Newton polyhedra and weighted oscillatory integrals with smooth phases

In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals, whose phases and weights are contained in a certain class of smooth functions including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.Comment: 67pages. arXiv admin note: text overlap with arXiv:1208.392

Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude

The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.Comment: 36 page

Meromorphy of local zeta functions in smooth model cases

It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general ($C^{\infty}$) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as $u(x,y)x^a y^b +$ flat function, where $u(0,0)\neq 0$ and $a,b$ are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each cases. Our results show new interesting phenomena in one of these cases. Actually, when $a<b$, local zeta functions can be meromorphically extended to the half-plane ${\rm Re}(s)>-1/a$ and their poles on the half-plane are contained in the set $\{-k/b:k\in\mathbb{N}$ with $k<b/a\}$.Comment: 21 pages, 2 figure

In vivo tracking transplanted cardiomyocytes derived from human induced pluripotent stem cells using nuclear medicine imaging

Introduction: Transplantation of human induced pluripotent stem cell-derived cardiomyocytes (iPSC-CMs) is a promising treatment for heart failure. Information on long-term cell engraftment after transplantation is clinically important. However, clinically applicable evaluation methods have not yet been established. Methods: In this study, to noninvasively assess transplanted cell engraftment, human SLC5A5, which encodes a sodium/iodide symporter (NIS) that transports radioactive tracers such as 125I, 18F-tetrafluoroborate (TFB), and 99mTc-pertechnetate (99mTcO4−), was transduced into human induced pluripotent stem cells (iPSCs), and nuclear medicine imaging was used to track engrafted human iPSC-CMs. Results: To evaluate the pluripotency of NIS-expressing human iPSCs, they were subcutaneously transplanted into immunodeficient rats. Teratomas were detected by 99mTcO4− single photon emission computed tomography (SPECT/CT) imaging. NIS expression and the uptake ability of 125I were maintained in purified human iPSC-CMs. NIS-expressing human iPSC-CMs transplanted into immunodeficient rats could be detected over time using 99mTcO4− SPECT/CT imaging. Unexpectedly, NIS expression affected cell proliferation of human iPSCs and iPSC-derived cells. Discussion: Such functionally designed iPSC-CMs have potential clinical applications as a noninvasive method of grafted cell evaluation, but further studies are needed to determine the effects of NIS transduction on cellular characteristics and functions

Prevalence of hypertension at high altitude: cross-sectional survey in Ladakh, Northern India 2007-2011

Objective: Prevalence of hypertension was examined in a widely dispersed (45 110 km2) representative group of Ladakhi in Northern India. The influence of hypoxic environment of wide-ranged altitude (2600–4900 m) and lifestyle change on hypertension was studied. Methods: 2800 participants (age 20–94 years) were enrolled. Systolic blood pressure ≥140 mm Hg and/or diastolic blood pressure of ≥90 mm Hg and/or taking current anti-hypertensive medicine was defined as hypertension. Height and weight for body mass index and SpO2 were examined. The rural population comprised six subdivisions with a distinct altitude, dietary and occupational pattern. Participants in the urban area of Leh consist of two groups, that is, migrants settled in Leh from the Changthang nomadic area, and dwellers born in Leh. The prevalence of hypertension in the two groups was compared with that in the farmers and nomads in rural areas. The effects of ageing, hypoxia, dwelling at high altitude, obesity, modernised occupation, dwelling in an urban area, and rural-to-urban migration to hypertension were analysed by multiple logistic regression. Results: The prevalence of hypertension was 37.0% in all participants and highest in migrants settled in Leh (48.3%), followed by dwellers born in Leh town (41.1%) compared with those in rural areas (33.5). The prevalence of hypertension in nomads (all: 27.7%, Tibetan/Ladakhi: 19.7/31.9%)) living at higher altitude (4000–4900 m) was relatively low. The associated factors with hypertension were ageing, overweight, dwelling at higher altitude, engagement in modernised sedentary occupations, dwelling in urban areas, and rural-to-urban migration. The effects of lifestyle change and dwelling at high altitude were independently associated with hypertension by multivariate analysis adjusted with confounding factors. Conclusions: Socioeconomic and cultural factors play a big role with the effect of high altitude itself on high prevalence of hypertension in highlanders in Ladakh

Asymptotic limit of oscillatory integrals with certain smooth phases (Regularity and Singularity for Partial Differential Equations with Conservation Laws)

"Regularity and Singularity for Partial Differential Equations with Conservation Laws". June 3～5, 2015. edited by Keiichi Kato, Mishio Kawashita, Masashi Misawa and Takayoshi Ogawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.Asymptotic limit of the oscillatory integral is explicitly computed in the case when its smooth phase contains some flat function

On the asymptotic expansion of oscillatory integrals with smooth phases in two dimensions (Several aspects of microlocal analysis)

"Several aspects of microlocal analysis". October 20～24, 2014. edited by Naofumi Honda, Yasunori Okada and Susumu Yamazaki. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed