554 research outputs found
Greek Astronomical Calendars and their Relation to the Athenian Civil Calendar
Several investigations have been devoted to the Athenian calendar and to the cycles of Meton and Kallippos. However, most authors have not clearly distinguished between true and mean lunar months, nor between astronomical calendars and the Athenian calendar. In investigating the Athenian calendar, many authors have made use of the regular successions of full and hollow months described by Geminos in his Isagoge, without first making sure that these months were in actual use at Athens. Discussion as to whether ‘the month' began with the astronomical New Moon or with the visibility of the crescent might have been avoided if the authors had realised that the word ‘month' has several meanings and that in every particular case the meaning has to be inferred from the context. Peasants or soldiers, far away from civilisation, would start their month with the visible crescent, astronomers would make it begin at the day of true or mean New Moon, and cities would adapt their festival calendar to the needs of the moment, intercalating or omitting days in such a way that the festivals can be held at the days prescribed by law or tradition. Of course, it may happen any time that a civil month coincides with the astronomical or with the observed lunar month, but in absence of definite evidence we never have the right to identify a civil month with an astronomical mont
Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate
quadratic equations in n variables have 2 n solutions over algebraically closed
fields. When only a small subset M of monomials appear in the equations
(fewnomial systems), the number of solutions may decrease dramatically. We
focus in this work on subsets of quadratic monomials M such that generic
systems with support M do not admit any solution at all. For these systems,
Hilbert's Nullstellensatz ensures the existence of algebraic certificates of
inconsistency. However, up to our knowledge all known bounds on the sizes of
such certificates -including those which take into account the Newton polytopes
of the polynomials- are exponential in n. Our main results show that if the
inequality 2|M| -- 2n \sqrt 1 + 8{\nu} -- 1 holds for a quadratic
fewnomial system -- where {\nu} is the matching number of a graph associated
with M, and |M| is the cardinality of M -- then there exists generically a
certificate of inconsistency of linear size (measured as the number of
coefficients in the ground field K). Moreover this certificate can be computed
within a polynomial number of arithmetic operations. Next, we evaluate how
often this inequality holds, and we give evidence that the probability that the
inequality is satisfied depends strongly on the number of squares. More
precisely, we show that if M is picked uniformly at random among the subsets of
n + k + 1 quadratic monomials containing at least (n 1/2+)
squares, then the probability that the inequality holds tends to 1 as n grows.
Interestingly, this phenomenon is related with the matching number of random
graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results
showing that certificates in inconsistency can be computed for systems with
more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201
Efficient numerical diagonalization of hermitian 3x3 matrices
A very common problem in science is the numerical diagonalization of
symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be
too inefficient if the number of matrices is large, we study several
alternatives. We consider optimized implementations of the Jacobi, QL, and
Cuppen algorithms and compare them with an analytical method relying on
Cardano's formula for the eigenvalues and on vector cross products for the
eigenvectors. Jacobi is the most accurate, but also the slowest method, while
QL and Cuppen are good general purpose algorithms. The analytical algorithm
outperforms the others by more than a factor of 2, but becomes inaccurate or
may even fail completely if the matrix entries differ greatly in magnitude.
This can mostly be circumvented by using a hybrid method, which falls back to
QL if conditions are such that the analytical calculation might become too
inaccurate. For all algorithms, we give an overview of the underlying
mathematical ideas, and present detailed benchmark results. C and Fortran
implementations of our code are available for download from
http://www.mpi-hd.mpg.de/~globes/3x3/ .Comment: 13 pages, no figures, new hybrid algorithm added, matches published
version, typo in Eq. (39) corrected; software library available at
http://www.mpi-hd.mpg.de/~globes/3x3
Integrable Quasiclassical Deformations of Cubic Curves
A general scheme for determining and studying hydrodynamic type systems
describing integrable deformations of algebraic curves is applied to cubic
curves. Lagrange resolvents of the theory of cubic equations are used to derive
and characterize these deformations.Comment: 24 page
Integrable Deformations of Algebraic Curves
A general scheme for determining and studying integrable deformations of
algebraic curves, based on the use of Lenard relations, is presented. We
emphasize the use of several types of dynamical variables : branches, power
sums and potentials.Comment: 10 Pages, Proceedings Workshop-Nonlinear Physics: Theory and
Experiment, Gallipoli 200
Stochastic Quantization of Topological Field Theory: Generalized Langevin Equation with Memory Kernel
We use the method of stochastic quantization in a topological field theory
defined in an Euclidean space, assuming a Langevin equation with a memory
kernel. We show that our procedure for the Abelian Chern-Simons theory
converges regardless of the nature of the Chern-Simons coefficient
Uniqueness of collinear solutions for the relativistic three-body problem
Continuing work initiated in an earlier publication [Yamada, Asada, Phys.
Rev. D 82, 104019 (2010)], we investigate collinear solutions to the general
relativistic three-body problem. We prove the uniqueness of the configuration
for given system parameters (the masses and the end-to-end length). First, we
show that the equation determining the distance ratio among the three masses,
which has been obtained as a seventh-order polynomial in the previous paper,
has at most three positive roots, which apparently provide three cases of the
distance ratio. It is found, however, that, even for such cases, there exists
one physically reasonable root and only one, because the remaining two positive
roots do not satisfy the slow motion assumption in the post-Newtonian
approximation and are thus discarded. This means that, especially for the
restricted three-body problem, exactly three positions of a third body are true
even at the post-Newtonian order. They are relativistic counterparts of the
Newtonian Lagrange points L1, L2 and L3. We show also that, for the same masses
and full length, the angular velocity of the post-Newtonian collinear
configuration is smaller than that for the Newtonian case. Provided that the
masses and angular rate are fixed, the relativistic end-to-end length is
shorter than the Newtonian one.Comment: 18 pages, 1 figure; typos corrected, text improved; accepted by PR
A Classification of Integrable Quasiclassical Deformations of Algebraic Curves
A previously introduced scheme for describing integrable deformations of of
algebraic curves is completed. Lenard relations are used to characterize and
classify these deformations in terms of hydrodynamic type systems. A general
solution of the compatibility conditions for consistent deformations is given
and expressions for the solutions of the corresponding Lenard relations are
provided.Comment: 21 page
Collinear solution to the general relativistic three-body problem
The three-body problem is reexamined in the framework of general relativity.
The Newtonian three-body problem admits Euler's collinear solution, where three
bodies move around the common center of mass with the same orbital period and
always line up. The solution is unstable. Hence it is unlikely that such a
simple configuration would exist owing to general relativistic forces dependent
not only on the masses but also on the velocity of each body. However, we show
that the collinear solution remains true with a correction to the spatial
separation between masses. Relativistic corrections to the Sun-Jupiter Lagrange
points L1, L2 and L3 are also evaluated.Comment: 12 pages, 2 figures, accepted for publication in PR
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