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Internship - Mathematics at:rekrutacja@cesj.comSubject:
integro-differential equationsDear all,in a modeling problem,
I have arrived to an integro-differentialequation, which I
write in (La)TeX as follows:partial_t f(x,t) = f(x,t) ( int x
f(x,t) dx - x )where f(x,t) is a probability distribution.
Now, I have been able toprove basically everything I need to,
but since in the following Iwill have to deal with more
complex equations of similar kind, I wouldlike to know whether
there is a standard method for dealing with suchequations. In
particular, I would like to solve initial-conditionproblems of
the form f(x,0)=g(x).Directions for reading are
welcome!Thanks,-- Stefano Subject: Unmapped territory?In the
following, 
===
 represents the congruence symbol.Given the
congruence b^n 
===
 m (mod x), where the values ofn, m and x are
known (all positive naturals > 1), what isthe easiest (= best?)
way of finding the first occurrence(if any) of b that satisfies
the equation?Actually, any occurrence would do, although the
first oneis deemed preferable.If this has been done before,
I'd appreciate some pointers.BTW, someone mentioned A Course
in Computational AlgebraicNumber Theory by Henri Cohen,
Springer Verlag, pages 36-38as a possible recursive solution
when x is a prime. Whatsay you, any improvements to
this?TIA,DanielSubject: Re: Unmapped territory?Daniel Dudley>
In the following, 
===
 represents the congruence symbol.> Given
the congruence b^n 
===
 m (mod x), where the values of> n, m and
x are known (all positive naturals > 1), what is> the easiest
(= best?) way of finding the first occurrence> (if any) of b
that satisfies the equation?> Actually, any occurrence would
do, although the first one> is deemed preferable.....I'm no
expert but I've heard that there is no known reasonably
smartalgorithm even to find a mere quadratic nonresidue modulo
a prime, i.e. analgo for the equationb^{(p-1)/2} 
===
 p-1 mod
p.You might try this mailing list for
pros:http://listserv.nodak.edu/archives/
nmbrthry.htmlLarrySubject: Re: Unmapped territory?Daniel
Dudley> In the following, 
===
 represents the congruence
symbol.> Given the congruence b^n 
===
 m (mod x), where the
values of> n, m and x are known (all positive naturals > 1),
what is> the easiest (= best?) way of finding the first
occurrence> (if any) of b that satisfies the equation?>
Actually, any occurrence would do, although the first one> is
deemed preferable.> ....> I'm no expert but I've heard that
there is no known reasonably> smart algorithm even to find a
mere quadratic nonresidue> modulo a prime, i.e. an algo for
the equation> b^{(p-1)/2} 
===
 p-1 mod p.I wouldn't know about
the expert part, but clearly you aresmart enough to guess
where I'm going with this. ;-)IMHO, a considerable amount of
effort should be devoted tofinding a smart algorithm that
resolves the equation, sincethis would probably provide (I
suspect) an almost constanttime solution to the prime
identification problem.> You might try this mailing list for
pros:> http://listserv.nodak.edu/archives/nmbrthry.htmlI
wasn't aware of this ML -- thanks for the tip,
Larry.DanielSubject: Re: Question on category
theoryEpigone-thread: feewermflilDear Tom,Thank you for your
answer.> This is a special case of the notion of discrete
