Applications of the Universal Surjectivity of the Cantor Set
Author(s): Yoav Benyamini
Reviewed work(s):
Source: The American Mathematical Monthly, Vol. 105, No. 9 (Nov., 1998), pp. 832-839
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2589212 .
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ApplicationsoftheUniversalSurjectivity
ofthe Cantor Set
Yoav Benyamini
course.Its
withtheCantorsetis usuallyin a basicrealanalysis
encounter
Ourfirst
makes
properties
counter-intuitive
and
seemingly
ofunusual
combination
striking
It
course.
in
the
introduced
notions
new
the
illustrating
examplefor
it theperfect
an
set
plays
Cantor
that
the
learns
to
appreciate
is onlymuchlaterthatthestudent
and is not just an artificial
role in manybranchesof mathematics,
important
thatcan arisein
pathologies
exhibit
the
possible
to
designed
especially
construct,
real
analysis.
of
thesystematic
development
of the Cantorset, its
In thisarticlewe discussone of the basic properties
metric
of
class
in
the
spaces:
compact
universality
surjective
imageof theCantorset,
space is a continuous
Theorem1. Everycompactmetric
setA
mapfromtheCantor
is a continuous
spaceK there
metric
i.e.,foreachcompact
ontoK.
[7,p. 226].It is a
[1]and Hausdorff
is due to Alexandroff
Thisclassicaltheorem
e.g.,
and
topology,
on
real
analysis
in
books
many
thatappears
standardtheorem
[5,p. 363]and [8,p. 127].
unrelated
of seemingly
can be appliedto a variety
We showhowthistheorem
results
is considWhen
each
of
the
and
analysis.
in topology,
geometry,
problems
as an ad
to
appear
seems
Theorem
the Alexandroff-Hausdorff
ered separately,
in
a
thread
common
there
is
that
realize
we
soon
however,
hoctrick.Puttogether,
Phrased
heuristia
method.
represent
actually
that
they
and
all theseapplications,
encode"compactsets
wayto "continuously
cally,thetheoremgivesa systematic
ofdata.
compact
perfect,totallydisconnected,
The Cantorset is the uniqueinfinite,
of
its
representations
can
use
any
we
metricspace. Usingthischaracterization,
to do so. For example,the standardproofof the
wheneverit is convenient
as an infinite
product
Theoremuses the representation
Alexandroff-Hausdorff
here
we
encounter
sets
that
Cantor
All
the
set.
Hnn=1Fn,whereeach Fn is a finite
the
real
line.
of
closed
subsets
are,however,
novelpartsare
Mostoftheresultsinthisarticleare known.The only(possibly)
in Sections4 and 5. It was the resultin Section4, and the reactionsof several
me to writethisarticle.
peopleto theproof,thatprompted
a spacefilling
curve,i.e.,
1. SPACE FILLING CURVES. We startbyconstructing
[0,11ontotheunitcube[0,1]d in
a continuous
thatmapstheunitinterval
function
thed-dimensional
spaceRd.
4 from
function
thereis a continuous
Theorem,
BytheAlexandroff-Hausdorff
theCantorset A ontothecompactmetricspace[0,1]d.
[0,1],andextend4 to
ConsiderA as theclassicalCantorsetintheunitinterval
The
bylinearinterpolation:
definedon thewholeinterval
a continuous
function
If (a, b) is one of these
of A is a countableunionof open intervals.
complement
832
SURJECIIVITY OF THE CANTOR SET
[November
itspointsin theformta + (1 - t)b for0 < t < 1, and define
intervals,
represent
k(ta + (1 - t)b) = t4P(a) + (1 - t)o(b).
takesitsvaluesin [0,1]d becausethecubeis convex,
and
The extended
function
on [0,1] as required.
it is easyto checkthatit is continuous
mapfromA to [0,1] is a common
oftheAlexandroff-Hausdorff
The extension
of theunit
stepin almostall theproofswe present.Notethattheonlyproperty
We thusobtain
cubethatwe used in theextension
was itsconvexity.
subsetof a lineartopologiCorollary2. Let K be a convex,compact,and metrizable
mapfrom[0, 1] ontoK. More
cal vectorspace V. Thenthereis a continuoussur]ective
generally,
ifK is notassumedto be convex,thenthereis a continuousmapfrom[0,1]
intoV whoseimagecontainsK.
