34,374 research outputs found
Quantum Computation as Geometry
Quantum computers hold great promise, but it remains a challenge to find
efficient quantum circuits that solve interesting computational problems. We
show that finding optimal quantum circuits is essentially equivalent to finding
the shortest path between two points in a certain curved geometry. By recasting
the problem of finding quantum circuits as a geometric problem, we open up the
possibility of using the mathematical techniques of Riemannian geometry to
suggest new quantum algorithms, or to prove limitations on the power of quantum
computers.Comment: 13 Pages, 1 Figur
Simplicial Ricci Flow
We construct a discrete form of Hamilton's Ricci flow (RF) equations for a
d-dimensional piecewise flat simplicial geometry, S. These new algebraic
equations are derived using the discrete formulation of Einstein's theory of
general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation
is naturally associated to each edge, L, of a simplicial lattice. In defining
this equation, we find it convenient to utilize both the simplicial lattice, S,
and its circumcentric dual lattice, S*. In particular, the RRF equation
associated to L is naturally defined on a d-dimensional hybrid block connecting
with its (d-1)-dimensional circumcentric dual cell, L*. We show that
this equation is expressed as the proportionality between (1) the simplicial
Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume
weighted average of the fractional rate of change of the edges, lambda in L*,
of the circumcentric dual lattice, S*, that are in the dual of L. The inherent
orthogonality between elements of S and their duals in S* provide a simple
geometric representation of Hamilton's RF equations. In this paper we utilize
the well established theories of Regge calculus, or equivalently discrete
exterior calculus, to construct these equations. We solve these equations for a
few illustrative examples.Comment: 34 pages, 10 figures, minor revisions, DOI included: Commun. Math.
Phy
An analytic relation for the thickness of accretion flows
We take the vertical distribution of the radial and azimuthal velocity into
account in spherical coordinates, and find that the analytic relation
c_{s0}/(v_K \Theta) = [(\gamma -1)/(2\gamma)]^{1/2} is valid for both
geometrically thin and thick accretion flows, where c_{s0} is the sound speed
on the equatorial plane, v_K is the Keplerian velocity, \Theta is the
half-opening angle of the flow, and \gamma is the adiabatic index.Comment: 4 pages, 2 figures, accepted by Science in China Series
Earth matter density uncertainty in atmospheric neutrino oscillations
That muon neutrinos oscillating into the mixture of tau neutrinos
and sterile neutrinos has been studied to explain the
atmospheric disappearance. In this scenario, the effect of Earth
matter is a key to determine the fraction of . Considering that the
Earth matter density has uncertainty and this uncertainty has significant
effects in some neutrino oscillation cases, such as the CP violation in very
long baseline neutrino oscillations and the day-night asymmetry for solar
neutrinos, we study the effects caused by this uncertainty in the above
atmospheric oscillation scenario. We find that this uncertainty
seems to have no significant effects and that the previous fitting results need
not to be modified fortunately.Comment: 7 pages, 1 figure, to appear in Phys. Rev.
The Stream-Stream Collision after the Tidal Disruption of a Star Around a Massive Black Hole
A star can be tidally disrupted around a massive black hole. It has been
known that the debris forms a precessing stream, which may collide with itself.
The stream collision is a key process determining the subsequent evolution of
the stellar debris: if the orbital energy is efficiently dissipated, the debris
will eventually form a circular disk (or torus). In this paper, we have
numerically studied such stream collision resulting from the encounter between
a 10^6 Msun black hole and a 1 Msun normal star with a pericenter radius of 100
Rsun. A simple treatment for radiative cooling has been adopted for both
optically thick and thin regions. We have found that approximately 10 to 15% of
the initial kinetic energy of the streams is converted into thermal energy
during the collision. The angular momentum of the incoming stream is increased
by a factor of 2 to 3, and such increase, together with the decrease in kinetic
energy, significantly helps the circularization process. Initial luminosity
burst due to the collision may reach as high as 10^41 erg/sec in 10^4 sec,
after which the luminosity increases again (but slowly this time) to a steady
value of a few 10^40 erg/sec in a few times of 10^5 sec. The radiation from the
system is expected to be close to Planckian with effective temperature of
\~10^5K.Comment: 19 pages including 12 figures; Accepted for publication in Ap
Active Semi-Supervised Learning Using Sampling Theory for Graph Signals
We consider the problem of offline, pool-based active semi-supervised
learning on graphs. This problem is important when the labeled data is scarce
and expensive whereas unlabeled data is easily available. The data points are
represented by the vertices of an undirected graph with the similarity between
them captured by the edge weights. Given a target number of nodes to label, the
goal is to choose those nodes that are most informative and then predict the
unknown labels. We propose a novel framework for this problem based on our
recent results on sampling theory for graph signals. A graph signal is a
real-valued function defined on each node of the graph. A notion of frequency
for such signals can be defined using the spectrum of the graph Laplacian
matrix. The sampling theory for graph signals aims to extend the traditional
Nyquist-Shannon sampling theory by allowing us to identify the class of graph
signals that can be reconstructed from their values on a subset of vertices.
This approach allows us to define a criterion for active learning based on
sampling set selection which aims at maximizing the frequency of the signals
that can be reconstructed from their samples on the set. Experiments show the
effectiveness of our method.Comment: 10 pages, 6 figures, To appear in KDD'1
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