• Bulletin of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1673-1685
• /
• 2023

Inspired by Hongjie Dong and Qi S. Zhang's article [3], we find that the analyticity in time for a smooth solution of the heat equation with exponential quadratic growth in the space variable can be extended to any complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature bounded below and the potential function being of at most quadratic growth. Therefore, our result holds on all gradient Ricci solitons. As a corollary, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with the similar growth condition. In addition, we also consider the solution in certain L^p spaces with p ∈ [2, +∞) and prove its analyticity with respect to time.

• Journal of the Korean Mathematical Society
• /
• v.37 no.6
• /
• pp.1071-1084
• /
• 2000

We show that a weak solution of the Cauchy problem for he nonlinear Schrodinger equation, {i∂(sub)t u + ∂$^2$(sub)x u = f(u,u), t∈(-T,T), x∈R, u(0,x) = ø(x).} in the negative solbolev space H(sup)s has a smoothing effect up to real analyticity if the initial data only have a single point singularity such as the Dirac delta measure. It is shown that for H(sup)s (R)(s>-3/4) data satisfying the condition (※Equations, See Full-text) the solution is analytic in both space and time variable. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [18] and previous work by Kato-Ogawa [12]. We give an improved new argument in the regularity argument.

• Asia Pacific Journal of Corpus Research
• /
• v.4 no.1
• /
• pp.1-23
• /
• 2023

The Japanese language, as a typical agglutinating language, permits large noun phrases (NP) containing ten or more morphemes. In this paper, we argue that the nature of the NP in Japanese is changing. Our data are drawn from the Balanced Corpus of Contemporary Written Japanese. We conduct a series of apparent-time studies of ongoing changes in complex NPs. We first examine the length of compound nouns, followed by the usage of bound suffixes. We then examine ongoing changes in complex NPs that contain genitive case markers. Finally, we examine noun incorporation. All of our studies show a trend towards shorter, less complex NPs. Furthermore, our results suggest that the usage rate of phrases that modify the noun inside the NP (compound nouns, bound nouns, NPs containing genitive case, noun incorporation) appears to be decreasing over time. On the other hand, the usage rate of modifying material outside of the NP (positional phrases, relative clauses) appears to be increasing over time. We conclude by suggesting that our results reflect a diachronic change of decreasing synthetic morphology and increasing analytic morphology. We end by pointing out the implications of this work on our understanding syntheticity and analyticity.