Sec, 9| Hausdorff Measures 329
An outer measure ,u* on subsets of a metric space X with the
property that ,u*(A U B) = MA + ,u*B whenever p(A, B) > 0 is
ii I 3 l E F 3? .
measure.
Probtems
51. Prove Proposition 41. [Let F be the set of f

320 lVlQaSLLadUIBLMasuLeLCIJAMZ
since B is measurable. Thus
,u*BZ,u*B>,u*B,
and the lemma follows by letting 6 + 0. I
WMwysawuhmteommeasutes
containedinan(iaawithtlresameoutermeasure.Thefollowirrg
. .1 l ['
32. Proposition: Let

222
CHAPTER 10. GENOME RE ARRANGEMENTS
Lemma 10.31 For every two unoriented UT-paths located on the same chromosome, there exists an oriented gray edge that joins these paths into a TYT-path.
In a search for an optimal capping, we first ignore the term /(

226
CHAPTER 10. GENOME REARRANGEMENTS
Genomic_Sort (II, F)
Construct the graph G = (2(11, F)
Close all IlT-paths in simple components of (7(11, F) (lemma 10.32)
Close all but one IIF-path in components having more than one ITF-path inside them
while G con

296
BIBLIOGRAPHY
[244] M. Muzio, A.M. Chinnaiyan, EC. Kischkel, K. O'Rourke, A. Shevchenko,
J. Ni, C. Scaffidi, J.D. Bretz, M. Zhang, R. Gentz, M. Mann, RH. Krammer,
M.E. Peter, and V.M. Dixit. FLICE, a novel FADD-homologous ICE/CED3-like protease, is rec

288
BIBLIOGRAPHY
[152] S. Hannenhalli, C. Chappey, E. Koonin, and RA. Pevzner. Genome sequence
comparison and scenarios for gene rearrangements: A test case. Genomics,
30:299-311, 1995.
[153] S. Hannenhalli and RA. Pevzner. Transforming men into mice (pol

256
CHAPTER 12. PROBLEMS
12.4 Sequencing
The simplest heuristic for the Shortest Superstring Problem is the GREEDY algorithm: repeatedly merge a pair of strings with maximum overlap until only one
string remains. Tarhio and Ukkonen, 1988 [333] defined the

CHAPTER 11. COMPUTATIONAL PROTEOMICS
246
A
7
11 \
A
15
D
18
21
24
30
0
38
0
o o
43
-o
o o
A
11 13
19
22
0
0
0
25
31 33
38
Figure 11.8: Aligning spectra. The shared peaks count reveals only D(0) = 3 matching peaks
on the main diagonal, while spectral al

268
CHAPTER 12. PROBLEMS
The well-known variant of sorting by transpositions is sorting by transpositions
p(i,i + l,i + 2) where the operation is an exchange of adjacent elements. A
simple bubble-sort algorithm solves this problem for linear permutations.

324 MeasuLILauduIeLMeasuLeLChaQJQ
41. Let ,u be a measure on an algebra (i, and let E be a u* measurable
set. Show that the measure it dened in Theorem 38 has the property that
B=y*BoraUBeCB.ThusagreesonBwiththemeasuregiVenbythe
4lLetEbeanyset.Ameasurable

Sec. *6] Inner Measure L23
Proof: If (A) is any disjoint sequence of sets from Ci with E C
U A, then E = U (A, n E), and so
E=ZuMJHasZum.
i=1 il
ThusE<p*E
HA is any set 'm a with W~E)<oo, then (A~E)s
H*LA~Eland
HAH*(A~EJSHA-tA~E)-/1(EDAJ

332 Measurmandlopology [Chao13
If X is any topological space, the smallest (Jr-algebra containing the
closed sets is called the class of Borel sets. Thus, if X is locally
compact,everyBairesetisaBorelset.TheconverseistruewhenX
is a locally c

Sec. 2] Measurable Functions 259
b.- If u is a measure on (R, dene [:l on (B by [1E = uE if E 8 (R and
[E = 00 ifE 3 OT. Then [1 is a measure on (B.
c. DenegenBbyEE=uEifE86Fand
EE=sup cfw_uA2ACE,A8(R
if E e (R. Then g is also a measure on (B

298 MeasuLLandMQLMeasuLe [Chan 12
extension to a measure on the algebra (1 generated by C if the follow-
ing conditions are satised:
i. IfasetCinCistheunionofanitedisjoinLcollecLionJrCiLof
sets in C, then uC = Z pCi.
ii. Ifa set C in (3 is the union ofa

262 Measure and Integration [Chap. 11
Let or e D, and choose a sequence 0)") from C with or < y" and
or = lim y". Then
Budgeted;
Th$P=Uwa~B;n)EacounmbmmndnwwgaMwa
null set. Let A = n 3;". Thenfginfyn=or on A, andA ~BaCP.
Thus f g or almost

Sec4] Product Measures :03
the range of F, we have mE = v[F 1(E)], with m Lebesgue measure. [Hintz
This is true for intervals, and the uniqueness part ofTheorem 8 can be used
to derive its truth in general]
b. Generalize to the case of cscontinuous cu

