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Acta Mathematica Sinica,Chinese Series
2019, Vol.35 Num.1
Online: 20190115
Articles
Articles
1
Ran GU, Ya Xiang YUAN
A Partial FirstOrder AffineScaling Method
We present a partial firstorder affinescaling method for solving smooth optimization with linear inequality constraints. At each iteration, the algorithm considers a subset of the constraints to reduce the complexity. We prove the global convergence of the algorithm for general smooth objective functions, and show it converges at sublinear rate when the objective function is quadratic. Numerical experiments indicate that our algorithm is efficient.
2019 Vol. 35 (1): 116 [
Abstract
] (
3
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1KB] [
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17
Alexander BULINSKI, Denis DIMITROV
Statistical Estimation of the Shannon Entropy
The behavior of the KozachenkoLeonenko estimates for the (differential) Shannon entropy is studied when the number of i.i.d. vectorvalued observations tends to infinity. The asymptotic unbiasedness and
L
^{2}
consistency of the estimates are established. The conditions employed involve the analogues of the HardyLittlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.
2019 Vol. 35 (1): 1746 [
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2
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47
Peter BEELEN, Mrinmoy DATTA, Sudhir R. GHORPADE
Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound
We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue for the socalled footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included.
2019 Vol. 35 (1): 4763 [
Abstract
] (
1
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288KB] (
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64
B. Daya REDDY
Modelling, Analysis and Computation in Plasticity
The typical problem in the mechanics of deformable solids comprises a mathematical model in the form of systems of partial differential equations or inequalities. Subsequent investigations are then concerned with analysis of the model to determine its wellposedness, followed by the development and implementation of algorithms to obtain approximate solutions to problems that are generally intractable in closed form. These processes of modelling, analysis, and computation are discussed with a focus on the behaviour of elasticplastic bodies; these are materials which exhibit pathdependence and irreversibility in their behaviour. The resulting variational problem is an inequality that is not of standard elliptic or parabolic type. Properties of this formulation are reviewed, as are the conditions under which fully discrete approximations converge. A solution algorithm, motivated by the predictorcorrector algorithms that are common in elastoplastic problems, is presented and its convergence properties summarized. An important extension of the conventional theory is that of straingradient plasticity, in which gradients of the plastic strain appear in the formulation, and which includes a length scale not present in the conventional theory. Some recent results for straingradient plasticity are presented, and the work concludes with a brief description of current investigations.
2019 Vol. 35 (1): 6482 [
Abstract
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1
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322KB] (
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83
Jacek BANASIAK
Analytic Fragmentation Semigroups and Classical Solutions to Coagulationfragmentation Equationsa Survey
In the paper we present a survey of recent results obtained by the author that concern discrete fragmentationcoagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentationcoagulation equations.
2019 Vol. 35 (1): 83104 [
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105
Shang Quan BU, Gang CAI
Periodic Solutions of Thirdorder Differential Equations with Finite Delay in Vectorvalued Functional Spaces
In this paper, we study the wellposedness of the thirdorder differential equation with finite delay (
P
_{3}
):
αu
"'(
t
) +
u
"(
t
)=
Au
(
t
) +
Bu
' (
t
) +
Fu
_{t}
+
f
(
t
)(
t
∈ T:=[0, 2
π
]) with periodic boundary conditions
u
(0)=
u
(2
π
),
u
' (0)=
u
' (2
π
),
u
"(0)=
u
" (2
π
), in periodic LebesgueBochner spaces
L
^{p}
(T;
X
) and periodic Besov spaces
B
_{p,q}
^{s}
(T;
X
), where
A
and
B
are closed linear operators on a Banach space
X
satisfying
D
(
A
) ∩
D
(
B
)≠ {0},
α
≠ 0 is a fixed constant and
F
is a bounded linear operator from
L
^{p}
([2
π
, 0];
X
) (resp.
B
_{p,q}
^{s}
([2
π
, 0];
X
)) into
X
,
u
_{t}
is given by
u
_{t}
(
s
)=
u
(
t
+
s
) when
s
∈[2
π
, 0]. Necessary and sufficient conditions for the
L
^{p}
wellposedness (resp.
B
_{p,q}
^{s}
wellposedness) of (
P
_{3}
) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.
2019 Vol. 35 (1): 105122 [
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] (
1
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0
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123
Li Xin CHENG, Long Fa SUN
Stability Characterizations of
∈
isometries on Certain Banach Spaces
Suppose that
X, Y
are two real Banach Spaces. We know that for a standard
∈
isometry
f
:
X
→
Y
, the weak stability formula holds and by applying the formula we can induce a closed subspace
N
of
Y
^{*}
. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard
∈
isometry to be stable in assuming that
N
is
ω
^{*}
closed in
Y
^{*}
. Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces. We also prove that if, in addition, the space
Y
is quasireflexive and hereditarily indecomposable, then
L
(
f
) ≡
span
[
f
(
X
)] contains a complemented linear isometric copy of
X
; Moreover, if
X
=
Y
, then for every
∈
isometry
f
:
X
→
X
, there exists a surjective linear isometry
S
:
X
→
X
such that
f

S
is uniformly bounded by 2
∈
on
X
.
2019 Vol. 35 (1): 123134 [
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] (
2
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219KB] (
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135
Hong Wei BI, Hui HE
A Treevalued Markov Process Associated with an Admissible Family of Branching Mechanisms
Let T ⊂ R be an interval. By studying an admissible family of branching mechanisms {
ψ
_{t}
,
t
∈ T} introduced in Li[
Ann. Probab
.,
42
, 4179 (2014)], we construct a decreasing LévyCRTvalued process {
T
_{t}
,
t
∈ T} by pruning Lévy trees accordingly such that for each
t
∈ T,
T
_{t}
is a
ψ
_{t}
Lévy tree. We also obtain an analogous process {
T
_{t}
^{*}
,
t
∈ T} by pruning a critical Lévy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {
T
_{t}
,
t
∈ T} at the ascension time
A
:=inf{
t
∈ T;
T
_{t}
is finite} can be represented by {
T
_{t}
^{*}
,
t
∈ T}. The results generalize those studied in Abraham and Delmas[
Ann. Probab
.,
40
, 11671211 (2012)].
2019 Vol. 35 (1): 135160 [
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] (
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1KB] [
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344KB] (
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