fibration, or more > accurately, discrete opfibration.What is
your mention about using term close in for this specialcase? I
use some easy notions of the category theory for the
PDEinvestigation (you can see some old results in [Prokhorova,
M.Factorization of nonlinear heat equation posed on Riemann
manifold //http://arxiv.org/abs/math.AP/0108001]). So I try to
avoid unnecessary(for my aids) terms and notions of the
category theory (most of PDEinvestigators don't familiar with
this theory, and I am not expert init too :-)> I'm not sure
what's meant by equivalent here,F:A->B is equivalent to
F':A'->B' if that there exist isomorphisms ofK G:A'->A,
H:B->B' such that GFH=F'. In my definition must be: there
exist isomorphism of K H:B->B' suchthat FH=F'.> A decent
reference for fibrations is Borceux's Handbook of >
Categorical Algebra (Cambridge University Press) - volume 2,
Ithink.Unfortunately our library don't has this books...> The
name dense is probably not a great choice, as it already hasan
> established meaning in category theoryOh, it is important
information for me. Thank you!What do you think about term
rich in? Or may be you have moreconvinient terms for my
notion?Yours,Marinahttp://vpro.convex.ru/Marina/Subject: Re:
Symmetric designs> I wanted too but the only copy our library
has is on loan till 2015.> This seems excessive even for a
university librarian.Ask a librarian. Perhaps the book can be
called in for you.Subject: This week in the mathematics arXiv
(18 Aug - 22 Aug)Here are this week's titles in the
mathematics arXiv, available at:
http://front.math.ucdavis.edu/
http://front.math.ucdavis.edu/submissionsThis week in the
mathematics arXiv may be freely redistributedwith attribution
and without modification.Titles in the mathematics arXiv (18
Aug - 22
Aug)-------------------------------------------------AG:
Algebraic Geometry----------------------math.AG/0308205
<http://front.math.ucdavis.edu/math.AG/0308205 Hal Schenck,
Gregory G. Smith: Syzygies of projective toric
varietiesmath.AG/0308201
<http://front.math.ucdavis.edu/math.AG/0308201 Ivan V.
Arzhantsev, Dmitri A. Timashev: On the canonical embeddings of
certain homogeneous spacesmath.AG/0308200
<http://front.math.ucdavis.edu/math.AG/0308200 E. Lupercio, M.
Poddar: The global McKay-Ruan correspondence via motivic
integrationmath.AG/0308198
<http://front.math.ucdavis.edu/math.AG/0308198 Christopher D.
Hacon: A derived category approach to generic
vanishingmath.AG/0308195
<http://front.math.ucdavis.edu/math.AG/0308195 Alfred Jungkai
Chen, Christopher D. Hacon: Varieties with $P_3(X)=4$ and
$q(X)=dim (X)$math.AG/0308191
<http://front.math.ucdavis.edu/math.AG/0308191 Shulim Kaliman,
Mikhail Zaidenberg: Venereau polynomials and related fiber
bundlesmath.AG/0308182
<http://front.math.ucdavis.edu/math.AG/0308182 Brendan
Hassett, Yuri Tschinkel: Universal torsors and Cox
ringsmath.AG/0308178
<http://front.math.ucdavis.edu/math.AG/0308178 Vik.S.Kulikov:
Full twists factorization formula for double number of
stringsmath.AG/0308173
<http://front.math.ucdavis.edu/math.AG/0308173 Anton Kapustin,
Dmitri Orlov: Lectures on Mirror Symmetry, Derived Categories,
and D-branesmath.AG/0308146
<http://front.math.ucdavis.edu/math.AG/0308146 Tamas Hausel:
Quaternionic Geometry of MatroidsAP: Analysis of
PDEs--------------------math.AP/0308204
<http://front.math.ucdavis.edu/math.AP/0308204 Oliver C.