Theorem[8, p. 129],
2 is a specialcase of the Hahn-Mazurkiewicz
Corollary
whichcharacterizes
thecontinuous
imagesof theinterval
[0,1] as theconnected
compactmetricspaces.
and locallyconnected
procedure
onlyusedthefactthattheCantorsetis closed.
Remark.The extension
function
definedon anyclosedsubsetA of the
real-valued
If f is a continuous
can be usedto extenditto all ofR. If A is bounded
realline,thesameprocedure
forx > b = max{t:t E A) byf(x) = f(b); use a
fromabove,definetheextension
we couldjustuse
similarformula
whenA is boundedfrombelow.Alternatively,
Theorem.
theTietzeExtension
Does there
2. A UNIVERSALCONVEX SET. Here is a questionin geometry:
thatevery
exista three-dimensional
compactconvexset B withthe property
to one of
subsetoftheunitsquareis congruent
compactconvextwo-dimensional
itsfaces?
ifthereis an affine
isometry
ofRd that
RecallthattwosetsinRd are congruent
H in Rd is said to support
a compact
takesone setontotheother.A hyperplane
determined
convexset B if B is containedin one oftheclosedhalf-spaces
byH,
in the formH = {x E Rd: f(x) = al,
and B touchesH. If H is represented
on Rd and a is a realnumber,
thenH supportsB if
wheref is a linearfunctional
x E B} = a. In thiscase we saythatH
eithermax{f(x):x E B) = a or min{tf(x):
is attained,i.e.,
(or minimum)
supportsB in the set F wherethismaximum
F = B n H; such sets F are called the faces of B.
faceof
ofeach two-dimensional
The answerto thequestionis no. The interior
of B, henceB can haveat
B is relatively
boundary
open in thetwo-dimensional
faces.But thereare uncountably
mostcountablymanytwo-dimensional
many
two-dimensional
compactconvexsubsetsof the unitsquare.For
noncongruent
triangles.
manynoncongruent
example,thesquarecontainsuncountably
set thatis
This topologicalargument
failsif we look fora four-dimensional
sets.In thiscase the boundaryis three-dimen"universal"fortwo-dimensional
to the existenceof uncountably
sional,and thereis no topologicalobstruction
theonlyobstruction?
faces.Butis thistopological
argument
manytwo-dimensional
factthatthisis indeedthecase,
R. Grzaslewicz
geometric
[6] provedthestriking
he proved:
setexists!Moregenerally,
andthatsucha "universal"
four-dimensional
Theorem 3. For each d ? 1 thereis a compact convexset B in Rd?2 withthe
unitcube is congruent
thateach compactconvexsubsetof thed-dimensional
property
to a face ofB.
1998]
SURJECTIVITYOF THE CANTORSET
833
The case d = 1 is elementary,
anditis instructive
tovisualizeitsincethehigher
dimensional
construction
followsthesameideas,and is moredifficult
to visualize.
The one-dimensional
unitcube is justtheinterval
[0,1],and itsconvexsubsets
are intervals
oflength1 with0 < 1 < 1; byan interval
oflength0 we meana single
point.
Represent
R3 as R2 x R, andwritethepointsinR3 as pairs(t,x),wheret E 2
and x E R. Let T be theunitcircleinR2,andletf be a continuous
from
function
T onto[0,1]. Put
G= {(t,x):t ET
and 0<x <f(t)}.
The set G is compact(sincef is continuous),
andwe takeB to be itsconvexhull.
