703,357 research outputs found
Quantum mechanics without spacetime II : noncommutative geometry and the free point particle
In a recent paper we have suggested that a formulation of quantum mechanics
should exist, which does not require the concept of time, and that the
appropriate mathematical language for such a formulation is noncommutative
differential geometry. In the present paper we discuss this formulation for the
free point particle, by introducing a commutation relation for a set of
noncommuting coordinates. The sought for background independent quantum
mechanics is derived from this commutation relation for the coordinates. We
propose that the basic equations are invariant under automorphisms which map
one set of coordinates to another- this is a natural generalization of
diffeomorphism invariance when one makes a transition to noncommutative
geometry. The background independent description becomes equivalent to standard
quantum mechanics if a spacetime manifold exists, because of the proposed
automorphism invariance. The suggested basic equations also give a quantum
gravitational description of the free particle.Comment: 8 page
On extensions of typical group actions
For every countable abelian group we find the set of all its subgroups
() such that a typical measure-preserving -action on a standard
atomless probability space can be extended to a free
measure-preserving -action on . The description of all
such pairs was made in purely group terms, in the language of the
dual , and -actions with discrete spectrum. As an application, we
answer a question when a typical -action can be extended to a -action
with some dynamic property, or to a -action at all. In particular, we offer
first examples of pairs satisfying both is countable abelian, and
a typical -action is not embeddable in a -action.Comment: 30 page
The bipartite Brill-Gordan locus and angular momentum
This paper is a sequel to math.AG/0411110. Let P denote the projective space
of degree d forms in n+1 variables. Let e denote an integer < d/2, and consider
the subvariety X of forms which factor as L^{d-e} M^e for some linear forms
L,M. In the language of our earlier paper, this is the Brill-Gordan locus
associated to the partition (d-e,e).
In this paper we calculate the Castelnuovo regularity of X precisely, and
moreover show that X is r-normal for r at least 3. In the case of binary forms,
we give a classical invariant-theoretic description of the defining equations
of this locus in terms of covariants of d-ics. Modulo standard cohomological
arguments, the proof crucially relies upon showing that certain 3j-symbols from
the quantum theory of angular momentum are nonzero.Comment: LaTeX, 37 page
Sequence to Sequence -- Video to Text
Real-world videos often have complex dynamics; and methods for generating
open-domain video descriptions should be sensitive to temporal structure and
allow both input (sequence of frames) and output (sequence of words) of
variable length. To approach this problem, we propose a novel end-to-end
sequence-to-sequence model to generate captions for videos. For this we exploit
recurrent neural networks, specifically LSTMs, which have demonstrated
state-of-the-art performance in image caption generation. Our LSTM model is
trained on video-sentence pairs and learns to associate a sequence of video
frames to a sequence of words in order to generate a description of the event
in the video clip. Our model naturally is able to learn the temporal structure
of the sequence of frames as well as the sequence model of the generated
sentences, i.e. a language model. We evaluate several variants of our model
that exploit different visual features on a standard set of YouTube videos and
two movie description datasets (M-VAD and MPII-MD).Comment: ICCV 2015 camera-ready. Includes code, project page and LSMDC
challenge result
Equivariant cohomology over Lie groupoids and Lie-Rinehart algebras
Using the language and terminology of relative homological algebra, in
particular that of derived functors, we introduce equivariant cohomology over a
general Lie-Rinehart algebra and equivariant de Rham cohomology over a locally
trivial Lie groupoid in terms of suitably defined monads (also known as
triples) and the associated standard constructions. This extends a
characterization of equivariant de Rham cohomology in terms of derived functors
developed earlier for the special case where the Lie groupoid is an ordinary
Lie group, viewed as a Lie groupoid with a single object; in that theory over a
Lie group, the ordinary Bott-Dupont-Shulman-Stasheff complex arises as an a
posteriori object. We prove that, given a locally trivial Lie groupoid G and a
smooth G-manifold f over the space B of objects of G, the resulting
G-equivariant de Rham theory of f boils down to the ordinary equivariant de
Rham theory of a vertex manifold relative to the corresponding vertex group,
for any vertex in the space B of objects of G; this implies that the
equivariant de Rham cohomology introduced here coincides with the stack de Rham
cohomology of the associated transformation groupoid whence this stack de Rham
cohomology can be characterized as a relative derived functor. We introduce a
notion of cone on a Lie-Rinehart algebra and in particular that of cone on a
Lie algebroid. This cone is an indispensable tool for the description of the
requisite monads.Comment: 47 page
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