Let $Omega$ be an open set in $R^n$, and $f in L^1_{loc}(Omega)$, such that foreach multiindex $alphain N^n$, $|alpha| = l$ f has weak derivative $D^alpha f$ in $L^p(Omega)$, with $1leq pleq infty$.
In general it is not true that $fin L^p(Omega)$, but it have to be true that $fin L^p_{loc}(Omega)$. How can be shown that?
I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, ldots, X_k$. Suppose I have observed that $sum_i X_i = n$. In this case I do not want to treat them as independent Bernoulli random variables. Instead, I would like something like the multinomial:
$P(X_1=x_1, ldots, X_k=x_k) = f(x_1, ldots, x_k; n, p_1, ldots, p_k) = frac{n!}{x_1! cdots x_k!} prod_{i=1}^k p_i^{x_i}$
but instead of the $x_i$ being nonnegative integers, I want them restricted to be either 0 or 1. I have been trying to see if the multivariate hypergeometric is appropriate, but I'm not sure.
Thanks in advance for any advice.
Is your graph topologically planar or non-planar, weighted or unweighted, directed or undirected?
Do you want an algorithm and/or a formula/bound?
For bounds on planar graphs, see Alt et al. On the number of simple cycles in planar graphs
For an algorithm, see the following paper. It incrementally builds k-cycles from (k-1)-cycles and (k-1)-paths without going through the rigourous task of computing the cycle space for the entire graph. It also handles duplicate avoidance.
Mathematics is an activity of investigation and exploration. Informally, both calculi and an algebras are tools which consist of sets of symbols and systems of rules (usually called axioms) for manipulating those symbols.
Calculi tend to be specified/defined/explored/used to answer questions of "calculation" or reckoning, in some very general sense. Calculi tend to be used to investigate properties of objects (i.e "What is the area under the curve?")
Algebras tend to be specified/defined/explored/used to answer questions about how different "things" are related, in some very general sense. Algebras tend to be used to study the relationship between objects. (i.e. "Is this equation 'the same' as that equation?")
I think it is safe to say that the term "algebra" today, carries a bit more meaning to most mathematicians than the general teram "calculus".
As examples:
The Calculus (as taught in high-school or undergraduate university), also known as "infinitesimal calculus", is a calculus focused on limits, functions, derivatives, integrals, and infinite series. It is chiefly concerned with calculations or answering questions about change. The Calculus uses the complex numbers (chiefly) as a foundation for this investigation.
Opening a book on computer science, you might find a "calculus of computation" which might involve symbols and rules which let one "calculate" or "discover" behavioral properties of a computer program. As a foundation, such a calculus might use "states" and "transitions", instead of the complex numbers, to ground the investigation.
Elementary Algebra (ie. high-school algebra) is, informally, the study of relationships of variables and structures (e.g. equations) arising from combining variables according to certain rules (i.e. performing "operations"). It uses the complex numbers as the basic foundation in which one could "check" or "verify" statements, but quickly one finds that "calculating with numbers" is not that useful (or practical) in investigating relationships between equations.
"The general theory of arithmetic operations is algebra: so we can also develop an algebra of set theory." - Concepts of Modern Mathematics, Ian Stewart
In that sense, Elementary Algebra is more "abstract" than arithmetic, and is often the subject where schools (specifically bad teachers) lose a student's interest and attention in mathematics. It is a tragedy, since it is exactly at Elementary Algebra that things get interesting.
In computer science or other engineering disciplines, you might find a "process algebra" when reasoning about how various states of a computer program relate to each other. We can ask questions like "is a specification of a collection of processes 'functionally equivalent' to another specification (i.e do they do the same thing? as in the case of a particular hardware design versus a software program)? The same "process algebra" could possibly be used to reason about how the various "states" of a garage door opener relate to each. Such an algebra might use states, transitions, and time as a foundation.
sigstop
There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^delta$. He showed that solvable Lie groups have the same homology with finite coefficients as their underlying discrete groups. Morel recently announced a proof of the Friedlander-Milnor conjecture that implies this is also true for complex algebraic Lie groups.
I've always been curious about this problem, and I've discussed it with a number of other mathematicians. One question that I haven't been able to sort out about it was to what extent the homology of a Lie group with coefficients in an arbitrary abelian group $A$ made discrete is determined, e.g. whether the homology groups being nonzero, or finitely generated, is independent of ZFC. This problem involves aspects of both manifolds and the projective resolution of modules, and hence it might be plausible that this is the case.
Are there any results, positive or negative, about the dependence of (co)homology of $G^delta$ on the underlying model of set theory?
Cher Michel, these rings are uncommon.
1) Over a local ring ALL projective modules are free : this is a celebrated theorem due to Kaplansky.
2) If $R$ is commutative noetherian and $Spec(R)$ is connected, every NON-finitely generated projective module is free. This is due to Bass in his article "Big projective modules are free" which you can download for free here
And now for the good news: the rings you are after are uncommon but they exist. Bass in the article just quoted shows that the ring $R=mathcal C([0,1])$ of continuous functions on the unit interval has all its finitely generated projective modules free. Nevertheless the ideal consisting of functions vanishing in a neighbourhood of zero (depending on the function) is projective, not finitely generated and not free. Bass attributes the result to Kaplansky.
I tested this question with Sage, and the experiment suggests a clear pattern of asymptotics. Most polynomials are irreducible. Of the reducible ones, a third are of course divisible by $x$ (Edit: If 0 coefficients are allowed; see below.) An $O(1/sqrt{d})$ fraction are each divisible by $x+1$ and $x-1$. That's because if $p$ is such a polynomial, then $p(x) bmod x+1$ is understood as a random walk in the integers, and the same for $x-1$. Of the others of degree $d$, an $O(1/d)$ fraction are divisible by certain quadratic polynomials such as $x^2+x+1$. The remainder $p(x) bmod x^2+x+1$ can be interpreted as a random walk in the triangular lattice in the plane. Addendum: Actually, the only polynomials that can behave this way are cyclotomic polynomials. Divisibility by any specific non-cyclotomic polynomial is exponentially rare.
It could be very hard to prove a picture like this, although I don't really know. There is a similar picture for integer matrices with bounded entries: There is a sequence of explanations for why they might be singular, beginning with that two rows might be proportional. It is still a big open problem to prove that these explanations give you the correct asymptotics for the number of singular matrices of this type, although there are great partial results by Tao-Vu and Rudelson-Vershynin.
