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Optimal Internet Worm Treatment Strategy
Based on the Two-Factor Model
Xiefei Yan and Yun Zou
The security threat posed by worms has steadily
increased in recent years. This paper discusses the
application of the optimal and sub-optimal Internet worm
control via Pontryagin’s maximum principle. To this end,
a control variable representing the optimal treatment
strategy for infectious hosts is introduced into the twofactor worm model. The numerical optimal control laws
are implemented by the multiple shooting method and the
sub-optimal solution is computed using genetic algorithms.
Simulation results demonstrate the effectiveness of the
proposed optimal and sub-optimal strategies. It also
provides a theoretical interpretation of the practical
experience that the maximum implementation of
treatment in the early stage is critically important in
controlling outbreaks of Internet worms. Furthermore,
our results show that the proposed sub-optimal control
can lead to performance close to the optimal control, but
with much simpler strategies for long periods of time in
practical use.
Keywords: Optimal control, worm control, epidemic
model, two-factor model, Pontryagin’s maximum
principle, genetic algorithm.
Manuscript received Feb. 10, 2007; revised Nov. 16, 2007.
This work was supported by the National Natural Science Foundation of China (Grant nos.
60474078, 60574015 and 60304001).
Xiefei Yan (phone: + 86 25 84315463, email: yanxiefei@hotmail.com) is with the School of
Automation, Southeast University, Nanjing, China; the Department of Automation, Nanjing
University of Science and Technology, Nanjing, China; the Postdoctoral Research Working
Station of Wuxi Seamless Oil Pipe Co., Ltd., Wuxi, China; and the Postdoctoral Research
Working Station of Wuxi National High-Tech Industrial Development Zone, Wuxi, China.
Yun Zou (email: zouyun@vip.163.com) is with the Department of Automation, Nanjing
University of Science and Technology, Nanjing, China.
ETRI Journal, Volume 30, Number 1, February 2008
I. Introduction
With the explosive growth of Internet applications, Internet
worms have become a major problem for the security of
Internet networks [1]. Worms, defined as autonomous
programs that spread through computer networks by searching,
attacking, and infecting remote computers automatically, have
been developed since the first Morris worm appeared in 1988
[2]. In 2001, the Code Red and Nimda worms infected
hundreds of thousands computers [3] and cost millions of
dollars in losses [4].
In order to understand the general characteristics of worm
propagation, epidemiological models have been employed in
many studies [1], [4]-[12]. From 1991 to 1993, Kephart, White,
and Chess of IBM performed a series of studies on viral
infection based on epidemiology susceptible-infectioussusceptible (SIS) models [6]-[8]. This model is based on
biological epidemiology and provides a qualitative
understanding of the spread of viruses under the assumption
that classical epidemic models are all homogeneous, which
means that an infected host is equally likely to infect any other
susceptible hosts. Based on the classical epidemic model, Zou
and others [5] considered the two-factor model to study the
propagation of the Code Red worm. Chen and others presented
a discrete-time worm model that considers the patching and
cleaning effect during worm propagation [9]. Wang and others
[10] introduced an analytic model to capture the impact of
underlying topology in computer viral propagation. Moreover,
Wang and Wang [11] proposed a model extending the classical
epidemic model by including two specific parameters,
infection delay and user vigilance time. Kim and others [12]
introduced an extension of the susceptible-infectious-removed
(SIR) model to simulate worm propagation in two different
Xiefei Yan et al.
81
network topologies. Qing and Wen [1] investigated the wormanti-worm (WAW) model in their survey on Internet worms.
Almost all related studies have been concerned with the
simulative and predictive functions of models, thus, providing
insight into the dynamics of worm propagation. However,
there are few studies that mathematically discuss dynamic
strategies against worm propagation.
Zou and others [4] provided a dynamic quarantine method
based on the principle “assume guilty before proven innocent.”
A dynamic quarantine system with a constant quarantine time
and a worm detection threshold was mathematically analyzed.
