For every t there is an explicitly given number k0 such that the equation 1k + 2k + + (x − 1)k= xk has no integer solutions x≥2 for all k0 for which the denominator of the kth Bernoulli number Bkhas at most t distinct prime factors.
For every t there is an explicitly given number k0 such that the equation 1k + 2k + + (x − 1)k= xk has no integer solutions x≥2 for all k0 for which the denominator of the kth Bernoulli number Bkhas at most t distinct prime factors.
It is known that Lr(E, C(K)), the space of all regular operators from E into C(K), is a Riesz space for all Riesz spaces E if and only if K is Stonian. We prove that this statement holds if E is replaced by C(K), where K is a compact space, the cardinal number of which satisfies a certain condition.
K = F(√d) is a formally real field and a totally positive quadratic extension of F. A decomposition theorem for quadratic forms in Fed (K) is given. The invariants r(q) and ud(KF) are defined and relations between the invariants βF(i), βK(i), ud(F), ud(K), l(F), l(K) are studied, using the theory of quadratic forms.
We show that Cp*(ℚ) and Cp*(T) are not linearly homeomorphic, thus answering a question of van Mill.
In the paper, an important tool from fixed point theory, the Banach contraction principle, is extended to the more general setting where the spaces are Hausdorff locally convex and sequentially complete with calibrations Γ and the maps are not necessarily Γ-continuous. Applications are given for Γ-strictly pseudo-contractive maps.
Let {Gj}jεJ be a finite set of finitely generated subgroups of the multiplicative group of complex numbers Cx. Write H=∩ jεJ Gj. Let n be a positive integer and aij a complex number for i = 1, ..., n and j ε J. Then there exists a set W with the following properties. The cardinality of W depends only on {Gj}jεJ and n. If, for each jεJ, α has a representation α = Σ in = 1a ijgij in elements gij of Gj, then α has a representation a= Σk=1n wkhk with wkεW, hk εH for k = 1,..., n. The theorem in this note gives information on such representations.
In [3] certain Laurent polynomials of 2F1 genus were called “Jacobi Laurent polynomials”. These Laurent polynomials belong to systems which are orthogonal with respect to a moment sequence ((a)n/(c)n)nεℤ where a, c are certain real numbers. Together with their confluent forms, belonging to systems which are orthogonal with respect to 1/(c)n)nεℤ respectively ((a)n)nεℤ, these Laurent polynomials will be called “classical”. The main purpose of this paper is to determine all the simple (see section 1) orthogonal systems of Laurent polynomials of which the members satisfy certain second order differential equations with polynomial coefficients, analogously to the well known characterization of S. Bochner [1] for ordinary polynomials.
Stable n-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves: Let C be a totally degenerate stable curve of genus g ≥ 2 over a field k. This means that C is a connected projective curve of arithmetic genus g satisfyingo (a) every irreducible component of C is a rational curve over κ. (b) every singular point of C is a κ-rational ordinary double point. (c) every nonsingular component L of C meets C−L in at least three points. It is always possible to find g singular points P1,..., Pg on C such that the blow up C of C at P1,..., Pg is a connected projective curve with the following properties:o (i) every irreducible component of C is isomorphic to Pk1 (ii) the components of C intersect in ordinary κ-rational double points (iii) the intersection graph of C is a tree.
The morphism φ : C → C is an isomorphism outside 2g regular points Q1, Q1′, Qg, Qg′ and identifies Qi with Qj′. is uniquely determined by the g pairs of regular κ-rational points (Qi, Qi′). A curve C satisfying (i)-(iii) together with n κ-rational regular points on it is called a n-pointed tree of projective lines. C is stable if on every component there are at least three points which are either singular or marked. The object of this paper is the classification of stable n-pointed trees. We prove in particular the existence of a fine moduli space Bn of stable n-pointed trees. The discussion above shows that there is a surjective map πB2g → Dg of B2g onto the closed subscheme Dg of the coarse moduli scheme Mg of stable curves of genus g corresponding to the totally degenerate curves. By the universal property of Mg, π is a (finite) morphism. π factors through B2g = B2g mod the action of the group of pair preserving permutations of 2g elements (a group of order 2gg, isomorphic to a wreath product of Sg and ℤ/2ℤ
The induced morphism π: B2g → Dg is an isomorphism on the open subscheme of irreducible curves in Dg, but in general there may be nonequivalent choices of g singular points on a totally degenerated curve for the above constructio
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：