I have an intuition that Average of twin prime pairs is always Abundant number except for 4 and 6.
For example:
12 < 1+2+3+4+6=16
18 < 1+2+3+6+9=21
...
But I can't prove this. Could you give me any good idea?
2010-08-22
I think that Any prime is a factor of average of twin prime pair.
Do you agree with me?
Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak non-commutative desingularization of $R$ (note that, a) there is a natural map $Rto A$ and b) in the commutative case, finite global dimension implies regularity, hence the name).
This concept imitates Van den Bergh's definition of non-commutative crepant resolution (NCCR). His definition arises from a proof of dimension $3$ case of Bondal-Orlov conjecture. This is a long story, but an excellent account of the reasons behind the definition can be found in Section 4 of this paper. For existence of NCCR in some high dimensions case, check out this.
Now, non-commutative crepant resolution does not always exist (the above papers proves the equivalence, in some cases, with existence of projective crepant resolutions, which exists rarely in high dimensions). What we do know from Hironaka, in characteristic $0$ at least, is that resolution of singularity exist. So:
Question: Does weak non-commutative desingularizations of $R$ (as specified in the first paragraph) always exist?
Some discussions (I am a beginner in this, so feel free to correct me):
1) Why weak? By Morita equivalence, to ensures the desingularization is an isomorphism on the regular locus one needs $M$ to be free on that locus of $R$.
2) If one requires extra conditions (like $M$ being an generator-cogenerator) then there are examples when such desingularization does not exist (in some paper by Iyama which I forgot the name). I have not seen an example in the generality above.
3) There are positive results in diemension $0,1$, but I care about normal rings, so we start in dimension $2$.
4) Some people like Van den Bergh or Lieven le Bruyn probably know. May be they are even on MO!
I would appreciate even heuristic reasons for one way or another.
EDIT: The question is now resolved, by Lieven's answer below (as expected (:). I will provide a little bit more details in case someone is interested: Van den Bergh and Stafford proved that, in characteristic $0$, if $A$ is a non-commutative crepant resolution, then $R$ has rational singularity. The definition of NCCR is stronger, but if, for example, $R$ is Gorenstein of dimension $2$, it coincides with my version. So a counterexample is something like $R=k[x,y,z]/(x^3+y^3+z^3)$, which is a non-rational hypersurface.
I am not sure whether I will answer your question properly. I just tell some facts which might be useful
I know there are following categories equivalence
Qcoh(X),Qcoh(X,t),Qcoh(aX,t)
X is a presheaf
Qcoh(X) is quasi coherent modules on X,t is a grothendieck topology. aX is the sheafification according to topology t of X.
If t is coarser than topology of effective descent, then those three categories are equivalent. Which means Qcoh(X)=Qcoh(X,t1)=Qcoh(X,t2)=Qcoh(aX,t1)=Qcoh(aX,t2) are the same if t1 t2 are coarser than effective descent topology. This fact shows that it is not necessary to consider sheaf but presheaf. Their descent theory can be described by category of quasi coherent sheaves.
Similar results applied to stack. These facts are indicated in Giraud's book,but not wrote it out. They were proved by Orlov in his paper quasi coherent sheaves in commutative and noncommutative geometry and Kontsevich-Rosenberg preprint in MPIM "noncommutative stack"
First, the (simple!) setup:
I have a Markov chain X t on some finite state space Ω with stationary distribution π, and a function f from Ω to R. I'd like to estimate the integral of f with respect to π, which I'll write E π (f). There are theorems which say that
$frac{1}{n} Sigma_{t=1}^{n} f(X_{t})$ converges to E π (f) as n goes to infinity.
Now, if the $X_{t}$ were iid, then the Berry-Esseen theorem would give error rates in terms of n and (say) the maximum value of f.
Are there similar theorems which give error rates in terms of n, the maximum value of f, and one (or several) of the frequently computed statistics of finite state Markov chains, like relaxation time, mixing time, covering time, etc?
I'm vaguely aware of Sanov-type theorems for Markov chains, which give large-deviation results, but not in terms of these sorts of quantities, and I don't see how to convert the bounds immediately. Alternatively, I'd be very happy if anyone can give a reference for places that people have actually computed the sorts of error terms that do show up in statements of Sanov's theorem for some simple random walks.
EDIT: Added Mark's comments, so that the question might actually make some sense now. In particular, fixed a missing f, and the rather more important mistake that in fact the CLT doesn't give any sort of quantitative bounds by itself.
FURTHER EDIT: I accepted D. Zare's answer, since it certainly works. If anybody is interested in this question, I have since seen a bunch of articles, the latest of which is 'Optimal Hoeffding Bounds for Discrete Reversible Markov Chains' by C. Leon, which are a bit more specialized to the Markov chain case. I have also been told that Brad Mann's thesis is worth reading on the subject, but haven't yet picked up a copy myself.
For graph complexes, which are families of graph [on a fixed number of vertices n] closed under the deletion of edges, there is a natural simplicial complex capturing that information. Specifically, this family forms a subcomplex of the simplex on $binom{n}{2}$ vertices, with each vertex corresponding to an edge and each face in the simplex corresponding to a particular graph [where the dimension of the face is precisely the number of edges in the graph minus 1.] The family being closed under deletion of edges translates nicely into the family (a set of faces) truly being a subcomplex. As a result, each monotone graph property corresponds to a tangible, often combinatorially interesting simplicial complex.
[For an excellent survey of results on graph complexes, see Simplicial Complexes of Graphs, by Bjorner's student, Jakob Jonsson.]
For pure dimensional hypergraphs, a similar subcomplex can be constructed. A (d-1)-dimensional pure hypergraph family will sit as a subcomplex inside a simplex of dimension $left(binom{n}{d}-1right)$, now with each vertex in the simplex corresponding to a (d-1)-dimensional face - so again, any monotone (pure) hypergraph property can be examined via topology of these complexes. Unfortunately, many properties that are monotone for graphs fail to be monotone for hypergraphs (not-connectivity in codimension 1 for pure hypergraphs in dimension 2 or higher immediately springs to mind, among others.)
Hypergraphs in general, however, need not be pure, and several interesting classes of them most certainly are not. A hypergraph complex is just any family of hypergraphs closed under the deletion of edges.
Is there any standard way of simplicially representing hypergraph complexes which are not pure? There is a somewhat poor way of doing it, it seems, by considering the simplex on $2^n$ vertices and allowing each face to correspond to a hypergraph which is not necessarily a clutter (i.e. no edges in the hypergraph are properly contained inside other edges.) This is unsatisfying to put it mildly, however, and I was hoping there was an alternate way of considering hypergraph complexes as a topological space. Any ideas or literature references?
Edit: Although I'm interested in a better geometric representation of hypergraph complexes for its own sake, the first main application would be for either "matchings" (or, alternately, noncrossing partitions) on hypergraphs. A matching on a graph is just a set of edges which are pairwise disjoint. The matching complex (of a graph G) sits as a subcomplex inside the complex of all graphs above, and consists of all such sets of edges. The general matching complex on n vertices is a special case of this, with $G=K_n$, the complete graph.
For a hypergraph G, a matching can be defined equivalently - a subset of edges (now possibly with more than 2 vertices per edge) which are pairwise disjoint. The complete hypergraph (which is NOT a clutter, as its edges are all subsets) has a matching complex whose maximal faces correspond to all partitions of $[n]={1,2,...,n}$. This isn't precisely a complex of the form suggested below (one closed under the removal of a face, then the addition of any set of subfaces.) However, it is closed under the removal of a face and the addition of any disjoint set of subfaces.
