A. In this paper we investigate relations between solutions to the minimal surface equation in Euclidean 3-space E 3 , the zero mean curvature equation in Lorentz-Minkowski 3-space L 3 and the Born-Infeld equation under Wick rotations. We prove that the existence conditions of real solutions and imaginary solutions after Wick rotations are written by symmetries of solutions, and reveal how real and imaginary solutions are transformed under Wick rotations. We also give a transformation theory for zero mean curvature surfaces containing lightlike lines with some symmetries. As an application, we give new correspondences among some solutions to the above equations by using the noncommutativity between Wick rotations and isometries in the ambient space.2010 Mathematics Subject Classification. Primary 53A10; Secondary 58J72, 53B30.
The main object of this paper is to derive a number of general double series identities and to apply each of these identities in order to deduce several hypergeometric reduction formulas for the Srivastava-Daoust double hypergeometric function. The results presented in this paper are based essentially upon some
Abstract. We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one-parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan's Identities and the Weierstrass-Enneper representation of maximal surfaces, we derive further non-trivial identities.
We study the intersection lattice of the arrangement A G of subspaces fixed by subgroups of a finite linear group G. When G is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of G. We generalize the notion of finite reflection groups. We say that a group G is generated (resp. strictly generated) in codimension k if it is generated by its elements that fix point-wise a subspace of codimension at most k (resp. precisely k).If G is generated in codimension two, we show that the intersection lattice of A G is atomic. We prove that the alternating subgroup Alt(W ) of a reflection group W is strictly generated in codimension two; moreover, the subspace arrangement of Alt(W ) is the truncation at rank two of the reflection arrangement A W .Further, we compute the intersection lattice of all finite subgroups of GL 3 (R), and moreover, we emphasize the groups that are "minimally generated in real codimension two", i.e, groups that are strictly generated in codimension two but have no real reflection representations. We also provide several examples of groups generated in higher codimension.
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