Sincetheconvexhullofa compactsetin a finite-dimensional
spaceis compact,
[4,
p. 22],B is compact.
Fix any 1 E [0,1], and choose a point toE T so that f(to) = 1. Then F
{(to,y): 0 ? y < f(to)} is a face of B, whichis an intervalof length1.
-
The proofof Theorem3 ford > 1 uses theAlexandroff-Hausdorff
Theorem.
Thereis a standardpreparatory
stepthatwe alwaysneed to takebeforewe can
a metricon the set of "data" that
applyTheorem1: we firstneed to introduce
makesthisseta compactmetricspace.
To thisend we introduce
the Hausdorff
metricdH on theset of all compact
subsetsofRd. For a boundedset A and any8 > 0, we denotethee-neighborhood
of A by A, = {x E Rd :dist(x, A) < e}. The Hausdorffdistance between two
boundedsetsA and B is then
dH(fA, B) = inf{e: B c A, and A c B}.
Intuitively,
dH(A,B) measureshowmucheach of the sets A or B needsto be
"blownup" so thatitcoverstheother.
We needtheclassicalBlaschkeSelectionTheorem:Thesetofall compact
convex
subsetsof a fixedcompactsubsetof
p. 64].
Rd
is compactundertheHausdorffmetric[4,
Theorem
ProofofGrzaslewicz's
ford > 1: ConsiderRd?2 to be theproductR2 x Rd,
and writethe pointsin Rd+2 as pairs(t,x), wheret E R2 and x c Rd. By the
thespaceK ofall compactconvexsubsetsoftheunit
BlaschkeSelectionTheorem,
cubein Rd is a compactmetricspace.It followsthatthereis a continuous
map 4
fromtheCantorset A ontoK.
Let T be theunitcirclein R2,and considerA to be a closedsubsetof T. The
graphof 4 can be visualizedas a subsetG ofRd+2:
G = {(t, x): t E/A and x
c
+(t)}.
of 4 thatG is compact.The desiredset B is the
It followsfromthecontinuity
convexhullof G, whichis compactas theconvexhullofthecompactset G.
Fix any compactconvexsubset A of the d-dimensional
cube. Since 4 is
thereis a pointtoE A suchthat0(to) = A. The setF = {(to,x): x c A}
surjective,
to A. To see thatF is a faceof B, let L c R2 be the line
is clearlycongruent
tangentto the circleT at to,and considerthe (d + 1)-dimensional
hyperplane
H = L X Rd. This hyperplanesupportsthe cylinderT x Rd in to x Rd. Since G is
inthiscylinder,
H supports
F is exactly
contained
itsconvexhullB. Moreover,
the
setofpointsin G whosefirst
coordinate
is toanditis closedandconvex.It follows
that H n B = F.
834
SURJECTIVITY OF THE CANTOR SET
[November
3. A THEOREM OF BANACH AND MAZUR. In this sectionwe presentone of
theearlyandbasicresultson thestructure
ofBanachspaces,dueto S. Banachand
S. Mazur([3] or [2,p. 185]).
Let K be a compactmetricspace,anddenotebyC(K) theBanachspaceofall
continuousreal-valuedfunctions
on K (withthe supremum
norm).A Banach
space X is saidto be linearly
isometric
to a subspaceofa Banachspace Y ifthere
is a linearisometry
fromX intoY, i.e., a linearoperatorT: X -* Y suchthat
= llxllxforeveryx E X.
IlTxlly
Theorem 4. EveryseparableBanach space is linearlyisometricto a subspace of
C[O,1].
The proofoftheBanach-Mazur
theorem
has twosteps:
to a subspaceof C(K)
isometric
Step1. EveryseparableBanachspace is linearly
vector
forsomeconvex,compact,and metrizable
subsetK of a lineartopological
space.
isometric
to a subspaceof C[O,1].