There's a slightly subtle point near here of which some people are not aware: that it is dangerous (perhaps even nonsensical) to compare algebraic numbers under various different completions. So, to talk about $Q_pcap bar Q$, you should be talking about a completion of $Q$ containing $Q_p$, not, e.g., a completion of $Q$ lying inside $C$. I don't think this is what is happening here, but some people may find this interesting.
Now, there are lots of isomorphisms floating around, so usually everything turns out just fine, but sometimes not. Here are two examples.
(1) The following fallacious argument that $e$ is transcendental is from a talk by Gouvêa, "Hensel's p-adic Numbers: early history" (originally due to Hensel himself).
The series expansion of $e^p$ converges in $Q_p$, thus $e$ is a solution to the equation $X^p=1+pepsilon$, where $epsilon$ is a $p$-adic unit. So $[Q_p(e):Q_p]=p$ (of course you need to argue that the polynomial is irreducible), and so $[Q(e):Q]ge p$. Since $p$ was arbitrary, $e$ must be transcendental over $Q$.
The fallacy is that even though the series for $e$ (and $e^p$) converges in $R$ and $Q_p$, the numbers they converge to are not the same.
(2) The following is from Koblitz's $p$-adic book, page 83 (with an example and some other fallacious arguments).
It is not true that if an infinite sum of rational numbers (a) converges $p$-adically to a rational number for some $p$ and (b) converges in the real topology to a rational number, then the rational numbers the two series converge to are the same!
I have proved a certain result for all 2-connected graphs apart from those that fit the following criteria:
They are "minimally 2-connected", that is, deleting any vertex will produce a graph which is no longer 2-connected, and
They have circumference less than $frac{n+2}{2}$, where $n$ is the number of vertices.
I have not been able to come up with an example of such a graph. Can anyone help?
Of course the best possible outcome would be that they do not exist!
The integers can indeed be defined in the rational field, but not in the real field.
The question can be made precise by introducing some tools of first order logic. What you are asking about is the definability of the integers inside the rationals. For example, if you might consider the rational field structure 〈Q,+,.,0,1〉 and inquire whether the integers are defined by a first order order formula in this structure. That is, is there a first order formula φ(x) such that this structure satisfies φ(x) if and only if x is an integer? The answer is yes, and this paper appears to be about investigating how complex the definition is.
The fact that Z is definable in Q impies that the theory of the rational field is not a decidable theory. That is, there can be no computable algorithm which correctly tells us whether a given statement holds or fails in the rational field. The reason is that if we had such an algorithm, then by using the definability of the integers, we would be able to tell whether or not an arithmetic statement held or failed in the natural numbers, and with this, we would be able to solve the halting problem, which is impossible.
This situation contrasts sharply with the real field 〈R,+,.,0,1〉, whose theory IS decidable. Indeed, Tarski proved that the theory of real-closed (ordered) fields 〈R,+,.,0,1,<〉 is decidable. It follows that neither the integers nor the rationals are first-order definable in the real ordered field.
(Lastly, let me point out that the particular suggestions that you make for the definition are not first order definitions, and may suffer the criticism, as in some of the comments, that they beg the question concerning how the integers are implicit in your structure. The concept of first-order definability seems to avoid these criticisms, while clarifying both the question and the answers.)
Elon Lindenstrauss explains in his talk at the MSRI in Fall 2008 (the relevant comment is at minute 41 of the video) that the set of large Fourier coefficients of a probability measure $mu$ on the torus ${mathbb T}^n$ respects the additive structure. More precisely, he defines
$$A_{delta} := lbrace b in {mathbb Z}^n mid |hat mu(b)| geq delta rbrace$$
and says that it is "morally" true that $A_{delta} - A_{delta} subset A_{delta^2}$. (Here, the difference of two subsets is defined to be the set of all possible differences of elements in the respective subsets.) The precise statement (according to Lindenstrauss) is a consequence of the Balog-Szemeredi-Gowers Lemma.
Can someone provide the precise statement or give some hint how the lemma can be used to obtain bounds on Fourier coefficients?
Background/Motivation: Inspired by an interesting question by Joel, I've been wondering about the relationship between two very natural ways to define the category of ''all models of T'' where T is a first-order theory.
Let us assume that T is a complete theory with infinite models. Then on the one hand we can define the category Mod(T) whose objects are all the models of T and whose morphisms are all homomorphisms in the sense of model theory -- i.e. functions $varphi: M rightarrow N$ such that
For any $n$-ary relation $R$ in the language of $T$, if $M models R^M(a_1, ldots, a_n)$, then $N models R^N(varphi(a_1), ldots, varphi(a_n))$;
and
$varphi(f^M(a_1, ldots, a_n)) = f^N(varphi(a_1), ldots, varphi(a_n))$ for any $n$-ary function symbol $f$ in the language of $T$.
Also, one can define another category Elem(T) whose objects are also all the models of T, but whose morphisms are only the elementary embeddings, that is, functions which preserve the truth of all first-order formulas. As a model theorist, I'm more used to thinking about the category Elem(T), and this latter category arises naturally if one cares about which sets are definable but one does not particularly care about which sets are definable by positive quantifier-free formulas.
Question: What sorts of category-theoretic properties automatically transfer from Mod(T) to Elem(T), or from Elem(T) to Mod(T)?
To be clear, by a ``category-theoretic property'' I mean something that is preserved by an equivalence of categories.
Another related question is:
Question: Suppose we have a set of category-theoretic properties which we know characterize all the categories C which are equivalent to Mod(T) for some T [or to Elem(T) for some T]. Can we use this to characterize the categories which are equivalent to Elem(T) [respectively, Mod(T)] for some T?
Here are a couple of basic facts I know, which may or may not be useful here. First of all, every category Elem(T) is equivalent to a category Mod(T') for some other theory T' -- namely, the "Morleyization'' T' of T, where we expand the language by adding new predicates for every definable set (and iterating $omega$ times), thereby forcing T' to have quantifier elimination. However, it is certainly not true that every category Mod(T) is equivalent to a category of the form Elem(T') -- for instance, Mod(T) might not have colimits of $omega$-directed chains, but Elem(T) always will (by Tarski's elementary chain theorem).
Addendum: As Joel pointed out, there is a third possible notion of ''morphism'' for this category: the ``strong homomorphisms'' $varphi: M rightarrow N$ which commute with the interpretations of function symbols and have the property that for any $n$-ary relation $R$ in the language of $T$, $$M models R^M(a_1, ldots, a_n) Leftrightarrow N models R^N(varphi(a_1), ldots, varphi(a_n)).$$
I'd also be interested to learn about any relationships between the category of models with strong homomorphisms and the other two categories above.
For an elliptic curve over any field $K$ of characteristic different from $2$, the Kummer sequence reads
$0 rightarrow E(K)/2E(K) stackrel{iota}{rightarrow} H^1(K,E[2]) rightarrow H^1(K,E)[2] rightarrow 0$.
In particular, $iota$ is an injection. Therefore $P in 2E(K) iff$ the image $[P]$
of $P$ in $E(K)/2E(K)$ is equal to zero $iff iota([P]) = 0$.
Moreover, since you have full $2$-torsion, $H^1(K,E[2]) cong (K^{times}/K^{times 2})^2$ and in this case there is a well-known explicit description of the Kummer map: for any point $P = (x,y)$ different from $(a,0)$ and $(b,0)$,
$iota(P) = (x-a,x - b) pmod{K^{times 2} times K^{times 2}}$:
see e.g. Proposition X.1.4 of Silverman's book. The result you want follows immediately from this, taking $P = (c,0)$.
