This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2008.10.23)  18 NOV 2008 22:48
entering extended mode
 %&-line parsing enabled.
**NumericalSgpsMan
(./NumericalSgpsMan.tex
LaTeX2e <2005/12/01>
Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov
ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i
nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp
eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia
n, swedish, ukenglish, loaded.
(/usr/share/texmf-texlive/tex/latex/base/report.cls
Document Class: report 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size11.clo
File: size11.clo 2005/09/16 v1.4f Standard LaTeX file (size option)
)
\c@part=\count79
\c@chapter=\count80
\c@section=\count81
\c@subsection=\count82
\c@subsubsection=\count83
\c@paragraph=\count84
\c@subparagraph=\count85
\c@figure=\count86
\c@table=\count87
\abovecaptionskip=\skip41
\belowcaptionskip=\skip42
\bibindent=\dimen102
)
(/usr/share/texmf-texlive/tex/latex/ltxmisc/a4wide.sty
Package: a4wide 1994/08/30

(/usr/share/texmf-texlive/tex/latex/ntgclass/a4.sty
Package: a4 2004/04/15 v1.2g A4 based page layout
))
(/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
Package: amssymb 2002/01/22 v2.2d

(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty
Package: amsfonts 2001/10/25 v2.2f
\@emptytoks=\toks14
\symAMSa=\mathgroup4
\symAMSb=\mathgroup5
LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
(Font)                  U/euf/m/n --> U/euf/b/n on input line 132.
))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
Package: inputenc 2006/05/05 v1.1b Input encoding file
\inpenc@prehook=\toks15
\inpenc@posthook=\toks16

(/usr/share/texmf-texlive/tex/latex/base/latin1.def
File: latin1.def 2006/05/05 v1.1b Input encoding file
))
(/usr/share/texmf-texlive/tex/latex/base/makeidx.sty
Package: makeidx 2000/03/29 v1.0m Standard LaTeX package
)
(/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty
Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)

(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty
Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
\KV@toks@=\toks17
)
(/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty
Package: graphics 2006/02/20 v1.0o Standard LaTeX Graphics (DPC,SPQR)

(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty
Package: trig 1999/03/16 v1.09 sin cos tan (DPC)
)
(/etc/texmf/tex/latex/config/graphics.cfg
File: graphics.cfg 2007/01/18 v1.5 graphics configuration of teTeX/TeXLive
)
Package graphics Info: Driver file: pdftex.def on input line 90.

(/usr/share/texmf-texlive/tex/latex/pdftex-def/pdftex.def
File: pdftex.def 2007/01/08 v0.04d Graphics/color for pdfTeX
\Gread@gobject=\count88
))
\Gin@req@height=\dimen103
\Gin@req@width=\dimen104
)
\@indexfile=\write3
\openout3 = `NumericalSgpsMan.idx'.


Writing index file NumericalSgpsMan.idx
(/usr/share/texmf-texlive/tex/latex/graphics/color.sty
Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC)

(/etc/texmf/tex/latex/config/color.cfg
File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive
)
Package color Info: Driver file: pdftex.def on input line 130.
)
(/usr/share/texmf-texlive/tex/latex/fancyvrb/fancyvrb.sty
Package: fancyvrb 1998/07/17

Style option: `fancyvrb' v2.6, with DG/SPQR fixes <1998/07/17> (tvz)
\FV@CodeLineNo=\count89
\FV@InFile=\read1
\FV@TabBox=\box26
\c@FancyVerbLine=\count90
\FV@StepNumber=\count91
\FV@OutFile=\write4

