Search results for "Complex number"
showing 8 items of 8 documents
Unirationality of Hurwitz spaces of coverings of degree <= 5
2011
Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{*}O_X/O_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3, 4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.
Humbert surfaces and the Kummer plane
2003
A Humbert surface is a hypersurface of the moduli space A 2 \mathcal A_2 of principally polarized abelian surfaces defined by an equation of the form a z 1 + b z 2 + c z 3 + d ( z 2 2 − z 1 z 3 ) + e = 0 az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0 with integers a , … , e a,\ldots ,e . We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.
Phasor total unbalance power: Formulation and some properties
2007
Unbalance power has been used to quantify unbalance phenomenon effects in the electric power systems for a long time. Expressions of this power developed by some different theories only give the value of its module. Therefore, total unbalance power values in electric power systems cannot be obtained by the unbalance power values of each of their loads. For that, it is necessary to express the unbalance power as a complex number. In this paper the unbalance power is expressed in phasor notation and some of their properties are described, applied to electric systems with balanced voltages. Results are verified by simulation and also practical comparative experiences are realized in actual ins…
From the theory of “congeneric surd equations” to “Segre's bicomplex numbers”
2015
We will study the historical pathway of the emergence of Tessarines or Bicomplex numbers, from their origin as "imaginary" solutions of irrational equations, to their insertion in the context of study of the algebras of hypercomplex numbers.
Multiplicative Loops of Quasifields Having Complex Numbers as Kernel
2017
We determine the multiplicative loops of locally compact connected 4-dimensional quasifields Q having the field of complex numbers as their kernel. In particular, we turn our attention to multiplicative loops which have either a normal subloop of dimension one or which contain a subgroup isomorphic to $$Spin_3({\mathbb {R}})$$ . Although the 4-dimensional semifields Q are known, their multiplicative loops have interesting Lie groups generated by left or right translations. We determine explicitly the quasifields Q which coordinatize locally compact translation planes of dimension 8 admitting an at least 16-dimensional Lie group as automorphism group.
Invariants of unipotent groups
1987
I’ll give a survey on the known results on finite generation of invariants for nonreductive groups, and some conjectures. You know that Hilbert’s 14th problem is solved for the invariants of reductive groups; see [12] for a survey. So the general case reduces to the case of unipotent groups. But in this case there are only a few results, some negative and some positive. I assume that k is an infinite field, say the complex numbers, but in most instances an arbitrary ring would do it.
Symplectic automorphisms of prime order on K3 surfaces
2006
The aim of this paper is to study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients. We determine the invariant sublattice and its perpendicular complement, and show that the latter coincides with the Coxeter-Todd lattice in the case of automorphism of order three. We also compute many explicit examples, with particular attention to elliptic fibrations.
Combinatorics of generalized Bethe equations
2012
A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over \({\mathbb{Z}^M}\), and on the other hand, they count integer points in certain M-dimensional polytopes.