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Acta Mathematica Sinica,Chinese Series
2018, Vol.34 Num.10
Online: 20181015
Articles
Articles
1485
Ikumitsu NAGASAKI
Estimates of the Isovariant BorsukUlam Constants of Connected Compact Lie Groups
The isovariant BorsukUlam constant
c
_{G}
of a compact Lie group
G
is defined to be the supremum of
c
∈ R such that the inequality
c
(dim
V
dim
V
^{G}
) ≤ dim
W
dim
W
^{G}
holds whenever there exists a
G
isovariant map
f
:
S
(
V
)→
S
(
W
) between
G
representation spheres. In this paper, we shall discuss some properties of
c
_{G}
and provide lower estimates of
c
_{G}
of connected compact Lie groups, which leads us to some BorsukUlam type results for isovariant maps. We also introduce and discuss the generalized isovariant BorsukUlam constant
č
_{G}
for more general smooth
G
actions on spheres. The result is considerably different from the case of linear actions.
2018 Vol. 34 (10): 14851500 [
Abstract
] (
5
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1KB] [
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276KB] (
10
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1501
Ji Gao YAN
Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables
In this paper, the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables are investigated. Some sufficient conditions for the convergence are provided. In addition, the MarcinkiewiczZygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables is obtained. The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.
2018 Vol. 34 (10): 15011516 [
Abstract
] (
2
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1KB] [
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227KB] (
3
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1517
San Ying SHI, Michel WEBER
On the Equation
n
_{1}
n
_{2}
=
n
_{3}
n
_{4}
Restricted to Factor Closed Sets
We study the number of solutions
N
(
B, F
) of the diophantine equation
n
_{1}
n
_{2}
=
n
_{3}
n
_{4}
, where 1 ≤
n
_{1}
≤
B
, 1 ≤
n
_{3}
≤
B
,
n
_{2}
,
n
_{4}
∈
F
and
F
⊂[1,
B
] is a factor closed set. We study more particularly the case when
F
={
m
=
p
_{1}
^{ε1}
…
p
_{k}
^{εk}
,
ε
_{j}
∈ {0, 1}, 1 ≤
j
≤
k
},
p
_{1}
,…,
p
_{k}
being distinct prime numbers.
2018 Vol. 34 (10): 15171530 [
Abstract
] (
2
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1KB] [
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229KB] (
11
)
1531
Yi WU, Xue Jun WANG, Andrew ROSALSKY
Complete Moment Convergence for Arrays of Rowwise Widely Orthant Dependent Random Variables
In this paper, complete moment convergence for widely orthant dependent random variables is investigated under some mild conditions. For arrays of rowwise widely orthant dependent random variables, the main results extend recent results on complete convergence to complete moment convergence. These results on complete moment convergence are shown to yield new results on complete integral convergence.
2018 Vol. 34 (10): 15311548 [
Abstract
] (
1
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1KB] [
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239KB] (
7
)
1549
Shao Peng XU, Shu Lin ZHOU
Zero Extension for the Biharmonic Equation
In this paper we present a necessary and sufficient condition to guarantee that the extended function of the solution by zero extension for the biharmonic equation in a smaller domain is still the solution of the corresponding extension problem in a larger domain. We prove the results under the frameworks of classical solutions and strong solutions.
2018 Vol. 34 (10): 15491562 [
Abstract
] (
4
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1KB] [
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198KB] (
8
)
1563
Valerii IVANOV, Alexey IVANOV
Generalized Logan's Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson's Inequality in
L
_{2}
(R
^{3}
)
We study Jackson's inequality between the best approximation of a function
f
∈
L
_{2}
(R
^{3}
) by entire functions of exponential spherical type and its generalized modulus of continuity. We prove Jackson's inequality with the exact constant and the optimal argument in the modulus of continuity. In particular, Jackson's inequality with the optimal parameters is obtained for classical modulus of continuity of order
r
and ThueMorse modulus of continuity of order
r
∈ N. These results are based on the solution of the generalized Logan problem for entire functions of exponential type. For it we construct a new quadrature formulas for entire functions of exponential type.
2018 Vol. 34 (10): 15631577 [
Abstract
] (
2
) [
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1KB] [
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250KB] (
2
)
1578
Jin CHENG, Zi Ting ZENG
Irreducible Wakimotolike Modules for the Lie Superalgebra
D
(2, 1;
α
)
By using the idea of Wakimoto's free field, we construct a class of representations for the Lie superalgebra
D
(2, 1;
α
) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter
λ
. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter
λ
satisfies (
λ
+
m
)(
λ
(1+
α
/
α
)≠
m
)=0 for any
m
∈ Z
_{+}
.
2018 Vol. 34 (10): 15781588 [
Abstract
] (
4
) [
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1KB] [
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221KB] (
5
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1589
Feng Yang HE, Ye Bin CHENG, Tie Jun TONG
Nonparametric Estimation of Extreme Conditional Quantiles with Functional Covariate
Estimation of the extreme conditional quantiles with functional covariate is an important problem in quantile regression. The existing methods, however, are only applicable for heavytailed distributions with a positive conditional tail index. In this paper, we propose a new framework for estimating the extreme conditional quantiles with functional covariate that combines the nonparametric modeling techniques and extreme value theory systematically. Our proposed method is widely applicable, no matter whether the conditional distribution of a response variable
Y
given a vector of functional covariates
X
is short, light or heavytailed. It thus enriches the existing literature.
2018 Vol. 34 (10): 15891610 [
Abstract
] (
2
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1KB] [
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466KB] (
7
)
1611
Li Nan ZHONG, Hao ZHAO, Wen Huai SHEN
Cohomology of the Universal Enveloping Algebras of Certain Bigraded Lie Algebras
Let
p
be an odd prime and
q
=2(
p
1). Up to total degree
t

s
< max{(5
p
^{3}
+ 6
p
^{2}
+ 6
p
+ 4)
q
10,
p
^{4}
q
}, the generators of
H
^{s,t}
(
U
(
L
)), the cohomology of the universal enveloping algebra of a bigraded Lie algebra
L
, are determined and their convergence is also verified. Furthermore our results reveal that this cohomology satisfies an analogous Poinćare duality property. This largely generalizes an earlier classical results due to J. P. May.
2018 Vol. 34 (10): 16111625 [
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] (
2
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1KB] [
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244KB] (
3
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1626
Qun LIU, Zhi Shan DONG
Large Deviations in Generalized Random Graphs with Node Weights
Generalized random graphs are considered where the presence or absence of an edge depends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a generalized random graph, where the node weights are deterministic under some regularity conditions, as well as chosen i.i.d. from a finite set with positive components. When the node weights are random variables, obstacles arise because the independence among edges no longer exists, our main tools are some results of large deviations for mixtures. After calculating, our results show that the corresponding rate functions for the deterministic case and the random case are very different.
2018 Vol. 34 (10): 16261634 [
Abstract
] (
1
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1KB] [
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187KB] (
9
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