The results show that in the dynamic quarantine system, a
worm still propagates according to traditional epidemic models,
but with a slower propagation speed and a higher epidemic
threshold. Kreidl and others [13] presented a feedback control
host-based autonomic defense system to protect the
information and functionality of a server. Ram and others [14]
studied a state space feedback control model to detect and
control the spread of these viruses by measuring the number of
connections an infected host makes. The objective of the
mechanism is to slow down the spreading velocity of a worm
by controlling (delaying) the total number of connections made
by an infected host. As expected, the model showed that the
sooner the infection is detected, the faster the reduction of the
spreading velocity. Kim and others [15], [16] investigated an
optimization model that takes into account the infection and
treatment costs. As in many studies on the optimal control of
epidemics [17]-[23], a control variable representing filtering
treatment on infectious hosts was introduced into the classical
SIS model.
However, the classical model is not suitable for modeling
propagation states of Internet worms. In the Internet, cleaning,
patching, and filtering countermeasures against worms will
remove both susceptible hosts and infectious hosts from
circulation, but the classical model only accounts for the
removal of infectious hosts. Moreover, the classical model
assumes that the infection rate is constant, which is not true for
a rampantly spreading Internet worm, such as the Code Red
worm [5]. For these reasons, Zou and others [5] introduced the
two-factor propagation model, which considers two factors.
One factor is the dynamic countermeasures taken by ISPs and
users; the other is the slowed down worm infection rate
because rampant propagation of a worm causes congestion and
troubles to some routers. This two-factor model is an extension
and supplement of the classical model and more accurately
reflects the propagation states of Internet worms [1].
In this study, we follow the idea of Kim and others about the
optimal control of treatment costs for Internet worms [15], [16]
and the works in epidemics control [21]-[23]. We further
discuss the optimal strategy associated with the treatment of
82
Xiefei Yan et al.
infectious hosts to prevent worm propagation based on the
two-factor model. A sub-optimal solution with a much simpler
strategy is also proposed. We introduce into this model one
control variable representing the rate of treatment to remove
infectious hosts from circulation by filtering or disconnection
from Internet.
The remainder of this paper is organized as follows.
Section II describes the two-factor worm model with control.
The analysis of optimization problems and the sub-optimal
solution are presented in sections III and IV, respectively. In
section V, we carry out a numerical simulation and briefly
discuss the influences of some model parameters. Finally, the
conclusions are summarized in section VI.
II. Optimization of Two-Factor Worm Model
In Zou’s two-factor worm propagation model [5], the host
population is divided into four epidemiological classes: S, I, R,
and Q (see Table 1). The model which incorporates control of
infectious hosts is given by the following nonlinear system of
differential equations:
S (t ) = − β (t ) S (t ) I (t ) − μS (t )( I (t ) + R (t )) ,
(1)
I(t ) = β (t ) S (t ) I (t ) − u (t ) I (t ) ,
(2)
R (t ) = u (t ) I (t ) ,
(3)
Q (t ) = μS (t )( I (t ) + R (t )) ,
(4)
with S(0), I(0), R(0), and Q(0) given.
In this model, human countermeasures result in the removal
of both susceptible and infectious computers from circulation.
During the course of Code Red propagation, an increasing
Table 1. Notation of model parameters.
Notation
Definition
S
Number of susceptible hosts at time t
I
Number of infectious hosts at time t
R
Q
I+R
Number of removed hosts
from the infectious population at time t
Number of removed hosts
from the susceptible population at time t
Number of infected hosts at time t
N
Total number of hosts, N=S+I+R+Q
β
Infection rate at time t
µ
Removal rate of susceptible hosts
ETRI Journal, Volume 30, Number 1, February 2008
number of people became aware of the worm and implemented
some countermeasures: cleaning compromised computers,
patching or upgrading susceptible computers, setting up filters to
block the worm traffic on firewalls or edge routers, or even
disconnecting their computers from Internet. Therefore, the
process of removing susceptible hosts is directly related to the
number of infected hosts at time t. β (t ) = β 0 [1 − I (t ) N ]η is
the decreased infection rate, not a constant rate, where β 0 is
the initial infection rate. The large-scale worm propagation
caused congestion and troubles to some Internet routers [5], [24],
which slowed down the Code Red scanning process. The
exponent η is used to adjust the infection rate sensitivity to the
number of infectious hosts I(t), and η = 0 means a constant
infection rate. We take η = 3 here.