Step2. C(K) is linearly
The Alexandroff-Hausdorff
Theoremis used in the secondstep,butwe also
a combination
sketchtheproofofthefirst
ofsomestandard
step,whichis actually
factsin functional
analysis.
Proofofstep1: Let X be a separableBanachspace,and let X* be itsdual.Every
elementx E X can be considered
on X* bytheformula
to be a function
x(x*) = x*(x)
(1)
forx* e X.
Of theseveraltopologiesthatmakeX* intoa lineartopological
vectorspace,
we use theo-(X*,X) (orweak*)topology.
on X* under
It is theweakesttopology
on X*
whichall theelementsof X are continuous
as functions
whenconsidered
bytheidentification
(1) [9,p. 66].
inthe
The closedunitball K of X* is convex,and itis compactandmetrizable
weak*topology:
* Compactness
is thecelebratedtheorem
ofAlaoglu[9,p. 66].
* Metrizability
of X [9,p. 68].
is an easyconsequenceoftheseparability
J of X intoC(K) by
UsingthisK we nowdefinean isometry
(J(x))(k)
= k(x)
foreveryx E X and k E K.
of
That A(x) is a continuous
fromthedefinition
function
on K foreach x, follows
theweak*topology.
The operatorJ is clearlylinear,andwe nowcheckthatitis an
For each k E K and x E X
isometry.
x llx,
I( J(x) ) ( k) I = |k(x) I < || k lix*llx llx < 11
wherethefirstinequality
ofthenormon X*, and the
followsfromthedefinition
that
< 1 fork in theunitball K of X*. It follows
secondfromthefactthatIIkIIx*
= sup{I(J(x))(k) |: k E K) <
||J(X) 11C(K)
lixIlx
foreveryx E X. The reverseinequality
followsfromtheHahn-BanachTheorem:
1998]
SURJECTIVITY OF THE CANTOR SET
835
Foreveryx E X itensuresthatthereis a pointkx E K suchthatkx(x) = lIxIlx.It
followsthat
11
J(X)
ICK) 2 (J(x))(kx)
= kx(x) = 1x lx.
Proofof step 2: Since K is a convex,compact,and metrizablespace, the
Alexandroff-Hausdorff
Theoremand Corollary2 yieldsa continuous
surjective
map 4: [0,1] -> K. The operatorS ofcomposition
withthis4, givenby
Sf(t) = f( +(t))
foreveryt E [0, 1]
is a linearoperatorfromC(K) intoC[0,1],and it is an isometry
because
11SfllC[o1]= sup{ff(o)(t)) |: t E [0, 1]) = sup{lf(k) 1:k E K} = IlfJIc(K),
wherethesecondequalityfollowsfromthesurjectivity
of 4.
4. A CONTINUOUS FUNCTION THAT INTERPOLATES EVERY BOUNDED
SEQUENCE. The following
theoremanswersa questionthatwas posedto me by
Dr. Moshe Leshno and by Professors
Allan Pinkusand VladimirLin. It was
motivated
byDr. Leshno'sworkon neuralnets.We denotetheset ofall integers
byZ.
Theorem 5. Thereis a real-valued,bounded,and continuousfunctionf on thereal
line R withtheproperty
thatfor each doublyinfinite
sequence y = (Yn)nE z of real
numberssatisfying
I < 1 forall n, thereis a pointt E R such that
IyYn
Yn= f (t + n)
forall n E Z.
Proof: Considerthe infiniteproductK = [- 1,1]Zof all doublyinfinitesequences
ofrealnumbers
z = (z
suchthatIznl? 1 forall n. ByTychonoff's
Theorem,
K is compactwhenequippedwiththeproducttopology,
and it is metrizable
as a
metricon K can be
productofa countablenumberofmetricspaces.(An explicit
definedby d(y,z) = E2-JnIyn- zn,, and the compactness
can thenbe proved
directly
bya standarddiagonalsubsequenceargument.)
