count the turns. It is also known as a one-dimensional
quantum walk or an Ising model at imaginary temperature. We solve
mathematically a problem by R.Feynman from 1965, which was to prove that the
model reproduces the usual quantum-mechanical free-particle kernel for large
time, small average velocity, and small lattice step. We compute the
small-lattice-step and the large-time limits, justifying heuristic derivations
by J.Narlikar from 1972 and by A.Ambainis et al. from 2001. For the first time
we observe and prove concentration of measure in the former limit. We perform
the second quantization of the model. The main tools are the Fourier transform
and the stationary phase method.Comment: 55 pages, 16 figure
Characterizing envelopes of moving rotational cones and applications in CNC machining
Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.RYC-2017-2264
A classification of smooth embeddings of 3-manifolds in 6-space
We work in the smooth category. If there are knotted embeddings S^n\to R^m,
which often happens for 2m<3n+4, then no concrete complete description of
embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint
unions of spheres. Let N be a closed connected orientable 3-manifold. Our main
result is the following description of the set Emb^6(N) of embeddings N\to R^6
up to isotopy.
The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in
H_1(N;Z) the Kreck invariant \eta_u : W^{-1}u \to Z_{d(u)} is bijective, where
d(u) is the divisibility of the projection of u to the free part of H_1(N;Z).
The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on
Emb^6(N) by embedded connected sum. It was proved that the orbit space of this
action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's
smoothing theory). The new part of our classification result is determination
of the orbits of the action. E. g. for N=RP^3 the action is free, while for
N=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that
for each knot l:S^3\to R^6 the embedding f#l is isotopic to f.
Our proof uses new approaches involving the Kreck modified surgery theory or
the Boechat-Haefliger formula for smoothing obstruction.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in
Math. Zei
Analysis and Synthesis of Digital Dyadic Sequences
We explore the space of matrix-generated (0, m, 2)-nets and (0, 2)-sequences
in base 2, also known as digital dyadic nets and sequences. In computer
graphics, they are arguably leading the competition for use in rendering. We
provide a complete characterization of the design space and count the possible
number of constructions with and without considering possible reorderings of
the point set. Based on this analysis, we then show that every digital dyadic
net can be reordered into a sequence, together with a corresponding algorithm.
Finally, we present a novel family of self-similar digital dyadic sequences, to
be named -sequences, that spans a subspace with fewer degrees of freedom.
Those -sequences are extremely efficient to sample and compute, and we
demonstrate their advantages over the classic Sobol (0, 2)-sequence.Comment: 17 pages, 11 figures. Minor improvement of exposition; references to
earlier proofs of Theorems 3.1 and 3.3 adde