Since the sage code was requested, here is an improved version:
maxdegree = 16
maxcyclo = 400
displayother = 11
R.<x> = ZZ[]
cyclos = {}
for k in xrange(1,maxcyclo+1):
c = cyclotomic_polynomial(k,x)
if c.degree() <= maxdegree: cyclos[k] = c
def tally(key):
if not key in counts: counts[key] = 0
counts[key] += 1
for degree in xrange(1,maxdegree+1):
print
counts = {}
total = 0
for n in xrange(2^degree):
total += 1
p = x^degree
for k in xrange(degree):
choice = (int(n)>>k)%2
p += (2*choice-1)*x^k
cdiv = False
for k in cyclos:
if not p%cyclos[k]:
tally('div by C(%2d)' % k)
cdiv = True
if cdiv: continue
f = factor(p)
if len(f) > 1:
if degree <= displayother: print p,'=',f
tally('other reducible')
else: tally('irreducible')
counts['total'] = total
print 'nDegree',degree
for key in sorted(counts): print '%s: %d' % (key,counts[key])
It is clearly true that the fraction of these polynomials that are divisible by a cyclotomic polynomial of degree $c$ decays as a power law, in fact as $O(1/d^{c/2})$. It is also clearly true that the fraction divisible by any other fixed polynomial decays exponentially. However, the more careful experiment found more exceptional factorizations than I thought. There are a lot of polynomials whose roots are close to the unit circle even though they are not on the unit circle. For instance $x^3+x+1$ is like this and comes in 8 versions (such as also $x^3-x^2+1$). If the number of these near misses grows fast enough, then the asymptotics that I suggested has to be adjusted, and the statistical problem is probably then even more difficult.
Per JSE's remark above, I misunderstood the original question to mean that the coefficients are in ${-1,0,1}$. If $0$ is not allowed, then congruence conditions develop that make it much more likely for a random polynomial to be irreducible. I replaced the code to reflect the actual question, although if anyone is interested the old code is still there in the edit history. (I personally think that the ternary question is at least as interesting.) In particular, if the degree is one less than a prime, then as Mark Meckes suggests below, the polynomial $p$ can only be divisible by a cyclotomic polynomial by being a cyclotomic polynomial.
Here is some typical output from the code:
Degree 14
div by C( 3): 1126
div by C( 5): 244
div by C( 6): 1126
div by C(10): 244
div by C(15): 19
div by C(30): 19
irreducible: 13310
other reducible: 378
total: 16384
(The total does not add up because a polynomial can be divisible by more than one cyclotomic polynomial.)
No, such $langle N', x'rangle$ is not constructible at all given only a description of $M$, even if you remove the requirement of polynomial time.
Suppose that $CONSTRUCT$ is such a deterministic turing machine outputting $langle N', x'rangle$. Note that by your requirement, $N'$ must never halt on input $x'$, no matter how much time $N'$ is allowed to run.
Let $M$ be your favorite deterministic turing machine accepting $text{coBHP}$. Write a new machine $M'$ as follows:
Input$langle N, x, 1^trangle$
let
ifand$x=x'$then output 1; that is, let$M'(langle N,x,1^trangle)=1$
else, let
output; that is, let$M'(langle N,x,1^trangle)= y$
That is, $M'$ uses Kleene's recursion theorem to construct a hard input for itself and uses the fact that $CONSTRUCT$ outputs never-halting machines to make that 'hard' input very easy (actually, now bounded by a constant), which is a contradiction. Therefore, $CONSTRUCT$ cannot exist.
The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but tried a random sample) came up with no actual applications but just lots of vague "This is really important."s. The closest actual application was on Wikipedia's page, which says:
The second order Euler–Cauchy equation appears in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates.
Is there a more direct application of this ODE? Ideally, I'd like something along the lines of deriving the corresponding ODE with constant coefficients from considering springs or pendula.
My motivation is pure and simple that I'd like to be able to say something in class a little more motivating than: "We study this ODE simply because we can actually write down a solution, and it's quite amazing that we can do so."
Assume we have a complete orthogonal system on a domain $D$, given by the eigenfunctions of the Laplacian on $D$. For example, the set ${e^{int}}$ on $[-pi, pi]$, or the spherical harmonics on the unit ball. Now consider a domain $D'$, which is "close" to D in some sense (the boundary of $D$ is close to the boundary of $D'$ in some suitable norm).
Are the eigenfunctions of the Laplacian on $D$ close, in some sense, to the eigenfunctions of the Laplacian on $D'$? Does knowing the basis of $D$ help approximate the basis of $D'$ ? Any known results along these or similar lines appreciated.
You may do better with an approach that mimics the likely characteristics, and then selects cards that meet the characteristics, and then resolves conflicts. Here is a possible approach:
Consider the gross characteristics of such a deck: number and distribution of costs, number of +n Buys +n Cards +n Actions, number of duration cards, number of attack cards.
Now start the build by choosing a cost distribution, say 2 2's, 3 3's, 2 4's, and 3 5's.
Choose 20 cards at random with cost distribution mirroring the target distribution. Now try a subset of 10 appropriate cards. Check their stats against the others, e.g. number of +1 Buys. If all the stats match up, then check to see how many pairs of cards are disallowed. By whatever means, determine which cards out of the ten chosen do not represent a good fit to a random desired deck, and replace those cards with appropriate choices from the remainder of the 20 cards. If possible, let the stats dictate the
replacement subset. Now evaluate the modified deck, and see how many of the stats are
out of whack. Chances are good that you will converge to an acceptable deck within a
few trials.
If you implement this and find contrarily that chances are bad on converging to a good deck, then try resolving conflicts using a subset of 30 cards instead of a subset of 20 cards. I believe that finding a good set of characteristics will give you a way of
generating many good random decks, rather than just considering how often individual cards and card pairs occur or do not occur in favored decks.
Gerhard "Ask Me About System Design" Paseman, 2010.04.06
Let me start by rigorously pose my question.
Let $K$ be an algebraically closed field of characteristic $2$, let $n$ be an even integer number, let $f(X) = X^n + T_1 X^{n-1} + cdots + T_n$, be the generic polynomial, that is, $T = (T_1, ldots, T_n)$ is a tuple of algebraically independent variables over $K$.