Note that, as Bjorn points out in his nice answer to the question, the finiteness of $K$ is not needed or used here. In my original version of this answer, I mentioned the fact that $K$ finite implies $H^1(K,E) = 0$ -- it seemed like it could be helpful! -- but the argument does not in fact use the surjectivity of $iota$, so is valid over any field of characteristic different from $2$ over which $E$ has full $2$-torsion.
I am looking for suitable algorithm to compute the eigenvalues and eigenvectors of a matrix. My matrix is sparse ( think of Finite Element Matrix), and it is very, very big ( think of hundreds of thousands or even million degrees of freedom).
The leading candidate for this task seems to be Lanczos algorithm.
The issue now is, how well Lanczos algorithm fare if the matrix is sparse? The reason I ask this is because I want to know if there are a lot of zero terms in a matrix, will Lanczos take advantage of this by storing only nonzero terms and operate on them? Since my matrix is big, I want to conserve as much memory space as possible.
For any Riemannian (in particular, for any Hermitean) manifold there is the decomposition: all forms are the direct sum of the harmonic ones, exact ones and those in the image of the conjugate operator of the de Rham differential. Every de Rham cohomology class is represented by a unique $d$-harmonic form.
For any complex (Hermitean) manifold there is a similar theory for the $barpartial$ operator, and we get a similar decomposition for each complex $({cal E}^{p,bullet},barpartial)$ where ${cal E}^{p,q}$ stands for complex valued smooth $(p,q)$-forms. See e.g. Chern, Complex manifolds. Every Dolbeault cohomology class is represented by a unique $barpartial$-harmonic form.
For general complex hermitean manifolds the above decompositions have nothing to do with one another. However, if the metric is Kaehler, some miracles happen:
the Laplacians of $d$ and of $barpartial$ coincide (more precisely, one is twice the other), so the spaces of harmonic forms coincide as well.
the $(p,q)$-projection of a $d$-harmonic form is again $d$-harmonic. Indeed, a local check shows that the $(p,q)$ projection operator commutes with the $d$-Laplacian. So each of the $(p,q)$ components of a $d$-harmonic form is closed and none is exact (if non-zero). So the $(p,q)$-decomposition of forms gives a $(p,q)$-decomposition of the cohomology classes (this does not exist for general complex manifolds). This is the Hodge decomposition.
since the conjugate of a $d$-harmonic form is again $d$-harmonic, the $(p,q)$ and the $(q,p)$ parts of the Hodge decomposition are conjugate to one another.
As a consequence of the above, the Hodge-to-de Rham spectral sequence degenerates in the first term. This implies that the $dd^c$-lemma holds for Kaehler manifolds, and hence, they are formal. See Deligne, Griffiths, Morgan, Sullivan, Real homotopy theory of Kaehler manifolds.
The cohomological Hodge decomposition holds also for arbitrary bimeromorphically Kaehler compact complex manifolds (in particular, for smooth, complete but not necessarily projective complex algebraic varieties). See e.g. Peters, Steenbrink, Mixed Hodge structures, p. 49.
It sounds to me like the OP is asking about the Diophantine Problem of Frobenius. This is as follows: let $(a_1,ldots,a_n)$ be positive integers which generate the unit ideal (i.e., their setwise gcd is $1$). The Frobenius number $f(a_1,ldots,a_n)$ is the largest positive integer $N$ such that there do not exist non-negative integers $x_1,ldots,x_n$ such that
$a_1 x_1 + ldots + a_n x_n = N$.
In the case of $n = 2$, the Frobenius number was explicitly computed by J.J. Sylvester (before Frobenius!): it is $a_1 a_2 - a_1 - a_2$, as the OP mentioned. Using this fact, it is a nice exercise to show by induction on $n$ that every sufficiently large integer $N$ can indeed be represented as a non-negative integer linear combination of the $a_i$'s.
Perhaps the two most famous results on the Frobenius problem are as follows:
I. Schur's Theorem: if we define
$r(a_1,ldots,a_n;N) = # {(x_1,ldots,x_n) in mathbb{N}^n | a_1 x_1 + ldots + a_n x_n = N}$
to be the number of representations of $N$, then as $N rightarrow infty$ we have
$r(a_1,ldots,a_n;N) = frac{N^{n-1}}{(a_1 cdots a_n) (n-1)!} + O(N^{n-2})$.
II. (Alfred) Brauer's theorem: for $1 leq i leq n$, put $e_i = operatorname{gcd}(a_1,ldots,a_i)$. Then
$f(a_1,ldots,a_n) leq sum_{i=2}^n a_i frac{e_{i-1}}{e_i} - sum_{i=1}^n a_i$,
with equality iff for all $i geq 2$, $frac{e_{i-1}}{e_i} a_i$ can be represented as a non-negative integer combination of the integers $(a_1,ldots,a_{i-1})$.
There have been on the order of a thousand papers written about various aspects of this problem and as well as a rather authoritative recent book:
Ramírez Alfonsín, J. L.
The Diophantine Frobenius problem.
Oxford Lecture Series in Mathematics and its Applications, 30. Oxford University Press, Oxford, 2005.
According to some form of Tannakian reconstruction, given a finite tensor category with a fiber functor to the category of vector spaces, one determines a Hopf algebra by considering tensor endomorphisms of the fiber functor. As far as I know, a similar procedure is used to reconstruct a group from its symmetric tensor category of representations.
I am curious about what happens if one is given a finite tensor category $mathcal{C}$ and a tensor functor $mathcal{C} to Rep(G)$ for $G$ a finite group. It follows that there should exist a Hopf algebra $H$ (by the previous reconstruction business applied to the composition of this tensor functor with the forgetful functor $Rep(G) to mathrm{Vect}$) and homomorphism $mathbb{C}[G] to H$.
Under what conditions will $H$ be a
semidirect product of $G$ with some Hopf algebra?
I hope this question ist still interesting for some people, as it is for me. I will first try to give a motivation for the "ghost map", i.e. the Witt polynomials, which, I think, is absent from Harder's article, and then give some historical remarks.
Motivation (partly inspired by the impressive article on Witt vectors in the German wikipedia): For this forget for a moment that we know the Teichmüller representatives. Imagine we have a strict $p$-ring $A$ with (perfect) residue ring $k$, that is, for any set-theoretic section $sigma: k rightarrow A$ of $A twoheadrightarrow k$, every element of A can be written as
$$sum_{i=0}^infty sigma(a_i) p^i$$
with unique $a_i$ in $k$. That's a set-theoretic bijection
$$k^mathbb{N} leftrightarrow A .$$
How to describe the ring structure on the "coordinates" on the left? It suffices to describe it mod $p^{n+1}$ for every n. The most naive "coordinate map" would be
$$k^{n+1} rightarrow A/p^{n+1} A$$
$$(a_0, ..., a_n) mapsto sigma(a_0) + sigma(a_1) p + ... + sigma(a_n)p^n .$$
To this, there would correspond the naiveWitt polynomial
$$NW(X_0, ..., X_{n}) = X_0 + pX_1 + ... + p^n X_n .$$
"Naive" because the above map induced by it depends on $sigma$. (We could even do worse and choose different $sigma$'s for each coordinate, still maintaining a bijection). Now do remember -- not the Teichmüller representatives, but the crucial fact from their construction: $a equiv b$ mod $p Rightarrow a^{p^i} equiv b^{p^i}$ mod $p^{i+1}$. All possible $sigma(a_0)$ are congruent mod $p$; we want something unique mod $p^{n+1}$, so why not write $sigma(a_0)^{p^n}$ in the 0-th coordinate. The first coordinate will be multiplied by $p$ anyway, so we only have to raise it to the $p^{n-1}$-th power to make it unique mod $p^{n+1}$. Upshot:
$$sigma(a_0)^{p^n} + sigma(a_1)^{p^{n-1}} p + ... + sigma(a_n) p^n in A/p^{n+1} A$$
is independent of $sigma$; in a way, it is a canonical representative in $A/p^{n+1} A$ of one element in $k^{n+1}$. And it is induced by the (non-naive) Witt polynomial
$$W(X_0, ..., X_n) = X_0^{p^n} + p X_1^{p^{n-1}} + ... + p^n X_n .$$
In other words: For fixed $(a_0, ..., a_n)$, whatever liftings $sigma_i$ one might choose, evaluating the Witt polynomial at $X_i = sigma_i(a_i)$ will give the same element in $A/p^{n+1} A$. So it looks like it might produce universal formulae for a ring structure. (Still, it is astounding that these turn out to be polynomials.)
