数理解析研究所講究録 1415 巻 2005 年 127-131

127

CONVERGENCE OF DYNAMICAL SYSTEMSWITH CONVEX LYAPUNOV FUNCTIONSSIMEON REICH AND ALEXANDER J.ZASLAVSK1
the convergence of several classes of dynamical systems
ABSTRACT. This is asurvey of recent results regarding
For each class we define an appropriate complete
spaces.
Banachwith convex Lyapunov functions in general(in the sense of Baire category) are convergent.
them
of
most
that
show
and
systems
dynamicalof
space
metric.
category, but also
the
first
only
of
is
not
systems
divergentof
In some cases the set$\sigma- \mathrm{p}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{s}$INTRODUCTION
topic in optimization theory
The study of minimization methods for convex functions is acentral bounded, closed and convexaondefinedIn this survey, we are given a continuous convex function
self-mapping : $Karrow K$ such that
subset $K$ of aBanach space $X$ , and a minimization algorithm is a
, the sequences$\in K$ . We show that for most of these algorithms$f(Ax)\leq f(x)$ for allof the elements of a comple.
most
that
say
we
.
When
valuesall
initialfortend to the inflmum of
have this property contains
which
pointsmetric space $X$ enjoy acertain property, we mean that the set of
Such an approach,generically.
holds
$X$
this
words,
property
everywhere dense subset of . In othera$X$ , has
$X$ and not just for asingle point in
space
whole
the
for
when a certain property is investigatedconcept
therecall15, 16]. We now
already been successfully applied in many areas of Analysis [1-6, 8, 11,
results.
refinedmore
even
obtaintous
will
enable
16]
which
13,
of porosity [6,$B_{d}(y,r)$
closed ball of center $y\in Y$ andtheLet $(Y, d)$ be acomplete metric space. We denote by$>0$ such
exist $\alpha\in(0, 1)$ and
$(Y,
d)$
there
if
in
is porous
radius $>0$ . We s that a subset$\in Y$ for which$y\inY$exists,there
and eachthat for each
$f$

$A$

$\{f(A^{n}x)\}_{n=1}^{\infty}$

$A$

$x$

$\mathrm{t}\mathrm{e}$$x\in I\zeta$$f$$G_{\delta}$

$E$

$r$

$r_{0}$$\subset Y$$\mathrm{a}\mathrm{y}$

$r$

$z$

$\in(0, r_{0}]$$B_{d}(z, \alpha r)\subset B_{d}(y, r)\backslash E$

union of porous subsets in $(Y, d)$ .
Asubset of the space is called -porous in ( , if it is a countable
is a finitethe first category. If
Since porous sets are nowhere dense, all -porous sets are of
cr-porous
of
class
fact,
the
In
dimensional Euclidean space, then a-porous sets are of Lebesgue measure 0.
the first
of
are
0and
measurehave
sets in such a space i much smaller than the class of sets which
category.
sets, note that if $E\subset Y$ is nowhere
To point out the difference between porous and nowhere dense.
$s>0$ such that $B_{d}(z, s)$dense, $y\in Y$ and $>0$ , then there are a point $z\in Y$ and a number
aconstantis$s=\alpha
r$
,
where
$(0, 1)$
If, however, is also porous, then for small enough we can choose
.
on
only
which depends
minimization methods studied in [7,Our paper is organized as follows. In Section 1we review thecontinuous.
In the second section is
uniformly
beto12, 13], where the convex function is assumed
to some examples.
assumed to be merely continuous [14], The third section is devoted
$Y$

$Y$

$\sigma$$d\rangle$$Y$$\sigma$$\mathrm{s}$$\subset B_{d}(y)$ $r)\backslash E$$r$$\alpha\in$

$r$

$E$

$E$

$f$

$f$

1. UNIFORMLY C0NT1.N UOUS LYAPUNOV FUNCT10NS$K\subset X$ is anonempty, bounded, closed andAssume that $(X, ||\cdot||)$ is aBanach space with norm
$Karrow R^{1}$ is aconvex uniformly continuous function. Set
convex subset of , and :$||\cdot||$

$X$

)

$f$

$\inf(f)=\inf\{f(x) :x \in K\}$ .

Typese by$A_{\mathcal{M}^{\mathrm{S}- \mathrm{I}}\mathrm{E}^{\mathrm{X}}}$

128
SIMEON REICH AND ALEXANDER J. ZASLAVSKI

Observe that this infimum is finite because $K$ is bounded and is uniformly continuous. We consider
the topological subspace $K\subset X$ with the relative topology Denote by the set of all self-mappings: $Karrow K$ such that$f$$\mathfrak{U}$$A$$f(Ax)$ $\leq f(x)$ for all x $\in K$ ,

(1.1)

.
the set of all continuous mappingsand byIn Example 2 of Section 3 we construct many such mappings.
byFor the set we define a metric :$A\in \mathfrak{U}$$\mathfrak{U}_{c}$$\mathfrak{U}\cross \mathfrak{U}arrow R^{1}$$\mathfrak{U}$$\rho$$\rho(A, B)=\sup\{||Ax-Bx||$:(12)
Clearly, the metric space
the set of all sequencesa boldface A. For the set

x$\in I\iota^{\nearrow}\}$, A, B$\in \mathfrak{U}$.

is a closed subset of it. Denote byis complete andwill occasionally be denoted by) A member of(respectively,
we consider the uniformity determined by the following base:$\mathfrak{M}$$\mathfrak{U}$$(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}1\mathrm{y}_{\mathrm{I}}\wedge\sigma_{\mathrm{D}\mathrm{t}_{e})}$$\mathfrak{U}_{\mathrm{c}}$$\mathfrak{M}$$\mathfrak{U}_{\mathrm{c}}$$\{A_{t}\}_{t=1}^{\infty}\subset \mathfrak{U}$$\mathfrak{M}$$E(N, \epsilon)=\{(\{A_{t}\}_{t=1}^{\infty}, \{B_{1}\}_{t=1}^{\infty})\in \mathfrak{M}\cross \mathfrak{M} :\rho(A_{t}, B_{t})\leq\epsilon, t=1, .

,

N\}$

is metrizable (by a metric. Clearly, the uniform space
where $N$ is a natural number and)complete.
and:
From the point of view of the theory of dynamical systems each element of describes a nonstationary
dynamical system with a Lyapunov function . Also, some optimization procedures in Hilbert and Banach(see the first example in Section 3 and [9, 10]).
spaces can be represented by elements ofthe value of the Lyapunov function
taken from the spacesequence
In [12] we show that for a genericfollowing two theorems.
theweobtainprecisely,
More
its
to
infimum.
tends
along all trajectories
which is a countable intersection of open everywhere dense
Theorem 1.1. There exists a set
the following assertion holds:
each
that
such
sets inforand a natural number $N$ such that for each
For each $>0$ , there exists a neighborhood ofand each $x\in K$ ,$\mathfrak{M}$$\epsilon>0$$\mathfrak{M}$$\cross \mathfrak{M}$$arrow R^{1}$$\rho_{w}$$\mathfrak{M}$$f$$9\mathrm{R}$$\Re f_{c}$$\mathcal{F}\subset \mathfrak{M}_{c}$$\mathrm{B}=\{B_{t}\}_{t=1}^{\infty}\in \mathcal{F}$$\mathfrak{M}_{c}$$U$$\mathrm{B}if\downarrow \mathfrak{M}_{c}$$\epsilon$$\mathrm{C}=\{C_{t}\}_{t=1}^{\infty}\in U$$f(C_{N} .C_{1}x) \leq\inf(f)+\epsilon$ .

a countable intersection of open everywhere dense
which
Theorem 1.2. There exists a set
holds:
assertion
followingthesuch that for eachsets
and a natural number $N$ such that for eachinneighborhoodaexists
there
,
each
of
For
$C\in U$ and each $x\in K$ ,$\dot{\mathrm{z}}s$$\mathcal{G}\subset \mathfrak{U}_{c}$$B\in \mathcal{G}$$i\mathfrak{s}i\mathfrak{U}_{c}$

$B$

$U$

$\epsilon>0$$\mathfrak{U}_{c}$$f(C^{N}x) \leq\inf(f)+\epsilon$.

The key auxiliary result which is used in the proofs of these theorems is the following proposition.
with the followfng property:
Proposition 1.1. There exists a mapping$x\in
K$ satisfying $f(x) \geq\inf(f)- 4-\epsilon$ , the inequality
each
that
such
, there is
for
Given
$A_{*}\in \mathfrak{U}_{c}$$\epsilon>0$$\delta(\epsilon)>0$$f(A_{*}x)\leq f(x)-\delta(\epsilon)$is true.

Remark t. 1. If there is $x_{\min}\in K$ for which $f(x_{\min})= \inf(f)$ , then we can set $A_{*}(x)=X \min$ for all$x\in K$ .
. For the set
and its closed subset
In the sequel we continue to study the metric space
already been
has
one
The
first
them
by
inducedtopologies
theand
we will consider two uniformities
subsetclosedisaClearly,.
bydenotedandweakcalledbeinduceswill
ittopologyThedefined.

$\mathfrak{M}$$\mathfrak{U}_{\mathrm{c}}$$(\mathfrak{U}, \rho)$$\mathfrak{M}_{c}$$\tau_{w}$ofwith the weak topology.
we also define a metric
For the set$\mathfrak{M}$$\mathfrak{M}$$\rho_{s}$:$\mathfrak{M}$$\cross \mathfrak{M}$$arrow R^{1}$$\rho_{s}(\{A_{1}\}_{t=1}^{\infty}, \{B_{t}\}_{t=1}^{\infty})=\sup\{\rho(A_{t}, B_{t}).

byt=1, 2, \ldots\})$$\{A_{t}\}_{t=1}^{\infty}$,$\{B_{t}\}_{t=1}^{\infty}\in \mathfrak{M}$.

12$
DYNAMICAL SYSTEMS. In the sequel we will
is a closed subset ofis complete andClearly, the metric space.
also study the metric space, it
is clearly stronger thanon M. Since
the topology induced by the metric
Denote bystrongandweakwith the relative
will be called strong. We consider the topological subspace$(\mathfrak{M}, \rho_{s})$$\mathfrak{M}_{\mathrm{c}}$$(\mathfrak{M}_{1}\rho_{s})$$(\Re \mathrm{t}_{c}, \rho_{s})$$\tau_{w}$$\tau_{\delta}$$\rho_{s}$$\tau_{s}$$\mathfrak{M}_{c}\subset \mathfrak{M}$topologies.
The following notion of normality was introduced in [7].
, there isis called normal if given
A mapping$f(x) \geq\inf(f)+\epsilon$ , the inequality$\epsilon>0$$A\in \mathfrak{U}$such that for each $x\in K$ satisfying$\delta(\epsilon)>0$$f(Ax)\leq f(x)-\delta(\epsilon)$is trueis called normal if given $>0$ , there is
A sequencesatisfying $f(x) \geq\inf(f)+\epsilon$ and each integer $t\geq 1$ , the inequality

such that for each$\delta(\epsilon)>0$$\epsilon$$\{A_{t}\}_{t=1}^{\infty}\in \mathfrak{M}$$f(A_{t}x)\leq f(x)$$\in K$$x$$-\delta(\epsilon)$holds.

is normal. This
and,spaces ,In [7] we show that a generic element taken from thealong anyfunction
Lyapunov
theofvaluesofsequence
is important because it turns out that the$K$ .
on(unrestricted) trajectory of such an element tends to the infimum of$(0, 1)$
deffine
,
,
For
. and
We can easily prove the following fact.
is also normal.
$(0, 1)$ ,and let A be normal. Then,
Proposition 1.2. $lei$$\mathfrak{U}$$\mathfrak{M}_{c}$$\mathfrak{M}$$\mathfrak{U}_{\mathrm{c}}$

$f$

$f$

$\mathrm{A}=\{At\}_{\star=1}^{\infty}$$\alpha\in$$\alpha \mathrm{A}+(1-\alpha)\mathrm{B}=\{\alpha A_{1}+(1-\alpha)B_{t}\}_{t=1}^{\infty}\in \mathfrak{M}$$\mathrm{B}=\{B_{t}\}^{\infty}t=1\in \mathfrak{M}$$\alpha \mathrm{A}+(1-\alpha)\mathrm{B}$$\mathrm{A}$$\alpha\in$$\mathrm{B}\in \mathfrak{M}$We now state the main results of [7].
there exists
be normaI and let
Theorem 1.3. Let$N$ such thatfor eachwith the strong topoiogy and a notural number
A in$The\gamma 1$$\epsilon>0$$\mathit{0}$$\mathrm{A}=\{A_{t}\}_{t=1}^{\infty}\in \mathfrak{M}$$\mathrm{c}$neighborhood$=\{C_{t}\}_{t=1}^{\infty}\in U$

$U$

of

, each$\mathfrak{M}$$x\in K$

, and each

$r$

$\{$

1, 2, .

$.\}arrow\{1,2, \ldots\}$,$f(C_{r(N)} \ldots C_{r(1)}x)\leq\inf(f)+\epsilon$.

. Then there exzsts a neighborh $oodU$ of
be normal and let
Theorem 1.4. Let
and each$N$ such thatfor eachwith the weak topology and a natural numberA$x\in K$ ,$\epsilon>0$$\mathrm{A}=\{A_{t}\}^{\infty}t=1\in \mathfrak{M}$$\mathrm{C}=\{C_{t}\}_{t=1}^{\infty}\in U$$rn\mathfrak{M}$$f(C_{N}.

.C_{1}x) \leq\inf(f)+\epsilon$.

which is a countable intersection of open everywhere dense
Theorem 1.5. There exists a set
table intersection of$a;td$
which is aasetwith the strong topology
subsets ofis normal
each
that
such
topology
strongthewitheverywhere dense subsets ofa countable intersection of open everywhere dense
which
Theorem 1.6. There exists a set
intersection of open everywhere dense subsets ofwhich is asubsets of and a set
is nomalsuch that eachto perturbed trajectories of a normal
In [13] we prove two theorems. The first one extends Theorem 1.3
practical significance [9, 10]considerableofobviously
sequence. The study of such trajectories isand let be positive. Then theoe exist a natural numberbeTheorem 1.7. Let: 1, . . $n\}arrow\{1,2, \}$ , and$n\geq n_{0}$ , each mapping
integereach
and a number $\gamma>0$ such that for
which satisfies
each sequence, $i=0$ , . . , $-1$ ,$\mathcal{F}\subset \mathfrak{M}$$\mathcal{F}_{c}\subset \mathcal{F}$$o\rho en$$cour\iota$$\cap \mathfrak{M}_{c}$$\mathfrak{M}$$\mathrm{A}\in \mathcal{F}$$\mathfrak{M}_{c}$$\mathcal{F}$$\subset \mathfrak{U}$$fs$$cou\mathfrak{s}?table$$\mathfrak{U}$$\mathcal{F}_{\mathrm{c}}\subset \mathcal{F}\cap \mathfrak{U}_{\mathrm{c}}$$\mathrm{A}\in \mathcal{F}$$\mathfrak{U}_{c}$$n_{0}$$\{At\}^{\infty}t=1\in \mathfrak{M}$$\mathrm{r}\iota \mathrm{o}rmal$$\epsilon$

$r$

$\{$

$)$

$\{x:\}_{i=0}^{n}\subset K$$\mathrm{n}$$||x_{j+1}-A_{r(,+1)}x_{\mathrm{i}}||\leq\gamma$, .
the inequality $f(x_{i}) \leq\inf(f)+\epsilon$ holds for $i=n_{0}$ ,and,For each of the spaces ,The second result of [13] improves upon Theorems 1.5 and 1.6.
its
of
one
each
subset such that, these theorems establish the existence of an everywhere dense
above
the
of
each
for
then
Lipschitzian,elements is normal. $\ln[13]$ we show that if the function is
not only of the first category) but also aiselements
normalallofsubsettheofcomplement
spaces, the-porous set$\ldots$$n$$\mathfrak{M}$$G_{\delta}$$\mathfrak{U}_{\mathrm{c}}$$f$$\sigma$$\mathfrak{M}_{c}$$\mathfrak{U}$130
SIMEON REICH AND ALEXANDER J. ZASLAVSKI

Theorem 1.8. Let$\mathcal{F}$be the set of all normal sequences in the space

F

$=$

{A

$\in \mathfrak{U}:\{A_{t}\}_{t=1}^{\infty}\in \mathcal{F}$$\mathfrak{M}$and letwhere $A_{t}=A$ , t $=1,$ 2, \ldots }.

is a -porous subset of
Lipschitzian. Then the complement of the set
Assume that the function
Mooeovef,
the complement.
-poroussubset
a
is
set
of
the
( , ps) and the complement ofsubset of
a-porousisand the complement of the setof the set $F$ is a -porous subset of$\mathcal{F}$$\sigma$$ft\mathrm{S}$$(\mathfrak{M}_{c}, \rho_{s})$$a$$\mathcal{F}\cap \mathfrak{M}_{c}$$\mathfrak{M}$$F\cap \mathfrak{U}_{\mathrm{c}}$$(\mathfrak{U}_{\mathrm{c}}, \rho)$$a$$(\mathfrak{U}, \rho)$$\sigma$.

2 CONTINUOUS LYAPUNOV FUNCTIONS
convexIn this section we continue to use the notation introduced in the previous sections, but the$Karrow R^{1}$ is assumed to be merely continuous and bounded from below.
function :whichequipped with the product topology and the space
We also consider the spaceof
product
theistopology)
weak
theis equipped with a pair of topologies. One of them (which is called
topology)
strongthecalledis(whichone
second
$K$
the
and
,
the topology of and the weak topology of.
is the product of the topology of $K$ and the strong topology ofis merely continuous, we are stil able to obtain two results in the direction of$\ln[14]$ , assuming thatThe
the previous sections. To achieve this, we change our point of view and consider a new framework.
maysystemdynamicalofour
trajectoryaofmain feature of this new framework is that the initial point
also vary
We now state the two main results of [14].
which a countable intersection of open (in the weak
Theorem 2.1. There exists a set,
such that for eachsubsets of
topology)
strong
(in
the
dense
everywhere
topology)
the following property holds:
with the weak topology andin, there exists a neighborhood of
For each,$N$ such thatnatural number
for each$f$$K\mathrm{x}$$\mathfrak{M}_{c}$$K\cross \mathfrak{U}_{\mathrm{c}}$$\mathfrak{M}_{c}$$\mathfrak{M}_{c}$$f$$\dot{\mathrm{z}}s$$\mathcal{F}\subset K\cross \mathfrak{M}_{c}$$(x, \{At\}_{t=1}^{\infty})\in \mathcal{F}$$K\cross \mathfrak{M}_{c}$$a$$K\cross \mathfrak{M}_{c}$$(x, \{A_{t}\}_{t=1}^{\infty})$$\mathcal{U}$$\epsilon>0$$(y, \{Bt\}_{t=1}^{\infty})\in \mathcal{U}$$f(B_{N}.

.

Theorem 2.2. There exists a set
such that for eachdense subsets of, there exists a neighborhood
For each,for each (y)B_{1}y) \leq\inf(f)+\epsilon$$\mathcal{F}\subset K\cross \mathfrak{U}_{c}$$K\mathrm{x}$.

which is a countable intersection of open everywhere, the following property holds:
and a natural number $N$ such that$(x,of A)$ in$(x, A)\in \mathcal{F}$$\mathfrak{U}_{c}$$\epsilon>0$$\mathcal{U}$$K\cross \mathfrak{U}_{c}$$B)\in \mathcal{U}$$f \langle B^{N}y)\leq\inf(f)+\epsilon$

3. EXAMPLES$Karrow K$
be a Banach space. In this section we present examples of continuous mappings :
Let
and
of
subsetis a bounded, closed and convex
satisfying $f(Ax)\leq f(x)$ for all $x\in K$ , whereis a convex function [12]:Example 1. Let : $Xarrow R^{1}$ be a convex, uniformly continuous function satisfying$A$$(X, ||\cdot||)$

$X$

$IC$

$f$

$Karrow R^{1}$$f$$f(x)arrow\infty$

as$||x||arrow\infty$.

.
Evidently, the function is bounded from below. For each real number , let $K_{c}=\{x\in X : f(x)\leq c\}$supposeWe
is bounded, closed and convex.
Clearly, the set
Fix a real number such that, namelythat the function is strictly convex on
$c$

$f$

$c$

$f$

$K_{c}$$K_{\mathrm{c}}\neq\emptyset$$h_{c}’$$f(\alpha x +(1-\alpha)y)<\alpha f(x)+(1-\alpha)f(y)$for all , $y\in K_{c}$ , $x\neq y$ , and all $\alpha\in(0,1)$ .
: $K_{c}arrow X$ be any continuous mapping. For eachLetfollowing minimization problem:
$x$

$V$

$f(z) arrow\min$ ,$z\in\{x+\alpha V(x) :$x\in K_{\mathrm{c}}$, there is a unique solution of the\alpha\in[0,1]\}$ .

131
DYNAMICAL SYSTEMS, we conclude that
This solution will be denoted by $Ax$ . Since $f(Ax)\leq f(x)$ for all
is continuous.
It is shown in [12] that the mapping$K-*R^{1}$ be a convex
Example 2. Let $K$ be a bounded, closed and convex subset of $X$ , and let .
$x0$ , $x_{1}\in K$ satisfying $f(x0)>f(x_{1})$ , we will
each
For
from
below
boundediscontinuous function whichconstruct a continuous mapping : $Karrow K$ such that $f(Ax)$ $\leq f(x)$ for all $x\in K$ and $Ax–x_{1}$ for all
in a neighborhood of .
andsuch thatIndeed, let , $x_{1}\in K$ with $f(x_{0})>f(x_{1})$ . There are numbers$A(K_{c})\subset K_{c}$$x\in K_{\mathrm{c}}$$A:K_{\mathrm{c}}arrow K_{c}$

$f$

$x$

$A$

$x_{0}$$r_{0}$$\epsilon_{0}$$x_{0}$$\mathrm{f}(\mathrm{x})-\epsilon_{0}>\mathrm{f}(\mathrm{x})$Now we define an open covering$\{Vx :

for all $x\in K$ satisfying $||x-x0||\leq r_{0}$ .
x \in K\}$ of $K$ Let $x\in K$ If $||x-x_{0}||<r_{0}$ , we set$V_{x}=\{y\in K : ||y-x0||<r_{0}\}$ and $a_{x}=x_{1}$If $||x-x_{0}||\geq r_{0}$ , then there is$r_{x}\in(0,4^{-1}r\mathrm{o})$$f(a_{x})\leq f(y)$and $a_{x}\in K$ such thatfor all $y\in\{z\in K : ||z-x||\leq r_{x}\}$ .

In this case, we set$V_{x}=\{y\in K : ||y-x||<\mathrm{r}_{x}\}$.

Clearly, $\cup\{V_{x} : x \in K\}=K$ . There is a continuous partition of unity
for all $x\in K$ ). For $y\in K$ we define(namely,$\{V_{x}\}_{x\in K}$.

$\{\phi_{x}\}_{x\in K}$

on$K\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{d}_{1}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$to$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\phi_{x}\subset V_{x}$$Ay= \sum\phi_{x}(y)a_{x}$.

$x\in K$

: $Karrow X$ , and it is continuous. Since $\sum_{x\in K}\phi_{x}(y)=1$ for
Evidently, the mapping is well defined,$A(K)\subset
K$ .
$K$
all $y\in K$ and is convex, we see that$y\in
K$ and that $Ay–x_{1}$ if $||y-x_{0}||\leq 4^{-1_{f}}0$ .
$f(Ay)\underline{<}f(y)$allfor
It is shown in [12] that
Acknowledgments. The work of the first author was partially supported by the Israel Science
Foundation founded by the Israel Academy of Sciences and Humanities, by the Fund for the Promotionof Research at the Technion, and by the Technion VPR Fund
$A$

$A$

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HAIFA, ISRAELDEPARThlENT OF MATHEMATICS, THB TBCHNION-ISRABL INSTITUT8 OF TECHNOLOGY, 32000
il-mail address: sreichStx. technion. . il., ajzaslQtx Technion$a\mathrm{r}\iota d$$\tau \mathrm{e}$$general\dot{\mathrm{t}}zed$$\mathrm{Z}\mathrm{a}\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{v}\mathrm{s}\mathrm{k}\dot{\mathrm{t}}$$il_{l}ty$$E$$\mathrm{a}\mathrm{c}$$\mathrm{a}\mathrm{c}$