111,574 research outputs found
A family of Thue equations involving powers of units of the simplest cubic fields
E. Thomas was one of the first to solve an infinite family of Thue equations,
when he considered the forms and
the family of equations , . This family is
associated to the family of the simplest cubic fields of
D. Shanks, being a root of . We introduce in this family a
second parameter by replacing the roots of the minimal polynomial
of by the -th powers of the roots and we effectively solve the
family of Thue equations that we obtain and which depends now on the two
parameters and .Comment: Expanded version (31p) of a paper to appear in the Journal de
Th\'eorie des Nombres de Bordeau
Extremal families of cubic Thue equations
We exactly determine the integral solutions to a previously untreated
infinite family of cubic Thue equations of the form with at least
such solutions. Our approach combines elementary arguments, with lower
bounds for linear forms in logarithms and lattice-basis reduction
Black Holes in Quasi-topological Gravity
We construct a new gravitational action which includes cubic curvature
interactions and which provides a useful toy model for the holographic study of
a three parameter family of four- and higher-dimensional CFT's. We also
investigate the black hole solutions of this new gravity theory. Further we
examine the equations of motion of quasi-topological gravity. While the full
equations in a general background are fourth-order in derivatives, we show that
the linearized equations describing gravitons propagating in the AdS vacua
match precisely the second-order equations of Einstein gravity.Comment: 33 pages, 4 figures; two references adde
Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons
Due to their explicit construction, Aloff-Wallach spaces are prominent in
flux compactifications. They carry G_2-structures and admit the G_2-instanton
equations, which are natural BPS equations for Yang-Mills instantons on
seven-manifolds and extremize a Chern-Simons-type functional. We consider the
Chern-Simons flow between different G_2-instantons on Aloff-Wallach spaces,
which is equivalent to Spin(7)-instantons on a cylinder over them. For a
general SU(3)-equivariant gauge connection, the generalized instanton equations
turn into gradient-flow equations on C^3 x R^2, with a particular cubic
superpotential. For the simplest member of the Aloff-Wallach family (with
3-Sasakian structure) we present an explicit instanton solution of tanh-like
shape.Comment: 1+17 pages, 1 figur
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