Let $Omega = $ { $omega_1, ldots, omega_m$} be a finite subset of $K$, let $f_i(X) = f(X) - omega_i$, and let $F_i$ be the splitting field of $f_i$ over $K(T)$ ($i=1,ldots, m$).
Question: For which $Omega$ the splitting fields $F_1, ldots, F_m$ are linearly disjoint over $K(T)$?
Remarks:
If the characteristic of $K$ is NOT $2$, or if $n$ is odd, then the splitting fields are linearly disjoint for arbitrary $Omega$. Thus, I pose the question the specific case of $p=2$ and $n$ even.
The answer cannot be ALWAYS, as in the previous remark. Indeed, one can show that if $p=n=2$, $m=4$, and $omega_1 + omega_2 + omega_3 + omega_4 = 0$, then the splitting fields are not linearly disjoint. In fact, if $p=n=2$, the answer is that the splitting fields are linearly disjoint if and only if the sum of any even number of elements of $Omega$ does not vanish.
How one proves 1 + 2: The linear disjointness of the splitting fields can be reduced to the linear independent of the discriminant as elements in $H^1(K,mathbb{Z}/2mathbb{Z})$. If $pnmid n$, then one can use ramification theory to achieve this, if $pneq 2$ but divides $n$, one can calculate this by hand using the formula given by the determinant of the Sylvester matrix. If $p=2$, I know of no formula for the discriminant in terms of the coefficients. However when $p=n=2$ situation is simple enough to do calculations and hence get 2.
Motivation: The linear disjointness of the splitting fields allows one to calculate a Galois group of a composite of polynomials, which in turn yields arithmetic features of the ring of polynomials over large finite fields. Let me not elaborate on that here
Every equation ought to be numbered in print publications or fixed-format electronic publications; if an equation is not important enough to be included as a numbered equation in the article, it ought not be included at all. As for "above" and "below", I've learned them contextually as meaning "prior" and "later" in the current article. I've never understood it to mean exactly one equation above or one equation down. In fact, I've even seen absolute and relative references together, as in "see equation 12 above." If an equation reference goes too far forwards or backwards, it makes sense to repeat the equation renumbered with a new number in this location. It's much easier to look at it on the same page rather than have to flip back and forth.
Relative references make some sense for fixed publication media such as printed copies of journals. Absolute references, such as pointers and index numbers and URLs, make more sense for variable view-model media such as electronic publications (HTML particularly).
I agree with the other answers (above, and below) for mathematical writings for print publications such as journals. However, there is an extra consideration for electronically published items in electronic journals or particularly in forums like this web-site, Mathoverflow.
Users have the option of controlling their viewing model on electronic publication systems and changing what appears at the "top" of their electronic page. They may choose chronological order in order to view comments in the same order they were submitted, allowing ease in understanding the flow of commentary. They may choose reverse chronological order, for example when they are revisiting a question just to see what the latest entries have been. They may also choose to order the results by popularity or relevance (with popularity of votes being an electronic self-selected polling of relevance by other readers).
On forums like Mathoverflow, references to other writer's contributions as "the answer above" or the "answer below" are rendered meaningless and confusing by the fact that the physical ordering of the answers is different for different readers and at different times. Reader preference can re-order the answers according to time submitted (oldest first, newest first) or by popularity (votes thus far); the popularity is evanescent as the number of votes will also change over time.
Certain options lead to difficulty in following threads. For example, comments tend to be initally shown in descending order of votes, destroying the temporal ordering of conversations or the ordering of comments spread out amongst multiple entry boxes. I would have expected that long comments spread out amongst multiple boxes would be discouraged; but they seem to be rather prevalent among mathoverflow. I find that I always have to click on the "show additional comments" button in order to be able to follow the unfolding of the comments and understand the conversation in the commentary.
Let $A$ be an algebra, $H$ a Hopf algebra, and
$$
beta_A: A to A otimes H, ~~~~~ a mapsto a^{(1)} otimes a^{(2)}
$$
a right $H$-coaction. This induces a right $H$-coaction on $A otimes A$ defined by
$$
beta_{A otimes A}: a otimes b mapsto a^{(1)} otimes b^{(1)} otimes a^{(2)}b^{(2)}.
$$
My question is: Does this restrict to a coaction on the universal calculus over $A$, namely to a $H$-coaction on the kernel of the multiplication map $m:A otimes A to A$? I feel this is a very simple question but I can't seem to find an answer.
If the construction does not work, does anyone know of a way to induce a coaction on the universal calculus over $A$ from $beta_{A}$?
There is a rather strange situation here because a russian mathematician, Alexander Dynin, is claiming to have solved it. Please, refer to this question of mine or this paper published in a reputable journal. Till now I have not seen any reaction from the community but I think that an analysis of the work of this author is overdue. Indeed, he has worked on this since 2009 as I can see from arxiv.
It is a general fact that a closed manifold of odd Euler characteristic cannot bound a compact manifold. This can be deduced pretty easily from the fact that a closed manifold of odd dimension has Euler characteristic zero (a consequence of Poincaré duality) as follows. Suppose N is the boundary of a compact manifold P. Let M be the double of P, the union of two copies of P glued along N. Then the Euler characteristics of M, N, and P are related by:
$chi(M)=chi(P)+chi(P)-chi(N)$
Thus $chi(M)$ and $chi(N)$ are congruent mod 2. If the dimension of N is even, then M is a closed manifold of odd dimension so $chi(M)=0$, hence $chi(N)$ is even. And if the dimension of N is odd then $chi(N)=0$ anyhow.
I should have put this in my book!
de la Harpe's book is quite nice and has an amazing bibliography, but it doesn't really prove any deep theorems (though it certainly discusses them!). Some other sources.
1) Bridson and Haefliger's book "Metric Spaces of Non-Positive Curvature". Very easy to read and covers a lot of ground.
2) Ghys and de la Harpe's book on hyperbolic groups. Another classic, but in French. If you look around the web, you can find English translations.
3) Cannon's survey "Geometric Group Theory" in the Handbook of Geometric Topology is very nice.
4) Bowditch's survey "A course on geometric group theory" is also very nice.
5) Bridson has written two beautiful surveys entitled "Non-Positive Curvature in Group Theory" and "The Geometry of the Word Problem". The latter was one of the first things I read in any depth.
6) Geoghegan's "Topological Methods in Group Theory" is very nice, with a more topological approach.
7) Mike Davis's "The Geometry and Topology of Coxeter Groups" is a bit specific, but covers a lot of important material in a nice way.
8) John Meier's book "Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups" is well-written and pretty gentle.
I'm programming an n-dimensional polynomial fitting function. It uses the basic concept of least squares and a design matrix.
For example, a quadratic fit on 2d data:
This is a row in the design Matrix:
$X={({1, x, x^2})}$
And this is the matrix formula I'm using:
$(X^T*X)^{-1}(X^T*Y)=Ax^2+Bx+C$
This is nothing special or complicated.
I'm at the point however where I'd like to introduce confidence intervals. How do I compute the confidence intervals from this regression technique?
$Astackrel{sim}{hookrightarrow}B$? Alternatively, using Oberdiek's stackrel.sty you could say something like
A mathrel{raisebox{2pt}{$stackrel[raisebox{1pt}{$sim$}]{}subset$}} B
and play a little with the raiseboxes so that this aligns more or less correctly (this depends on your final font, and your publisher's typographer is not going to love you for this...)
It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable separately is a rational function.
a holomorphic function in each variable separately is holomorphic in all its variables.
A complete analytic function can be single-valued or multiple-valued according as it does not have, or does have, branch points. The algebraic functions are examples of the latter.
Here is my question: is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?
A necessary and sufficient condition (but I do not feel satisfied with that) for the locally compact length space $X$ to have a locally compact completion is that there exists some $r>0$ such that each ball of radius $r$ in $X$ is totally bounded.
In fact, if the condition holds closed balls of radius $r/2$ in $overline{X}$ are compact.
On the other hand, suppose that $overline{X}$ is locally compact. Then, as it is a complete length space, it is proper (this is called the Hopf-Rinow Theorem in the book by Bridson and Haefliger). This should imply that balls of any radius in $X$ are totally bounded.
The main reason why I am not satisfied with it is that the proof that the condition is sufficient does not use that $X$ is a length space, so this is not really the answer to what you asked. I thought it might be relevant, anyway...
The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but tried a random sample) came up with no actual applications but just lots of vague "This is really important."s. The closest actual application was on Wikipedia's page, which says:
The second order Euler–Cauchy equation appears in a number of physics and engineering applications, such as when solving Laplace's equation in polar coordinates.
Is there a more direct application of this ODE? Ideally, I'd like something along the lines of deriving the corresponding ODE with constant coefficients from considering springs or pendula.
My motivation is pure and simple that I'd like to be able to say something in class a little more motivating than: "We study this ODE simply because we can actually write down a solution, and it's quite amazing that we can do so."
I am trying to build a mathematical model of the availability of a file in a distributed file-system. The system works like this: a node $x$ stores a file $f$ (encoed using erasure codes) at $rb$ remotes nodes, $r$ is the replication-factor where $b$ is a constant. Erasure-coded files have the property that the file can be restored $iff$ at least $b$ of the remote nodes are available and return its part of the file.
The simplest approach to this is to assume that all remote nodes are independent of each other and have the same availability $p$. With these assumptions the availability of a file becomes $$pf = sum_{i=b}^{rb} binom{rb}{i} p^i(1 - p)^{rb - i}$$
Unfortunately these assumptions can introduce a non-neligible error, as shown by this paper: http://deim.urv.cat/~lluis.pamies/uploads/Main/icpp09-paper.pdf.
The other extreme is to calculate the probability of each possible combination of availaible/non-available node and take the sum of all these outcomes (which is sort of what they suggest in the paper above, just more formally than what I just described). This approach is more correct but has a computational cost of $O(2^{rb})$.
Do you guys have any ideas of a good approximation which introduces less error than the binomial distribution-aproximation but with better computational cost than $O(2^{rb})$?
You can assume that the availability-data of each node is a set of tuples consisting of $(measurement-date, node measuring, node being measured, succes/failure-bit)$
In the study of (finite-dimensional?) paracompact and locally compact (?) spaces there is Verdier's topological duality theorem, expressed in terms of a dualizing complex (which is built up from a sheafification process using duals of compactly-supported cohomologies of open subspaces, or something like that). It is pure topology, having nothing to do with ringed spaces (just like the orientation sheaf!). In the special case of smooth (paracompact) manifolds, this recovers the orientation sheaf up to a shift on each connected component. It is analogous to the fact that the super-abstract dualizing complexes in Grothendieck duality for (quasi-)coherent cohomology collapses to the old friend "top-degree relative differentials" (up to shifting) in the smooth case.
But that's all just fancy mumbo-jumbo which puts the orientation sheaf into a broader perspective (like many duality theories for cohomologies). This does not qualify as a good way to initially "define" the orientation sheaf, much as appealing to Grothendieck duality would be a strange (and even circular, from some viewpoints) way to "define" top-degree relative differentials in the smooth case. To get a real theorem out, we have to put some content in.
It seems more illuminating at a basic level to understand how the orientation sheaf is constructed using punctured neighborhoods along the lines of Emerton's comment or the oriented double cover as in David Roberts' answer, and how one can remove some orientability hypotheses in some classical results on "constant coefficient" cohomology by instead allowing coefficients in the locally constant sheaf given by the orientation sheaf. And likewise to understand why the constant sheaf associated to $mathbf{Z}(1) = ker(exp)$ has $n$th tensor power that serves as an orientation sheaf on a complex manifold of dimension $n$ (and so the natural isomorphism $mathbf{C}(1) = mathbf{C}$ via multiplication explains the absence of needing to choose orientations for various cohomological calculations on complex manifolds (very relevant if one is to have a hope to translating things into algebraic geometry).
Let $P$ be the space of all real monic polynomials of degree $n$; it is isomorphic to $mathbb{R}^n$. There is a hypersurface $Delta$ in $P$, cut out by the equation of the discriminant. As you cross $Delta$, the number of real roots goes up or down by $2$. Also, every time you cross $Delta$, the discriminant switches signs.
When $n$ is $2$ or $3$, then $P setminus Delta$ only has two connected components. On one of these components, all roots are real, and on the other two roots are imaginary. So you can tell which component you are in just be looking at the sign of the discriminant. Once $n$ is $4$ or larger, there are more than two connected components to $P setminus Delta$. So the sign of the discriminant can't tell you which component you are in. I don't know a formula for the number of connected components of $P setminus Delta$, but one could be extracted from the description of the toplogy of $(P, Delta)$ in Gelfand, Kapranov and Zelevinsky, Discriminants, Resultants and Multidimensional Determinants.
One way to interpret your question is
Is there a polynomial $F$ on $P$ which is positive on the polynomials with $n$ real roots, and negative otherwise?
The answer is "no". The variety $Delta$ is irreducible (by the Horn uniformization). By a standard lemma, any nonempty open subset of the real points of $Delta$ is Zariski dense in $Delta$. So, if $F$ vanishes on the part of $Delta$ which forms the boundary of {Polynomials with all roots real}, then $F$ also vanishes on all of $Delta$. Working a little harder, we can show that if $F$ vanishes to odd order on the boundary of {Polynomials with all roots real}, then it also vanishes to odd order on the other connected components of $Delta(mathbb{R})$.
On the other hand, there are many good ways to determine the number of real roots by polynomial computations, such as Sturm's method.
This should really be a comment on Marco Radeschi's answer from Feb 22 involving the area formula for spherical triangles, but since I'm new here I don't have the reputation to leave comments yet.
In reply to Igor's comment (on Marco's answer) wondering about an analogous proof for the area formula of hyperbolic triangles: there is one along similar lines, and you're rescued from non-compactness by the fact that asymptotic triangles have finite area. In particular, the proof in the spherical case relies on the fact that the area of a double wedge with angle $alpha$ is proportional to $alpha$; in the hyperbolic case, you need to replace the double wedge with a doubly asymptotic triangle (one vertex in the hyperbolic plane and two vertices on the ideal boundary) and show that if the angle at the finite vertex is $alpha$, then the area is proportional to $pi - alpha$. That follows from similar arguments to those in the spherical case (show that the area function depends affinely on $alpha$ and use what you know about the cases $alpha=0,pi$).
Once you have that, then everything follows from the picture below, since you know the area of the triply asymptotic triangle and of the three (yellow, red, blue) doubly asymptotic triangles.
(That picture is slightly modified from p. 221 of this book, which has the whole proof in more detail.)
I do recreational math from time to time, and I was wondering about a couple of graph enumeration issues.
First, is it possible to enumerate all simple graphs with a given degree sequence?
Second, is it possible to enumerate all valid degree sequences for simple graphs with a given number of vertices?
Based on my wikipedia surfing, we can use the Erdos-Gallai theorem to determine if a degree sequence is valid, but this doesn't really lend itself to enumerating valid degree sequences efficiently. Similarly, we can use the Havel-Hakimi algorithm to construct at least one graph for a given valid degree sequence, but this doesn't help to enumerate all graphs for that degree sequence.
My (admittedly uneducated) guess is that it might be possible to work backwards using the Havel-Hakimi condition to construct graphs by building them up in different ways. Any insight would be appreciated :D
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure which I do not understand.
Suppose we have finitely presented monoid with unity $M$ with two generators say $g_1,g_2$. Lets relations for this monoid would be $Rel = {g_1^2 = e , g_2^2 = g_2 }$ where $e$ is unit element of monoid. So we have monoid $M$ to be quotient of free monoid $F$ by relations $M = F/ Rel$. $M$ is infinite. Words in $M$ has structure "$stststststst...$" etc. Rather boring ;-)
Now I want to define ring $G$ over such monoid. Lets play with field of real numbers R, as a background field. So we have ring $G = R[M]$. Suppose we are able to find such element, let's call it $g_3$ in R[M] that the following equations are satisfied:
(1) $g_i g_j = c_{ijk}g_k$ where $i,j,k=1,2,3$ just like in Lie algebra structure.
Note that $g_3$ is not element of monoid $M$ but is element of ring $G$. Also there is no antisymmetry relations for $c_{ijk}$. Then note, that from (1) we have that every element in $R[M]$ is linear combination of set of "generators" ${ e,g_i } , i=1,2,3$.
In one sentence within ring $R[M]$ we have some structure which allows us to easily compute every polynomial formula as it after some evaluations may be always turned into linear combination of generators. But such generators of $R[M]$ are different that generators of base monoid $M$.
Do You know any references where I may find examples of such structures? How they are called? They are examples of what? Are there any computer algebra systems which compute with such structures?
Additional remarks:
@ Darji - "just like Lie algebra" is about formal structure. It reminds me definition of Lie algebra, but of course $c_{ijk}$ is not antisymmetric nor Jacobi identity is satisfied so of course it is not Lie algebra.
@ Darij - Of course in general there is no associativity. In case I am interesting in this structure is associative, as it follow from simple algebra monoid which is associative, and by R[M] I mean formal combinations $sum a_i g_i$ and combinations of its multiplies as in section "two simple examples" in http://en.wikipedia.org/wiki/Group_ring
So we have noncommutative ring over monoid which is associative, has unity, and $c_{ijk}$ in j,k has both symmetric and antisymmetric components.
Further clarifications:
Structure I tried to describe consists of multiplicative monoid, and ring over it in reals. In this ring every polynomial has linear decomposition in "basis" $g_i$, somehow as in vector space. In ring every ring element allows such decompositions ( but not every linear combination of $g_i$ is ring element so it is not linear space). What is that? Do You know examples of such structures?
@Scott: You are right I am very bad English writer. Thank You for being so polite. So I will wrote it in the most explicit way I can.
I have finitely presented noncommutative monoid with unity and two generators $g_1,g_2$: M = F/Rel where $Rel = {g_1^2 = e , g_2^2 = g_2 }$, $e$ is unit element and $F$ is free monoid over two generators. Because of relations $Rel$ every elemet in monoid has form for example $g = g_1g_2g_1g_2...g_1g_2$ ( alternating finite sequence with subscripts 1212... or 2121...). Different monoid elements contains different number of multiplications. It is very simple although infinite multiplicative structure.