Let 4 be a surjective
continuous
mappingfromtheCantorset A ontoK. The
on K is definedin sucha waythatforeach fixedn, thenthcoordinate
topology
function
function
on A, and it is clearly
real-valued
(4)(.))n of 4) is a continuous
boundedin absolutevaluebyone.
A as a closedsubsetof [0,1/2].It followsthatA + n and A + m
We identify
are disjointforn = m, and we firstdefinethe function
f on the closedsubset
A = U{A +n: n E Z}ofRby
f(t + n) = (4Ot))n
fort EAA and n E Z.
The functionf is well definedand continuouson A, and we extendit to a
boundedcontinuous
on all of R by linearinterpolation
function
(or by Tietze's
Extension
Theorem).
The extendedfunction
(whichwe continueto denoteby f) is the required
function.Indeed, givenany y = (Yn) E K, thereis a point toE/A such that
0(to) = y, i.e., ((t0))n = yn for all n. Then the definitionof f ensures that
f(to + n) = Ynforall n.
836
SURJECTIVITY OF THE CANTOR SET
[November
5. VARIATIONS ON SECTION 4. The specificbound 1 on the sequences in the
previoustheorem
can be replacedbyanyotherfixedbound.Somecommonbound
is, however,
necessary,
and it is impossible
to findone continuous
function
that
interpolates
all boundeddoublyinfinite
sequences.It is evenimpossible
to finda
continuous
function
thatinterpolates
all constant
sequences,i.e.,a function
f with
theproperty
thatforeveryrealnumbera thereis a pointt forwhichf(t + n) = a
foreveryn. Indeed,sucha function
wouldhaveto takeeveryreal a as a valueat
some pointin the compactinterval[0,1], whichis impossiblefora continuous
function.
On theotherhand,thesameproofshowsthatif{Mn}n
E z are arbitrary
positive
thenthereis a continuous
numbers,
function
f on R suchthatify = (Yn)nE z is a
sequenceofrealnumbers
satisfying
I < Mn forall n, thenthereis pointt E R
IyYn
suchthatYn= f(t + n) forall n E Z. (Justreplacethe product[-1, l]z in the
M=
].)
proofbytheproductln=[-Mn,
all one-sidedbounded
In particular
it followsthatit is possibleto interpolate
Moreprecisely,
sequencesofrealnumbers.
Thereis a continuousreal-valuedfunctionf on the real line R, such thatfor each
boundedsequence of real numbersy = (Yn)n2,
0 thereis a point t E R such that
f(t + n) = Ynforall n 20.
Indeed,let f be thefunction
constructed
abovewithMn= n. Givenanybounded
sequence (Yn)n2
?0' choose a positiveintegerk such that yn
I < k forall n > 0, and
finda point s e R such that f(s + m) = 0 for m < k, and such that f(s + m)
Ym-k forall m ? k. Then take t = s + k.
=
In thenextvariation
we interpolate
continuous
functions
ratherthansequences,
and we consideronlyone of manyresultsof thistype.Let 9 be a familyof
continuous
real-valued
functions
on theunitinterval
[0,1]. Underwhatconditions
on 9 can one finda continuousreal-valuedfunctiong on the unit square
[0,1] x [0,1],suchthateach f E 7 can be realizedas a horizontal
sectionof g?
Moreprecisely,
we lookfora function
g suchthatforeach f E 7 thereis a point
s E [0,1] with
f(t) =g(t,s)
forall0 < t < 1.
