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Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions , called its hyperation, in such a way that , and for all α, β.Hyperations are a refinement of the Veblen hierarchy of f. Moreover, if f is normal and has a well-behaved left-inverse g called a left adjoint, then g can be assigned a cohyperation , which is a family of initial functions such that is a left adjoint to for all ξ. 相似文献
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Let V be a 6-dimensional vector space over a field , let f be a nondegenerate alternating bilinear form on V and let denote the symplectic group associated with . The group has a natural action on the third exterior power of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field . The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of that are contained in a fixed algebraic closure of . In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of on . 相似文献
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Filippo Bracci Manuel D. Contreras Santiago Díaz-Madrigal 《Journal de Mathématiques Pures et Appliquées》2017,107(1):78-99
Let , be two one-parameter semigroups of holomorphic self-maps of the unit disk . Let be a homeomorphism. We prove that, if for all , then f extends to a homeomorphism of outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of ). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk. 相似文献
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Vignon Oussa 《Journal of Functional Analysis》2018,274(4):1202-1254
In this work, we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable Lie groups. In contrast to other well-known techniques such as the coorbit theory, the generalized coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Moreover, our scheme produces explicit constructions of frames with precise frame bounds. As an illustration of the scope of our results, we highlight that a large class of representations which naturally occur in wavelet theory and time–frequency analysis is handled by our scheme. For example, the affine group, the generalized Heisenberg groups, the shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are a few examples of groups whose irreducible representations are handled by our method. The class of representations studied in this work is described as follows. Let G be a simply connected, connected, completely solvable Lie group with Lie algebra . Next, let π be an infinite-dimensional unitary irreducible representation of G obtained by inducing a character from a closed normal subgroup of G. Additionally, we assume that , is a closed subgroup of G, is a fixed Haar measure on the solvable Lie group M and there exists a linear functional such that the representation is realized as acting in . Making no assumption on the integrability of , we describe explicitly a discrete subset Γ of G and a vector such that is a tight frame for . We also construct compactly supported smooth functions s and discrete subsets such that is a frame for . 相似文献
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Jaiung Jun 《Journal of Pure and Applied Algebra》2018,222(8):2063-2088
We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (hyperfield valuation). In particular, we classify valuations on the semifield (the max-plus semifield of rational numbers) and also valuations on the ‘function field’ (the semifield of rational functions over ) which are trivial on . We construct and study the abstract curve associated to in relation to the projective line over the field with one element and the tropical projective line. Finally, we discuss possible connections to tropical curves and Berkovich's theory of analytic spaces. 相似文献
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《Stochastic Processes and their Applications》2005,115(2):275-298
In this paper, we consider a uniformly ergodic Markov process valued in a measurable subset E of with the unique invariant measure , where the density f is unknown. We establish the large deviation estimations for the nonparametric kernel density estimator in and for , and the asymptotic optimality in the Bahadur sense. These generalize the known results in the i.i.d. case. 相似文献