Finally, if $k$ is perfect, this coordinate map is still bijective, and we can and will normalise it. Depending on whether one considers the normalisation
$(a_0, ..., a_n) mapsto (a_0^{p^{-n}}, ..., a_n^{p^{-n}})$ or
$(a_0, ..., a_n) mapsto (a_0^{p^{-n}}, a_1^{p^{1-n}}, ..., a_n)$
to be more natural, in the limit one will look at the element
$sum_{i=0}^infty tau (a_i^{p^{-i}}) p^i$ or
$sum_{i=0}^infty tau (a_i) p^i$
as a natural representative in $A$ of the coordinates $(a_0, a_1, ...)$ -- where the map $tau: a mapsto lim sigma (a^{p^{-i}})^{p^i}$ is independent of $sigma$ and turns out to be the Teichmüller map, which we have thus generalised by forgetting it for a while.
History: The relevant papers are
(Beware, obsolete notation: "perfekt" $sim$ complete; "(un)vollkommen" = (im)perfect)
In (1), a structure theory of complete discretely valued fields had already been done (!), although more complicated. As olli_jvn has already said, Witt was mainly working on generalising Artin-Schreier theory to what is now Artin-Schreier-Witt theory as well as constructing cyclic algebras of degree $p^n$. This is also what Schmid did in (2). This paper was discussed in an Arbeitsgemeinschaft led by Witt with participants Hasse, Teichmüller, Schmid and others. On the first page of (2), there is a note added during correction that Witt has found a "neues Kalkül" which simplifies Schmid's results. -- Hazewinkel notes (p. 5) that the Witt polynomials turn up in (3) and suggests that this might have inspired Witt, however if one looks where they come from here, one reads (p. 155 = p. 57 in Teichmüller's Collected Works):
"Tatsächlich ergibt sich das Verfahren
aus einem Formalismus, den H. L.
Schmid und E. Witt zu ganz anderen
Zwecken aufgestellt haben."
and then the Witt polynomials appear, and the summation polynomials (at least, mod p) are deduced from them. On the first page of (3), Teichmüller writes that this work was inspired by the mentioned Arbeitsgemeinschaft. In the introductions to (4) and (5), Witt and Teichmüller credit each other with realising the use of the "neues Kalkül" for the structure theory of complete discretely valued fields in the unequal characteristic case. As Hazewinkel writes (p. 9, in accordance with Witt's introduction in (4)), a decisive inspiration for Witt had been the "summation" polynomials that occured in (2), which are constructed recursively (in building an algebra of degree $p^n$ recursively by adding Artin-Schreier-like $p$-layers), and which for $n = 1$ reduce to a plain sum. Indeed, on p. 111 of (2), there are polynomials $z_nu$ which in today's notation would be Witt's $S_nu - X_nu - Y_nu$, defined recursively with the help of a polynomial $f_nu$ to be found on p. 112, which is nothing else than the Witt polynomial $W_nu$ in slightly different normalization. So presumably the timeline is:
Schmid presents his paper in the Arbeitsgemeinschaft (before January 1936) $rightarrow$ Witt finds general Witt vector "Kalkül" (January 1936) $rightarrow$ Witt and Teichmüller independently realise that this gives a structure theory of complete discretely valued fields with perfect residue field; Teichmüller finds sum and product polynomials (mod p) as well as Teichmüller representatives and reduction of the general case to the case of perfect residue field (January-February 1936) $rightarrow$ Witt works out his whole theory, Witt and Teichmüller agree to put the perfect case among all the other applications into (4), a detailed treatment of the imperfect case in (5).
Personally, I think the answer to this question is largely going to depend on one's particularly interests (whether they lie in algebra, analysis, topology, or whatever). This can be seen from many of the previous posts.
That being said, I do think that more number theory would be a great addition to the undergraduate curriculum. Many students take an introductory number theory course (or skip it because they learned it all in high school) and then don't do any more. There are lots of great areas of number theory which don't require too much background. P-adics would be great (Gouvea even laments in his book that p-adics aren't taught earlier - so maybe such a course should use his book). One could teach a basic semester of algebraic number theory, or a course in elliptic curves (following Silverman and Tate, for example). Both of these require no more than a basic course in undergraduate algebra. You can probably find these courses at many top universities, but they usually aren't emphasized as much to undergraduates. The reason why I think that these would be good is because number theory is a particularly beautiful area of math, and by getting glimpses of modern number theory early on, students get to see how beautiful is the math that's ahead of them.
(Another possibility is to have a course on Ireland and Rosen's book A Classical Introduction to Modern Number Theory. Princeton had a junior seminar on this book, for example.)
I also think Riemann surfaces are a very beautiful topic which should be taught early on and aren't too complicated in their most basic form. For, you get to see the deep geometrical theory lying behind the $e^{2 i pi}=1$ and the ambiguity of complex square roots which you learned about when you were younger. It shows the student that there can be very deep ideas lying behind a simple observation, and it shows the beauty and deep understanding that modern mathematics can lead you to.
The answers to question (1) for the 2 body problem are fine, and complete enough.
Regarding (2). The 3 body problem (and N-body) with p =3 is significantly simpler
than with $p ne 3$. The added simplicity is due to the
occurenc of an additional integral which comes out of the Lagrange Jacobi identity
for the evolution of the total moment of inertia $I$. This identity asserts
that $d^2 I/ dt^2 = 4 H + (4 - 2(p-1)) U$ where $H = K - U$ is the total energy,
with $K$ the kinetic energy and $U$ the NEGATIVE of the total potential energy,
a function which is homogeneous of degree $p-1$. When $p =3$ we get
$d^2 I/ dt^2 = 4H = const.$!.
(The total moment of inertia
is a the squared norm relative to the ``mass inner product' and as such
measures the total size of the system. )
For details on this Lagrange-Jacobi identity and its
use see the first sections of my paper
`Hyperbolic Pants fit a three-body problem', Ergodic Theory and Dynamical Systems, Volume 25, - June 2005, 921-947, which you can also find on my web site
http://count.ucsc.edu/~rmont/papers/list.html
or on the arXivs. Also see the references there.
For a study of choreographies with various $p$- potentials see the paper by Fujiwara et al.
`Choreographic Three Bodies on the Lemniscate':J. Phys. A: Math. Gen. 36 (21 March 2003) 2791-2800, available on his web site ( or the ArXivs).
http://www.clas.kitasato-u.ac.jp/~fujiwara/nBody/nbody.html
The discoverer of the figure eight, Cris Moore, in his beautiful
2 page paper `Braids and Classical Gravity'
(which Casselman should have a ref. to) found numerically, and argues
convincingly that as one increases $p$ more and more ``braid types'' (and hence choreographies) appear. It is known that
all possible braid types (and so choreography types) occur as soon as $p =2$.
The cases $p ge 2$ are often called ``strong-force potentials'' and from the
variational perspective are much simpler than $p < 2$ for the reason that
with the strong force potentials all collision paths have infinite action.
This fact regarding action is surprising, since with the strong force condition in
force it seems that almost all bounded solutions end in collision. This "seems" is a
theorem for the 2-body problem, and for the negative energy three body problem
when $p=2$.
Woe is me! I'm again resorting to this forum to ask a silly question.
Here is the example I had in mind: observe the (complex) curve $y^3=x^2(x-1)$. In attempt to normalize this curve, I've begun by blowing it up once at the origin. Of the two affines resulting from the blow up, the $x=ys$ affine is the one that will meet the strict transform. Indeed the strict transform will be $mathbb{C}[x,y,s]/x=ys,y=s^2(x-1)$. I've always assumed the natural morphism from the strict transform to the original curve is a finite one (otherwise using blow-ups to desingularize would be an odd concept!). This would imply that the above ring is integral over $mathbb{C}[x,y]/y^3-x^2(x-1)$. But I've been staring at this, and staring at this, and by the life of me I can't come up with a monic polynomial that $s=frac{x}{y}$ would satisfy over this ring.
Is the strict transform a finite morphism?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do homological algebra for commutative monoids, but first let me explain some background, my motivation, and articulate more precisely what I am after.
A Quillen Model structure on a category has three classes of morphisms: fibrations, weak equivalences, and cofibrations. This structure allows one to do many advanced homotopical constructions mimicking the homotopy theory of (nice) topological spaces. There is a notion of Quillen equivalence between model categories which consists of a particular adjunction between the two model categories in question. This gives you "equivalent homotopy theories" for the two model categories in question.
The usual Model structure on simplicial sets has fibrations the Kan fibrations, the weak-equivalences are the maps which induces isomorphisms of homotopy groups, and the cofibrations are the (levelwise) inclusions. This is equivalent to the usual model category of topological spaces, and the Quillen equivalence is realized by the adjoint pair of functors: the geometric realization functor and the singular functor.
Quillen model categories are also useful for doing homological algebra, and particularly for working with derived categories. For reasonable abelian categories there are several nice (Quillen equivalent) model category structures on the category of (possibly bounded) chain complexes which one can use which reproduce the derived category. More precisely the homotopy category of the model category is the derived category and the "homotopical constructions" I mentioned above, in this case, correspond to the notion of (total) derived functor.
This story is further enriched by the Dold-Kan correspondence which is an equivalence between the categories of positively graded chain complexes of, say, abelian groups and the category of simplicial abelian groups, a.k.a. simplicial sets which are also abelian group objects. This in turn is Quillen equivalent to a model category of topological abelian groups.
Previously I asked a question on MO about doing homological algebra for commutative monoids. I got many fascinating and exciting answers. Some were more or less "here is something you might try", some were more like "here is a bit that people have done, but the full theory hasn't been studied". After hearing those answers I'm much more excited about the field with one element. But still, in the end none of the answers really had what I was after. Then Reid Barton asked his MO question.
This got me thinking about commutative monoids and simplicial commutative monoids again. I had some nice observations which have lead me to the question at hand.
The first is that while a simplicial abelian group is automatically a Kan simplicial set (i.e. if you forget the abelian group structure what have sitting in front of you is a Kan simplicial set) this is not the case for simplicial commutative monoids. A simplicial commutative monoid does not have to be a Kan simplicial set.
Next, I realized that the (normalized!) Dold-Kan correspondence still seems to work. You can go from a simplicial commutative monoid to a "complex" of monoids, ( and back again It is just an adjunction. Thanks, Reid, for pointing this out!).
If your simplicial commutative monoid is also a Kan complex, then under the Dold-Kan correspondence you get a complex where the bottom object is a commutative monoid, but all the rest are abelian groups (this was pointed out to me by Reid Barton in a conversation we had recently). Thus the theory of topological commutative monoids (which was one of the suggested answers to my previous question), which has a nice model category structure (see Clark Barwick's answer to an MO question), should be equivalent to a theory of chain complexes of this type. It shouldn't model arbitrary complexes of commutative monoids.
If you have a simplicial set which is not necessarily a Kan complex you can still define the naive simplicial homotopy "groups" $pi_iX$. I put "goups" in quotes because for a general simplicial set these are just pointed sets. If your complex is Kan, these are automatically groups (for $i>0$). If your simplicial set is a simplicial abelian group, these are abelian groups (for all $i$) and they are precisely the homology groups of the chain complex you get under the Dold-Kan correspondence. If your simplicial set is a commutative monoid, but not Kan, then they are commutative monoids and are again the "homology" monoids of the chain complex you get under the Dold-Kan correspondence.
All this suggests that there should be a Quillen model structure on simplicial commutative monoids in which the weak equivalences are the $pi_{bullet} $-isomorphisms, where here $pi_{bullet} $ denotes the naive simplicial version, i.e. these are commutative monoids, not groups. I'm sure if such a thing was well known then it would have been mentioned as an answer to my previous question. I'd really like to see something that generalizes the usual theory of abelian groups. That way if we worked with simplicial abelian groups and construct derived functors we would just reproduce the old answers. As a stepping stone, there should be a companion model structure on simplicial sets which, I think, is more likely to be well known.
One of the properties that I think this hypothetical model structure on simplicial sets should have is that every simplicial set is fibrant, not just the Kan simplicial sets.
Question: Is there a model structure on simplicial sets in which every simplicial set is fibrant, and such that the weak equivalences between the Kan complexes are exactly the usual weak equivalences? Specifically can the weak equivalences be taken to be those maps which induce $pi_*$-isomorphism, where these are the naive simplicial homotopy sets?
If this model structure exists, I'd like to know as much as possible about it. If you have any references to the literature, I'd appreciate those too, but the main question is as it stands.
There are many formulas for atmospheric pressure on earth, but how does gas behave in free space?
I am thinking about why stars form. I am guessing that the gas density will influence pressure, as well as gravitational force. So both factors must determine if a given gas mass in a given volume in space will collapse into a star.
So a more well defined question would be: what is the relation between mass and volume in space (let's think about hydrogen only) so that this mass will collapse into a star (or at least forms a spherical body)?
A second question is: given a mass and volume so that the gas collapses into a spherical body, what is the pressure vs. distance from the center of the sphere?
Quite interesting I think!
Any ideas?
Let $C$ be an algebra. Let $E = C^{otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = Cotimes C^{op}otimes Cotimes C^{op}cdotsotimes C otimes C^{op}quad$ ($2n$ copies of $C$, half of them "op").]
Let $F_{even}$ be the $E$ module with underlying vector space $C^{otimes n}$, where the $(2i)$-th factor of $E$ acts on the left side of the $i$-th factor of $F_{even}$, and the $(2i+1)$-th factor of $E$ acts on the right side (i.e. left action of $C^{op}$) of the $i$-th factor of $F_{even}$. ($0 le i le n-1$ .)
Define $F_{odd}$ similarly (same underlying vector space $C^{otimes n}$), but with the $(2i)$-th factor of $E$ acting on the left side of the $i$-th factor of $F_{odd}$, and the $(2i+1)$-th factor of $E$ acting on the right side of the
$(i+1)$-th (modulo $n$) factor of $F_{odd}$. (Note that "modulo $n$" means the $n$-th factor is the 0-th factor.)
I'm interested in the derived $Hom_E(F_{even}, F_{odd})$. We can think of this as living in a disk with its boundary divided into $2n$ segments, alternating between incoming and outgoing segments. For $n=1$ this is just the Hochschild cohomology of $C$. For $n=2$ it's what one might associate to a saddle bordism in a 1+1-dimensional TQFT, or rather, the space in which the TQFT invariant of the saddle bordism lives.
Does $Hom_E(F_{even}, F_{odd})$ have a name? Is it mentioned in the literature anywhere?
Some of the many (semi)standard references are below (with no claims to completeness or representativeness, if that's a word -- just the first references that came to mind). My feeling is the subject is still very much in its infancy however, for example one would like to know the standard package of nonabelian Hodge theory results for singular curves (geometry of Higgs bundles and local systems, Hitchin fibration, its self-duality etc) and there are partial results but no complete picture as far as I know.
Caporaso, Lucia A compactification of the universal Picard variety over the moduli space of stable curves. J. Amer. Math. Soc. 7 (1994), no. 3, 589--660.
Pandharipande, Rahul A compactification over $overline {M}_g$ of the universal moduli space of slope-semistable vector bundles. J. Amer. Math. Soc. 9 (1996), no. 2, 425--471.
Seshadri, C. S. Moduli spaces of torsion free sheaves on nodal curves and generalisations. I. Moduli spaces and vector bundles, 484--505, London Math. Soc. Lecture Note Ser., 359, Cambridge Univ. Press, Cambridge, 2009.
(and earlier papers of his)
arXiv:1001.3868 Title: Autoduality of compactified Jacobians for curves with plane singularities
Authors: D.Arinkin
--see this reference for refs to the vast literature by Altman-Kleiman and Esteves-Kleiman on compactified Jacobians
Kausz, Ivan A Gieseker type degeneration of moduli stacks of vector bundles on curves. Trans. Amer. Math. Soc. 357 (2005), no. 12, 4897--4955 (electronic).
Schmitt, Alexander H. W. Singular principal $G$-bundles on nodal curves. J. Eur. Math. Soc. (JEMS) 7 (2005), no. 2, 215--251.
(and earlier papers of his)
To revisit Bruce's earlier question, it might be useful to suggest a more sceptical alternative to Ben's answer and the related comments. I doubt that there will be a "correct" version of the CDE-triangle that has enough breadth to take in the variety of analogues that have emerged in representation theory, though it's obviously important to make the assumptions precise about underlying fields, local rings, etc. See also the conference paper: MR2184010 (2006g:17027) 17B67 (17B10)
Ekedahl, Torsten (S-STOC),
Kac-Moody algebras and the cde-triangle.
Noncommutative geometry and representation theory in mathematical physics, 49–58, Contemp.
Math., 391, Amer. Math. Soc., Providence, RI, 2005.
For me the essential ingredients to start with are suitable module categories (or more general analogues) in which three distinctive types of objects play a leading role: (1) simple modules, which are "small" and basic but often hard to get at directly; (2) indecomposable projectives (or injectives ...) which are also naturally present in suitable categories and usually have just finitely many composition factors, but are typically "large" and messy to study; (3) intermediate objects, easier to construct and often having known dimensions (if finite) or "formal" characters. The idea then is to express (say) a projective cover of a simple module formally, perhaps via a filtration, in terms of the intermediate objects. To make this interesting, when all composition factor multiplicities are finite or otherwise controllable, one wants a kind of "reciprocity" between multiplicities of simples in intermediate objects and multiplicities of the latter in the big modules. Here is some brief history followed by a sample of more recent instances. (It does get a bit long...)
(1) Elie Cartan actually studied what we now call finite dimensional associative algebras and emphasized the importance of knowing the composition factor multiplicities of indecomposable projectives (now dubbed Cartan invariants). For a finite group with $r$ classes of elements having orders not divisible by a given prime $p$, you get an $r times r$ matrix $C$ (over a large enough field).
(2) Work by Richard Brauer and his Toronto student Cecil Nesbitt after 1937 introduced in the finite group setting (for a prime $p$ dividing the group order and large enough fields) an $s times r$ decomposition matrix $D$ showing how to express the $s$ ordinary irreducible characters (= number of classes) as formal sums of $p$-modular irreducible characters (counting composition factor multiplicities in reduction mod $p$ for any suitable lattice in a module). Then $C = D^{t} D$ (so $C$ is symmetric), where the transpose of $D$ shows how to express a projective (lifted to characteristic 0) as a combination of ordinary characters.
These ideas were exposed by Curtis-Reiner (1962) in Section 83, etc. Brauer was studying $p$-blocks, which show up in the block decompositions of the matrices.
(3) Following Swan's formalism, Serre (1971) formulated the more abstract cde-triangle in his part III, using homomorphisms between various Grothendieck groups. This was further codified by Curtis-Reiner in their later 1981 book, Section 18, with a lot of attention to the rings and fields involved.
(4) Having studied the old CR book in a 1963-64 course at Yale taught by Jacobson (!), I later tried to adapt $C = D^{t} D$ to modular representations of Lie algebras of simple algebraic groups (working just in prime characteristic). The f.d. restricted enveloping algebra imitates a finite group algebra in some ways. Here the "intermediate" objects were f.d. analogues of Verma modules. At first I studied blocks and multiplicities but not filtrations. This rough version appeared in J. Algebra (1971) and inspired Verma's introduction of affine Weyl groups relative to $p$, as well as much more sophisticated work by Jantzen treating filtrations of projectives, plus action of a maximal torus in the group.
(5) Work by Bernstein-Gelfand-Gelfand in the early 1970s was partly inspired by Jantzen's work and by my 1971 paper, leading to their "BGG category" and "BGG reciprocity" in 1976. Then Kazhdan-Lusztig theory for finite and affine Weyl groups came into play, etc.
(6) Eventually some but not all of the ideas spread elsewhere in representation theory, including the work by Alvany Rocha and her thesis advisor Nolan Wallach
and much other work on Kac-Moody algebras (recently by Arakawa-Fiebig for the mysterious critical level in the affine case). Plus early work by Dan Nakano on other modular Lie algebra settings, recent work on rational Cherednik algebras, and so on.
(7) The most fruitful general formulation for some purposes was given in a 1988 Crelle paper by Cline-Parshall-Scott on highest weight categories and quasi-hereditary algebras. Other general settings were proposed by Ron Irving and by Apoorva Khare. It's hard though to find just one common framework.
On the category CatSet of usual set based categories,
there is a "folk" model structure, as described on the first page of
Model structures for homotopy of internal categories
by T. Everaert, R.W. Kieboom and T. Van der Linden. Namely: in
CatSet, ws are weak equivalences, cs are functors injective on
objects, fs are functors with the lifting property for isomorphisms.
wfs are then precisely the full faithful functors surjective on
objects.
Is there's any nice sense in which this model
category structure on CatSet is unique?
Your third question has a very well known answer here:
What means, that although a big time difference had to exist, there is a much smaller one, and in the opposite direction. This is considered the effect of the dark matter.
Our Sun is around 30000 ly from the galactic center, the edge of the galaxy is around 50000 ly, so we can read on the diagram, that the actual speed difference is around 15 km/s, which is 1/20000 of the speed of light. This causes a (not really big) time dilation due to the special relativity.
There is another source of the time dilation, which is caused by the gravity of the galaxy, which differs in the case of us and in the case of somebody living on the outer edge. On such weak gravitation (i.e. for from any black hole, neutron star, etc) it practically depends on the newtonian gravitational acceleration.
But this acceleration is very small - it takes a half galactic year (some hundred millions of year) to revert the orbital speed of the Sun around the galactic core! So, we can consider that actually negligible.
Thus only the special relativistic effect remains. 15 km/s is around 1/20000 of the speed of the light. In cases of speed which are much smaller as the light speed, we can use the speed dilation formula 1/(2*(v/c)^2). Substituting 1/20000 into this formula, we get 1:800 000 000 .
For laymans, we can say, it takes around 25 years for the clocks of this people living on the outer edge of our galaxy to dilate 1 second.
The following answer is for finite groups.
