Volume 113, Issue 2
May 2003, pages 91-212
pp 91-98 May 2003
Some functional equations originating from number theory
We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.
pp 99-137 May 2003
This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK_{k}. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren_{G} is the break in the upper ramification filtration ofG = Gal(E/K) defined by$$G^{n_G } \ne G^{n_{G + \delta } } = 1,\forall \delta \in \mathbb{R}_{_ \ne ^ > 0} $$. Next, we study the basic properties of the idealf(E/K) inO_{k} in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]).
After reviewing the Artin charactera_{G} : G → ℂ ofG := Gal(E/K) and Artin representationsA_{g} G → G →GL(V) corresponding toa_{G} : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5)$$\mathfrak{f}_{Artin} (\chi _\rho ) = \mathfrak{p}_K^{dim_\mathbb{C} (V)\left[ {n_{G/\ker (\rho )} + 1} \right]} $$ where Χ_{gr}: G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then$$\mathfrak{f}_{Artin} (\chi _\rho ) = \mathfrak{p}_K^{[E^{\ker (\rho )^ \cdot :K} ](n_{G/\ker (\rho )} + 1)} $$ where Gal(E^{ker(ρ)}/E^{ker(ρ)•}) is any maximal abelian normal subgroup of Gal(E^{ker(ρ)}/K) containing Gal(E^{ker(ρ)}/K)′, and the break n_{G/ker(ρ)} in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]).
We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A_{G} ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then$$A_G \simeq \sum\limits_N {\left[ {(E^N )^ \bullet :K} \right]\left( {n_{G/N} + 1} \right) \times \sum\limits_{\left[ \infty \right] \sim \in V_N } {Ind_{\pi _N^{ - 1} ((G/N)^ \bullet )}^G (\omega \circ \pi _N |_{\pi _N^{ - 1} ((G/N) \bullet )} )} } $$Kâzim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V_{n} denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^{•} → ℂ^{Χ} satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_{δ}} = ω =(G/N) and$$\bigcap\limits_\delta {\ker (\omega _\delta ) = \left\langle {1_{G/N} } \right\rangle } $$ where δ runs over R((G/N)^{•}/(G/N)), a fixed given complete system of representatives of (G/N)^{•}/(G/N), by declaring that ω_{1} ∼ ω_{2} if and only if ω_{1}= ω_{2,δ} for some δ ∈ R((G/N)^{•}/(G/N)).
Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.
pp 139-152 May 2003
The Jacobian of a nonorientable Klein surface
Pablo Arés-Gastesi Indranil Biswas
Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surfaceY is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms onY quotiented by the torsion-free part of the first integral homology ofY. Denote byX the double cover ofY given by orientation. The Jacobian ofY is identified with the space of all degree zero holomorphic line bundlesL overX with the property thatL is isomorphic to σ*/-L, where σ is the involution ofX.
pp 153-163 May 2003
Reduction theory for a rational function field
LetG be a split reductive group over a finite field F_{q}. LetF = F_{q}(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.
pp 165-169 May 2003
A note on absolute summability factors
In this paper, by using an almost increasing and δ-quasi-monotone sequence, a general theorem on φ- ¦ C, α ¦_{k} summability factors, which generalizes a result of Bor [3] on φ ¦C, 1¦_{k} summability factors, has been proved under weaker and more general conditions.
pp 171-178 May 2003
Wavelet subspaces invariant under groups of translation operators
Biswaranjan Behera Shobha Madan
We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.
pp 179-182 May 2003
Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.
pp 183-187 May 2003
Fixed point of multivalued mapping in uniform spaces
In this paper we prove some new fixed point theorems for multivalued mappings on orbitally complete uniform spaces.
pp 189-193 May 2003
The Weierstrass—Enneper representation using hodographic coordinates on a minimal surface
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate ρ∈ D ⊂ ℂ and σ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.
pp 195-205 May 2003
Positive solutions of singular boundary value problem of negative exponent Emden—Fowler equation
This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden-Fowler equation. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C^{1}[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.
pp 207-212 May 2003
Bounds on the phase velocity in the linear instability of viscous shear flow problem in the β-plane
Results obtained by Joseph(J. Fluid Mech.33 (1968) 617) for the viscous parallel shear flow problem are extended to the problem of viscous parallel, shear flow problem in the beta plane and a sufficient condition for stability has also been derived.
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