Here's a differential equation that seems very beautiful to me: f'(f*(Exp(kt)f(x))) = s(t)f'(x) f is a function of a real variable. f' is the derivative of f. f* is the inverse of f. Exp(kt) is e to the power kt. s(t) can be any function of t. I'd like to know if there are any solutions f(x) other than f(x)=a(x^n) and f(x) = Exp(bx). I would be extremely grateful for any assistance. Eugene Shubert http://www.everythingimportant.org ==== Could someone prove/disprove this: If max clique size of a graph omega(G)=3 and there are no two triangles in this graph having a common edge, then the Lovasz number theta(~G)=3 ? Stas http://www.busygin.dp.ua ==== what are Bogomolny equations? ==== > what are Bogomolny equations? > They are equations describing magnetic monopoles. They were especially popular in the late 80s. For instance you could look in the book bib{MR89k:53067}{book}{ author={Atiyah, Michael}, title={The geometry and dynamics of magnetic monopoles}, series={M. B. Porter Lectures}, publisher={Princeton University Press}, place={Princeton, NJ}, date={1988}, pages={viii+134}, isbn={0-691-08480-7}, review={MR 89k:53067}, } Math. Review also lists more than a 100 references to Bogomolny. ==== > what are Bogomolny equations? > case you cannot get it I have a short introduction to monopoles (which are solutions of the Bogomolny equations) at http://front.math.ucdavis.edu/math-ph/0101035 Michael ==== > what are Bogomolny equations? Nucl Physics in 1974. In this he discusses 3 relativistic field equations overdetermined first order system implies the full second order field equations. The first order solutions saturate a lower bound (given by a topological charge) on the energy of the system (Bogomolny inequality) and are related to the existence of a non trivial supersymmetry (which was undiscovered at Peter Ruback ==== > > in finite dimensions, it is no problem to show that a closed (algebraic) > subgroup of a Lie group is in fact a Lie subgroup. > Since the proofs I know of require local compactness, which fails in > infinite dimension, my question is: > Does anyone know of a counterexample for this in the infinite dimensional > case? > > Frederick Examples are well-known, many and varied. Here is perhaps one of the most convincing - and simplest - among them. $L^2(0,1)$ with the norm topology, and take as $H$ the group formed by all functions $f$ whose range is the set of integers a.e. Then $G$ is the nicest infinite dimensional Lie group imaginable, that. At the same time, $H$, which is a closed connected topological subgroup, fails to form a Lie group in any possible sense of the word no matter how you weaken it, because $H$ contains no non-trivial one-parameter subgroups. There is of course a great variety of differing approaches to infinite dimensional Lie group theory beyond the Banach-Lie case (check out the work by H. Omori, J. Leslie, J. Milnor, P. Michor and co-authors, and W. Wojtynski, among others). However everyone seems to agree that a connected Lie group must satisfy a regularity condition in the sense that it contains at least some one-parameter subgroups, e.g. enough of those in order for their union to generate the connected component of identity, or something of the sort. Having said that, I hasten to add that even as I work on this posting, one of the big unresolved problems concerning infinite dimensional Lie groups modeled on sequentially complete locally convex spaces other than Banach spaces (e.g., Fr.8echet spaces) remains this: does every group $G$, which is at the same time a manifold modeled on a locally convex space as above in such a way that the group operations are smooth, admit an exponential map from its Lie algebra (defined in a natural way)? If the answer turns out to be `yes,' then of course there will be exponential subgroups in existence in every such $G$. But, strictly speaking, nothing at this point prevents the existence of an example of a group which is a Fr.8echet (say) manifold, with smooth multiplication and invertion, and without any one-parameter subgroups whatsoever. As far as I know, even an abelian such example has not been yet ruled out. But of course even if this sort of pathological object does exist, hardly anyone would refer to the sorry creature as a `Lie group.' (`Fake Lie' would rather be like it.) Vladimir Pestov ==== Is there a dense subset S of R^2 such that for every pair of points p,q in S the Euclideean distance d(p,q) is a rational number ? ==== > Also, it seems a good improvement for Lovasz's > theta(~G) to calculate t=theta(~H) and consider > phi(G) = (1+sqrt(1+8t))/2 instead as a tighter > max clique size bound. Any thoughts? Can you explain why this woud be an improved bound, and why another transformation like the standard substitution of each vertex with some large clique wouldn't be as good an improvement? Jim ==== > > Also, it seems a good improvement for Lovasz's > theta(~G) to calculate t=theta(~H) and consider > phi(G) = (1+sqrt(1+8t))/2 instead as a tighter > max clique size bound. Any thoughts? > > Can you explain why this woud be an improved bound, > and why another transformation like the standard substitution > of each vertex with some large clique wouldn't be as good > an improvement? > Jim First off, substitution of a vertex by a larger clique is equivalent to assigning the vertex weight w equal to the clique size. So, you just assign each vertex the weight w and this only multiplies theta by w, nothing more. Of course it doesn't provide any improvement. As for the considered transformation, it destroys imperfect structures in a graph. Generalizing, consider m-th level involution transforming all K_{m+1} subgraphs to vertices providing an edge iff two of them induce a complete subgraph in the original graph. At that, the maximum clique size becomes omega!/((m+1)!(omega-m-1)!). Of course, ~theta provides exact clique number at least when m=omega-1, since the graph becomes a set of isolated vertices, each corresponding to a maximum clique of the original graph, so no imperfection is possible. But I assume the bound m=omega-1 is not tight here. In fact, I've not found so far any graph with omega=3, for which the 1-st level involution (edges->vertices) doesn't make ~theta=omega. Comparing to Lovasz-Shrijver lift-and-project or Pasechnik-de Klerk copositivity framework for theta improvement, the complete subgraph involution looks better because its dimensionality growth is slower. Indeed, the m-th level involution involves only those m-tuples of vertices corresponding to cliques of the original graph, while the mentioned frameworks consider all m-tuples of vertices at the same improvement level. Best, Stas http://www.busygin.dp.ua ==== It seems that the examples (with which I'm familiar) of second-kind linear Fredholm problems are defined either over intervals or over boundaries~$partialOmega$, where the region~$Omegasubsetreals^d$ has positive $d$-dimensional Lebesgue measure. Could somebody point me to examples of problems that are defined over such regions~$Omega$ themselves? This request may sound a little ill-defined, since (using a change of variables) a problem over~$partialOmega$ can be rewritten as a problem over a $d-1$-dimensional region of~$mathbb{R}^{d-1}$ having positive $d-1$-dimensional Lebesgue measure. What I'm interested in is problems whose initial or natural formulation is over regions~$Omega$ as described in the first paragraph. -- Art Werschulz (8-{)} Metaphors be with you. -- bumper sticker GCS/M (GAT): d? -p+ c++ l u+(-) e--- m* s n+ h f g+ w+ t++ r- y? Internet: agw@cs.columbia.eduWWW ATTnet: Columbia U. (212) 939-7060, Fordham U. (212) 636-6325 ==== I am smoothing then differentiating position data in Matlab. Then, I calculate and plot the fft spectra of raw and smoothed, position, velocity, and acceleration data (velocity and acceleration are derived from raw and also smoothed data). Could the spectra from smooth data ever be greater (peaks) than those from raw data? That is what I am finding in many instances, but it seems erroneous relative to my understanding of the filtering: 2nd-order no-lag butterworth filter 8hz cutoff (after 2 passes) . The data are from human hand movement during pointing (Expected signal power is generally below 8 hz, but some Parkinsonās subjects might have tremor up to 15 hz, hence, the analysis of fft spectra). ==== I may be applying Fourier Integral Operators to seismic ray theory (and probably the transport equation portion) for my Ph.D. thesis and would like to know of any good online or printed recent or standard references in the subject and I guess also in its subset pseudodifferential operators. David ==== Dan Lior > One widely used definition for adjoint is as follows: Let V, W be finite dimensional inner product spaces and T:V--W a > linear transformation. The adjoint of T is the unique transformation > T*:W-->V that satisfies; > = for every x in V and y in W. Of course the existence and uniqueness of T* takes requires some > argument. Another definition of adjoint doesn't require that V and W > be any more than mere vector spaces. It goes like this: Let V,W be vector spaces and T:V--W a linear transformation. The > adjoint of T is the transformation T*:W'-->V' defined by; > T*(a) = a o T. Here, V' denotes dual of V, o denotes composition and a is a typical > element of W'. This latter mapping is more properly called the _transpose_ of T. But an inner product gives rise to an isomorphism of a finite-dimensional space onto its dual; if we thereby identify the two spaces, then the adjoint is identical (up to mere notation) to the transpose. LH ==== >I wonder if there has been any recent work in >modified equation methods (introduced by Warming >and Hyett in the 1970s for CFD) when the >discrete solution is nodally exact. >[...] > > I am not sure what you want (for starters, what is 'nodally exact'?) > but you might be interested in Chapter IX, Backward Error Analysis > and Structure Preservation, of Hairer, Lubich and Wanner: Geometric > Numerical Integration, Springer, 2002. > > Here, the authors discuss theorems like: > - the modified equation of a (time) reversible method is reversible > - the modified equation of a symplectic method is Hamiltonian > > Jitse Niesen Many thanks for the reference. Actually I am familiar with those theorems. They are not of use in my research, since the systems I am dealing with are strongly dissipative. But I found what I needed in an AMS book titled Chaotic Numerics, the proceedings of a 1993 workshop on nondeterministic dynamical systems. The qualifier nodally exact (NE) is a subset of nodally superconvergent (NS). A discretization is NS if the nodes are superconvergent locations (aka Barlow points when the discretization is FEM based). NE only holds under very restrictive conditions and never in 2D or 3D, but is is a good departure point for iterative parameter selection in multiple dimensions. ==== Fields-sponsored Mini-Conference and Workshop on Concentration Phenomenon, Transformation Groups, and Ramsey Theory will be held from The Workshop Web page: http://www.fields.utoronto.ca/programs/scientific/03-04/cgr/ The workshop will consist of a series of lectures, assuming no specific background, and concerning dynamics and geometry of `large,' `infinite-dimensional' groups, in particular interactions between the phenomenon of concentration of measure on high-dimensional structures, actions of large groups on compact spaces, and combinatorial Ramsey-type results. `Large' groups at the centre of attention incude various groups of automorphisms of measure spaces and measurable equivalence relations, groups of homeomorphisms and isometries, groups of automorphisms of various countable structures (graphs, Boolean algebras...), etc. The concepts, results and techniques from this area could benefit mathematicians working in a broad variety of disciplines. There will be also an open problem discussion session, and possibly a session for short contributed talks. Confirmed lecturers: Thierry Giordano (University of Ottawa) S. Solecki (Univ. of Illinois at UC) S. Todorcevic (Paris VII) V.V. Uspenskij (Ohio University) A. Vershik (St. Petersburg, Steklov Institute) Benjy Weiss (Hebrew University) Funding support is available for graduate student to partially cover local expenses and travel. Some funding may also be available to postdocs depending on budget constraints. Apply for funding via the w/shop web page above. A block of rooms has been reserved at: Quality Hotel Downtown, 290 Rideau St. Ottawa, Ontario K1N 5Y3. Reservations: 1-800-359-4827. The special conference rate: $CAN 105.00 + 12 % tax, single, double, or triple room share. Workshop organizers: Thierry Giordano, David Handelman and Vladimir Pestov (University of Ottawa). ==== The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-25.abs.html Title: Near-group categories Author(s): Jacob Siehler Abstract: We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules, and an explicit construction of matrix solutions to the pentagon equations in the cases where we establish existence. Many of these also support (braided) commutative and tortile structures and we indicate when this is possible. Small examples are presented in detail. AMS Classification Numbers. Primary: 18D10 Keywords: Monoidal categories, braided categories Author(s) address(es): Department of Mathematics, Virginia Tech Blacksburg, VA 24061-0123, USA ==== The following UBASIC program correctly identifies divisors of phi(N), without having to factor N or know the value of phi(N), for all the values of N that I have tried it on. Admittedly this program is not practical and the amount of work done is far more than simply just factoring N, but I believe it is of theoretical interest, and maybe someone may be able to improve upon it to make it practical, or maybe the idea has been already thought of and rejected. Any comments on this will be welcome. The theoretical basis for the algorithm has title Possible test for divisor of Phi(N), for anyone who might be interested. This program test a bunch of trial divisors of phi(N) up to the limit specified in line 100, so only those trial divisors less than this value will be returned. It seems to clearly draw a distinction between the divisors of phi(N) and the non-divisors, though admittedly after far too much work to make this practical. However the program following this compromises on certain parameters of the program, such as the size of the factor base and the number of different values of A to try. It correctly identifies with high probability that the primes 2, 3, 83, 103 and 599 are divisors of phi(N) when N = 2^103 -1. The amount of work needed to do this is much more reasonable, but still factoring N would still probably be faster. I only present the algorithm of these programs to demonstrate that one can theoretically test for divisors of phi(N) without first having to factor N. Perhaps it is not practical, but maybe someone better than me can make it so, or it may lead to other useful algorithms. 