In characteristic zero, the group algebra is semisimple, so there are finitely many simple representations. These representations correspond to the blocks in the decomposition as a product of matrix algebras.
Here is a way to determine the number:
Start with the field $K$, adjoin the $g$th roots of unity ($g = |G|$) to get $L$, and consider the Galois group $Gamma_K=L/K$. This is a subgroup of the multiplicative group of the integers mod $g$. Then $sigma_t in Gamma_K$ corresponding to $t in (mathbb{Z}/gmathbb{Z})^*$ acts on $G$ by raising $x in G$ to the $t$-th power. The dimension of the space of class functions constant on $Gamma_K$-orbits is the number of simple $K$-representations.
As for characteristic p, this is modular representation theory, The number of irreducibles is the number of $p$-regular conjugacy classes (where $p$-regular means the period is prime to $p$), when the field contains the $g$th roots of unity for $g$ the order of the group. See, e.g., Serre's Linear Representations of FInite Groups. My guess is that it should be true even without the assumption on the field being sufficiently large.
Background
Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 rightarrow F rightarrow G rightarrow H rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 rightarrow F(U) rightarrow G(U) rightarrow H(U) rightarrow 0$ is exact for any open set $U$. My solution to this involved the axiom of choice in (what seems to be) an essential way.
Essentially, you are asking to $G(U) rightarrow H(U)$ to be surjective when you only know that $G rightarrow H$ is locally surjective. Ordinarily, you might not be able to glue the local preimages of sections in $H(U)$ together into a section of $G(U)$, but since $F$ is flasque, you can extend the difference on overlaps to a global section. This observation deals with gluing finitely many local preimages together. Zorn's lemma enters in to show that you can actually glue things together even if the open cover of $U$ is infinite.
Now, I have not really studied sheaf cohomology, but the idea I have is that it detects the failure of the global sections functor to be right exact. So if you can't even show sheaf cohomology vanishes for flasque sheaves without the axiom of choice, it seems like a lot of the machinery of cohomology would go out the window.
Now, just on the set theoretic level, it seems like there is something interesting going on here. Essentially the axiom of choice is a local-global statement (although I had never thought of it this way before this problem), namely that if $f:X rightarrow Y$ is a surjection you can find a way to glue the preimages $f^{-1}({y})$ of a surjection together to form a section of the map $f$.
This brings me to my
Questions
Can the above mentioned exercise in Hartshorne be proven without the axiom of choice?
How much homological machinery depends on choice?
Have any reverse mathematicians taken a look at sheaf cohomology as a subject to be "deconstructed"?
Have any constructive set theorists thought about using cohomological technology to talk about the extent to which choice fails in their brand of intuitionistic set theory? (it seems like topos models of such set theories might make the connection to sheaves and their cohomology very strong!)
My google-fu is quite weak, but searches for "reverse mathematics cohomology" didn't seem to bring anything up.
Dear saurav, yes the restriction is closed and you don't have at all to assume $k$ algebraically closed.
Reminder: For a scheme $S$ of finite type over $k$, the set $T_0$ of closed points of $T$ is very dense in $T$, i.e. dense in every closed subset of $T$ .
Here then is the statement you need :
Let $f:Xto Y$ be a closed morphism between schemes of finite type over a field $k$. Then the restriction $f_0:X_0to Y_0$ to the subspaces of respective closed points is closed.
Proof:
1) We may assume $X$ and $Y$ reduced. We have to show that, for $F$ closed in $X$, the subset $f(Fcap X_0)$ is closed in $Y_0$. By endowing $F$ and $f(F)$ with their reduced scheme structure we can assume $F=X, f(F)=Y$ and we have reduced (!) the problem to showing that $Y_0=f(X_0)$ : call this "closed surjectivity".
2) Here is why "closed surjectivity" is true. Take any closed point $y_0in Y_0$. It has a residual field $kappa (y_0)$ which is a finite extension of $k$.The fibre $Z=f^{-1}(y_0)$ is a closed non-empty subscheme of $X$ (recall that after our reduction $f$ is assumed surjective!).
But $Z$ is also a $kappa (y_0)$-scheme of finite type. Hence it has a closed point $x_0$, since those closed points are even very dense in $Z$. Since $Z$ is closed in $X$, $x_0$ is closed also in $X$ i.e. $x_0in X_0$. So we have shown "closed surjectivity" and everything is proved.
(Everything I used is contained in EGA I)
"What is the most effective way to learn mathematics?"
I have been trying to answer this question for myself, and one measure I've taken towards this goal is to record all of my mathematical reading, work, and random thoughts in a journal. I highly recommend the practice as it has been very illuminating to me since I started a few months ago. Reviewing my previous readings allows me to ascertain how much math I actually end up retaining from my study sessions, and keeping all of my work in one place (as opposed to throwaway scrap paper) allows me to spot any particularly common mistakes.
So far, I've found that my memory is far more tenuous than I had previously assumed. I'd look at last month's entries and realize that I'd only retained 20% of what I had learned; fine details being especially prone to slippage. Yet from analyzing my mistakes, I've also found that those very details are much more crucial than I had thought.
The result of all of this is that I've started to shift my focus from "learning new math rapidly" (which has been my focus since I am still an undergraduate) to "winning the uphill battle against memory loss." From this new perspective, the old adage: "the only way to learn mathematics is through doing" begins to make a lot more sense. While active learning is far from any cure to forgetfulness, given my own mnemonic capabilities I have come to see that it would probably be a better long-term investment to spend a month on fully working and understanding a chapter, than to spend the same time blazing through several chapters but skipping the exercises (having done both.)
I emphasize again that this is my own conclusion based on my own characteristics, and that is precisely why I recommend everyone to find their own answer to this question by keeping their own math notebook.
Let $W^{k,p}$ denote the sobolev space of all $k$-times weak-differentiable $L^p$ functions such that each derivative is $L^p$ and $W_0^{k,p}(Omega)$ be the closure of $mathcal C^infty_c(Omega)$ in this space.
Given $Omegasubsetsubsetmathbb R^N$, a Caccioppoli set $Esubset Omega$, $Usubseteq Omega$ open and $win W_0^{1,p}(Omega)$. Can we conclude in the case of
$$ int_E div(eta w) = 0 qquadforall;etainmathcal C^1_c(U;mathbb R^N) $$
that there exists a sequence $w_ninmathcal C^1_c(Omega)$ with $mathcal H^{N-1}(Ucappartial^*E cap [w_nneq 0])=0$ which converges to $w$ in $W^{1,p}(Omega)$? (Where $[w_nneq 0]=Omegasetminus w_n^{-1}(0)$ and $mathcal H^{N-1}$ is the $N-1$-dimensional Hausdorff-measure).
The assertion seems naturally to me, because for $winmathcal C_c^1(Omega)$ the equality would per definition of a caccippoli set imply
$$ int_{partial^*E} weta nu_E = 0 qquadforall;etainmathcal C^1_c(U,mathbb R^N)$$
where $nu_E$ is the inner normal of $E$ existing almost-everywhere on the reduced boundary. This would imply that $w=0$ $mathcal H^{N-1}$-almost everywhere on the reduced boundary, wouldn't it? Just the same is by the divergence-theorem true if $partial E$ is lipschitz and $w$ sobolev, because in that case the trace of $w$ on $partial E$ is well defined, which means that there exists a sequence of $mathcal C_c^1(Omega)$ functions vanashing on $partial Ecap U$ approximating $w$ in $W^{1,p}(Omega)$.
But I didn't find informations about this special case, except in the case of $win W_0^{1,p}(Omega)cap L^infty(Omega)$ in which this seems also to be correct -- see Divergence-Measure Fields, Sets of Finite Perimeter, and Conservation Laws, by Gui-Qiang Chen, Monica Torres in Arch. Rational Mech. Anal. 175, 2005. They do actually not need $W^{1,p}$ but it works in that case...