No file fancyvrb.cfg.
) (/usr/share/texmf-texlive/tex/latex/pslatex/pslatex.sty
Package: pslatex 1996/07/24 v1.2 pslatex emulation (DPC)
LaTeX Font Info:    Redeclaring symbol font `operators' on input line 65.
LaTeX Font Info:    Overwriting symbol font `operators' in version `normal'
(Font)                  OT1/cmr/m/n --> OT1/ptmcm/m/n on input line 65.
LaTeX Font Info:    Overwriting symbol font `operators' in version `bold'
(Font)                  OT1/cmr/bx/n --> OT1/ptmcm/m/n on input line 65.
LaTeX Font Info:    Redeclaring symbol font `letters' on input line 66.
LaTeX Font Info:    Overwriting symbol font `letters' in version `normal'
(Font)                  OML/cmm/m/it --> OML/ptmcm/m/it on input line 66.
LaTeX Font Info:    Overwriting symbol font `letters' in version `bold'
(Font)                  OML/cmm/b/it --> OML/ptmcm/m/it on input line 66.
LaTeX Font Info:    Redeclaring symbol font `symbols' on input line 67.
LaTeX Font Info:    Overwriting symbol font `symbols' in version `normal'
(Font)                  OMS/cmsy/m/n --> OMS/pzccm/m/n on input line 67.
LaTeX Font Info:    Overwriting symbol font `symbols' in version `bold'
(Font)                  OMS/cmsy/b/n --> OMS/pzccm/m/n on input line 67.
LaTeX Font Info:    Redeclaring symbol font `largesymbols' on input line 68.
LaTeX Font Info:    Overwriting symbol font `largesymbols' in version `normal'
(Font)                  OMX/cmex/m/n --> OMX/psycm/m/n on input line 68.
LaTeX Font Info:    Overwriting symbol font `largesymbols' in version `bold'
(Font)                  OMX/cmex/m/n --> OMX/psycm/m/n on input line 68.
\symbold=\mathgroup6
\symitalic=\mathgroup7
LaTeX Font Info:    Redeclaring math alphabet \mathbf on input line 74.
LaTeX Font Info:    Overwriting math alphabet `\mathbf' in version `normal'
(Font)                  OT1/cmr/bx/n --> OT1/ptm/bx/n on input line 74.
LaTeX Font Info:    Overwriting math alphabet `\mathbf' in version `bold'
(Font)                  OT1/cmr/bx/n --> OT1/ptm/bx/n on input line 74.
LaTeX Font Info:    Redeclaring math alphabet \mathit on input line 75.
LaTeX Font Info:    Overwriting math alphabet `\mathit' in version `normal'
(Font)                  OT1/cmr/m/it --> OT1/ptm/m/it on input line 75.
LaTeX Font Info:    Overwriting math alphabet `\mathit' in version `bold'
(Font)                  OT1/cmr/bx/it --> OT1/ptm/m/it on input line 75.
)
(/usr/share/texmf-texlive/tex/latex/hyperref/hyperref.sty
Package: hyperref 2007/02/07 v6.75r Hypertext links for LaTeX
\@linkdim=\dimen105
\Hy@linkcounter=\count92
\Hy@pagecounter=\count93

(/usr/share/texmf-texlive/tex/latex/hyperref/pd1enc.def
File: pd1enc.def 2007/02/07 v6.75r Hyperref: PDFDocEncoding definition (HO)
)
(/etc/texmf/tex/latex/config/hyperref.cfg
File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
)
(/usr/share/texmf-texlive/tex/latex/oberdiek/kvoptions.sty
Package: kvoptions 2006/08/22 v2.4 Connects package keyval with LaTeX options (
HO)
)
Package hyperref Info: Option `bookmarks' set `false' on input line 2238.
Package hyperref Info: Option `colorlinks' set `true' on input line 2238.
Package hyperref Info: Option `breaklinks' set `true' on input line 2238.
Package hyperref Info: Hyper figures OFF on input line 2288.
Package hyperref Info: Link nesting OFF on input line 2293.
Package hyperref Info: Hyper index ON on input line 2296.
Package hyperref Info: Plain pages OFF on input line 2303.
Package hyperref Info: Backreferencing ON on input line 2306.

Implicit mode ON; LaTeX internals redefined
Package hyperref Info: Bookmarks OFF on input line 2450.
(/usr/share/texmf-texlive/tex/latex/hyperref/backref.sty
Package: backref 2006/10/06 v1.27 Bibliographical back referencing
)
(/usr/share/texmf-texlive/tex/latex/ltxmisc/url.sty
\Urlmuskip=\muskip10
Package: url 2005/06/27  ver 3.2  Verb mode for urls, etc.
)
LaTeX Info: Redefining \url on input line 2599.
\Fld@menulength=\count94
\Field@Width=\dimen106
\Fld@charsize=\dimen107
\Choice@toks=\toks18
\Field@toks=\toks19
Package hyperref Info: Hyper figures OFF on input line 3102.
Package hyperref Info: Link nesting OFF on input line 3107.
Package hyperref Info: Hyper index ON on input line 3110.
Package hyperref Info: backreferencing ON on input line 3115.
Package hyperref Info: Link coloring ON on input line 3120.
\Hy@abspage=\count95
\c@Item=\count96
\c@Hfootnote=\count97
)
*hyperref using driver hpdftex*
(/usr/share/texmf-texlive/tex/latex/hyperref/hpdftex.def
File: hpdftex.def 2007/02/07 v6.75r Hyperref driver for pdfTeX
\Fld@listcount=\count98
)
\pagenrlog=\write5
\openout5 = `NumericalSgpsMan.pnr'.


(./NumericalSgpsMan.aux)
\openout1 = `NumericalSgpsMan.aux'.

LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 44.
LaTeX Font Info:    ... okay on input line 44.
LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 44.
LaTeX Font Info:    ... okay on input line 44.
LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 44.
LaTeX Font Info:    ... okay on input line 44.
LaTeX Font Info:    Checking defaults for OMS/pzccm/m/n on input line 44.
LaTeX Font Info:    Try loading font information for OMS+pzccm on input line 44
.
 (/usr/share/texmf-texlive/tex/latex/psnfss/omspzccm.fd
File: omspzccm.fd 2000/01/03 Fontinst v1.801 font definitions for OMS/pzccm.
)
LaTeX Font Info:    ... okay on input line 44.
LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 44.
LaTeX Font Info:    ... okay on input line 44.
LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 44.
LaTeX Font Info:    ... okay on input line 44.
LaTeX Font Info:    Checking defaults for PD1/pdf/m/n on input line 44.
LaTeX Font Info:    ... okay on input line 44.
 (/usr/share/texmf/tex/context/base/supp-pdf.tex
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count99
\scratchdimen=\dimen108
\scratchbox=\box27
\nofMPsegments=\count100
\nofMParguments=\count101
\everyMPshowfont=\toks20
\MPscratchCnt=\count102
\MPscratchDim=\dimen109
\MPnumerator=\count103
\everyMPtoPDFconversion=\toks21
)
Package hyperref Info: Link coloring ON on input line 44.
 (/usr/share/texmf-texlive/tex/latex/hyperref/nameref.sty
Package: nameref 2006/12/27 v2.28 Cross-referencing by name of section

(/usr/share/texmf-texlive/tex/latex/oberdiek/refcount.sty
Package: refcount 2006/02/20 v3.0 Data extraction from references (HO)
)
\c@section@level=\count104
)
LaTeX Info: Redefining \ref on input line 44.
LaTeX Info: Redefining \pageref on input line 44.
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <24.88> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 48.
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <12> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 53.
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <10> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 61.

Underfull \hbox (badness 10000) in paragraph at lines 59--68

 []

[1

{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <14.4> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 72.
LaTeX Font Info:    Font shape `OMS/ptm/m/n' in size <10> not available
(Font)              Font shape `OMS/pzccm/m/n' tried instead on input line 74.
LaTeX Font Info:    Font shape `OT1/phv/m/n' will be
(Font)              scaled to size 8.99994pt on input line 77.

Underfull \hbox (badness 10000) in paragraph at lines 77--79

 []


Underfull \hbox (badness 10000) in paragraph at lines 91--95

 []


Underfull \hbox (badness 10000) in paragraph at lines 97--100

 []

[2] (./NumericalSgpsMan.toc
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <10.95> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 1.
LaTeX Font Info:    Try loading font information for OT1+ptmcm on input line 3.


(/usr/share/texmf-texlive/tex/latex/psnfss/ot1ptmcm.fd
File: ot1ptmcm.fd 2000/01/03 Fontinst v1.801 font definitions for OT1/ptmcm.
)
LaTeX Font Info:    Try loading font information for OML+ptmcm on input line 3.


(/usr/share/texmf-texlive/tex/latex/psnfss/omlptmcm.fd
File: omlptmcm.fd 2000/01/03 Fontinst v1.801 font definitions for OML/ptmcm.
)
LaTeX Font Info:    Try loading font information for OMX+psycm on input line 3.


(/usr/share/texmf-texlive/tex/latex/psnfss/omxpsycm.fd
File: omxpsycm.fd 2000/01/03 Fontinst v1.801 font definitions for OMX/psycm.
)
LaTeX Font Info:    Try loading font information for U+msa on input line 3.

(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2002/01/19 v2.2g AMS font definitions
)
LaTeX Font Info:    Try loading font information for U+msb on input line 3.

(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2002/01/19 v2.2g AMS font definitions
)
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <8> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 3.
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <6> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 3.
 [3

] [4])
\tf@toc=\write6
\openout6 = `NumericalSgpsMan.toc'.

 [5]
Chapter 1.
LaTeX Font Info:    Font shape `OT1/ptm/bx/n' in size <20.74> not available
(Font)              Font shape `OT1/ptm/b/n' tried instead on input line 108.
[6

] [7] [8]
Chapter 2.
LaTeX Font Info:    Font shape `OT1/phv/m/n' will be
(Font)              scaled to size 9.85492pt on input line 217.

Underfull \hbox (badness 10000) in paragraph at lines 245--248

 []

LaTeX Font Info:    Try loading font information for OML+ptm on input line 253.