10 ' Uses least absolute residues to reduce factor base size 20 ' N = prm(30) * prm(50) = 113 * 229 30 ' phi(N) = 2^6 * 3 * 7 * 19 40 ' Finds all divisors of phi(N) 50 word 4:cls 60 dim Fb(10000),Td(2,2000):' max array size DO NOT EXCEED 70 N=prm(30)*prm(50) 80 Fb_sz=fnCalc_FB_Sz(N):' calc factor base size 90 A_sz=fnCalc_A_Sz(Fb_sz):' calc # of A to use for each X 100 Td_sz=50:' # of trial divisors X of phi(N) to try 110 print N = ;N 120 print Factor base = primes 2 to ;prm(Fb_sz) 130 print # of bases A: ;A_sz 140 print Testing primes 2 to;prm(Td_sz) 150 ' *** Generate, Test & Score Trial Divisors X of phi(N) *** 160 for I=0 to Td_sz-1 170 X=prm(I) 180 if X>(N-1)2 then cancel for:goto 360 190 Bs=fnClr_FB(Fb_sz) 200 Td(0,I)=X 210 ' generate various residues & factor over FB() 220 for A=2 to A_sz+2 230 ' if gcd(A,N)>1 then goto 230 240 R=modpow(A,X,N):Rr=N-R:if Rr(N-1)2 then cancel for:goto 330 310 if Fb(J)=0 then Score=Score+10 320 next J 330 Td(1,I)=Score 340 next I 350 ' sort factor base by score & print out 360 Bs=fnDsc_Sort(Td_sz) 370 print Possible divisors of phi(N): 380 for I=0 to 19 390 print Td(0,I),Td(1,I) 400 next I 410 end 420 fnNum_PdivN(P,N) 430 ' Expected # of times P is a factor of N random integers 440 local Sum,Sx 450 if P<2 then goto 500 460 Sum=0:Sx=N 470 Sx=SxP:if Sx=0 then goto 500 480 Sum=Sum+Sx 490 goto 470 500 return(Sum) 510 fnDsc_Sort(N) 520 ' Descending sort 1st N-1 elements of array TD() 530 local I,Flg,Tmp0,Tmp1 540 Flg=0 550 for I=0 to N-2 560 if Td(1,I)>=Td(1,I+1) then goto 610 570 Flg=1 580 Tmp0=Td(0,I+1):Tmp1=Td(1,I+1) 590 Td(0,I+1)=Td(0,I):Td(1,I+1)=Td(1,I) 600 Td(0,I)=Tmp0:Td(1,I)=Tmp1 610 next I 620 if Flg=1 then goto 540 630 return(0) 640 fnFact_FB(R,N) 650 ' Factors R over factor base FB() up to Prm(N) 660 local I,U 670 if abs(R)<2 then goto 770 680 for I=1 to N-1 690 U=prm(I) 700 if R@U>0 then goto 750 710 R=RU 720 Fb(I)=Fb(I)+1 730 if R=1 then cancel for:goto 770 740 goto 700 750 next I 760 if R>1 then Fb(0)=Fb(0)+1 770 return(R) 780 fnClr_FB(N) 790 ' Clears FB() array up to element N-1 800 local I 810 for I=0 to N-1 820 Fb(I)=0 830 next I 840 return(0) 850 fnCalc_FB_Sz(N) 860 ' Calc factor base size, if use least absolute 870 ' residues R, max R ~ (N-1)/2, which may be prime 880 local I,Pmax 890 Pmax=(N-1)2 900 for I=2 to 12251 910 if prm(I)>Pmax then cancel for:goto 940 920 next I 930 I=1 940 return(I-1) 950 fnCalc_A_Sz(Fb_sz) 960 ' Calc # of A's to use to ensure each prime of factor 970 ' base has probability of occurence = at least once 980 local A 990 A=2*prm(Fb_sz) 1000 if A>N-1 then A=N-1 1010 return(A) The following output is obtained - after several minutes of computation. Note that the true divisors of phi(N) all have high scores, while the non-divisors all have score of 0. N = 25877 Factor base = primes 2 to 12923 # of bases A: 25846 Testing primes 2 to 229 Possible divisors of phi(N): 19 12120 7 9340 2 7190 3 7020 1 0 5 0 11 0 13 0 17 0 23 0 29 0 31 0 37 0 41 0 43 0 47 0 53 0 59 0 61 0 67 0 The following program compromises on the parameters of the previous program, and thus is more fuzzy in its results. However it still identifies within the top 10 scores many divisors of phi(N). 10 ' Identifies prime factors 2, 3, 83, 103 & 599 of M103 20 word 10:cls 30 dim Fb(5000),Td(2,1000):' max array size DO NOT EXCEED 40 N=2^103-1 50 Fb_sz=200 60 A_sz=3*prm(Fb_sz) 70 Td_sz=150:' # of trial divisors X of phi(N) to try 80 print N = ;N 90 print Factor base = primes 2 to ;prm(Fb_sz) 100 print # of bases A: ;A_sz 110 print Testing primes 2 to;prm(Td_sz) 120 ' *** Generate, Test & Score Trial Divisors X of phi(N) *** 130 for I=0 to Td_sz-1 140 X=prm(I) 150 if X>(N-1)2 then cancel for:goto 370 160 Bs=fnClr_FB(Fb_sz) 170 Td(0,I)=X 180 ' generate various residues & factor over FB() 190 for A=2 to A_sz+2 200 ' if gcd(A,N)>1 then goto 230 210 R=modpow(A,X*X,N):Rr=N-R:if Rr(N-1)2 then cancel for:goto 340 280 Ex=fnNum_PdivN(prm(J),A_sz) 290 Dif=abs(Fb(J)-Ex) 300 if Ex>0 then Wdif=Dif/Ex else Wdif=Dif 310 Score=Score+Wdif 320 if Fb(J)=0 then Score=Score+50 330 next J 340 Td(1,I)=Score 350 next I 360 ' sort factor base by score & print out 370 Bs=fnDsc_Sort(Td_sz) 380 print Possible divisors of phi(N): 390 for I=0 to 19 400 print Td(0,I),Td(1,I) 410 next I 420 end 430 fnNum_PdivN(P,N) 440 ' Expected # of times P is a factor of N random integers 450 local Sum,Sx 460 if P<2 then goto 510 470 Sum=0:Sx=N 480 Sx=SxP:if Sx=0 then goto 510 490 Sum=Sum+Sx 500 goto 480 510 return(Sum) 520 fnDsc_Sort(N) 530 ' Descending sort 1st N-1 elements of array TD() 540 local I,Flg,Tmp0,Tmp1 550 Flg=0 560 for I=0 to N-2 570 if Td(1,I)>=Td(1,I+1) then goto 620 580 Flg=1 590 Tmp0=Td(0,I+1):Tmp1=Td(1,I+1) 600 Td(0,I+1)=Td(0,I):Td(1,I+1)=Td(1,I) 610 Td(0,I)=Tmp0:Td(1,I)=Tmp1 620 next I 630 if Flg=1 then goto 550 640 return(0) 650 fnFact_FB(R,N) 660 ' Factors R over factor base FB() up to Prm(N) 670 local I,U 680 if abs(R)<2 then goto 780 690 for I=1 to N-1 700 U=prm(I) 710 if R@U>0 then goto 760 720 R=RU 730 Fb(I)=Fb(I)+1 740 if R=1 then cancel for:goto 780 750 goto 710 760 next I 770 if R>1 then Fb(0)=Fb(0)+1 780 return(R) 790 fnClr_FB(N) 800 ' Clears FB() array up to element N-1 810 local I 820 for I=0 to N-1 830 Fb(I)=0 840 next I 850 return(0) This is the output. Note there is no longer a clear distinction between divisors and non-divisors of phi(N). N = 10141204801825835211973625643007 Factor base = primes 2 to 1223 # of bases A: 3669 Testing primes 2 to 863 Possible divisors of phi(N): 3 1083.4524295400385285135 103 604.9451914092674068615 2 597.4499131113493295557 619 372.83237058645136445 823 337.2740150815111568052 347 317.5110992872035363556 379 315.6857063919037966168 83 314.7626013083207306273 811 313.7072659523984019686 599 311.3255338542056126114 67 311.2123353310085655291 167 310.1772617882111763595 401 273.8717707457288138118 257 272.6599300837469459347 457 271.5376592974496433302 499 268.4911567131769994752 179 267.7067183016312623204 433 267.5523987468723091826 31 266.9582656858247732207 787 265.7991150192255056549 ==== My apologies, my first post had an error in the first program listed. Line 230 was commented out and should be uncommented and the goto destination should be changed to 260. This line was originally intended to exclude values of A that were multiples of factors of N, as that would be cheating. When I was experimenting with larger vales of N, this line was no longer needed so I commented it out, but forgot to put it back for the small values of N in this program. The results are not changed much, the non-divisors have a small non-zero score, now, but much smaller than the divisors. Original: 230 ' if gcd(A,N)>1 then goto 230 New: 230 if gcd(A,N)>1 then goto 260 ==== Schatten-$p$ operator. Write $|F|_p$ for the Schatten-$p$ norm of $F$. Is following true, or do you know a counterexample? $|F (A+B)^{1/p}|_p^p le |F A^{1/p}|_p^p + |F B^{1/p} |_p^p$ The cases $p = 1$, $p = 2$ and $p = infty$ (with $A^{1/infty} = A^0 = identity$) are easy to prove, but what about other values of $p$? I have with no result, so any ideas will be welcome. Markus Sigg Approved: Daniel Grayson, dan@math.uiuc.edu, moderator for sci.math.research ==== Ops, please remove the exponents. It should read $|F (A+B)^{1/p}|_p le |F A^{1/p}|_p + |F B^{1/p} |_p$ Or, which means a second question: Keep the exponents, and restrict $p$ to $1 le p le 2$. Markus Sigg ==== > > A mirror site for Abramowitz and Stegun exists at: > > http://jove.prohosting.com/~skripty/ > > - Tom Willis 12. Struve Functions and Eelated Functions ..... 495 Nice to know that they're elated, but isn't it spelt with one e? GC -- ==== Is there anything updated and yet comprehensive in mathematical logic??? Something like Shoenfield's 1967 Mathematical Logic, but updated with the last 35 years of FOM development. Even Barwise's Handbook of Mathematical Logic misses out on the last 25 years of FOM. For set theory, Jech has a wonderfully up-to-date 3rd edition of his Set Theory textbook just published this year, but besides Adamowicz's Logic of Mathematics: A Modern Course of Classical Logic published in 2001, I can't seem to find a corresponding comprehensive graduate-level mathematical logic text. (Adamowicz doesn't seem popular, though I know not why.) ==== > Is there anything updated and yet comprehensive in mathematical > logic??? Something like Shoenfield's 1967 Mathematical Logic, but > updated with the last 35 years of FOM development. Even Barwise's > Handbook of Mathematical Logic misses out on the last 25 years of > FOM. > > For set theory, Jech has a wonderfully up-to-date 3rd edition of his > Set Theory textbook just published this year, but besides > Adamowicz's Logic of Mathematics: A Modern Course of Classical Logic > published in 2001, I can't seem to find a corresponding comprehensive > graduate-level mathematical logic text. (Adamowicz doesn't seem > popular, though I know not why.) Published in 1995, Richard Hodel's An Introduction to Mathematical Logic is a pretty comprehensive graduate-level textbook and has references going right up to 1993, but I really don't know if it touches the last 35 years of FOM development. Perhaps you want to be more specific as to the developments you are referring to. Jim ==== > Is there anything updated and yet comprehensive in mathematical > logic??? Something like Shoenfield's 1967 Mathematical Logic, but Sorry to make a 2nd post, but I thought I'd point out something about the Hodel book that relates to the above comment of yours: The author acknowledges Shoenfield's lectures and calls Shoenfield his mathematical uncle (i.e. close to mathematical father.) So since you are looking for a book something like Shoenfield's perhaps this one will serve as a modern sequel to it (but I'm unfamiliar with Schoenfield's book.) Jim ==== What is known about the following game? There are n non-cooperating players (where the value of n is fixed and is public knowledge), each of whom secretly selects a positive integer. All the numbers are then revealed, and the winner is the player with the smallest number not chosen by any other player (if such a player exists; otherwise there is no winner). -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences ==== I saw this contest in a magazine a long time ago. I beleive the winner chose 2. Unsure how many people participated, maybe between 20 and 50. Michel > What is known about the following game? There are n non-cooperating > players (where the value of n is fixed and is public knowledge), each > of whom secretly selects a positive integer. All the numbers are then > revealed, and the winner is the player with the smallest number not > chosen by any other player (if such a player exists; otherwise there > is no winner). -- ==== > I saw this contest in a magazine a long time ago. I beleive the > winner chose 2. Unsure how many people participated, maybe between > 20 and 50. > > Michel As I understand the problem, a strategy is sought which should work for all participants. This strategy can't be of the type choose k which fixed k, because if all participants would play with this strategy, no one would win. So some type of randomness has to be included. I think an optimal strategy would be one which wins one play out of every n plays in average. A better result cannot be expected if you do not know what the other players will do. Choosing one of the numbers {1,...,n} by chance is the first idea coming to mind. However, this does not win one out of n plays, because it still will happen that all players choose the same number (in average every n^(n-1) plays). And a player who chooses the number 1 at every play would would win against players playing this strategy. So the numbers {1,...,n} have to be chosen with certain probabilities, where the small numbers have to be preferred. What is the optimal probability distribution? Markus ==== > What is known about the following game? There are n non-cooperating players > (where the value of n is fixed and is public knowledge), each of whom secretly > selects a positive integer. All the numbers are then revealed, and the winner > is the player with the smallest number not chosen by any other player (if such > a player exists; otherwise there is no winner). and Michel Grim replied: > I saw this contest in a magazine a long time ago. I beleive the > winner chose 2. Unsure how many people participated, maybe between > 20 and 50. When played anonymously like that, a player could 'spoil' all small numbers by sending in bogus entries. E.g. if he chooses 7 as his real entry, he sends in, under psuedonyms, two 1s, two 2s, ... , two 6s. If everyone entering knows this strategy, it would just become a matter of who has enough time to fill out the most entries. So just changing the *how* people participate in the game changes the game itself. Bob H ==== I have convinced myself that, with three players, an equillibrium has each player use 1 or 2 with probability .5. However, I think things get more complicated for more players. I think the game is unaffected if we require that all numbers be used be between 1 and n, the number of players. This is NOT a game in which the complexity comes from an infinite number of options. In fact, I suspect that the numbers could be restricted to something like 1 to n/2 without changing things. An interesting problem! ==== > I have convinced myself that, with three players, an equillibrium >has each player use 1 or 2 with probability .5. However, I think things >get more complicated for more players. I think the game is unaffected if we require that all numbers be used >be between 1 and n, the number of players. This is NOT a game in which >the complexity comes from an infinite number of options. In fact, I suspect >that the numbers could be restricted to something like 1 to n/2 without >changing things. If I am one of three players, and the other two are choosing 1 or 2 with probability .5, I can always choose 3 and win half the time. -M- ==== > I have convinced myself that, with three players, an equillibrium >has each player use 1 or 2 with probability .5. No. If you are in a three-player game and both opponents are using this mixed strategy, they will choose the same number with probability 1/2, and so if you choose 3 you will win with probability 1/2. For the three-player case, I believe an equilibrium strategy must choose j with probability p_j satisfying the equations p_j^2 + (1-p_1-...-p_{j+1})^2 = (1-p_1-...-p_j)^2 , j = 1,2,3... together with, of course, sum_j p_j = 1 and all p_j > 0. This is because if two players are using this mixed strategy, the third will have no reason to switch iff his probability of winning is the same for all pure strategies, and his probability of winning by choosing j is sum_{i=1}^{j-1} p_i^2 + (1-p_1-...-p_j)^2 (i.e. he wins if both opponents choose the same number < j or both choose any numbers > j). This system of equations looks rather formidable to solve; for n > 3 players, it will be even more complicated. In any case, there can't be an equilibrium strategy that involves sometimes using m and never using m+1, because then each player would have an incentive to use m+1 instead of m (if the opponents never use m+1, then m+1 would win in every case where m would win, and would also win in some cases where m would not). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== >For the three-player case, I believe an equilibrium strategy must >choose j with probability p_j satisfying the equations >p_j^2 + (1-p_1-...-p_{j+1})^2 = (1-p_1-...-p_j)^2 , j = 1,2,3... >together with, of course, sum_j p_j = 1 and all p_j > 0. Or rather, this would be required for a symmetric equilibrium where all three players use the same mixed strategy. >In any case, there can't be an equilibrium strategy that involves >sometimes using m and never using m+1, because then each player would >have an incentive to use m+1 instead of m (if the opponents never use >m+1, then m+1 would win in every case where m would win, and would also >win in some cases where m would not). I should be more precise: there can't be a Nash equilibrium where no player ever uses m+1, at least two players sometimes use m, and it is possible for one of those players to win with m. In the three-player case, there is a Nash equilibrium with player 1 always choosing 1, player 2 choosing 2 and player 3 choosing 3. Here player 1 wins all the time, but neither player 2 nor 3 can improve his/her own probability by a unilateral change in strategy, although of course players 2 and 3 together can improve both their chances by sometimes choosing 1. This shows one of the pitfalls of the notion of Nash equilibrium in multi-player games. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== >>For the three-player case, I believe an equilibrium strategy must >>choose j with probability p_j satisfying the equations >>p_j^2 + (1-p_1-...-p_{j+1})^2 = (1-p_1-...-p_j)^2 , j = 1,2,3... >>together with, of course, sum_j p_j = 1 and all p_j > 0. >Or rather, this would be required for a symmetric equilibrium where >all three players use the same mixed strategy. A solution is p_j = (1-r) r^(j-1) where r is the real root of the polynomial z^3+z^2+z-1, approximately 0.54368901269207636157. This gives each player probability r^2 (or approximately 0.29559774252208477098) of winning. Note that this is a Nash equilibrium since if any two players use this mixed strategy, the third player has probability r^2 of winning no matter what he does, and therefore has no incentive to deviate from the given strategy. Other Nash equilibria, where one player (say A) always chooses 1, are rather interesting: once it is known that A will be choosing 1, B and C are in a kind of Prisoner's Dilemma situation. They could each achieve probability 1/4 of winning if both used the strategy (p_1,p_2) = (1/2, 1/2). But if one used (0,1) while the other used (1/2, 1/2), the one using (0,1) would raise his/her winning probability to 1/2. So they can end up in a Nash equilibrium where neither ever chooses 1 and A always wins. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ==== Here are this week's titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modification. Titles in the mathematics arXiv (21 Jul - 25 Jul) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0307294 three-dimensional local rings math.AC/0307281 Anthony Iarrobino: Ancestor ideals of vector spaces of forms, and level algebras AG: Algebraic Geometry ---------------------- math.AG/0307326 S. V. Shadrin: Intersections in genus 3 and the Boussinesq hierarchy math.AG/0307325 hep-th/0307167 Brian Forbes: Open string mirror maps from Picard- Fuchs equations on relative cohomology math.AG/0307301 Gavin Brown, Alessio Corti, Francesco Zucconi: Birational Geometry of 3-fold Mori Fibre Spaces math.AG/0307299 Montserrat Teixidor i Bigas: Subbundles of maximal degree math.AG/0307298 Montserrat Teixidor i Bigas: Rank two vector bundles with canonical determinant math.AG/0307296 E.Artal, J.Carmona, J.I.Cogolludo, M.Marco: Topology and combinatorics of real line arrangements math.AG/0307260 P. M. H. Wilson: Sectional curvatures of Kahler moduli AP: Analysis of PDEs -------------------- math.AP/0307295 M. C. Lopes Filho, H. J. Nussenzveig Lopes, G. V. Planas: On the inviscid limit for 2D incompressible flow with Navier friction condition math.AP/0307291 Adam Sikora: Riesz transform, Gaussian bounds and the method of wave equation math.AP/0307289 Terence Tao: Global well-posedness of the Benjamin-Ono equation in H^1(R) math.AP/0307262 Wenxiong Chen, Congming Li, Biao Ou: Qualitative Properties of Solutions for an Integral Equation math.AP/0307253 Gunther Uhlmann, Andras Vasy: Fixed energy inverse problem for exponentially decreasing potentials AT: Algebraic Topology ---------------------- math.AT/0307327 Philippe Gaucher: The homotopy branching space of a flow CA: Classical Analysis and ODEs ------------------------------- math.CA/0307323 Joaquim Bruna, Alexander Olevskii, Alexander Ulanovskii: Completeness in $L^1(R)$ of discrete translates CO: Combinatorics ----------------- math.CO/0307315 Michel Lassalle, Michael Schlosser: An analytic formula for Macdonald polynomials math.CO/0307292 Denis Chebikin, Pavlo Pylyavskyy: Two bijections between G-parking functions and spanning trees math.CO/0307280 Jessica Sidman: Defining equations of subspace arrangements embedded in reflection arrangements math.CO/0307271 Lauren K. Williams: Enumeration of totally positive Grassmann cells math.CO/0307269 Paul Terwilliger, Chih-wen Weng: Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra math.CO/0307252 Ioana Dumitriu, Etienne Rassart: Path counting and random matrix theory DG: Differential Geometry ------------------------- math.DG/0307309 Masaaki Umehara, Kotaro Yamada: Maximal surfaces with singularities in Minkowski space math.DG/0307303 Denis Kochan, Pavol Severa: Differential gorms, differential worms math.DG/0307293 Albert Chau, Oliver C. Schnuerer: Stability of gradient Kaehler-Ricci solitons math.DG/0307288 Jian Song: The alpha-Invariant on Toric Fano Manifolds math.DG/0307286 Lars Andersson: Bel--Robinson energy and constant mean curvature foliations math.DG/0307282 Iakovos Androulidakis: Connections and holonomy for extensions of Lie groupoids math.DG/0307278 P.T. Chrusciel, R. Bartnik: Boundary value problems for Dirac--type equations, with applications math.DG/0307275 Lei Ni: A monotonicity formula on complete Kahler manifolds with nonnegative bisectional curvature math.DG/0307273 Josef Dorfmeister, Junichi Inoguchi, Magdalena Toda: Weierstra{ss} type representation of timelike surfaces with constant mean curvature math.DG/0307272 Magdalena Toda: Weierstrass-type Representation of Weakly Regular Pseudospherical Surfaces in Euclidean Space math.DG/0307270 Magdalena Toda: Initial Value Problems of the Sine-Gordon Equation and Geometric Solutions math.DG/0307267 Kenro Furutani: A Kaehler structure on the punctured cotangent bundle of the Cayley projective plane math.DG/0307266 Kenro Furutani: Quantization of the Geodesic flow on Quaternion Projective Spaces math.DG/0307261 Sarah Hansoul, Pierre B. A. Lecomte: Affine representations of Lie algebras and geometric interpretation in the case of smooth manifolds nlin.SI/0307021 Claudio Bartocci, Gregorio Falqui, Marco Pedroni: A geometric approach to the separability of the Neumann-Rosochatius system DS: Dynamical Systems --------------------- math.DS/0307316 C. M. Carballo, C. A. Morales: Omega-limit sets close to singular-hyperbolic attractors math.DS/0307290 A. J. Roberts: A step towards holistic discretisation of stochastic partial differential equations math.DS/0307259 Charles Holton, Charles Radin, Lorenzo Sadun: Conjugacies for Tiling Dynamical Systems FA: Functional Analysis ----------------------- math.FA/0307317 Gestur Olafsson, Darrin Speegle: Groups, Wavelets, and Wavelet Sets math.FA/0307312 Daniel M. Pellegrino: Almost summing mappings math.FA/0307311 Daniel M. Pellegrino: Cotype and nonlinear absolutely summing mappings math.FA/0307285 Daniel M. Pellegrino: On ideals of polynomials and their applications math.FA/0307274 Sandrine Grellier & Mohammad Kacim: Multilinear Hankel operator GR: Group Theory ---------------- math.GR/0307321 Henry Cohn, Christopher Umans: A group-theoretic approach to fast matrix multiplication GT: Geometric Topology ---------------------- math.GT/0307314 Ian Hambleton, Matthias Kreck: Homotopy self-equivalences of 4-manifolds math.GT/0307302 Alexander Barchechat: Minimal Triangulations of Reducible 3-Manifolds math.GT/0307297 Ian Hambleton, Mihail Tanase: Permutations, isotropy and smooth cyclic group actions on definite 4-manifolds math.GT/0307283 Ulrich Oertel: Incompressible maps of surfaces and Dehn filling math.GT/0307276 Ulrich Oertel, Jacek Swiatkowski: A contamination carrying criterion for branched surfaces math.GT/0307254 Greg Friedman: Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata LO: Logic --------- math.LO/0307284 William McCune, Ranganathan Padmanabhan, Robert Veroff: Yet Another Single Law for Lattices MP: Mathematical Physics ------------------------ nlin.SI/0307042 Vladimir Dorodnitsyn, Roman Kozlov, Pavel Winternitz: Continuous symmetries of Lagrangians and exact solutions of discrete equations math-ph/0307050 Yuri G. Kondratiev, Maria Jo~ao Oliveira: Invariant measures for Glauber dynamics of continuous systems math-ph/0307049 Fei Wang: Note on the asymptotic approximation of a double integral with an angular spectrum representation nlin.SI/0307026 P.G.Grinevich, P.M.Santini: The initial boundary value problem on the segment for the Nonlinear Schrodinger equation; the algebro-geometric approach. I math-ph/0307048 Claudio D'Antoni, Gerardo Morsella, Rainer Verch: Scaling algebras for charged fields and short-distance analysis for localizable and topological charges math-ph/0307047 Dariusz Chruscinski: Quantum Mechanics of Damped Systems II. Damping and Parabolic Potential Barrier math-ph/0307046 Elliott H. Lieb, Michael Loss: Existence of Atoms and Molecules in Non-Relativistic Quantum Electrodynamics hep-th/0307199 Michael Forger, Hartmann Romer: Currents and the Energy-Momentum Tensor in Classical Field Theory: A fresh look at an Old Problem math-ph/0307045 Chin-Sheng Wu: The Comparison between the Infinitesimal Operators for SU(3) and Boson Operators in Cartan-Weyl Basis hep-th/0307186 Henriette Elvang, Predrag Cvitanovi'c, Anthony D. Kennedy: Diagrammatic Young Projection Operators for U(n) math-ph/0307044 Andreas U. Schmidt: Mathematics of the Quantum Zeno Effect math-ph/0307043 N.G.Marchuk, S.E.Martynova: Notions of determinant, spectrum, and Hermitian conjugation of Clifford algebra elements math-ph/0307042 N.G.Marchuk: A coordinateless form of the Dirac equation math-ph/0307041 J. Guerrero, J.L. Jaramillo, V. Aldaya: Group-cohomology refinement to classify $G$-symplectic manifolds math-ph/0307040 S. V. Lototsky, B. L. Rozovskii: Time Evolution of a Passive Scalar in a Turbulent Incompressible Gaussian Velocity Field math-ph/0307039 E. G. Kalnins, J. M. Kress, W. Miller Jr, P. Winternitz: Superintegrable Systems in Darboux spaces math-ph/0307038 Peter Michael Jack: Physical Space as a Quaternion Structure, I: Maxwell Equations. A Brief Note hep-th/0307141 A. Mikovic: String Theory and Quantum Spin Networks math-ph/0307037 O. Babelon: Equations in dual variables for Whittaker functions hep-th/0306287 Roberto Zucchini: Global Aspects of Abelian and Center Projections in SU(2) Gauge Theory NA: Numerical Analysis ---------------------- math.NA/0307313 Fabricio Macia: Wigner measures in the discrete setting: high-frequency analysis of sampling & reconstruction operators NT: Number Theory ----------------- math.NT/0307322 Tim Dokchitser: LLL & Abc math.NT/0307308 Jonathan Sondow: An irrationality measure for Liouville numbers and conditional measures for Euler's constant math.NT/0307300 Joel Bellaiche: Augmentation du niveau pour U(3) (Level-Raising for U(3)) math.NT/0307279 Werner Georg Nowak: Primitive lattice points inside an ellipse math.NT/0307264 Takashi Aoki, Yasuo Ohno: Sum relations for multiple zeta values and connection formulas for the Gauss hypergeometric functions OC: Optimization and Control ---------------------------- math.OC/0307305 Steven J. Benson, Todd S. Munson: Flexible Complementarity Solvers for Large-Scale Applications PR: Probability Theory ---------------------- math.PR/0307310 Itai Benjamini, Zhen-Qing Chen, Steffen Rohde: Boundary Trace of Reflecting Brownian Motions math.PR/0307307 Alexander Gnedin, Jim Pitman: Regenerative Composition Structures math.PR/0307287 Jon Warren, Shinzo Watanabe: On Spectra of Noises associated with Harris flows math.PR/0307265 V. P. Maslov: Approximation probabilities, the law of quasistable markets, and phase transitions from the condensed state QA: Quantum Algebra ------------------- math.QA/0307324 Michael F. Mueller-Bahns, Nikolai Neumaier: Invariant Star Products of Wick Type: Classification and Quantum Momentum Mappings math.QA/0307306 A.A.Stolin, P.P.Kulish, E.V.Damaskinsky: On construction of universal twist element from $R$-matrix math.QA/0307277 Philippe Bonneau, Daniel Sternheimer: Topological Hopf algebras, quantum groups and deformation quantization math.QA/0307263 RA: Rings and Algebras ---------------------- math.RA/0307320 P. Ara, M.A. Gonzalez-Barroso, K.R. Goodearl, E. Pardo: Fractional skew monoid rings math.RA/0307304 Peter Jorgensen: Linear free resolutions over non-commutative algebras math.RA/0307258 Bangming Deng, Jie Du: On bases of quantized enveloping algebras math.RA/0307257 Bangming Deng, Jie Du: Monomial bases for quantum affine sl_n math.RA/0307256 Bangming Deng, Jie Du: Frobenius morphisms and representations of algebras math.RA/0307255 Shouchuan Zhang: Duality Theorem and Drinfeld Double in Braided Tensor Categories RT: Representation Theory ------------------------- math.RT/0307268 G. Lusztig: Character sheaves on disconnected groups, II SG: Symplectic Geometry ----------------------- math.SG/0307319 Ping Xu: Momentum Maps and Morita Equivalence math.SG/0307318 Jose Agapito: A weighted version of quantization commutes with reduction principle for a toric manifold -- / Greg Kuperberg (UC Davis) / / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that's fit to e-print * ==== Here are this week's titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modification. Titles in the mathematics arXiv (28 Jul - 1 Aug) ------------------------------------------------ AC: Commutative Algebra ----------------------- math.AC/0307403 Sara Faridi: Cohen-Macaulay Properties of Square-Free Monomial Ideals AG: Algebraic Geometry ---------------------- math.AG/0307398 Kang Zuo, Eckart Viehweg: Complex multiplication, Griffiths-Yukawa couplings, and rigidy for families of hypersurfaces math.AG/0307387 Andrea D'Agnolo, Pietro Polesello: Stacks of twisted modules and integral transforms math.AG/0307386 Artur Elezi: Virtual Class of Zero Loci and Mirror Theorems math.AG/0307378 Marta Casanellas, Elena Drozd, Robin Hartshorne: Gorenstein Liaison and ACM Sheaves math.AG/0307369 Thierry Zell: Topology of definable Hausdorff limits math.AG/0307366 Ricardo Garcia Lopez: Microlocalization and stationary phase math.AG/0307361 Hans-Christian v. Bothmer: Last syzygies of 1-generic spaces math.AG/0307355 Viacheslav V. Nikulin: On Correspondences of a K3 Surface with itself I AP: Analysis of PDEs -------------------- math.AP/0307406 Guenther Hoermann: First-order hyperbolic pseudodifferential equations with generalized symbols math.AP/0307400 Xavier Carvajal: Well-posedness for a higher order nonlinear Schrodinger equation in Sobolev spaces of negative indices math.AP/0307397 Chu-Pin Lo: A Blowup Problem of Reaction Diffusion Equation Related to the Diffusion Induced Blowup Phenomenon math.AP/0307377 Lubomira G. Softova: Poincar'e problem for a kind of parabolic operators math.AP/0307344 Chongsheng Cao, Edriss S. Titi, Mohammed Ziane: A ``horizontal hyper--diffusion $3-D$ thermocline planetary geostrophic model: well-posedness and long time behavior AT: Algebraic Topology ---------------------- math.AT/0307339 Wolfgang Pitsch, Jerome Scherer: Homology fibrations and group-completion revisited CA: Classical Analysis and ODEs ------------------------------- math.CA/0307372 Timoteo Carletti, Gabriele Villari: A note on existence and uniqueness of limit cycles for Li'enard systems math.CA/0307348 meaningful functions CO: Combinatorics ----------------- math.CO/0307405 Jeremy L. Martin: On the topology of multigraph picture spaces math.CO/0307401 Narad Rampersad: Words avoiding 7/3-powers and the Thue-Morse morphism math.CO/0307399 Martin Klazar: On the least exponential growth admitting uncountably many closed permutation classes math.CO/0307380 Alina Vdovina: Groups, periodic planes and hyperbolic buildings math.CO/0307370 David Orden, Francisco Santos, Brigitte Servatius, Herman Servatius: Combinatorial pseudo-Triangulations math.CO/0307365 Narad Rampersad: A note on non-repetitive colourings of planar graphs math.CO/0307363 Narad Rampersad: A note on avoidable words in squarefree ternary words math.CO/0307359 Denis Chebikin: Graph powers and k-ordered Hamiltonicity math.CO/0307357 Svante Linusson, Johan Wastlund: A proof of a conjecture of Buck, Chan and Robbins on the random assignment problem math.CO/0307350 Jesus De Loera, David Haws, Raymond Hemmecke, Peter Huggins, Bernd Sturmfels, Ruriko Yoshida: Short Rational Functions for Toric Algebra and Applications math.CO/0307347 Ruth Haas, David Orden, Guenter Rote, Francisco Santos, Brigitte Servatius, Herman Servatius, Diane Souvaine, Ileana Streinu, Walter Whiteley: Planar Minimally Rigid Graphs and Pseudo-Triangulations cond-mat/0307606 J. Bouttier, P. Di Francesco, E. Guitter: Statistics of planar graphs viewed from a vertex: A study via labeled trees CV: Complex Variables --------------------- math.CV/0307335 H Gaussier, A Sukhov: Wong-Rosay Theorem in almost complex manifolds math.CV/0307334 H Gaussier, A Sukhov: Estimates of the Kobayashi metric on almost complex manifolds math.CV/0307332 B Coupet, H Gaussier, A Sukhov: Riemann maps in almost complex manifolds DG: Differential Geometry ------------------------- hep-th/0307285 Marco Matone: The Affine Connection of Supersymmetric SO(N)/Sp(N) Theories gr-qc/0307117 David Maxwell: Solutions of the Einstein Constraint Equations with Apparent Horizon Boundary math.DG/0307375 M.L. Barberis, I. Dotti: Complex structures on affine motion groups math.DG/0307374 Boris Dubrovin: On almost duality for Frobenius manifolds math.DG/0307373 Kiyonori Gomi: Equivariant smooth Deligne cohomology math.DG/0307368 C. Jang, K. Park, P.E. Parker: PseudoH-type 2-step nilpotent Lie groups math.DG/0307360 Ilka Agricola, Thomas Friedrich: Killing spinors in supergravity with 4-fluxes math.DG/0307338 Lars Andersson: Constant mean curvature foliations of simplicial flat spacetimes DS: Dynamical Systems --------------------- math.DS/0307394 Chu-Pin Lo, Nedialko S. Nedialkov, Juan-Ming Yuan: Classification of Steadily Rotating Spiral Waves for the Kinematic Model math.DS/0307389 Lennard F. Bakker: Quasiperiodic Flows and Algebraic Number Fields math.DS/0307384 Idris Assani, Zoltan Buczolich, Daniel Mauldin: An $L^1$ counting problem in ergodic theory math.DS/0307379 Xianghong Gong: Existence of divergent Birkhoff normal forms of Hamiltonian functions math.DS/0307371 Lasse Rempe: A Landing Theorem for Periodic Rays of Exponential Maps math.DS/0307329 C. Azevedo, P. Ontaneda: On the fixed homogeneous circle problem FA: Functional Analysis ----------------------- math.FA/0307367 Ken Dykema, Nate Strawn: Manifold structure of spaces of spherical tight frames math.FA/0307337 Daniel M. Pellegrino: A remark on absolutely summing multilinear mappings GM: General Mathematics ----------------------- math.GM/0307395 Vaclav Studeny: Functional Equation of the Rate of Inflation GR: Group Theory ---------------- math.GR/0307362 D. Kotschick: Quasi-homomorphisms and stable lengths in mapping class groups math.GR/0307345 Arturo Magidin: Capability of certain nilpotent products of cyclic groups GT: Geometric Topology ---------------------- math.GT/0307396 Gwenael Massuyeau: Cohomology rings, Rochlin function, linking pairing and the Goussarov-Habiro theory of 3-manifolds math.GT/0307382 Benjamin A. Burton: Face pairing graphs and 3-manifold enumeration math.GT/0307340 Paolo Ghiggini: Tight Contact structures on Seifert Manifolds over $T^2$ with one singular fibre math.GT/0307328 Greg Friedman: Alexander polynomials of non-locally-flat knots KT: K-Theory and Homology ------------------------- math.KT/0307354 J.M. Casas, M. Ladra, T. Pirashvili: Triple Cohomology of Lie-Rinehart Algebras and the Canonical Class of Associative Algebras LO: Logic --------- math.LO/0307388 Vinay Deolalikar, Joel David Hamkins, Ralf-Dieter Schindler: P is not equal to NP intersect coNP for Infinite Time Turing Machines MG: Metric Geometry ------------------- math.MG/0307342 Marius Buliga: Tangent bundles to sub-Riemannian groups MP: Mathematical Physics ------------------------ quant-ph/0307232 R. M. Cavalcanti, P. Giacconi, R. Soldati: Decay in a uniform field: An exactly solvable model quant-ph/0307206 Asoka Biswas, G. S. Agarwal: Strong subadditivity inequality for quantum math-ph/0307065 Debasis Biswas, Asoke P. Chattopadhyay: Generalised definitions of certain functions and their uses math-ph/0307064 Zhenquan Li, A.J. Roberts: Low-dimensional modelling of a generalized Burgers equation math-ph/0307063 N.S. Witte: Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as $tau$-Functions -- Spectrum Singularity case math-ph/0307062 Ivan Veselic': Integrated density of states and Wegner estimates for random Schrodinger Operators math-ph/0307061 Bernhard G. Bodmann: A lower bound for the Wehrl entropy of quantum spin with sharp high-spin asymptotics cond-mat/0307698 Petr Jizba, Toshihico Arimitsu: On observability of Renyi's entropy cond-mat/0307649 Malte Henkel, Alan Picone, Michel Pleimling, Jeremie Unterberger: Local scale invariance and its applications to strongly anisotropic critical phenomena math-ph/0307060 Jaroslaw Wawrzycki: Generally covariant Quantum Mechanics math-ph/0307059 N.P. Landsman: Functorial quantization and the Guillemin-Sternberg conjecture math-ph/0307058 Frederic Lesage, Jorgen Rasmussen: SLE-type growth processes and the Yang-Lee singularity math-ph/0307057 Anna Jencova: Flat connections and Wigner-Yanase-Dyson metrics math-ph/0307056 Michele Correggi, Gianfausto Dell'Antonio: Rotating Singular Perturbations of the Laplacian math-ph/0307055 P.M. Bleher, A.B.J. Kuijlaars: Random matrices with external source and multiple orthogonal polynomials math-ph/0307054 K.Thirulogasanthar, G.Honnouvo: Coherent states labeled by the iterates of a complex function math-ph/0307053 Christian Gerard, Christian Jaekel: Thermal Quantum Fields with Spatially Cut-off Interactions in 1+1 Space-time Dimensions math-ph/0307052 Bertrand Eynard: Large N expansion of the 2-matrix model, multicut case math-ph/0307051 Tom Michoel, Bruno Nachtergaele: The large-spin asymptotics of the ferromagnetic XXZ chain hep-th/0307235 Conformally Invariant Quantum Field Theory gr-qc/0307103 Sergiu I. Vacaru: Exact Solutions with Noncommutative Symmetries in Einstein and Gauge Gravity NT: Number Theory ----------------- math.NT/0307376 David Goss: Applications of non-Archimedean integration to the $L$-series of $tau$-sheaves math.NT/0307352 Pieter Moree, Huib Hommersom: Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients OC: Optimization and Control ---------------------------- math.OC/0307333 Ivar Ekeland: A duality theory for some non-convex functions of matrices math.OC/0307331 Paolo d'Alessandro: A new conical internal evolutive LP algorithm PR: Probability Theory ---------------------- math.PR/0307353 Wendelin Werner: Conformal restriction and related questions math.PR/0307346 D. Khoshnevisan, D. A. Levin, P. J. Mendez-Hernandez: On Dynamical Gaussian Random Walks math.PR/0307336 Fabio Martinelli, Alistair Sinclair, Dror Weitz: Glauber dynamics on trees:Boundary conditions and mixing time math.PR/0307330 Wlodzimierz Bryc, Amir Dembo, Tiefeng Jiang: Spectral measure of large random Hankel, Markov and Toeplitz matrices QA: Quantum Algebra ------------------- math.QA/0307402 I. Heckenberger, S. Kolb: De Rham Complex for Quantized Irreducible Flag Manifolds math.QA/0307393 Yu. I. Manin: Functional equations for quantum theta functions math.QA/0307391 N. Aizawa, P. S. Isaac: Weak Hopf algebras corresponding to $U_q[sl_n]$ math.QA/0307381 Alexander V. Karabegov: On Dequantization of Fedosov's Deformation Quantization math.QA/0307364 James Conant, Ferenc Gerlits, Karen Vogtmann: Cut vertices in commutative graphs math.QA/0307356 Vadim V. Borzov, Eugene V. Damaskinsky: Generalized coherent states for q-oscillator connected with q-Hermite polynomials math.QA/0307351 S. Majid: Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere hep-th/0307168 G.A. Goldin, S. Majid: On the Fock space for nonrelativistic anyon fields and braided tensor products RA: Rings and Algebras ---------------------- math.RA/0307392 Osamu Iyama: The relationship between homological properties and representation theoretic realization of artin algebras math.RA/0307385 Gert K. Pedersen, Francesc Perera: Inverse limits of rings and Multiplier rings RT: Representation Theory ------------------------- math.RT/0307390 Fr'ed'eric Latour: Representations of rational Cherednik algebras of rank 1 in positive characteristic math.RT/0307383 Anthony Henderson: Representations of wreath products on cohomology of De Concini-Procesi compactifications math.RT/0307349 Konstanze Rietsch: An introduction to perverse sheaves math.RT/0307343 Mark Davidson, Gestur Olafsson: The Generalized Segal-Bargmann transform and Special Functions SG: Symplectic Geometry ----------------------- math.SG/0307404 Lisa Jeffrey, Nan-Kuo Ho: The volume of the moduli space of flat connections on a nonorientable 2-manifold math.SG/0307358 Junho Lee: Counting Curves in Elliptic Surfaces by Symplectic Methods math.SG/0307341 Paolo Lisca, Andras I. Stipsicz: Seifert fibered contact three--manifolds via surgery -- / Greg Kuperberg (UC Davis) / / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math that's fit to e-print * ==== > I'm curious to know if there is anyone studying the group structure of > first questions is on connectedness. Also, is there a theory on > infinite-dimensional Lie groups? need to go as far as suggested by other posters and use Fr.8echet-Lie groups. The traditional theory of Banach-Lie groups, developed in the 1930s, works very nicely here. And Banach-Lie groups are as close to finite dimensional Lie groups as possible in many respects, the only difference being that the Lie-Cartan theorem not necessarily works, but cohomology theory of central extensions provides a very satisfactory picture of when a Lie algebra can be enlarged to a Lie group. As an introductory reference, I'd suggest Bourbaki's Lie Groups and Lie Algebras volumes, where the entire theory is developed for Banach-Lie groups right from the start, read in conjunction with the indispensable monograph by Pierre de la Harpe, Classical Banach-Lie Notes in Math., volume 285. cheers, Vladimir Pestov ==== >Densely ordered means ordered in such a way that between >any two elements there is another. >A corollary is that having a countable dense subset >suffices for your purpose. -- Mike Hardy By dense subset do you mean topologically dense subset? Then I've counterexample. Let S = Rx{0,1} ordered lexicographically (r,a) <= (s,b) when r < s or r = s, a <= b S is the double pointed line. Write r_a for (r,a). A countable dense subset, that's also densely ordered, is Qx{0}. S is also first countable, but does S embed into the reals R ? If so, then S would have to be second countable, which I doubt as there are uncountably many (r_0, s_1) = [r_1, s_0] where r < s. Note: for [r_1, s_0] to be an open set, ie a union of open base sets, there has to be an open base set for which r_1 is the first element. ---- ==== > A theorem of Cantor says any two densely ordered sets > without endpoints that are countable are order-isomorphic. > Densely ordered means ordered in such a way that between > any two elements there is another. > > A corollary is that having a countable dense subset > suffices for your purpose. -- Mike Hardy William Elliot (mars@agora.rdrop.com) answered: > By dense subset do you mean topologically dense subset? If B is a linearly ordered set and A is a subset of B, then to say that A is dense in B means that between any two members of B there is a member of A. I think that's the same as topologically dense if you put the order topology on B. > Then I've counterexample. > > Let S = Rx{0,1} ordered lexicographically > (r,a) <= (s,b) when r < s or r = s, a <= b > S is the double pointed line. Write r_a for (r,a). > > A countable dense subset, that's also densely ordered, is Qx{0}. Is that dense? Let's see ... between (0, 0.1) and (0, 0.2) there is no member of Qx{0}, so Qx{0} is not dense according to the definition I stated above. It's also not topologically dense, since the interval from (0, 0.1) to (0, 0.2) is an open set that does not intersect Qx{0}. So this is not actually a counterexample. Mike Hardy ==== > A theorem of Cantor says any two densely ordered sets > without endpoints that are countable are order-isomorphic. > Densely ordered means ordered in such a way that between > any two elements there is another. > A corollary is that having a countable dense subset > suffices for your purpose. -- Mike Hardy William Elliot (mars@agora.rdrop.com) answered: > Then I've counterexample. > > Let S = Rx{0,1} ordered lexicographically > (r,a) <= (s,b) when r < s or r = s, a <= b > S is the double pointed line. Write r_a for (r,a). > > A countable dense subset, that's also densely ordered, is Qx{0}. I answered: > Is that dense? Let's see ... between (0, 0.1) and (0, 0.2) > there is no member of Qx{0}, so Qx{0} is not dense according to > the definition I stated above. It's also not topologically dense, > since the interval from (0, 0.1) to (0, 0.2) is an open set that > does not intersect Qx{0}. So this is not actually a counterexample. Oh .... I see that you had curly braces: {0,1}, denoting a set with two members. I had read this as [0,1], with square brackets, denoting the closed unit interval. It is not the case that strictly between any two members of your linarly ordered set Qx{0,1} (with curly braces) there is a member of the subset Qx{0}, so in that sense the subset is not dense in the larger set. I didn't have in mind topological denseness necessarily. Mike Hardy