Are at all the statements above for $winmathcal C^1_c$ or $partial E$ lipschitz correct? What is with the case of $w$ only in $W^{1,p}$ and $E$ only caccioppoli?
EGA IV$_4$, 19.3.8 (and 19.3.6); this addresses openness upstairs without properness, and (as an immediate consequence) the openness downstairs if $f$ is proper (which I assume you meant to require).
The general intuition is that openness holds upstairs for many properties, and so then holds downstairs when map is proper. As for proving openness upstairs, the rough idea is to first prove constructibility results, and then refine to openness by using behavior under generization. But it's a long story, since there are many kinds of properties one can imagine wanting to deal with. These sorts of things are developed in an extraordinarily systematic and comprehensive manner in EGA IV$_3$, sections 9, 11, 12 (especially section 12 for the niftiest stuff).
If you can check that the curve is geometrically irreducible, then you may try using the Hurwitz formula (you may use the formula in any case, but you would have to be more careful with the conclusions if the curve is not irreducible). Assuming that your example is geometrically integral, projection onto one of the axis is a morphism of degree 23. If you know that the morphism ramifies in at least 2*23+h smooth points, then the curve has genus at least 1+h/2. If you know the multiplicities of the ramification or if you know the behaviour of ramification at the singular points, then you can deduce more accurate information.
EDIT: here is an expanded, more computational, version. Write an equation f defining your curve C as a polynomial of degree d in y whose coefficients are polynomials in x. Thus the morphism $(x,y) to x$ is a morphism of degree d to $mathbb{A}^1$. We know that this induces a finite morphism from a smooth projective model C' of C to $mathbb{P}^1$ and we use the Hurwitz formula to compute the (arithmetic) genus g of C': $2g-2=-2d+r$, where r is the degree of the ramification divisor. The better you can estimate r from below, the better the approximation to the geometric genus of C.
Clearly, the points where f=df/dy=0 but $df/dx neq 0$ are smooth points of C that are ramified for the projection to $mathbb{A}^1$ and hence contribute to the Hurwitz formula (ramification occurs in C' since such points are smooth and hence C and C' are locally isomorphic here).
Computationally we find the resultant in y of (f,df/dy) (thus you get a polynomial in x) and purge out of it the roots it has in common with the resultant in y of (df/dx,df,dy). Denote this final polynomial by R; the number of distinct roots of R is a lower bound for r. If you are more careful, you might do a little better than this without having to worry about the singular points of C, but for "generic" projections this is optimal.
It seems to me that all this requires is the ability to compute resultants of polynomials in two variables of relatively small degree as well as factoring polynomials in a single variable (and computing gcd's and quotients of polynomials in one variable).
This can further be improved to a better and better lower bound by analyzing more carefully the structure of the ramification at the singular points of C and what happens at infinity: some of the points that we threw out could in fact contribute to the arithmetic genus of C', and there could be ramification "at infinity". I leave this as an exercise!
Both the temperature and the density of a toy universe can, in a way, be linked to the age of the universe ($t$). Certain properties of the universe are dependent upon a scale factor $a(t)$ that is a function of time. The expansion of such a universe can be written in the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, generally written as
$$ds^2 = dt^2 - a(t)^2(dSigma^2)$$
where, in spherical coordinates, $dSigma$ is given by
$$dSigma^2 = frac{dr^2}{1-kr^2} + r^2dtheta^2 + r^2sin^2theta dphi^2$$
where $k$ is a constant. This shows that, in a toy universe, $r$ and $dr$ is proportional to $a(t)$, and so in a perfectly spherical toy universe, you could relate the volume to $t$. We know that, for a sphere,
$$V=frac{4}{3}pi r^3$$
and because $dSigma^2$ (and by extension $r^2$) is related to $a(t)^2$, we can relate volume to time. If you knew the mass-energy of the universe, you could then relate the change in density to $t$. Unfortunately, our universe does not appear to be perfectly spherical, and so this may not apply.
You can relate something else to the age of the universe: the temperature of the cosmic microwave background. It has cooled over time, and is presently at about 2.7 Kelvin. While the temperature of the CMB may not be governed by the scale factor $a(t)$, it is certainly governed by time, and thus one you could in fact write the temperature of the universe as function of time. Unfortunately, I haven't been able to find data on the evolution of the CMB, but I can try, and perhaps this will lead you to an answer.
To summarize: In a perfectly spherical universe, assuming you know the matter-energy content and the scale factor $a(t)$, you could write an equation for the density of the universe. In any universe, you can also write an equation for the temperature of the CMB - the "temperature of the universe".
I hope this helps.
Source for FLRW metric: http://www.universe-galaxies-stars.com/Robertson-Walker_coordinates.html
Suppose X is a smooth projective variety, say over $mathbb{Q}$ for simplicity. Let $F$ be a finite extension of $mathbb{Q}$. Let $mathrm {Ch}^{r}(X/F)$ denote the Chow group of codimension $r$ algebraic cycles defined over $F$. A conjecture of Tate asserts that the cycle class map from $mathrm{Ch}^r(X/F)$ to $H^{2r}_{et}(X)(r)$ is injective, with image in the subspace fixed by $mathrm{Gal}(overline{mathbb{Q}}/F)$. In particular, the dimension of $mathrm{Ch}^r(X/F)$ should be uniformly bounded (by the $2r^{mathrm{th}}$ Betti number of $X(mathbb{C})$) as $F$ varies.
On the other hand, let $mathrm{Ch}^{r}(X/F)_0$ denote the kernel of the cycle class map, which is to say the group of homologically trivial cycles of codimension $r$ modulo rational equivalence. The dimension of this guy is predicted by Beilinson and Bloch to be given as the order of vanishing of $L(s,H^{2r-1}(X/F))$ at its central critical point. Now, the order of vanishing of this L-function can by made to increase very rapidly as $F$ varies; for example, one could (in some circumstances) choose $F$ to be an abelian extension such that each of the twists $L(s,H^{2r-1}(X/mathbb{Q})times chi)$ has root number $-1$ for $chi$ varying over characters of $mathrm{Gal}(F/mathbb{Q})$. When $X$ is an elliptic curve and $r$=1, this phenomenon has been confirmed in a variety of situations: for $mathbb{Z}/p^{n}mathbb{Z}$-towers over imaginary quadratic fields (Cornut-Vatsal), for Hilbert class fields of imaginary quadratic fields (Templier), and for towers of Kummer extensions (Darmon-Tian). However, for higher dimensional varieties and higher codimension cycles, the relevant L-functions aren't even well understood.
My question: is there a "conceptual" reason why there should have lots of homologically trivial cycle classes over extensions of the base field? In other words, if you believe certain conjectures about L-functions, then this is not hard to guess, but I am looking for some motivation which avoids L-functions.
(Edited in response to a comment of moonface.)
My internet access at the moment is limited & sluggish, so I haven't been able to look up all the details; but I think your reasoning is correct. Certainly my impression is that duals in the sense beloved by (S)MC people only work for finite-dimensional Banach spaces.
By the way, for arbitrary Banach spaces the first map you describe wants to land in the injective tensor product, while the second eants to come out of the projective tensor product. Thus the failure to get categorical duals for inf-dim Banach spaces is surely related to, though perhaps neither implying nor implied by, the following old result which I think is due to Grothendieck: if X is a Banach space and, for each Banach space E, the usual tensor product of X with E (in the category Vect) has a unique Banach completion, then X is finite-dimensional. More pithily, the only nuclear Banach spaces are the finite-dimensional ones.