(/usr/share/texmf-texlive/tex/latex/psnfss/omlptm.fd
File: omlptm.fd 
)
LaTeX Font Info:    Font shape `OML/ptm/m/n' in size <10.95> not available
(Font)              Font shape `OML/cmm/m/it' tried instead on input line 253.
 [9

]
Underfull \hbox (badness 10000) in paragraph at lines 293--296

 []


Underfull \hbox (badness 10000) in paragraph at lines 310--313

 []


Underfull \hbox (badness 10000) in paragraph at lines 328--358

 []

[10]
Underfull \hbox (badness 10000) in paragraph at lines 396--432

 []

[11]
Underfull \hbox (badness 10000) in paragraph at lines 458--461

 []


Underfull \hbox (badness 10000) in paragraph at lines 482--485

 []

[12]
Underfull \hbox (badness 10000) in paragraph at lines 507--510

 []


Underfull \hbox (badness 1688) in paragraph at lines 512--514
[]\OT1/ptm/m/n/10.95 Tests whether a list [] of in-te-gers may rep-re-sent the 
Ap[]ery list of a nu-mer-i-cal semi-
 []


Underfull \hbox (badness 1931) in paragraph at lines 512--514
\OT1/ptm/m/n/10.95 group. It re-turns \OT1/pcr/m/n/10.95 true \OT1/ptm/m/n/10.9
5 when the pe-ri-odic func-tion rep-re-sented by [] is sub-ad-di-tive (see
 []


Underfull \hbox (badness 10000) in paragraph at lines 525--528

 []


Underfull \hbox (badness 10000) in paragraph at lines 548--551

 []

[13] [14]
Chapter 3.

Underfull \hbox (badness 10000) in paragraph at lines 587--590

 []


Underfull \hbox (badness 10000) in paragraph at lines 606--615

 []


Underfull \hbox (badness 4479) in paragraph at lines 617--618
[][] \OT1/ptm/m/n/10.95 is a nu-mer-i-cal semi-group. \OT1/pcr/m/n/10.95 Genera
torsOfNumericalSemigroup \OT1/ptm/m/n/10.95 re-turns a set of
 []


Underfull \hbox (badness 2707) in paragraph at lines 617--618
\OT1/ptm/m/n/10.95 gen-er-a-tors of \OT1/pcr/m/n/10.95 S\OT1/ptm/m/n/10.95 , wh
ich may not be min-i-mal. \OT1/pcr/m/n/10.95 GeneratorsOfNumericalSemigroupNC \
OT1/ptm/m/n/10.95 re-
 []


Underfull \hbox (badness 5431) in paragraph at lines 617--618
\OT1/ptm/m/n/10.95 turns the set of gen-er-a-tors recorded in \OT1/pcr/m/n/10.9
5 S!.generators\OT1/ptm/m/n/10.95 , which may not be min-i-mal.
 []

[15

]
Underfull \hbox (badness 10000) in paragraph at lines 637--640

 []


Underfull \hbox (badness 10000) in paragraph at lines 655--658

 []


Underfull \hbox (badness 10000) in paragraph at lines 674--677

 []


Underfull \hbox (badness 10000) in paragraph at lines 693--696

 []


Underfull \hbox (badness 2293) in paragraph at lines 698--699
\OT1/ptm/m/n/10.95 should be in-stalled and also \OT1/pcr/m/n/10.95 Evince \OT1
/ptm/m/n/10.95 ([][]\OT1/pcr/m/n/10.95 http://www.gnome.org/projects/evince/[][
]\OT1/ptm/m/n/10.95 ) or \OT1/pcr/m/n/10.95 ggv
 []


Underfull \hbox (badness 10000) in paragraph at lines 705--708

 []

[16]
Underfull \hbox (badness 10000) in paragraph at lines 736--739

 []


Underfull \hbox (badness 10000) in paragraph at lines 754--757

 []


Underfull \hbox (badness 10000) in paragraph at lines 766--769

 []

[17]
Underfull \hbox (badness 10000) in paragraph at lines 793--796

 []


Underfull \hbox (badness 10000) in paragraph at lines 810--813

 []


Underfull \hbox (badness 10000) in paragraph at lines 832--835

 []

[18]
Chapter 4.

Underfull \hbox (badness 10000) in paragraph at lines 864--867

 []


Underfull \hbox (badness 10000) in paragraph at lines 887--890

 []

[19

pdfTeX warning (ext4): destination with the same identifier (name{L.X7969F7F27A
AF0BF1}) has been already used, duplicate ignored
<to be read again> 
                   \endgroup \set@typeset@protect 
l.905     [ [ 3, 1, 0 ], [ 0, 0, 2 ] ] ]
                                        ]
Underfull \hbox (badness 10000) in paragraph at lines 916--919

 []

[20]
Chapter 5.

