We investigate the asymptotic distribution of perturbed geometric progression
{
θ
k
x
+
γ
k
}
given by
θ
∈
(
-
∞
,
-
1
)
∪
(
1
,
∞
)
and
γ
1
,
γ
2
,
⋯
∈
R
. We prove that the discrepancy obeys the ...law of the iterated logarithm with limsup constant sensitively depending on
θ
and
{
γ
k
}
.
In the previous work, we proved the law of the iterated logarithm for a subsequence of geometric progression
and showed that the speed of convergence toward the uniform distribution is faster than ...that of the original sequence. It is natural to ask if the speed becomes faster again if we take a subsequence of the subsequence. In this note, we give a negative answer to this question by giving a counterexample.
For
θ
∈
(
-
∞
,
-
1
)
∪
(
1
,
∞
)
and for almost every
x
, it is known that the sequence
{
θ
k
x
}
is uniformly distributed modulo 1. The speed of convergence sensitively depends on the algebraic ...nature of
θ
. In this paper we prove that such dependence vanishes if we perturb the sequence by adding the irrational rotation
{
κ
γ
}
. The speed becomes identical with that of the sequence of uniformly distributed independent random variables.
We fabricated a cascade arc discharge source with a channel diameter of 8 mm for various applications to virtual vacuum interface (plasma window). Stationary helium arc plasma up to a discharge ...current of 100 A was generated, and the performance as plasma window was investigated. The pressure ratio of high-pressure discharge section to low-pressure vacuum side was above 70 at a 50-A He discharge with a gas flow rate of 0.2 L/min. To characterize the plasma at the anode electrode, emissions of vacuum UV and visible wavelengths were observed. Spectral analysis of He I continuum spectra and H-<inline-formula> <tex-math notation="LaTeX">\beta </tex-math></inline-formula> Stark broadening showed that the plasma temperature was around 0.9 eV and density was <inline-formula> <tex-math notation="LaTeX">3.4\times 10^{13} </tex-math></inline-formula> cm −3 for a 100-A He discharge with a gas flow rate of 0.05 L/min.
It is proved that two types of discrepancies of the sequence {
θ
n
x
} obey the law of the iterated logarithm with the same constant. The appearing constants are calculated explicitly for most of
θ
> ...1.
Tropical grasses mainly utilized in southwestern Japan exhibited that the dry matter yield is increased as the growth stage. However, the nutritive value, such as crude protein (CP) and non-fibrous ...carbohydrate (NFC) decreased strikingly. Therefore, breeding for the new grass or crops with high potential as protein source is an urgent problem. The aim of the study were to investigate the dry matter yield of two soybean cultivars ('Miyakonojo' and 'Williams82') with Rhodes grass as living mulch plant after 1st cutting and to investigate the nutritive value of the silage. The growth of the Rhodes grass hampered the growth of soybean at the initial and the subsequent growth stages. Hence, the dry matter yield of soybean ranged from 9.8 kg/10a to 26.6 kg/10a, correspond to about 3 to 10% of total dry matter yield. The nutritive value of EE, CA and CP in Rhodes grass with soybean silage higher than those of Rhodes grass silage. These results suggest that the addition of soybean biomass could be useful for increasing the nutrition value of silage. Moreover, it is necessary to choose an appropriate sowing date to increase dry matter yield of soybean biomass in southwestern of Japan.
Although the law of the iterated logarithm for discrepancies of geometric progression is proved, the constants appearing there are not concretely evaluated for small ratios. We consider the case when ...the ratio is a power root of a small rational number, and make a conjecture on the concrete value of the constant. We give several examples which are consistent with the conjecture.
subsp.
TUA4408L has the ability to grow and ferment soymilk and is able to modulate the innate immune response of intestinal epithelial cells
. These two properties prompt us to evaluate whether the ...soymilk fermented with the TUA4408L strain can induce beneficial immunomodulatory effects
. For this purpose, pigs were selected as a preclinical model. The studies performed here demonstrated that the
subsp.
TUA4408L-fermented soymilk (TUA4408L FSM) reduced blood markers of inflammation and differentially regulated the expression of inflammatory and regulatory cytokines in the intestinal mucosa. These immunological changes induced by the TUA4408L FSM were associated to an enhanced resistance to pathogenic
and an improved grow performance and meat quality of pigs. The experiments and analysis in our study indicate that the immunobiotic TUA4408L FSM could be an interesting non-dairy functional food to beneficially modulate the intestinal immune system, improve protection against pathogens and reduce inflammatory damage. The preclinical study carried out here in pigs could have a better correlation in humans, compared to a rodent model. However, the clinical relevance of these findings still needs to be confirmed by further research, for example, in controlled human challenge studies.
It is known that the discrepancy
D
N
{
k
x
}
of the sequence
{
k
x
}
satisfies
N
D
N
{
k
x
}
=
O
(
(
log
N
)
(
log
log
N
)
1
+
ε
)
a.e. for all
ε
>
0
, but not for
ε
=
0
. For
n
k
=
θ
k
,
θ
>
1
we ...have
N
D
N
{
n
k
x
}
≦
(
Σ
θ
+
ε
)
(
2
N
log
log
N
)
1
/
2
a.e. for some
0
<
Σ
θ
<
∞
and
N
≧
N
0
if
ε
>
0
, but not for
ε
<
0
. In this paper we prove, extending results of Aistleitner–Larcher
6
, that for any sufficiently smooth intermediate speed
Ψ
(
N
)
between
(
log
N
)
(
log
log
N
)
1
+
ε
and
(
N
log
log
N
)
1
/
2
and for any
Σ
>
0
, there exists a sequence
{
n
k
}
of positive integers such that
N
D
N
{
n
k
x
}
≦
(
Σ
+
ε
)
Ψ
(
N
)
eventually holds a.e. for
ε
>
0
, but not for
ε
<
0
. We also consider a similar problem on the growth of trigonometric sums.
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