Then I consider monoid ring over reals $R[M]$. Every element in $R[M]$ has form:
(1) $t = r_1g_1 + r_2g_2 r_3g_1g_2+r_4g_2g_1+ r_5g_1g_2g_1 + ...+r_p g_ig_kg_i...g_s+ ...$ and so on. $r_i in R$ and $g_i in M$.
Note that in general monomial element $g_ig_k...g_s$ every subscript has value in ${1,2}$ and no two following each other subscripts are the same ( they alternate like in sequence like $1212..$ or $2121..$. Of course this is standard ring definition.
In structure, I would like to describe You here, I have strange additional property: there is element $g_3$ in ring $R[M]$ ( but it is not monoid element!) which allows following decomposition:
For every $r in R[M]$ we have
$r= r_0 e + r_1 g_1 +r_2 g_2 +r_3 g_3$
Look: there are only four terms in decomposition, even if You decompose general ring element in the form of (1). However after such decomposition I may only multiply such elements and not add them. So in fact decomposition as above, I trying to treat as some kind of "parametrization" of ring elements. Is this interesting?
As far as I know this is not standard ring property - maybe I am wrong. If I think about for example polynomial ring (that in simple case is real ring over multiplicative monoid generated by one generator $x$) such decomposition is not possible.
So I ask You if that structure was described in literature? Is it special kind of some known structure? Where to find something about it?
Thank You all for Your remarks!
Assume you have a proper filter $F$ that avoids both $b$ and $neg b$. Then, you could consider the filter generated by $Fcup{b}$ - which is to say the smallest filter $F'$ containing $F$ and $b$.
Since $F$ was a proper filter it follows that $0notin F$.
If $0in F'$, then this means that there is some $fin F'$ such that $bwedge f = 0$. Now, $neg b=0veeneg b=(bwedge f)veeneg b=(bveeneg b)wedge(fveeneg b)=1wedge(fveeneg b)=fveeneg b$. Thus $f≤neg b$, which means that $neg bin F'$.
Since $neg bin F'$, either $neg bin F$ or $neg b$ may be acquired by meets and upwards closures from $Fcup{b}$. Say $bwedge f≤neg b$ for some $fin F$. Then $bwedge f= bwedge fwedgeneg b = bwedgeneg bwedge f = 0wedge f = 0$ for an $fin F$ and by the above argument, we derive $neg bin F$. This is a contradiction, from which we can derive that $0notin F$.
Hence, $0notin F'$, and thus $F'$ is a proper ideal strictly containing $F$.
Given a geodesic metric space $X$ together with a choice of midpoints
$m:Xtimes Xrightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive curvature condition is satisfied:
$d(m(x,y),m(x,z))le frac{d(y,z)}{2}$ for all $x,y,zin X$ .
This is just a special case of the CAT(0) inequality for the "triangle" $x,y,z$.
Lets call such a space a M-space.
Such a space needn't be CAT(0), as the example $(mathbb{R}^n,d^1)$ shows, where $d^1$ is the $l^1$ metric. The choice of midpoints is given by $m(x,y)=frac{x+y}{2}$. It also needn't be unique geodesic.
But this space can be equipped with another metric, that makes it a CAT(0) space.
So my question is: Is every group, that acts properly, isometrically and cocompactly on a M-space already a CAT(0)-group?
The shortest (in terms of weight) path, constrained to have exactly n (or at most n) edges, can be found in polynomial time. For instance, given your graph $G=(V,E)$ make an expanded graph $H$ that has as its vertices the pairs $(v,i)$ where $vin G$ and $0le ile n-1$. Draw an edge in $H$ from $(v,i)$ to $(w,i+1)$ whenever $G$ has an edge from $v$ to $w$, with the same weight. Then the shortest $n$-edge path in $G$ from $v$ to $w$ is the same as the shortest path in $H$ from $(v,0)$ to $(w,n)$. To look for paths in $G$ that are at most $n$ edges long, add to $H$ edges of weight zero from $(v,i)$ to $(v,i+1)$.
However, these paths allow repeated vertices and edges. If repetitions are disallowed, and $G$ has $n+1$ vertices, then the shortest length-$n$ path is just a Traveling salesman path, so of course it's NP-complete.
Edit: Somehow I totally misread the question. I talked about the group algebra $mathbb C[G]$, which is not at all the same as the character ring $R(G)$. Over $mathbb C$ (or any other field of characteristic 0), $R(G)$ is naturally a subalgebra of $(mathbb C[G])^*$, which is the algebra of functions on $G$ with pointwise multiplication, and now the comultiplication encodes the group structure. On the other hand, it is not a subbialgebra: the coproduct of a class function is not a class function.
Anyway, original post below, with the obviously wrong things struck out. So it's really an answer to Kevin, rather than anything else.
Well, it depends on what you mean by "$R(G)$". I won't address TK duality, and most of what I'll say is essentially a follow-up to Kevin's answer, rather than an answer in its own right. Also, I'm only going to address finite groups and their finite-dimensional representations. Also, for me the word "ring" means (associative, unital, noncommutative) "$mathbb C$-algebra".
Recall that a complex representation of $G$ is the same as an algebra representation of $mathbb C[G]$. Let $R$ be a ring. As Kevin says, it's in general impossible to define an $R$-module structure on $Motimes N$ when $M,N$ are $R$-modules. (When $R$ is abelian, which is not the case here, one can define a tensor product $M otimes_R N$, but that's not the tensor product of representations anyway.) What would a tensor product of modules require? It would require a rule that assigns to each $rin R$ and each pair $M,N$ of $R$-modules an endomorphism of $Motimes N$, of course, and we should impose all sorts of axioms that force the tensor product to be well-behaved. Among other things, it's much easier if the endomorphism is an element of the tensor product $text{End}(M) otimes text{End}(N) subseteq text{End}(Motimes N)$. And we already have some distinguished elements of $text{End}(M)$ and $text{End}(N)$, namely the action of $R$.
So one way to try to construct a well-behaved tensor product on the category of $R$-modules is to find a nice map $Delta: R to Rotimes R$. Then the axioms for this map that assure that the tensor product is good are that $Delta$ be an algebra homomorphism, and that it be "coassociative": $(text{id}otimes Delta)circ Delta = (Delta otimes text{id})circ Delta)$. Let's suppose that there's also a distinguished "trivial" representation $epsilon: R to text{End}(mathbb C) = mathbb C$; if this is to be the monoidal unit, then we'd need $(text{id}otimes epsilon) circ Delta = text{id} = (epsilon otimes text{id})circ Delta$. The maps $Delta, epsilon$ satisfying these axioms define on $R$ the structure of a bialgebra.