Underfull \hbox (badness 10000) in paragraph at lines 957--960

 []


Underfull \hbox (badness 10000) in paragraph at lines 982--985

 []

[21

]
Underfull \hbox (badness 10000) in paragraph at lines 1012--1015

 []


Underfull \hbox (badness 10000) in paragraph at lines 1039--1042

 []

[22]
Underfull \hbox (badness 10000) in paragraph at lines 1079--1082

 []


Underfull \hbox (badness 10000) in paragraph at lines 1100--1103

 []


Underfull \hbox (badness 10000) in paragraph at lines 1125--1128

 []

[23] [24]
Chapter 6.

Underfull \hbox (badness 10000) in paragraph at lines 1190--1193

 []


Underfull \hbox (badness 10000) in paragraph at lines 1211--1214

 []

[25

pdfTeX warning (ext4): destination with the same identifier (name{L.X83C597EC7F
AA1C0F}) has been already used, duplicate ignored
<to be read again> 
                   \endgroup \set@typeset@protect 
l.1216  \mbox{\texttt{\slshape s}}
                                   is a numerical semigroup. The output is t...
]
Underfull \hbox (badness 10000) in paragraph at lines 1232--1235

 []


Underfull \hbox (badness 10000) in paragraph at lines 1253--1256

 []


Underfull \hbox (badness 10000) in paragraph at lines 1274--1277

 []


Underfull \hbox (badness 10000) in paragraph at lines 1294--1297

 []

[26] [27]
Chapter 7.

Underfull \hbox (badness 10000) in paragraph at lines 1330--1333

 []


Underfull \hbox (badness 10000) in paragraph at lines 1359--1362

 []

[28

pdfTeX warning (ext4): destination with the same identifier (name{L.X83C2F0CF82
5B3869}) has been already used, duplicate ignored
<to be read again> 
                   \endgroup \set@typeset@protect 
l.1370   gap> IsIdealOfNumericalSemigroup(2);
                                             ]
Underfull \hbox (badness 10000) in paragraph at lines 1382--1385

 []


Underfull \hbox (badness 10000) in paragraph at lines 1405--1411

 []


Underfull \hbox (badness 1194) in paragraph at lines 1415--1417
[]\OT1/ptm/m/n/10.95 The out-put of \OT1/pcr/m/n/10.95 GeneratorsOfIdealOfNumer
icalSemigroup \OT1/ptm/m/n/10.95 is a sys-tem of gen-er-a-tors of
 []


Underfull \hbox (badness 3138) in paragraph at lines 1415--1417
\OT1/ptm/m/n/10.95 the ideal. If the min-i-mal sys-tem of gen-er-a-tors is know
n, then it is used as out-put.
 []


Underfull \hbox (badness 10000) in paragraph at lines 1438--1441

 []

[29]
Underfull \hbox (badness 10000) in paragraph at lines 1461--1464

 []


Underfull \hbox (badness 10000) in paragraph at lines 1489--1492

 []


Underfull \hbox (badness 10000) in paragraph at lines 1520--1523

 []

[30]
Underfull \hbox (badness 10000) in paragraph at lines 1553--1556

 []


Underfull \hbox (badness 10000) in paragraph at lines 1578--1581

 []

[31]
Underfull \hbox (badness 10000) in paragraph at lines 1611--1614

 []


Underfull \hbox (badness 10000) in paragraph at lines 1640--1643

 []

[32]
Underfull \hbox (badness 10000) in paragraph at lines 1671--1674

 []


Underfull \hbox (badness 10000) in paragraph at lines 1693--1696

 []


Underfull \hbox (badness 10000) in paragraph at lines 1716--1719

 []


Underfull \hbox (badness 10000) in paragraph at lines 1738--1741

 []

[33]
Underfull \hbox (badness 10000) in paragraph at lines 1757--1760

 []


Underfull \hbox (badness 10000) in paragraph at lines 1789--1792

 []

[34]
Underfull \hbox (badness 10000) in paragraph at lines 1817--1820

 []


Underfull \hbox (badness 10000) in paragraph at lines 1857--1860

 []

[35]
Underfull \hbox (badness 10000) in paragraph at lines 1886--1889

 []


Underfull \hbox (badness 10000) in paragraph at lines 1909--1912

 []

[36]
Chapter 8.