Recall thatthe modulusof continuity
continu(Of(e) of a real-valueduniformly
ous function
f, definedon somemetricspace,is givenby
(of (8)
=
sup{|f(x) -f(y)
:
<
d(x, y) ?}
and (f( e) --* 0 as e -> 0. A family7 of uniformly
continuous
functions
is called
function
w)(e), withw)(e) - 0 as e -O 0, such
equicontinuousifthereis a positive
that Wf(e) < w(e) forall f E 7 Since a continuousfunctionon the unitsquare is
sectionsis
boundedand uniformly
it followsthatitsset ofhorizontal
continuous,
bounded(i.e., theyare boundedbya commonbound),and
necessarily
uniformly
are sufficient
It turnsoutthattheseconditions
as well:
equicontinuous.
boundedand equicontinuousset of continuous
Theorem 6. Let 9 be a uniformly
on theunitinterval.Thenthereis a continuous
real-valuedfunctions
functiong on the
sectionofg.
unitsquare,such thateach functionin Sr can be realizedas a horizontal
1998]
SURJECTIVITYOF THE CANTORSET
837
Proof: BytheAscoli-ArzelaTheorem[9,p. 369],theclosureK (in thesupremum
norm)of the set S9 is compact.By the Alexandroff-Hausdorff
Theoremand
Corollary
2, thereis a continuous
map P fromtheinterval
[0,1] intoC[0,1],whose
imagecontainsK. Considerthisinterval
to be theinterval
[0,1] on they-axis,and
definea function
g on [0,1] x [0,1] byg(t,s) = +(s)(t) fort,s E [0,1]. It follows
fromthe continuity
of + thatg is continuous,
and the horizontal
sectionsof g
containK, hencealso 9
6. A VERY SLOWLY CONVERGENT SEQUENCE OF CONTINUOUS FUNC-
a sequenceofcontinuous
TIONS. W. Rudinconstructed
real-valued
functions
on
the unitintervalthatconvergespointwiseto zero, but does so at an arbitrarily
slow
rateat thedifferent
pointsoftheinterval
[10].Moreprecisely,
Theorem 7. Thereis a uniformly
boundedsequence of strictly
positivecontinuous
functions(f,,)= 1 on [0, 1] withtheproperty
that
(i) f,(x) -> 0 foreveryx E [0, 1].
(ii) For each unboundedsequence (An) of positivenumbersthereis a point
x E [0,1] at whichlimsupn ,00 nfn(X)= Co.
In Rudin'soriginal
proof,thefn'sare firstdefinedon theclassicalCantorsetin
[0,1] byexplicitformulas
thatuse theternary
of thepointsin the
representation
set. Theyare thenextendedto all of [0,1] by linearinterpolation.
We use the
Cantorset in a different
way,namely,by applyingthe Alexandroff-Hausdorff
Theorem.
The functions
in ourproofsatisfy
thestronger
condition
thattheseriesEfn(x)
foreveryx. Simplevariations
of the proofcould giveeven stronger
converges
conditions
thatwouldmaketheconvergence
of thesequencefn(x)to zero seem
evenfaster.
Proof: Considertheset K ofnumerical
sequencesa = (an) givenby
{a: 4- ?< a< ? forall n,and a ?<1).
One checkseasilythatK is a closedconvexsubsetof thecompactmetricspace
[0,1]N, and hence K is compact.By the Alexandroff-Hausdorff
Theoremand
Corollary
2, thereis a surjective
map 4 from[0,1] ontoK. We thendefine
K=
fn(x)
=
(O(X))n,
the nthcoordinateof +(x). The functions
and theysatisfy
f,, are continuous,
<
4-n
E
1
1].
for
x
and
2
every
[0,
fn(x)
Efn(x)
Let (A,2) be any unboundedsequence of positivenumbers,and choose a
thatAnj4-J
subsequence(nj) withtheproperty
2 j foreach j. The sequence
(4-i ifn = n
4-n otherwise
satisfiesa,1? 4-n forall n (because n1 ? j), and also Ea,1 < 1 (because Z4-'
+4-j
=
2T4-
< 1). Thus (a,)
e K.
Since 4 is surjective,
thereis an x E [0,1] such thatfn(x) = (4)(x))n = a,n.
Hence Anj'fn(x) = A.n*(O)(X))tj =
n4
2 j -* oo.