By the way, the map is called "$Delta$" because if $G$ is a group (or monoid) and $R = mathbb C[G]$, then the map $R to Rotimes R$ given on the basis $G$ by the diagonal map $Delta: g mapsto gotimes g$ is such a structure.
Then here's a cool fact. Define an element $rin R$ to be grouplike if $Delta(r) = rotimes r$. Then the grouplike elements are a multiplicative submonoid of $R$. And when $R = C[G]$, the grouplike elements are precisely $G$.
So my answer to your question is that "the additional information contained in $R(G)$ as opposed to the character table" is its bialgebra structure.
Let $(S,mathcal{L})$ be a linear space and $q$ be a prime power such that
then for every point $e in S$, there are at most $q^2$ lines in $mathcal{L}$ not containing $e$.
edit - 'How do I prove the above?' is my question.
By 'linear space', I mean a pair $(S,mathcal{L})$ such that $S$ is a finite set of points, and $mathcal{L}$ is a set of subsets of $S$, or 'lines', so that any two points lie on a unique line, and any two distinct lines intersect in at most one point.
I arrived at this problem from matroid theory, but it's essentially a combinatorial problem about incidence structures, so I have phrased it as such.
The $q^2$ here is best possible - equality will hold when the linear space is a projective plane over a $q$-element field.
The De Bruijn-Erdos theorem, as well as various results from the literature on linear spaces, give lower bounds for numbers of lines, but I can't find upper bounds anywhere.
My feeling is that Charles is on the right track with the answer above. But rather than looking for a counterexample, I think we should have a go at correcting the original question.
Now I'm not quite sure over which rings the next statements work, possibly only over rings over a field of char 0. Perhaps someone knows the details better than I, but to make it work will probably require working with simplicial algebras as these carry a model structure over any ring.
The cochains of X carry a dg-algebra structure A. Since ΩX is the homotopy pullback of • → X ← • and taking cochains should preserve the relevant (co)limits (can someone help me here), then the cochains ring of ΩX is the homotopy pushout of k ← A → k, that is, the derived tensor product. We can then take cohomology.
For the next bit we probably do need characteristic 0. The cochains ring will be rather large, so to keep track of things we could take the cohomology, but remember the higher operations. Then as an infinity ring the cohomology H(A) will be quasi-isomorphic to A (which isn't necessarily true if we don't remember the higher operations). Then with that in mind we can calculate the derived form of k ⊗_H(A) k. Its cohomology should be the cohomology of the loop space.
It would be nice to have a counterexample though, how about complements of links, the cohomology rings aren't so bad to calculate (only depending on the number of links over the rationals at least). What about the loop spaces?
A manifold is locally $mathbb R^n$. An orbifold is locally $mathbb R^n/{text{finite group}}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 rightrightarrows G_0$?
For example, the Lie algebroid determines a distribution on $G_0$, and I think that it is locally integrable? What extra structure "local" structure of the groupoid is there (e.g. this distribution loses the data about the automorphisms of a point).
Finally, is the right notion of "local structure" well-behaved under equivalences of groupoids? If it is, then I really should change the title of the question to "What is the local structure of a smooth stack?".
This is a summary of what I've learned about this question based on the answers of the other commenters.
[*] Any positive distribution defines a positive Radon measure.
I had naively assumed a result for distributions like The Hahn Decomposition Theorem[1] for measures, i.e. I assumed that a distribution could be expressed as the difference of two positive distributions. If it could be, then applying Theorem [*] would yield the result that any distribution is a signed measure.
However, this is not the case. The derivative of the delta function, i.e. δ', satisfies
δ'(f) = -f'(0). This is not a measure. I can't find any way of proving it's not the difference of two positive distributions, other than by contradiction using the above result.
First off, I would be skeptical of the claim that computer programs "may prove any theorem in elementary Euclidean geometry", simply because it is so wide and general that is prone to be false. Secondly, I am not directly an expert in this field myself, but I hope my references are not too much off.
However, modern automatic geometric theorem proving is definitely capable of dealing with a large number of geometric problems, including those which involve geometric inequalities. The older methods (going back to Wu), translate a geometric statement is translated into an implication of the form
$$ bigwedge_{i=1}^n f_i(x_1,ldots,x_m) implies f_0(x_1,ldots,x_m)$$
where the $f_i$ are polynomials. From this, with various methods one then obtains a prove of the statement, or a counter example. I am suppressing here that often you need to specify further side conditions for a proof to be possible, e.g. that a triangle is non-degenerate; in fact, Wu's approach and the Gröbner basis even allow deducing sufficient conditions to make a theorem true in retrospect. The Wikipedia page for Wu's method gives some more details and a few references to relevant papers; you can easily google more.
Anyway, this allows encoding things like multiple points being collinear, points being contained on a circle, intersection of lines, perpendicularity of lines, and so on. However, this does indeed not allow encoding inequalities effectively; e.g. just specifying that a point is 'inside' a triangle, or that one value is less than another, in general is not possible.
But since Wu's original work, there have been many advances. If one looks a bit closer, then one notices that the above techniques actually prove theorems about complex geometry, as we are arguing about zeros of polynomials, and this all happens over an algebraically closed field. But we are usually interested in real geometry only. There are surprisingly many classical theorems from "real" geometry which remain true in the complex case, and this somewhat surprising (and as far as I know rather mysterious) fact ensures that nevertheless Wu's method and its relatives are quite successful. Still, people have worked on overcoming this limitation, as well as that of inequalities.
One approach is described "A New Approach for Automatic Theorem Proving in Real Geometry", by Dolzmann, Sturm & Weispfenning, who use quantifier elimination (from logic) to prove theorems in real geometry (as the title suggests), using their Reduce package Redlog. They use that, for example, to prove Pedoe's inequality. I am, however, not sure if this can be used to prove the Erdős–Mordell inequality; one could ask them. I think Sturm wrote his PhD thesis on the subject.