Underfull \hbox (badness 10000) in paragraph at lines 1954--1957

 []


Underfull \hbox (badness 10000) in paragraph at lines 1977--1980

 []

[37

pdfTeX warning (ext4): destination with the same identifier (name{L.X7D2E70FC82
D979D3}) has been already used, duplicate ignored
<to be read again> 
                   \endgroup \set@typeset@protect 
l.1982  \mbox{\texttt{\slshape S}}
                                   is a numerical semigroup.]
Underfull \hbox (badness 10000) in paragraph at lines 2000--2003

 []


Underfull \hbox (badness 10000) in paragraph at lines 2035--2038

 []

[38]
Underfull \hbox (badness 10000) in paragraph at lines 2060--2063

 []


Underfull \hbox (badness 10000) in paragraph at lines 2083--2086

 []

[39]
Chapter 9.

Underfull \hbox (badness 10000) in paragraph at lines 2126--2129

 []

[40

]
Underfull \hbox (badness 10000) in paragraph at lines 2146--2149

 []


Underfull \hbox (badness 10000) in paragraph at lines 2166--2169

 []


Underfull \hbox (badness 10000) in paragraph at lines 2186--2189

 []


Underfull \hbox (badness 10000) in paragraph at lines 2205--2208

 []


Underfull \hbox (badness 10000) in paragraph at lines 2224--2227

 []

[41]
Underfull \hbox (badness 10000) in paragraph at lines 2244--2247

 []


Underfull \hbox (badness 10000) in paragraph at lines 2263--2266

 []


Underfull \hbox (badness 10000) in paragraph at lines 2282--2285

 []

[42]
Appendix A.

Underfull \hbox (badness 10000) in paragraph at lines 2321--2324

 []


Underfull \hbox (badness 10000) in paragraph at lines 2343--2346

 []


Underfull \hbox (badness 10000) in paragraph at lines 2363--2366

 []

[43

]
Underfull \hbox (badness 10000) in paragraph at lines 2386--2389

 []

[44]
Appendix B.

Underfull \hbox (badness 10000) in paragraph at lines 2417--2420

 []


Underfull \hbox (badness 10000) in paragraph at lines 2434--2437

 []


Underfull \hbox (badness 10000) in paragraph at lines 2451--2454

 []


Underfull \hbox (badness 10000) in paragraph at lines 2468--2471

 []

[45

pdfTeX warning (ext4): destination with the same identifier (name{L.X86746B487B
54A2D6}) has been already used, duplicate ignored
<to be read again> 
                   \endgroup \set@typeset@protect 
l.2473  R
         eturns a ``random" proportionally modular numerical semigroup (see ...
]
Underfull \hbox (badness 10000) in paragraph at lines 2485--2488

 []

[46]
Appendix C.

Underfull \hbox (badness 10000) in paragraph at lines 2516--2519

 []


Underfull \hbox (badness 10000) in paragraph at lines 2529--2530
[]\OT1/ptm/m/n/10.95 The in-ter-face also al-lows to ap-ply some func-tions to 
the semi-
 []


Underfull \hbox (badness 10000) in paragraph at lines 2529--2530
\OT1/ptm/m/n/10.95 group, namely \OT1/pcr/m/n/10.95 AperyListOfNumericalSemigro
upWRTElement \OT1/ptm/m/n/10.95 ([][]3.1.5[][]),
 []


Underfull \hbox (badness 4739) in paragraph at lines 2529--2530
\OT1/pcr/m/n/10.95 GapsOfNumericalSemigroup \OT1/ptm/m/n/10.95 ([][]3.3.1[][]),
 \OT1/pcr/m/n/10.95 FrobeniusNumberOfNumericalSemigroup \OT1/ptm/m/n/10.95 ([][
]3.2.1[][]) and
 []

<xnumsgp.png, id=635, 399.4925pt x 300.12125pt>
File: xnumsgp.png Graphic file (type png)
 <use xnumsgp.png>
(./NumericalSgpsMan.bbl (./NumericalSgpsMan.brf)
\tf@brf=\write7
\openout7 = `NumericalSgpsMan.brf'.

 [47

] [48 <./xnumsgp.png>]
[49

]) (./NumericalSgpsMan.ind [50]
Underfull \hbox (badness 10000) in paragraph at lines 5--7
[]\OT1/pcr/m/n/10.95 AnIrreducible\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Numeri
cal\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Semigroup\OT1/ptm/m/n/10.95 -\OT1/pcr
/m/n/10.95 With\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 7--8
[]\OT1/pcr/m/n/10.95 AperyListOfNumericalSemigroupAsGraph\OT1/ptm/m/n/10.95 ,
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 8--10
[]\OT1/pcr/m/n/10.95 AperyListOfNumericalSemigroupWRTElement\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 30--32
[]\OT1/pcr/m/n/10.95 Delta\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Set\OT1/ptm/m/
n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Factorizat
ions\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Element\OT1/ptm/m/n/10.95 -\OT1/pcr/
m/n/10.95 W\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 R\OT1/ptm/m/n/10.95 -\OT1/pcr
/m/n/10.95 T\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 32--34
[]\OT1/pcr/m/n/10.95 DifferenceOfIdealsOfNumericalSemigroup\OT1/ptm/m/n/10.95 ,