Schnuerer: Regularity and splitting of directed minimal
conesCA: Classical Analysis and
ODEs-------------------------------math.CA/0308159
<http://front.math.ucdavis.edu/math.CA/0308159CO:
Combinatorics-----------------cs.CR/0308027
<http://front.math.ucdavis.edu/cs.CR/0308027 Liam Wagner: A
Comparison of Secret Sharing Schemes Based on Latin Squares
and RSADG: Differential
Geometry-------------------------math.DG/0308203
<http://front.math.ucdavis.edu/math.DG/0308203 Harish
Seshadri: Positive scalar curvature and minimal
hypersurfacesmath.DG/0308197
<http://front.math.ucdavis.edu/math.DG/0308197 Ai-Ko Liu:
Family Switching Formula and the $-n$ Exceptional Rational
Curvesmath.DG/0308196
<http://front.math.ucdavis.edu/math.DG/0308196 Ai-Ko Liu: A
Note on Curve Counting Scheme in an Algebraic Family and The
Admissible Decomposition Classesmath.DG/0308184
<http://front.math.ucdavis.edu/math.DG/0308184 Eric Leichtnam,
Paolo Piazza: Etale Groupoids, eta invariants and index
theorymath.DG/0308179
<http://front.math.ucdavis.edu/math.DG/0308179 Theodore
Voronov: A criterion equivalent to the Helmholtz criterion in
the inverse problem of calculus of variationsmath.DG/0308152
<http://front.math.ucdavis.edu/math.DG/0308152 Boris Dubrovin,
Youjin Zhang: Virasoro Symmetries of the Extended Toda
Hierarchymath.DG/0308149
<http://front.math.ucdavis.edu/math.DG/0308149 Vasile Oproiu,
Dumitru Daniel Porosniuc: A Kaehler Einstein structure on the
cotangent bundle of a Riemannian manifoldDS: Dynamical
Systems---------------------chao-dyn/9906036
<http://front.math.ucdavis.edu/chao-dyn/9906036 Andrei
Sobolevskii: Aubry-Mather theory and idempotent eigenfunctions
of the Bellman operatorchao-dyn/9906035
<http://front.math.ucdavis.edu/chao-dyn/9906035 Andrei
Sobolevskii: On periodic solutions of a Hamilton-Jacobi
equation with periodic forcingmath.DS/0308158
<http://front.math.ucdavis.edu/math.DS/0308158 Claire
Anantharaman-Delaroche: Cohomology of property T groupoids and
applicationsFA: Functional
Analysis-----------------------math.FA/0308181
<http://front.math.ucdavis.edu/math.FA/0308181 M.Bekker, E.
Tsekanovskii: On Periodic Matrix-Valued Weyl-Titchmarsh
Functionsmath.FA/0308169
<http://front.math.ucdavis.edu/math.FA/0308169 Volker Runde:
Applications of operator spaces to abstract harmonic
analysismath.FA/0308166
<http://front.math.ucdavis.edu/math.FA/0308166 Guy Cohen,
Stephane Gaubert, Jean-Pierre Quadrat, Ivan Singer: Max-plus
convex sets and functionsmath.FA/0308145
<http://front.math.ucdavis.edu/math.FA/0308145 Daniel M.
Pellegrino: Relations between the different concepts of
summability of multilinear mappings between Banach spacesGM:
General Mathematics-----------------------math.GM/0308170
<http://front.math.ucdavis.edu/math.GM/0308170 Dr. W.B.
Vasantha Kandasamy: Smarandache Linear Algebramath.GM/0308153
<http://front.math.ucdavis.edu/math.GM/0308153 J Gerard Wolff:
Mathematics and Logic as Information Compression by Multiple
Alignment, Unification and Searchmath.GM/0308144
<http://front.math.ucdavis.edu/math.GM/0308144 Jody R. Miotke,
Laura Rebollo-Neira: Oversampling and transmission of hidden
informationGN: General
Topology--------------------math.GN/0308162
<http://front.math.ucdavis.edu/math.GN/0308162 K. P. Hart, B.
