4,805 research outputs found
Wedges, Cones, Cosmic Strings, and the Reality of Vacuum Energy
One of J. Stuart Dowker's most significant achievements has been to observe
that the theory of diffraction by wedges developed a century ago by Sommerfeld
and others provided the key to solving two problems of great interest in
general-relativistic quantum field theory during the last quarter of the
twentieth century: the vacuum energy associated with an infinitely thin,
straight cosmic string, and (after an interchange of time with a space
coordinate) the apparent vacuum energy of empty space as viewed by an
accelerating observer. In a sense the string problem is more elementary than
the wedge, since Sommerfeld's technique was to relate the wedge problem to that
of a conical manifold by the method of images. Indeed, Minkowski space, as well
as all cone and wedge problems, are related by images to an infinitely sheeted
master manifold, which we call Dowker space. We review the research in this
area and exhibit in detail the vacuum expectation values of the energy density
and pressure of a scalar field in Dowker space and the cone and wedge spaces
that result from it. We point out that the (vanishing) vacuum energy of
Minkowski space results, from the point of view of Dowker space, from the
quantization of angular modes, in precisely the way that the Casimir energy of
a toroidal closed universe results from the quantization of Fourier modes; we
hope that this understanding dispels any lingering doubts about the reality of
cosmological vacuum energy.Comment: 28 pages, 16 figures. Special volume in honor of J. S. Dowke
Memories of Mary Ellen Rudin
An invited collective remembrance celebrating Mary Ellen Rudin's lif
Sparse Nerves in Practice
Topological data analysis combines machine learning with methods from
algebraic topology. Persistent homology, a method to characterize topological
features occurring in data at multiple scales is of particular interest. A
major obstacle to the wide-spread use of persistent homology is its
computational complexity. In order to be able to calculate persistent homology
of large datasets, a number of approximations can be applied in order to reduce
its complexity. We propose algorithms for calculation of approximate sparse
nerves for classes of Dowker dissimilarities including all finite Dowker
dissimilarities and Dowker dissimilarities whose homology is Cech persistent
homology. All other sparsification methods and software packages that we are
aware of calculate persistent homology with either an additive or a
multiplicative interleaving. In dowker_homology, we allow for any
non-decreasing interleaving function . We analyze the computational
complexity of the algorithms and present some benchmarks. For Euclidean data in
dimensions larger than three, the sizes of simplicial complexes we create are
in general smaller than the ones created by SimBa. Especially when calculating
persistent homology in higher homology dimensions, the differences can become
substantial
Zero modes, entropy bounds and partition functions
Some recent finite temperature calculations arising in the investigation of
the Verlinde-Cardy relation are re-analysed. Some remarks are also made about
temperature inversion symmetry.Comment: 12 pages, JyTe
Heat kernel asymptotics: more special case calculations
Special case calculations are presented, which can be used to put
restrictions on the general form of heat kernel coefficients for transmittal
boundary conditions and for generalized bag boundary conditions.Comment: Invited talk at International Meeting on Quantum Gravity and Spectral
Geometry, Naples, Italy, 2-6 July 2001. 9 pages, LaTe
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