The control variable u(t) is a bounded Lebesgue integrable
function [20]. It represents the rate of treatment for removing
infectious hosts from circulation by filtering or disconnection
from Internet at time t . Furthermore, from an epidemiological
modeling point of view, the transfer rate u(t)I(t) corresponds to
an exponential waiting time e-ut as the fraction that is still in the
infectious class t units after entering class R and to 1 u as the
mean waiting time [25].
Remark 1. The two-factor model is suitable for worms that
propagate without the topology constraint [5]. The topology
constraint means that an infectious host may not be able to
directly reach and infect an arbitrary susceptible host. It needs
to infect several hosts on the route to the target before it can
reach the target. Most worms, such as Code Red, have no
topology constraint.
III. Optimal Control Problems
(6)
Ω
Here Ω = {u ∈ L1 (0, t f ) | 0 ≤ a ≤ u ≤ b ≤ 1} , and a and b are
fixed positive constants.
Pontryagin’s maximum principle [27], [28] provides the
necessary conditions for an optimal control problem. This
principle converts (1)-(3), (5), and (6) into a problem of
minimizing pointwise a Hamiltonian, H, with respect to u:
H = BI (t ) +
3
C 2
u (t ) + ∑ λi f i ,
2
i =1
(7)
where fi (i=1, 2, 3) is the right hand side of the differential
equation of (1)-(3), respectively. By applying Pontryagin’s
maximum principle, we have the following adjoint equations:
dλ1
∂H
= ρλ1 −
, λ1 (t f ) = 0,
dt
∂S
#
dλ3
∂H
= ρλ3 −
, λ3 (t f ) = 0.
dt
∂Q
That is,
⎡
⎛
⎣⎢
⎝
λ1 = ρλ1 + λ1 ⎢ β 0 ⎜1 −
η
⎤
I ⎞
⎟ I + μ ( I + R)⎥
N⎠
⎦⎥
η
λ2 = − B + ρλ2 − λ3u
tf
(5)
subject to (1)-(3), where tf is the final time. This performance
specification takes into account the cost of infection, the
numbers of infectious hosts, and the cost of control. The total
cost includes not only the cost for every infected host but also
the cost of reduced system performance, increased network
delay [15], organization, management, cooperation, and so on;
hence, the cost function should be nonlinear. In this paper, a
quadratic function is implemented to measure the control cost
by referenced to Kim and others [15], [16] and many other
studies in epidemics control [18]-[23]. Furthermore, the cost
function is written with a discount factor e-ρt with ρ>0 to obtain
ETRI Journal, Volume 30, Number 1, February 2008
J (u * ) = min J (u ) .
I ⎞
⎛
− λ 2 β 0 ⎜1 − ⎟ I ,
N
⎝
⎠
The problem is to minimize the cost function
C
J (u ) = ∫ [ BI (t ) + u 2 (t )]e − ρt dt
2
0
higher benefit than the present in intertemporal model (See
references in Kim and others [15] and some economic
applications in [26] and [27]). The coefficients B and C are
weight factors related to the size and importance of the two
parts of the objective function. We seek to find an optimal
control u*, such that
η
η −1
⎡ ⎛
⎤
I ⎞
I ⎞ SI
⎛
+ λ1 ⎢ β 0 ⎜1 − ⎟ S − β 0η ⎜1 − ⎟
+ μS ⎥
⎝ N⎠ N
⎢⎣ ⎝ N ⎠
⎥⎦
η
η −1
⎡ ⎛
⎤
I ⎞
I ⎞ SI
⎛
− λ2 ⎢ β 0 ⎜1 − ⎟ S − β 0η ⎜1 − ⎟
− u ⎥,
⎝ N⎠ N
⎢⎣ ⎝ N ⎠
⎥⎦
λ3 = ρλ3 + λ1 μS ,
(8)
with transversality conditions
λi (t f ) = 0 ,
i = 1, 2 , 3 .
(9)
By the bound in Ω = {u ∈ L1 (0, t f ) | 0 ≤ a ≤ u ≤ b ≤ 1} , the
optimal control is given by
Xiefei Yan et al.
83
1
⎧
⎫
u (t ) = min ⎨max{a, (λ2 − λ3 ) I }, b ⎬ ,
C
⎩
⎭
(10)
π =R
In this work, the search for a global minimum of J ′(π ) is
implemented by an extended method based on genetic
algorithms [29]. This extension admits the constraints
ti + Δ i = t i +1 , t N + Δ N = t f , ti ≥ 0 , Δ i ≥ 0, and u i ∈ Ω .
which is derived from the condition
∂H
= 0.
∂u
Remark 2. Due to the a priori boundedness of the state and
adjoint functions and the resulting Lipschitz structure of the
ordinary differential equations (ODEs), we obtain the
uniqueness of the optimal control for small tf [20]. The
uniqueness of the optimal control pair follows from the
uniqueness of the optimality system, which consists of (1)-(3),
(8), and (9) with characterization (10). There is a restriction on
the length of the time interval in order to guarantee the
uniqueness of the optimality system. This restriction on the
length of the time interval is due to the opposite time
orientations of (1)-(3), (8), and (9). The state problem has initial
values, and the adjoint problem has final values. This
restriction is very common in control problems [18], [20]-[23].
Remark 3. The problem previously described is a two-point
boundary value problem (TPBVP), with specified initial
condition for state equations (1)-(3) and terminal boundary
conditions (9) for adjoint equation (8). It can be numerically
solved by the multiple shooting method [19], [30], [31].
IV. A Sub-optimal Solution
In this section, a sub-optimal solution is obtained by genetic
algorithms following the ideas of [19] and [23]. Let the class of
admissible controls be restricted to the collection of functions
of type
⎧⎪u i
u (t ) = ⎨
⎪⎩ 0
if t ∈ I i ,
(11)
otherwise,
where u i ∈ Ω are constants, and Ii are closed intervals
[t i , ti + Δ i ] , such that I i ∩ I j = ∅ if i ≠ j . Therefore, the
restricted class admissible controls consist of pulses of height ui
and width Δ i , starting at time ti.
In order to simplify the actual policy implementation, we
assume that the two controls are of pulsed form. Let u(t)=uN(t)
be admissible controls with N pulses, characterized
by (t1 , Δ1 , u 1 , t 2 , Δ 2 , u 2 , " , t N , Δ N , u N ) . Therefore, the suboptimal control problem is to minimize
J ′(t1 , Δ1 , u , t 2 , Δ 2 , u , " , t N , Δ N , u ) := J [u (⋅)] ,
1
2
N
N
where J [⋅,⋅] is the same as defined in (5). Let
π = (t1 , Δ1 , u , t 2 , Δ 2 , u , " , t N , Δ N , u ) ∈ R
1
84
Xiefei Yan et al.
2
N
for the case of N pulses. The sub-optimal control problem is
simply min
J ′(π ) .
3N
V. Numerical Simulation
In this section, we investigate the numerical solution of the
optimal and sub-optimal control in the two-factor worm model.
The optimal control is obtained by solving the optimality
conditions, consisting of twelve ODEs from the state and
adjoint equations. This TPBVP problem is solved using the
improved multiple shooting method (see [19], [30], and [31])
as follows:
Step 1. Select n nodes to divide the interval [0, tf], so that
0 = t1 ≤ t 2 ≤ " ≤ t n = t f .
Step 2. Choose yi := [ x(t i ), λ (t i )] , i = 1, ", n .
Step 3. Integrate (1) and (5) for each time interval [t i , ti +1 )
using yi as the initial conditions and obtaining
y (ti −1 ) = [ x(ti −1 ), λ (ti −1 )] .
Step 4. Compute h := [hi ] , where hi := yi − y (ti ) .
Step 5. If h is sufficiently close to a minimum value, stop;
otherwise, modify the initial conditions yi for the next iteration
(using, for instance, the Newton-Raphson algorithm to
make hi := yinew − y (t i ) = 0 ) and go to step 3.
Table 2. Baseline values of parameters
Parameters
Values
N
1 million
I(0)
9989N/10,000
R(0)
N/1,000
S(0)
N/10,000
β0
0.35/N
η
3
µ
0.06/N
ρ
0.01
Final time, tf
140 hours
Time step duration, dt
1 hour
Upper bound for control
0.50
Lower bound for control
0.05
Weight associated with I:B
5
Weight associated with u:C
600
3N
ETRI Journal, Volume 30, Number 1, February 2008
Figure 1 shows the time dependent optimal and sub-optimal
control laws for the case in which C=600. The control u is
plotted as a function of time. To minimize the number of
infectious hosts (I), the optimal control stays at its upper bound
for about 39 days and then steadily decreases to its lower bound
over the remaining simulation time. The sub-optimal control law
stays at the upper bound during the first time interval.