J. van der Steeg: Span, chainability and the continua H* and
I_uGR: Group Theory----------------math.GR/0308199
<http://front.math.ucdavis.edu/math.GR/0308199 Peter
Brinkmann: Dynamics of free group automorphismsmath.GR/0308171
<http://front.math.ucdavis.edu/math.GR/0308171 Inna Bumagin:
The conjugacy problem for relatively hyperbolic groupsGT:
Geometric Topology----------------------math.GT/0308168
<http://front.math.ucdavis.edu/math.GT/0308168 Efstratia
Kalfagianni, Xiao-Song Lin: Knot adjacency and
satellitesmath.GT/0308155
<http://front.math.ucdavis.edu/math.GT/0308155 Martin
Scharlemann: Thin position in the theory of classical
knotsmath.GT/0308151
<http://front.math.ucdavis.edu/math.GT/0308151 Magnus
Jacobsson: Khovanov's conjecture over Z[c]math.GT/0308147
<http://front.math.ucdavis.edu/math.GT/0308147 Sadayoshi
Kojima, Shigeru Mizushima, Ser Peow Tan: Circle packings on
surfaces with projective structures and uniformizationKT:
K-Theory and Homology-------------------------math.KT/0308161
<http://front.math.ucdavis.edu/math.KT/0308161 Alan L. Carey,
John Phillips, Fyodor A Sukochev: Spectral flow in Fredholm
modules, eta invariants and the JLO cocycleLO:
Logic---------math.LO/0308177
<http://front.math.ucdavis.edu/math.LO/0308177 Giovanni Panti:
Generic substitutionsmath.LO/0308148
<http://front.math.ucdavis.edu/math.LO/0308148 Ewa Graczynska,
Dietmar Schweigert: HyperquasivarietiesMG: Metric
Geometry-------------------math.MG/0308187
<http://front.math.ucdavis.edu/math.MG/0308187 Francois
Fillastre: Spaces of Polygons to Spaces of Polyhedra Following
Bavard, Ghys and Thurstonmath.MG/0308186
<http://front.math.ucdavis.edu/math.MG/0308186 Julian Pfeifle,
Gunter M. Ziegler: On the Monotone Upper Bound ProblemMP:
Mathematical Physics------------------------quant-ph/0308094
<http://front.math.ucdavis.edu/quant-ph/0308094 Fabio
Dell'Anno, Silvio De Siena, Fabrizio Illuminati: Structure of
multiphoton quantum optics. II. Bipartite systems, physical
processes, and heterodyne squeezed statesquant-ph/0308081
<http://front.math.ucdavis.edu/quant-ph/0308081 Fabio
Dell'Anno, Silvio De Siena, Fabrizio Illuminati: Structure of
multiphoton quantum optics. I. Canonical formalism and
homodyne squeezed statesphysics/0307117
<http://front.math.ucdavis.edu/physics/0307117 O. V. Usatenko,
V. A. Yampol'skii, K. E. Kechedzhy, S. S. Mel'nyk: Symbolic
stochastic dynamical systems viewed as binary N-step Markov
chainsmath-ph/0308025
<http://front.math.ucdavis.edu/math-ph/0308025 A.G.Ramm:
Comments on the letter of P.Sabatier An erroneous
statementmath-ph/0308024
<http://front.math.ucdavis.edu/math-ph/0308024 John R.
Ockendon Yair Zarmi: Representation of solutions of linear
PDE's with constant coefficients as superpositions of
solutions in lower dimensionsmath-ph/0308023
<http://front.math.ucdavis.edu/math-ph/0308023 Michael
Aizenman, Alex Elgart, Sergei Naboko, Jeffrey H. Schenker,
Gunter Stolz: Moment Analysis for Localization in Random
Schroedinger Operatorsmath-ph/0308022
<http://front.math.ucdavis.edu/math-ph/0308022 M. Przanowski,
M. Skulimowski: Linear Hilbertian manifold
domainsgr-qc/0308057
<http://front.math.ucdavis.edu/gr-qc/0308057 Christian G.
Boehmer: General Relativistic Static Fluid Solutions with
Cosmological Constantmath-ph/0308021
<http://front.math.ucdavis.edu/math-ph/0308021 P. Gilkey, K.
Kirsten, JH. Park: Heat content asymptotics for operators of
Laplace type with spectral boundary conditionsphysics/0308050
<http://front.math.ucdavis.edu/physics/0308050 J.E. Avron, O.
Gat, O. Kenneth, U. Sivan: Optimal rotations of deformable
bodies and orbits in magnetic fieldsmath-ph/0308020
<http://front.math.ucdavis.edu/math-ph/0308020
K.Thirulogasanthar, A.L.Hohoueto: Vector Coherent States on
Clifford algebrasmath-ph/0308019
<http://front.math.ucdavis.edu/math-ph/0308019 I.M.Krichever,
S.P.Novikov: 2D Toda Chain, Commuting Difference Operators and
Holomorphic Bundlesmath-ph/0308018
<http://front.math.ucdavis.edu/math-ph/0308018 Jaedong Choi,
Soon-Tae Hong: Warped product approach to universe with
non-smooth scale factormath-ph/0308017
<http://front.math.ucdavis.edu/math-ph/0308017 Valeri V.