Sec *6J Inner Measure 321
contained in E U F. Thus
*(E U F) 2 *(G U H) = MG U H)
proving the rst inequaty.
If p*F = 00, the second inequality is trivial. If *F < 00, let A
beinGwithpA~(EUFj)<oo.SinceA~EC|A~(EUF)jUF,
wehave
,u*(A ~ E) S *(A ~

Sec. *3] The LebesgueStieltjes Integral 299
b. For a given set E we have u+E = ,u*E if and only if there is a
p*-measurable set A D E with MA = *E.
c. Showthatp+Ep*EforeVeryEifandonlyifp*isregular.
d. Show that an outer measure [1* is regular if and onl

306 MeasuLLandJlumLMeasure [Chantal
Then each 9,- is a nonnegative measurable function, and
g=2m-
Thus g is measurable, and by the corollary of the Monotone Con-
vergence Theorem (11.L49, we haye
JawZJ%@
=Z X VlEi)
;1 x v(E).
ons

Sec. 1] Measure Spaces 25]
imply A e (B. Thus Lebesgue measure is complete, while Lebesgue
measure restricted to the o-algebra of Borel sets is not complete. The
fooyrrgjar0positioirwhoseproofislefttothereader(Problemj
shows that each measure Space can

294
BIBLIOGRAPHY
[219] V.I. Levenshtein. Binary codes capable of correcting deletions, insertions
and reversals. Soviet Physics Doklady, 6:707-710, 1966.
[220] B. Lewin. Genes VII. Oxford University Press, 1999.
[221] M. Li, B. Ma, and L. Wang. Finding si

298
BIBLIOGRAPHY
[268] P.A. Pevzner. DNA statistics, overlapping word paradox and Conway equation. In H.A. Lim, J.W. Fickett, C.R. Cantor, and RJ. Robbins, editors,
Proceedings of the Second International Conference on Bioinformatics, Supercomputing, and

266
CHAPTER 12. PROBLEMS
Problem 12.92 Estimate the expected number of questions in the "Twenty Questions Game with a Liar" if the probability of a false answer is p^\Problem 12.93 Design experimental and computational protocols to find all alternatively

230
CHAPTER 11. COMPUTATIONAL PROTEOMICS
fications (such as phosphorylation and glycosylation) and genetic variations (Gooley and Packer, 1997 [134], Krishna and Wold, 1998 [207]). For example, at least
1,000 kinases exist in the human genome, indicating

CHAPTER 11. COMPUTATIONAL PROTEOMICS
236
1
h
I
Sequence
Figure 11.5: Noise in a spectrum generates many "false" edges and vertices in the spectrum
graph and disguises edges corresponding to the real peptide sequence. Sequence reconstruction
corresponds to

10.14. CAPS AND TAILS
221
between cycles and paths in (3(11, F). We say that a vertex TTJ is inside a component U of 7T if j E p m jn, C^maa;]- An intrachromosomal component for genomes
II and F is called a real component if it has neither a II-cap nor a

226
CHAPTER 10. GENOME REARRANGEMENTS
Genomic_Sort (II, F)
Construct the graph G = (2(11, F)
Close all IlT-paths in simple components of (7(11, F) (lemma 10.32)
Close all but one IIF-path in components having more than one ITF-path inside them
while G con

220
CHAPTER 10. GENOME
REARRANGEMENTS
for 0 < j , k < 2N - 1. Genomes Ft and f are co-tailed, since T(ft) = T(f) =
U ^ o " 1 ! " 1 ) 2 ^ ^ - There exist (27V)! different cappings of F, each capping defined
by the distribution of 2N caps of Ft in f. Denote

Chapter 12
Problems
12.1 Introduction
Molecular biology has motivated many interesting combinatorial problems. A few
years ago Pevzner and Waterman, 1995 [275] compiled a collection of 57 open
problems in computational molecular biology. Just five years l

242
CHAPTER 11. COMPUTATIONAL PROTEOMICS
(52 0 5i)(0) + (52 Si)(5) is an upper bound on the number of shared peaks
between Si and "mutated" S2The other set of correlations between spectra of mutated peptides is captured
by the reverse spectral convolution

RUNNING HEAD: Adaptive Response
1
Adaptive Response
[Name]
[University]
[Date]
RUNNING HEAD: Adaptive Response
2
Scenario 1
Tonsilitis
Pathophysiology of Tonsilitis
The two masses found at the back of the throat are called tonsils. The main functions of
t

RUNNING HEAD: Leader Traits
1
Leadership Traits
[Name]
[University]
[Date]
RUNNING HEAD: Leader Traits
2
Introduction
A good leader is important in an organization because it leads multiple groups and
departments to mold its role and work. To make an orga

RUNNING HEAD: Manager and Leader
1
Manager and Leader
[Name]
[University]
[Date]
RUNNING HEAD: Manager and Leader
2
A manager may have different roles in the organization and most of it would be
supervision. There are different levels of management in lar

RUNNING HEAD: Data Mining
1
Data Mining
[Name]
[University]
[Date]
RUNNING HEAD: Data Mining
2
The most common diseases in indifferent groups of people is cardiovascular disease.
According to the data gathered, heart diseases is the most common cause of d

Running Head: Hypertension
1
Name
University
Date
Running Head: Hypertension
2
Hypertension or elevated blood pressure may develop over the years. The disease may affect almost
everyone, most especially if the person has poor lifestyle habit. High blood p