26,608 research outputs found
Polynomials with symmetric zeros
Polynomials whose zeros are symmetric either to the real line or to the unit
circle are very important in mathematics and physics. We can classify them into
three main classes: the self-conjugate polynomials, whose zeros are symmetric
to the real line; the self-inversive polynomials, whose zeros are symmetric to
the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric
by an inversion with respect to the unit circle followed by a reflection in the
real line. Real self-reciprocal polynomials are simultaneously self-conjugate
and self-inversive so that their zeros are symmetric to both the real line and
the unit circle. In this survey, we present a short review of these
polynomials, focusing on the distribution of their zeros.Comment: Keywords: Self-inversive polynomials, self-reciprocal polynomials,
Pisot and Salem polynomials, M\"obius transformations, knot theory, Bethe
equation
Solution of Supplee's submarine paradox through special and general relativity
In 1989 Supplee described an apparent relativistic paradox on which a
submarine seems to sink to observers at rest within the ocean, but it rather
seems to float in the submarine proper frame. In this letter, we show that the
paradox arises from a misuse of the Archimedes principle in the relativistic
case. Considering first the special relativity, we show that any relativistic
force field can be written in the Lorentz form, so that it can always be
decomposed into a \emph{static} (electric-like) and a \emph{dynamic}
(magnetic-like) part. These gravitomagnetic effects provide a relativistic
formulation of Archimedes principle, from which the paradox is explained.
Besides, if the curved spacetime on the vicinity of the Earth is taken into
account, we show that the gravitational force exerted by Earth on a moving body
must increase with the speed of the body. The submarine paradox is then
analyzed again with this speed-dependent gravitational force.Comment: Final version. 7 pages, 2 figures, Keywords: Supplee's submarine
paradox, theory of relativity, gravitomagnetism, Archimedes principle,
Lorentz forc
Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems
The formal derivatives of the Yang-Baxter equation with respect to its
spectral parameters, evaluated at some fixed point of these parameters, provide
us with two systems of differential equations. The derivatives of the
matrix elements, however, can be regarded as independent variables and
eliminated from the systems, after which two systems of polynomial equations
are obtained in place. In general, these polynomial systems have a non-zero
Hilbert dimension, which means that not all elements of the matrix can be
fixed through them. Nonetheless, the remaining unknowns can be found by solving
a few number of simple differential equations that arise as consistency
conditions of the method. The branches of the solutions can also be easily
analyzed by this method, which ensures the uniqueness and generality of the
solutions. In this work we considered the Yang-Baxter equation for two-state
systems, up to the eight-vertex model. This differential approach allowed us to
solve the Yang-Baxter equation in a systematic way and also to completely
classify its regular solutions.Comment: Final version. 40 pages, 3 tables. Keywords: Yang-Baxter Equation,
Lattice Integrable Models, Eight-Vertex Model, Bethe Ansatz, Differential and
Algebraic Geometr
On the integrability of halo dipoles in gravity
We stress that halo dipole components are nontrivial in core-halo systems in
both Newton's gravity and General Relativity. To this end, we extend a recent
exact relativistic model to include also a halo dipole component. Next, we
consider orbits evolving in the inner vacuum between a monopolar core and a
pure halo dipole and find that, while the Newtonian dynamics is integrable, its
relativistic counterpart is chaotic. This shows that chaoticity due only to
halo dipoles is an intrinsic relativistic gravitational effect.Comment: 9 pages, REVTEX, two postscript figures include
Thin-disk models in an Integrable Weyl-Dirac theory
We construct a class of static, axially symmetric solutions representing
razor-thin disks of matter in an Integrable Weyl-Dirac theory proposed in
Found. Phys. 29, 1303 (1999). The main differences between these solutions and
the corresponding general relativistic one are analyzed, focusing on the
behavior of physical observables (rotation curves of test particles, density
and pressure profiles). We consider the case in which test particles move on
Weyl geodesics. The same rotation curve can be obtained from many different
solutions of the Weyl-Dirac theory, although some of these solutions present
strong qualitative differences with respect to the usual general relativistic
model (such as the appearance a ring-like density profile). In particular, for
typical galactic parameters all rotation curves of the Weyl-Dirac model present
Keplerian fall-off. As a consequence, we conclude that a more thorough analysis
of the problem requires the determination of the gauge function on
galactic scales, as well as restrictions on the test-particle behavior under
the action of the additional fields introduced by this theory.Comment: 18 pages, 3 figures; accepted in General Relativity and Gravitatio
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