Dvoeglazov: Is the Space-Time Non-Commutativity Simply
Non-Commutativity of Derivatives?math-ph/0308011
<http://front.math.ucdavis.edu/math-ph/0308011 Andrzej J.
Maciejewski, Maria Przybylska, Jacques-Arthur Weil:
Non-integrability of the generalised spring-pendulum
problemhep-th/0308093
<http://front.math.ucdavis.edu/hep-th/0308093 S.P. Gavrilov,
D.M. Gitman, A.A. Smirnov, B.L. Voronov: Dirac fermions in a
magnetic-solenoid fieldNA: Numerical
Analysis----------------------physics/0308049
<http://front.math.ucdavis.edu/physics/0308049 Andrei G.
Borisov, Sergei V. Shabanov: An application of interpolating
scaling functions to wave packet propagationNT: Number
Theory-----------------math.NT/0308202
<http://front.math.ucdavis.edu/math.NT/0308202 Adrian Vasiu: A
motivic conjecture of Milnemath.NT/0308194
<http://front.math.ucdavis.edu/math.NT/0308194 J. Browkin, J.
Brzezinski, W. Holsztynski, P. Kurlberg: On the congruence
ax+by = 1 modulo xymath.NT/0308165
<http://front.math.ucdavis.edu/math.NT/0308165 Romyar Sharifi:
Massey Products and Ideal Class GroupsOA: Operator
Algebras---------------------math.OA/0308192
<http://front.math.ucdavis.edu/math.OA/0308192 Hanne Schultz:
Non-commutative Polynomials of Independent Gaussian Random
Matrices. The Real and Symplectic CasesOC: Optimization and
Control----------------------------math.OC/0308206
<http://front.math.ucdavis.edu/math.OC/0308206 Ivar Ekeland:
An optimal matching problemmath.OC/0308172
<http://front.math.ucdavis.edu/math.OC/0308172 Delfim F. M.
Torres: On the Constancy of the Pontryagin Hamiltonian for
Autonomous ProblemsPR: Probability
Theory----------------------math.PR/0308193
<http://front.math.ucdavis.edu/math.PR/0308193 Volker Betz,
Herbert Spohn: A central limit theorem for Gibbs measures
relative to Brownian motionmath.PR/0308190
<http://front.math.ucdavis.edu/math.PR/0308190 Olivier Garet:
Central limit theorems in Random cluster and Potts
Modelsmath.PR/0308175
<http://front.math.ucdavis.edu/math.PR/0308175 Nils Berglund,
Barbara Gentz: On the noise-induced passage through an
unstable periodic orbit I: Two-level modelmath.PR/0308164
<http://front.math.ucdavis.edu/math.PR/0308164 Wendelin
Werner: SLEs as boundaries of clusters of Brownian
loopsmath.PR/0308157
<http://front.math.ucdavis.edu/math.PR/0308157 Harold Widom:
On Asymptotics for the Airy Processmath.PR/0308154
<http://front.math.ucdavis.edu/math.PR/0308154 Eddy
Mayer-Wolf, Alexander Roitershtein, Ofer Zeitouni: Limit
theorems for one-dimensional transient random walks in Markov
environmentsmath.PR/0308150
<http://front.math.ucdavis.edu/math.PR/0308150 Wolfgang
Koenig, Peter Moerters: Brownian intersection local times:
Exponential moments and law of large massesQA: Quantum
Algebra-------------------hep-th/0308130
<http://front.math.ucdavis.edu/hep-th/0308130 Jae-Suk Park:
Flat family of QFTs and quantization of
d-algebrasmath.QA/0308189
<http://front.math.ucdavis.edu/math.QA/0308189 Pierre
Bieliavsky, Philippe Bonneau, Yoshiaki Maeda: Universal
Deformation Formulae, Symplectic Lie groups and Symmetric
Spacesmath.QA/0308188
<http://front.math.ucdavis.edu/math.QA/0308188 Pierre
Bieliavsky, Philippe Bonneau, Yoshiaki Maeda: Universal
Deformation Formulae for Three-Dimensional Solvable Lie
groupsmath.QA/0308176
<http://front.math.ucdavis.edu/math.QA/0308176 Dong-Uy Shin:
Crystal bases and monomials for
$U_q(G_2)$-modulesmath.QA/0308163
<http://front.math.ucdavis.edu/math.QA/0308163 Mikhail
Karasev: Intrinsic Dynamics of Manifolds: Quantum Paths,
Holonomy, and Trajectory Localizationcond-mat/0308267
<http://front.math.ucdavis.edu/cond-mat/0308267 Christian