 []


Underfull \hbox (badness 10000) in paragraph at lines 38--40
[]\OT1/pcr/m/n/10.95 Elasticity\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/pt
m/m/n/10.95 -\OT1/pcr/m/n/10.95 Factorizations\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/
10.95 Element\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 W\OT1/ptm/m/n/10.95 -\OT1/p
cr/m/n/10.95 R\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 T\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 44--46
[]\OT1/pcr/m/n/10.95 Factorizations\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Eleme
nt\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 W\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.9
5 R\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 T\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Numerical\OT1/ptm/m/n/10.95 -
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 47--49
[]\OT1/pcr/m/n/10.95 FortenTruncatedNCForNumericalSemigroups\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 56--58
[]\OT1/pcr/m/n/10.95 GeneratorsOfIdealOfNumericalSemigroup\OT1/ptm/m/n/10.95 ,
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 58--60
[]\OT1/pcr/m/n/10.95 GeneratorsOfIdealOfNumericalSemigroupNC\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 62--64
[]\OT1/pcr/m/n/10.95 Graph\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Associated\OT1
/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 To\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Ele
ment\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 In\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/1
0.95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 67--69
[]\OT1/pcr/m/n/10.95 Hilbert\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Function\OT1
/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Ide
al\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 73--75
[]\OT1/pcr/m/n/10.95 IntersectionIdealsOfNumericalSemigroup\OT1/ptm/m/n/10.95 ,

 []


Underfull \hbox (badness 10000) in paragraph at lines 76--78
[]\OT1/pcr/m/n/10.95 Irreducible\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Numerica
l\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Semigroups\OT1/ptm/m/n/10.95 -\OT1/pcr/
m/n/10.95 With\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 81--83
[]\OT1/pcr/m/n/10.95 IsGraded\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Associated\
OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Ring\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.9
5 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 90--92
[]\OT1/pcr/m/n/10.95 IsNumericalSemigroupByFundamentalGaps\OT1/ptm/m/n/10.95 ,
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 95--97
[]\OT1/pcr/m/n/10.95 IsNumericalSemigroupByMinimalGenerators\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 99--101
[]\OT1/pcr/m/n/10.95 IsNumerical\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Semigrou
p\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 By\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.9
5 Sub\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Additive\OT1/ptm/m/n/10.95 -
 []

[51

]
Underfull \hbox (badness 10000) in paragraph at lines 101--103
[]\OT1/pcr/m/n/10.95 IsProportionally\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Mod
ular\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 109--111
[]\OT1/pcr/m/n/10.95 Lengths\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m
/n/10.95 -\OT1/pcr/m/n/10.95 Factorizations\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Element\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 W\OT1/ptm/m/n/10.95 -\OT1/pcr/
m/n/10.95 R\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 T\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 115--117
[]\OT1/pcr/m/n/10.95 Maximum\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Degree\OT1/p
tm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Eleme
nt\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 W\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.9
5 R\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 T\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 119--121
[]\OT1/pcr/m/n/10.95 Minimal\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Arf\OT1/ptm/
m/n/10.95 -\OT1/pcr/m/n/10.95 Generating\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 
System\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n
/10.95 Arf\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 121--123
[]\OT1/pcr/m/n/10.95 Minimal\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Generating\O
T1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 System\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Ideal\OT1/ptm/m/n/10.95 -\OT1/pcr/m
/n/10.95 Of\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 123--125
[]\OT1/pcr/m/n/10.95 Minimal\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Generating\O
T1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 System\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 125--127
[]\OT1/pcr/m/n/10.95 Minimal\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 M\OT1/ptm/m/
n/10.95 -\OT1/pcr/m/n/10.95 E\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 D\OT1/ptm/m
/n/10.95 -\OT1/pcr/m/n/10.95 Generating\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 S
ystem\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/
10.95 M\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 E\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n
/10.95 D\OT1/ptm/m/n/10.95 -
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 127--129
[]\OT1/pcr/m/n/10.95 MinimalPresentationOfNumericalSemigroup\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 141--143
[]\OT1/pcr/m/n/10.95 NumericalSemigroupByMinimalGenerators\OT1/ptm/m/n/10.95 ,
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 143--145
[]\OT1/pcr/m/n/10.95 NumericalSemigroupByMinimalGeneratorsNC\OT1/ptm/m/n/10.95 
,
 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 147--149
[]\OT1/pcr/m/n/10.95 NumericalSemigroupBySubAdditiveFunction\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 149--151
[]\OT1/pcr/m/n/10.95 NumericalSemigroupsWithFrobeniusNumber\OT1/ptm/m/n/10.95 ,