Remark.As observedbyRudin,thefunctions
in Theorem7 can be chosento be
Theoremto approximate
each fn by a
polynomials.
Indeed,use Weierstrass'
polynomialpl up to 2min{fn(x):x E [0,1]) (whichis strictly
positive).Then
fn/2 < Pn < 3fn/2.
838
SURJECTIVITY OF THE CANTOR SET
[November
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
P. Alexandroff,
Uber stetigeAbbildungenkompakterRaume, Math. Ann. 96 (1927) 555-571.
S. Banach, The'oriedes Ope'rations
Line'aires,MonografjeMatematyczne,Warsaw,1932.
S. Banach and S. Mazur, Zur Theorie der linearenDimension,StudiaMath. 4 (1933) 100-112.
H. G. Eggleston,Convexity,
CambridgeUniversityPress, 1958.
R. Engelking,GeneralTopology,PWN, Warsaw, 1977.
R. Grzaslewicz,A universalconvexset in Euclidean spaces, Colloq. Math. 45 (1981) 41-44.
F. Hausdorff,Set Theory,Third Edition,Chelsea, 1978.
J. G. Hockingand G. S. Young, Topology,Dover, 1988.
W. Rudin, FunctionalAnalysis,McGraw-Hill,1973.
W. Rudin,A veryslowlyconvergentsequence of continuousfunctions,Proc. Amer. Math. Soc. 39
(1973) 647-648.
YOAV BENYAMINIreceivedhis Ph.D. in 1974 at the HebrewUniversity
in Jerusalem.
He is a
professor
ofmathematics
at theTechnion,
IsraelInstitute
ofTechnology,
andhe hasvisitedYale, Ohio
StateUniversity,
TheUniversity
ofTexasatAustin,
andWeizmann
Institute.
He is interested
invarious
aspectsoffunctional
analysis,
especially
thegeometry
ofBanachspaces.
Technion-IsraelInstitute
of Technology,
Haifa 32000, Israel
[email protected]
From the MONTHLY 100 years ago...
The Annals of Mathematics. Edited by Wm. H. Echols. Published
of Virginia.Bi-Monthly,
under the auspices of the University
price $2.00
per year in advance.
containsthe following
The October(1897) numberof the AnnlalsofMathematics
articles:The AnalyticalRepresentationon a Power of PrimeNumbersof Letters
witha Discussionof the Linera Group,byDr. L. E. Dickson; Note on Integraland
Integro-Geometrico
Series,by Prof.Edward Drake Roe; Note upon a Representation in Space of the Ellipses Drawn by an Ellipsograph,by Prof. E. M. Blake.
B. F. F.
The Cosmopolitan. An InternationalIllustratedMonthlyMagazine.
Edited by JohnBrisbenWalker.Price, $1.00 per year in advance. Single
number,10 cents.Irvington-on-the-Hudson.
The principalarticlesof the Februarynumberare: The Selectionof One's Life
Work,by E. BenjaminAndrews;The Great ElectricTrust,by FrancisLynde;and
Personnelof the SupremeCourt,by Nannie-BilleMaury.
TheAmericanMonthlyReviewof Reviews. An InternationalIllustrated
MonthlyMagazine. Edited by Dr. AlbertShaw. Price,$2.50 per year in
advance. Single number,25 cents.The Review of ReviewsCo., 13 Astor
Place, New York.
Cuba, Hawaii, and China furnishthe principaltopicsdiscussededitoriallyin the
ReviewofReviewsforFebruary.There are also a fewparagraphs
Americatn
Monthly
of pointed commenton currentdomesticpolitics-the factionaldifferencesbetween Ohio Republicans and the swellingtide of Crokerismin the Democratic
party.The editor gives his views on Tammany'sattitudetowardthe New York
rapid-transit
problemand on the recklessexpenditureof canal-improvement
funds
by the Republicanbosses of the State.
MONTHLY5 (1898) 34
1998]
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