Korff: The twisted XXZ chain at roots of unity revisitedRA:
Rings and Algebras----------------------math.RA/0308160
<http://front.math.ucdavis.edu/math.RA/0308160 Samer
Al-Ashhab: A Class of Strongly Homotopy Lie Algebras with
Simplified sh-Lie StructuresRT: Representation
Theory-------------------------math.RT/0308156
<http://front.math.ucdavis.edu/math.RT/0308156 Peter Magyar:
Littelmann paths for the basic representation of an affine Lie
algebraSG: Symplectic
Geometry-----------------------math.SG/0308185
<http://front.math.ucdavis.edu/math.SG/0308185 Yi-Jen Lee:
Non-contractible periodic orbits, Gromov invariants, and
Floer-theoretic torsionsmath.SG/0308183
<http://front.math.ucdavis.edu/math.SG/0308183 Frederic
Bourgeois, Yakov Eliashberg, Helmut Hofer, Kris Wysocki,
Eduard Zehnder: Compactness results in Symplectic Field
Theorymath.SG/0308180
<http://front.math.ucdavis.edu/math.SG/0308180 Alberto S.
Cattaneo: On the integration of Poisson manifolds, Lie
algebroids, and coisotropic submanifoldsmath.SG/0308174
<http://front.math.ucdavis.edu/math.SG/0308174 Leonardo
Macarini: Hofer-Zehnder capacity of standard cotangent
bundlesmath.SG/0308167
<http://front.math.ucdavis.edu/math.SG/0308167 Augustin
Banyaga: Examples of non $d_{omega}$-exact locally conformal
symplectic forms-- / Greg Kuperberg (UC Davis) / / Visit the
Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the
math that's fit to e-print *Followup-to: posterSubject: [CFP]
Mathematical Models and Methods in Applied SciencesDear
Friends and Colleagues,The Mathematical Models and Methods in
Applied Sciences (M3AS)provides a medium of exchange for
scientists engaged in appliedsciences (physics, biology,
economy, finance, social sciences, naturaland technological
sciences) where there exists a non-trivial interplaybetween
mathematics, mathematical modeling of complex systems
andmathematical and computer methods oriented towards the
qualitative andquantitative analysis of mathematical
models.The principal areas of interest of this journal
are:1.Mathematical modeling of systems in applied
sciences.2.Mathematical methods for the qualitative and
quantitative analysisof models of mathematical physics and
technological sciences.3.Numerical and computer treatment of
mathematical models or realsystems.Special attention will be
paid to the analysis of nonlinearities andstochastic
aspects.Within the above limitation, scientists in all fields
which employmathematics are encouraged to submit research and
review papers to thejournal. Both theoretical and applied
papers will be considered forpublication. High quality,
novelty of the content and potential forthe applications to
modern problems in applied sciences and technologywill be the
guidelines for the selection of papers to be published inthe
journal.If you are interested in submitting your work to M3AS
to be consideredfor publication, please consult or email any
of the Editors, NicolaBellomo: nicola.bellomo@polito.it or
Fanco Brezzi:brezzi@dragon.imati.cnr.it .For more details on
the journal, please
visit:http://www.worldscinet.com/m3as.htmlSincerely,The
Editorial BoardM3ASSubject: Re: Euclid's BooksLitnbimi> Any
idea how much of Euclid's books that lay the> foundation of
geometry can be attributted to his> own pen?> I always thought
it was all him, but my brother> laughed at me!Well, conics were
known to Menaechmus around 350 BC, but Euclid and laterwriters
added some results and equivalent definitions. Pythagoras's
theorem(due to some anonymous Pythogorean) is even
earlier.LHSubject: [Graph theory] How should this be
called?(This question has also been posted in graphnet.
