17,332 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
Robustness of the O() universality class
We calculate the critical exponents for Lorentz-violating O()
scalar field theories by using two independent methods. In
the first situation we renormalize a massless theory by utilizing normalization
conditions. An identical task is fulfilled in the second case in a massive
version of the same theory, previously renormalized in the BPHZ method in four
dimensions. We show that although the renormalization constants, the
and anomalous dimensions acquire Lorentz-violating quantum corrections, the
outcome for the critical exponents in both methods are identical and
furthermore they are equal to their Lorentz-invariant counterparts. Finally we
generalize the last two results for all loop levels and we provide symmetry
arguments for justifying the latter
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