 []


Overfull \hbox (4.20505pt too wide) in paragraph at lines 159--161
[]\OT1/pcr/m/n/10.95 ProportionallyModularNumericalSemigroup\OT1/ptm/m/n/10.95 
,
 []


Underfull \hbox (badness 10000) in paragraph at lines 170--172
[]\OT1/pcr/m/n/10.95 Random\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 List\OT1/ptm/
m/n/10.95 -\OT1/pcr/m/n/10.95 Representing\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.9
5 Sub\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Additive\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 174--176
[]\OT1/pcr/m/n/10.95 Random\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Proportionall
y\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Modular\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n
/10.95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 176--178
[]\OT1/pcr/m/n/10.95 ReductionNumberIdealNumericalSemigroup\OT1/ptm/m/n/10.95 ,

 []


Underfull \hbox (badness 10000) in paragraph at lines 178--180
[]\OT1/pcr/m/n/10.95 Remove\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Minimal\OT1/p
tm/m/n/10.95 -\OT1/pcr/m/n/10.95 Generator\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.9
5 From\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Numerical\OT1/ptm/m/n/10.95 -
 []


Underfull \hbox (badness 10000) in paragraph at lines 181--183
[]\OT1/pcr/m/n/10.95 RepresentsPeriodicSubAdditiveFunction\OT1/ptm/m/n/10.95 ,
 []


Underfull \hbox (badness 10000) in paragraph at lines 183--185
[]\OT1/pcr/m/n/10.95 Represents\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Small\OT1
/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Elements\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.
95 Of\OT1/ptm/m/n/10.95 -\OT1/pcr/m/n/10.95 Numerical\OT1/ptm/m/n/10.95 -
 []


Overfull \hbox (9.78952pt too wide) in paragraph at lines 188--190
[]\OT1/pcr/m/n/10.95 SmallElementsOfIdealOfNumericalSemigroup\OT1/ptm/m/n/10.95
 ,
 []


Underfull \hbox (badness 10000) in paragraph at lines 198--200
[]\OT1/pcr/m/n/10.95 TranslationOfIdealOfNumericalSemigroup\OT1/ptm/m/n/10.95 ,

 []

[52]) (./NumericalSgpsMan.aux) ) 
Here is how much of TeX's memory you used:
 4491 strings out of 94080
 64320 string characters out of 1165310
 120353 words of memory out of 1500000
 7371 multiletter control sequences out of 10000+50000
 49982 words of font info for 97 fonts, out of 1200000 for 2000
 645 hyphenation exceptions out of 8191
 27i,7n,27p,790b,614s stack positions out of 5000i,500n,6000p,200000b,5000s
{/usr/share/texmf-texlive/fonts/enc/dvips/base/
8r.enc}</usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmmi10.pfb></usr/share/
texmf-texlive/fonts/type1/bluesky/cm/cmr10.pfb></usr/share/texmf-texlive/fonts/
type1/bluesky/cm/cmsy10.pfb></usr/share/texmf-texlive/fonts/type1/public/cmex/f
mex9.pfb></usr/share/texmf-texlive/fonts/type1/bluesky/ams/msam10.pfb></usr/sha
re/texmf-texlive/fonts/type1/bluesky/ams/msbm10.pfb></usr/share/texmf-texlive/f
onts/type1/urw/courier/ucrr8a.pfb></usr/share/texmf-texlive/fonts/type1/urw/cou
rier/ucrro8a.pfb></usr/share/texmf-texlive/fonts/type1/urw/helvetic/uhvr8a.pfb>
</usr/share/texmf-texlive/fonts/type1/urw/symbol/usyr.pfb></usr/share/texmf-tex
live/fonts/type1/urw/times/utmb8a.pfb></usr/share/texmf-texlive/fonts/type1/urw
/times/utmr8a.pfb></usr/share/texmf-texlive/fonts/type1/urw/times/utmr8a.pfb></
usr/share/texmf-texlive/fonts/type1/urw/times/utmri8a.pfb>
Output written on NumericalSgpsMan.pdf (52 pages, 339974 bytes).
PDF statistics:
 919 PDF objects out of 1000 (max. 8388607)
 308 named destinations out of 1000 (max. 131072)
 22 words of extra memory for PDF output out of 10000 (max. 10000000)