However, since I do not get any response thus far, I would
like to seek the suggestion of readers in this group as
well)Dear newsgroup reader,Please forgive me if this question
sounds trivial to you. I have done somegoogle and mathworld
search on my question but I cannot find the answer.Given any
graph G, its clique number is defined as the size of the
largestclique in the graph. There is a theorem by Matula
stating that in a classof random graph, the clique number lies
in a short interval for almostevery graph.My question is: what
happens if we replace clique by regular graph?How should the
quantitymax_r ( G has a r-regular graph as a subgraph )be
called? Is there any known result on this property for
randomgraphs?Thank you!Subject: Re: [Graph theory] How should
this be called?> (This question has also been posted in
graphnet. However, since I do not> get any response thus far,
I would like to seek the suggestion of> readers in this group
as well)> Dear newsgroup reader,> Please forgive me if this
question sounds trivial to you. I have done some> google and
mathworld search on my question but I cannot find the answer.>
Given any graph G, its clique number is defined as the size of
the largest> clique in the graph. There is a theorem by Matula
stating that in a class> of random graph, the clique number
lies in a short interval for almost> every graph.> My question
is: what happens if we replace clique by regular graph?> How
should the quantity> max_r ( G has a r-regular graph as a
subgraph )> be called? Is there any known result on this
property for random> graphs?> Thank you!A spanning r-regular
graph is called an r-factor. So maybe yourquantity could be
called an r-subfactor?HTH,Felix.Subject: Tree
EmbeddingEpigone-thread: drirbeiplingHiCan anyone please give
me a reference to the complexity of thefollowing
problem?----------------------------Consider a weighted tree T
with |V| nodes and |E| edges. All the nodesand edges have
weights assigned to them. (edge(i, j) has weight areew_{i,j},
node i has weight vw_i)Consider a complete graph G of N nodes,
where each edge (i,j) hasweight l_{i,j) and each node has a
capacity C > any vw_i in T.1. cosider |V| <= N. Find a 1-to-1
mapping M of the nodes of T to thenodes of G such that,
sum_{i, j} ew_{i, j} * l_{M(i), M(j)}is minimized. (M(i) = x
means that the tree node i maps to the graphnode x)2. consider
|V| > N. Find a many-to-one mapping M such that 1. sum of
weights of all the nodes in T mapped to a node in Gis less
than C 2. sum_{i,j} ew_{i, j} * l_{M(i), M(j)} is
minimizedThanks,CharubakSubject: Re: Number of cyclic digraphs
without sources nor sinks> A cyclic digraph is a digraph in
which every vertex is in a cycle. In> other words, cyclic
digraps are non-acyclic digraphs.> In a digraph, a source is a
vertex with in-set of size zero; a sink is> a vertex with
out-set of size zero.> I would like to know the number of
labeled cyclic digraphs on n> vertices without sources nor
sinks.The corresponding sequence is now in the OEIS as
sequence number A086193Brendan McKay and Vladeta Jovovic have
given formulas:Vladeta Jovovic's formula is
sum((-1)^(n-r)*binomial(n,r)*(2^(r-1)-1)^r*(2^r-1)^(n-r),
r=0..n).The first few values
are:0,1,18,1699,592260,754179301,3562635108438,
63770601591579079,4405870283636411477640,
1190873924687350003735546441,
1270602397076493907445608866890778,
5381240610642043789096251476993474339179--Edwin Clark