In fact, at the beginning of the simulation time, both the
optimal control and sub-optimal controls remain at their upper
bounds in order to remove as many infectious hosts as possible.
The steady decrease to the lower bound in the optimal control
case and maintenance at a proper value in the sub-optimal case
are determined by the balance between the cost of the worm
and the control.
Comparisons of the number of infectious hosts under the
0.50
Optimal control
Sub-optimal control
0.45
Control laws
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0
20
40
60
80
Time (days)
100
120
Fig. 1. Optimal and sub-optimal control laws.
ETRI Journal, Volume 30, Number 1, February 2008
140
Number of infectious hosts
Under optimal control
Under sub-optimal control
800
600
400
200
0
0
20
40
60
80
Time (days)
100
120
140
(a)
1000
Number of infectious hosts
Remark 4. Obtaining the ideal weights is very difficult in
practice. It requires a lot of work on data mining, analysis, and
fitting. The weights in the simulations here are only of
theoretical interest, and are used to illustrate the control
strategies proposed in this paper. We will briefly discuss cases
with various values of C later in this section.
1000
Under optimal control
Under constant control (u≡0.40)
800
600
400
200
0
0
20
40
60
80
Time (days)
100
120
140
(b)
5
3.0
Number of infectious hosts
In simulations, we studied the impact of the optimal process
of removal of infectious host during the outbreak of the Code
Red worm. The optimal control was computed using the above
iterative algorithm with 70 nodes up to tolerance 10-5 for h .
We set the upper bound of u to 0.50 according to our
reasonable assumption that it takes an average of at least two
hours to cure an infectious host (cleaning, patching, filtering, or
even disconnecting from the Internet). The sub-optimal control
was computed for the case in which N=3 and for specified time
intervals: Δ1 = 40 , Δ 2 = 60 , and Δ 3 = 40 . Parameters such as
population size for the genetic algorithm, generation gap, and
others were adjusted ad hoc [19], [23]. The model parameters
are from published data [5] (see Table 2). In addition,
considering the two weight factors associated with I and u, we
chose B=5 and C=600 to illustrate the optimal strategies.
x 10
Under lower bound control (u≡0.05)
Under constant control (u≡0.20)
2.5
2.0
1.5
1.0
0.5
0.0
0
20
40
60
80
100
120
140
Time (days)
(c)
Fig. 2. Number of infectious hosts under optimal control, suboptimal control, constant control, and lower bound control.
optimal control, sub-optimal control, and the constant controls
(u ≡ 0.20,0.40) throughout the simulated time are shown in
Fig. 2. As expected, in the early phase of the breakout, keeping
the control at its upper bound directly slows down the increase
in the number of the infectious hosts. The optimal control is
much more effective for controlling the outbreak and
decreasing the duration of the epidemic, which is the time
interval from the beginning until there is no emergence of new
infectious hosts. Furthermore, Fig. 2(c) demonstrates that if the
lower bound control (u ≡ 0.05) and constant control with
u ≡ 0.20 were implemented throughout the simulation time,
the number of infectious hosts reached about 2.61×105 and
Xiefei Yan et al.
85
0.50
x 104
Under optimal control
Under sub-optimal control
3.0
Cost
2.5
2.0
1.5
1.0
10
0.40
0.35
0.30
0.25
0.20
0.15
0.5
0
µ=0.02/N
µ=0.06/N
µ=0.18/N
0.45
3.5
Optimal control law
4.0
0
20
40
x 104
60
80
Time (days)
(a)
100
120
0.10
140
0
20
40
60
80
Time (days)
100
120
140
Fig. 5. Comparison of optimal control laws for different values of µ.
Under constant control (u≡0.40)
0.50
6
4
2
0
C=300
C=600
C=900
0.45
0
20
40
60
80
Time (days)
(b)
100
120
140
Fig. 3. Cost of infection (a) under optimal and sub-optimal
control and (b) under constant control.
Optimal control law
Cost
8
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0
20
40
60
80
100
120
140
Time (days)
Fig. 6. Comparison of optimal control laws for different values of C.
