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be an integer or, generally, a Pisot number. Put
be a piecewise linear transformation whose slopes have the form
with positive integers
. We give a sufficient condition for
to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.
Let b ⩾ 2 be an integer. Among other results we establish, in a quantitative form, that any sufficiently large integer which is not a multiple of b cannot simultaneously be divisible only by very small primes and have very few nonzero digits in its representation in base b.
We present high resolution molecular line observations of dusty AGN and starburst in nearby luminous infrared galaxies (LIRGs), VV 114 (band 3/4/7) and NGC 1614 (band 3/6/7/9), with ALMA. Multi-frequency imaging from 4.8 GHz to 691 GHz of NGC 1614 allows us to study spatial properties of the radio-to-FIR continuum and multiple CO transitions, and we find the CO excitation up to Jupp = 6 can be explained by a single ISM model powered by nuclear starbursts. Our processing line imaging survey for VV 114 detected at least 30 molecular lines which show different chemical composition from region to region. Multi-molecule imaging helps us to diagnose the chemical differences of dusty ISM, while multi-transition imaging allows us to investigate gas physical conditions affected by nuclear activities directly.
We study the digits of $\beta $-expansions in the case where $\beta $ is a Salem number. We introduce new upper bounds for the numbers of occurrences of consecutive 0s in the expansion of 1. We also give lower bounds for the numbers of non-zero digits in the $\beta $-expansions of algebraic numbers. As applications, we give criteria for transcendence of the values of power series at certain algebraic points.
Borel conjectured that all algebraic irrational numbers are normal in base 2. However, very little is known about this problem. We improve the lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.
We evaluated the absorption and metabolism of palatinose in rats by the carbohydrate load test and the 13C- and H2-breath tests. We compared the results of these tests with those of sucrose, since sucrose is an isomer of palatinose and generally known to be degraded and absorbed from the small intestine. In the carbohydrate load test, blood glucose and plasma insulin levels after oral administration of palatinose rose more gradually and reached a maximum that was lower than that after sucrose administration. In the 13C-breath test, rats were orally administrated [1-13C]sucrose or [1-13C]palatinose and housed in a chamber. The expired air in the chamber was collected, and the level of 13CO2 in the expired air was measured at appropriate intervals for 360 min. The value of time taken to reach the maximum concentration for expired 13CO2 from [1-13Cglucose] ([1-13Cglc]) and [1-13Cfructose] ([1-13Cfru]) palatinose was significantly longer than that from [1-13Cglc] and [1-13Cfru]sucrose, respectively. The value of area under the curve (AUC) for [1-13Cglc]palatinose was larger than that for [1-13Cglc]sucrose, but AUC for [1-13Cfru] showed no difference between palatinose and sucrose. In the H2-breath test, the concentration of H2 in the expired air was measured for 420 min. H2 was hardly detected with both palatinose and sucrose and no significant difference was observed between the two groups. These results suggest that palatinose is utilised in vivo at a rate equal to that of sucrose.
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