Another paper to look at is "A Practical Program of Automated Proving for a Class of Geometric Inequalities" by Lu Yang and Ju Zhang. There they describe a Maple package "Bottema" (unfortunately, this does not seem to be available on the net, at least I couldn't find it). They use it to prove a load of inequalities, and give many examples involving inequalities. To give you a flavor, here is an example (which they proved using their package):
Example 4. By $m_a$, $m_b$, $m_c$ and $2 s$
denote the three medians and perimeter
of a triangle, show that
$$frac{1}{m_a}+frac{1}{m_b}+frac{1}{m_c}geq
> frac{5}{2}.$$
And here is another excerpt (I included it as it also goes back to Erdős).
A. Oppenheim studied the following inequality in order
to answer a problem proposed by P. Erdös.
Example 9. Let $a$, $b$, $c$ and $m_a$, $m_b$, $m_c$ denote the side
lengths and medians of a triangle,
respectively. If $c = min{a, b, c}$, then
$$2m_a+2m_b+2m_c leq 2a+2b+(3sqrt{3}-4)c.$$
This all does not answer your question about the Erdős–Mordell inequality. And I am afraid I don't know the answer! But I hope my above explanations at least made it plausible that the answer could be yes, however vague that statement is.
I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps group schemes) over a non-algebraically closed field (resp. a "nice" ring, say a Dedekind domain).
Are there analogous results to Lie's theorems in the case of algebraic groups (perhaps even arbitrary group schemes)? I am aware of Jantzen's book on representations of algebraic groups, but if I remember correctly, he does everything over an algebraically closed base field. I have not studied the book in detail to convince myself that the arguments there will carry over to the non-algebraically closed case.
I suppose the Borel-Bott-Weil-(Schmidt) construction of highest weights using sections of cohomology groups of line bundles may be generalized to a more arithmetic setting (as Jantzen has done in his book). Is there any progress in this direction beyond algebraic groups, say to include a "nice" class of group schemes? I am more curious of the case of classical groups.
Concerning more general group schemes, I have looked up parts of SGA3, but I did not find any clearly stated results connecting the Lie algebra of a group scheme (as defined there using universal properties) to the underlying group scheme.
A more general and more loaded question: to what extent is a smooth scheme determined by its tangent space at a distinguished point. I am aware of the notion of jet schemes, are there some important or at least neat results in this area anyone would like to share?
Thanks in advance.
I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
for every $m$-rectifiable varifold $mu$ exists a $m$-rectifiable set $E$ in $mathbb R^n$, meaning $E=E_0 cup bigcup_{kinmathbb N} E_k$ with $mathcal H^m(E_0)=0$ and $E_ksubseteq F_k$ for some $mathcal C^1$-manifolds $F_k$ of dimension $m$, and a non-negativ function $thetain L^1_{text{loc}}(mathcal H^m|_E)$ such that $mu=theta mathcal H^m|_E$. This is a characterisation of $m$-recitifiable varifolds.
The first variation $deltamu$ of a varifold $mu$ is for $etainmathcal C^1_c(mathbb R^n;mathbb R^n)$ given by
$$deltamu(eta)=int div_mueta,dmu,$$
where $ div_mu(eta)(x) = sum_{i=1}^n tau_i^T(x)cdot Deta(x)cdot tau_i(x)$ where $tau_i(x)$ is a orthogonal basis of the tangentspace of $mu$ in $x$, which coinsidence $mu$-almost everywhere with $T_xF_i$ for $xin E_isubseteq F_i$ as above. So $div_mueta(x)$ is just the divergence in the manifold $F_i$, with $xin E_isubseteq F_i$.
We say $mu$ has an locally bounded first variation, if for all $Omega'subseteq Omega$ there exists $c(Omega')<infty$ such that
$$ deltamu(eta) le C(Omega',Omega) Vert etaVert_{L^infty(Omega)} qquadforall;etainmathcal C^1_c(Omega'). $$
See for more explanation for example http://eom.springer.de/G/g130040.htm.
For a $mathcal C^2$-manifold $M$ in $mathbb R^n$ with mean curvature $H_M$ the first variation is
$$ delta M(eta)=-int_M H_M cdot eta ,dvol_M -int_{partial M} tau_0 cdot eta ,dvol_{partial M} qquadforall;etainmathcal C_c^1(mathbb R^n)$$
with the inner normal $tau_0in T_xMcap(T_xpartial M)^bot$ and where the mean curvature is the trace of the second fundamental form $A$ by the meaning of $H_M(x)=sum_{i=1}^m A_x(tau_i,tau_i)$ in the normal space of $M$. As obviouse in this case the first variation is locally bounded.
Though most people seem to advise against reading Mac Lane's "Categories for the working mathematician", and neither did I read it from start to finish, looking at it, I find that it is highly worth trying to read some parts because it's extremely very well written: to the best of my knowledge this book is second to none in his field (of course, nowadays you have also Borceux's book, but its scope is something different). So if your mathematical interests force you to use categories, you'll have to consult Mac Lane's again and again. Starting to read it directly is a way to get an idea of where to find things when you need them.
Ok, there is a problem because, for instance, it begins with "metacategories" (chapter I, section 1), so when you arrive to real categories (section 2) you may be already completely lost. Hence, I asked myself: what parts of Mac Lane did I really have used in my own work and find them useful, worth reading or consulting, or are unavoidable in the language of categories? The following is my own selection of some chapters and sections of Mac Lane's book, based only on my personal tastes and biases (the selection is from the first edition, but, if I'm not wrong, the only difference is that the second one has an extra chapter on monoidal and braided categories and functors near the end):
Chapter I: 2, 3, 4, 5, 8.
Chapter II: 2, 3.
Chapter III: 1, 2, 3, 4, 5
Chapter IV: 1, 2, 4.
Chapter V: 1, 4, 5.
Chapter VI: 1, 2.
Chapter VII: 1, 3, 4, 5, 6.
Chapter VIII: 1, 2, 3, 4.
Chapter IX: 1, 2.
In general, I've tried to avoid both too abstract issues, logical foundations and too specific or specialized matters.
This kind of reading, of course, raises the problem of encountering terms you haven't seen defined before, or results you haven't studied. But in this cases, I think there's no harm in going to the index and find where the term is defined, or taking the result you haven't seen before on faith. Trying to do some exercises is of course necessary and the historical notes at the end of the chapters are interesting too.