0.50
implement than the optimal control. Moreover, the results
demonstrate that, in order to obtain a minimum cost, the
0.40
maximum implementation of control in the early stage is critically
0.35
important in controlling the outbreak of an Internet worm.
0.30
Figures 4 to 6 illustrate how the optimal control strategies
0.25
β0=0.30/N
depend on the parameters β 0 , μ , and C, respectively. The
0.20
β0=0.35/N
value
of β 0 , which denotes the transmission rate of infections,
0.15
β0=0.40/N
varies from network to network, depending on many factors
0.10
0
20
40
60
80
100
120
140
including the security conditions. In Fig. 4, the control is
Time (days)
plotted as a function of time for the three different values
Fig. 4. Comparison of optimal control laws for different values of β0.
of β 0 : 0.30/N, 0.35/N, and 0.40/N. Other parameters for these
three cases are presented in Table 2. It shows that the control
5.04×104, respectively. In addition, we noticed that
plays an increasing role as β 0 increases. This is an expected
implementation of the constant control with u ≡ 0.40 can
result because as β 0 increases, the number of new infectious
also effectively control outbreaks (see Fig. 2(b)).
hosts increases, too. On the contrary, the level of control
However, let us investigate the cost incurred by this constant
increases as μ or C decreases, as shown in Figs. 5 and 6,
control, as shown in Fig. 3(b). Comparing this cost and the
costs incurred by the optimal and sub-optimal controls (Fig. 3(a)), respectively. As μ decreases, the number of susceptible hosts
the costs incurred by the optimal control and sub-optimal removed by users decreases, too. So the level of control
control are much lower than that of the constant control increases due to the increase in the number of newly infected
(u ≡ 0.40) . Moreover, the cost incurred by the sub-optimal hosts. Moreover, as C decreases, the cost of control decreases,
control is very close to that incurred by the optimal control, so the level of control increases to play a more significant role
despite the fact that the sub-optimal control is much easier to to remove more infectious hosts.
Optimal control law
0.45
86
Xiefei Yan et al.
ETRI Journal, Volume 30, Number 1, February 2008
VI. Conclusion
To better prepare us against future Internet worms, optimal
control theory has been applied for worm control in this study.
One control variable representing the rate of treatment to
remove infectious hosts from circulation has been incorporated
into the optimization two-factor model. Simulation results give
a theoretical interpretation to the practical experience that
maximum implementation of treatment in the early stage is
critically important in controlling outbreaks of Internet worms
with a minimum cost. As was mentioned in [17], the ideal
time-varying optimal strategy might not be easy to apply in
practice. Nevertheless, it provides a reference basis on which to
design practical quasi-optimal control strategies or policies and
assess their effectiveness. However, the proposed sub-optimal
control in this paper can provide much simpler strategies,
which, as verified by numerical simulations, can also lead to
performance very similar to that achieved by the optimal
control. In practical implementations, it may be considerably
easier to adopt sub-optimal strategies.
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Xiefei Yan received the BS degree in
information and computational mathematics
from the Nanjing University of Science and
Technology in 2003, and the PhD degrees in
automatic control engineering from the Nanjing
University of Science and Technology in 2007.
He was a Research Assistant in the Department
of Building Service Engineering at the Hong Kong Polytechnic
University in 2005. Currently, he is a Postdoctoral Researcher in
School of Automation, Southeast University and Postdoctoral
Research Working Station of Wuxi Seamless Oil Pipe Co., Ltd. and
Postdoctoral Research Working Station of Wuxi National High-tech
Industrial Development Zone, China. His current research interests
include industrial control system, industrial control network,
optimization, emergency control, epidemic control, and isolation
control.
Yun Zou received the BS degree in
mathematics from the Northwestern University,
Xian, China, in 1983, and the M.S. and PhD
degrees in automatic control engineering from
Nanjing University of Science and Technology,
Nanjing, China, 1987, and 1990, respectively.
He is currently a Professor of Electrical
Engineering at Nanjing University of Science and Technology. His
current research interests include differential-algebraic equation
systems, two-dimensional systems, singular perturbations, transient
stability of power systems, and power market. Dr. Zou is a
Mathematical Reviewer of the journal Mathematical Reviews and a
Member of American Mathematical Society.
88
Xiefei Yan et al.
ETRI Journal, Volume 30, Number 1, February 2008
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