• 대한수학회보
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• 제59권5호
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• pp.1093-1103
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• 2022

In this paper, we establish an evaluation formula to calculate the Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces (H, B, 𝜈). To do this we first derive a translation theorem for the Wiener integral of functionals associated with operators in 𝓛(B), the Banach space of bounded linear operators from B to itself. We then apply the translation theorem to establish an integration by parts formula for the Wiener integral of functionals combined with operators in 𝓛(B). We finally apply this parts formula to evaluate the Wiener integral of certain polynomials in terms of natural dual pairings.

• 대한수학회보
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• 제26권2호
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• pp.151-158
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• 1989

Let (H, B, .nu.) be an abstract Wiener space where H is a separable Hilbert space with the inner product <.,.> and the norm vertical bar . vertical bar=.root.<.,.>, which is densely and continuously imbedded into a separable Banach space B with the norm ∥.∥ , and .nu. is a probability measure on the Borel .sigma.-algebra B(B) of B which satisfies (Fig.) where $B^{*}$ is the topological dual of B and (.,.) is the natural dual pairing between B and $B^{*}$. We will regard $B^{*}$.contnd.H.contnd.B in the natural way. Thus we have =(y, x) for all y in $B^{*}$ and x in H. Let $R^{n}$ and C denote the n-dimensional Euclidean space and the complex numbers respectively.ctively.

• 원예과학기술지
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• 제33권1호
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• pp.83-92
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• 2015

xBrassicoraphanus line BB#5, a new synthetic intergeneric hybrid between Brassica rapa L. ssp. pekinensis and Raphanus sativus L. var. rafiphera induced by N-methyl-N-nitroso-urethane mutagenesis in microspore culture, shows high seed fertility and morphological uniformity. Dual-color fluorescence in situ hybridization (FISH) using 5S and 45S rDNA probes and genomic in situ hybridization (GISH) using B. rapa genomic DNA probe were carried out to analyze the chromosome composition and the meiosis pairing pattern compared to its parental lines. The somatic chromosome complement is 2n = 38, which consists of 17 metacentric and two submetacentric chromosomes with lengths of 2.18 to $5.01{\mu}m$. FISH karyotype analysis showed five and eight pairs of 5S and 45S rDNA loci. GISH meiosis pairing analysis showed that 19 complete bivalents were most frequent and accounted for 42% of the 100 pollen mother cells examined. Based on chromosome number, size, morphology, rDNA distribution, and meiosis pairing pattern, both parental genomes of B. rapa and R. sativus appear to exist in xBrassicoraphanus line BB#5, demonstrating its genome integrity. Such stable chromosome constitutions and meiotic pairing patterns in somatic and meiotic cells are very rare in natural and synthetic intergeneric hybrids. Chromosomal studies and genetic and phenotypic changes in allopolyploids a re discussed. The results p resented h erein will b e usef ul f or f urther g enomic s tudy o f xBrassicoraphanus lines and their improvement as promising new breeding varieties.