nt.number theory - Spacing of zeros of zeta function on the critical line

I believe $Lambda(k)$ is the lower bound that arises from assuming the $k$th moment is predicted by random matrix theory (equation (1.13)) with the explicit constants $b(h,k)$ derived by Hughes. The method seems to require knowledge of the constants $b(k,k)$. Indeed at the top of page 5 he discusses the work of Hall in the case $h=3$, for which the value of $Lambda$ derived (equation (1.17)) occurs in the table (1.18) as $Lambda(3)$.

differential equations - Are there alternative proofs for existence/uniqueness of ODE solutions?

A very interesting kind of existence (though not uniqueness) proofs are proofs that use one of the various fixed point theorems and the tools from fixed point theory: The Schauder fixed point theorem can be used to prove Peano's existence theorem or simple existence theorems for boundary value problems. The theory of Brouwer-degree of certain mappings (not just between manifolds but also between banach spaces) can be used to prove several existence theorems, for example existence of periodic solutions for certain ODEs.

Granas / Dugundji - Fixed point theory is a very good (and very densely written) book about all kinds of fixed point theorems and one of the main application that frequently occurs throughout the book are existence theorems for differential equations.

Then there is the variational approach: Finding minima of functionals often is the same thing as finding solutions of PDEs (Euler-Lagrange-equations !). So very much of functional analysis can be applied to prove various existence theorems for PDEs. For example have a look at Guisti - Direct methods in the calculus of variations. Any book on the finite element method (Braess comes to mind, but I'm not sure at the moment if it is in english or in german... may be there are two versions?) will show you how Hilbert space methods can be applied to prove existence theorems.

The whole theory of distributions was invented for dealing with (linear) PDEs, their (weak) solutions and the regularity theory of these solutions. The Ehrenpreis-Malgrange-theorem is a very strong existence theorem that says that all linear PDEs with constant coefficient have distributional solutions. In Fact there exist (tempered) Green's functions for every such PDE.

And there is of course a more heavy machinery too: Morse theory and generalizations of it were used for (invented for?) the proof of the Arnol'd conjecture which also shows the existence of certain periodic solutions of differential equations.

lie algebras - sl(2)-modules...

You can't assume the module is irreducible, since many aren't!

However, if you want to learn something about modules of $sl_2$ it helps to make the following observations:

1. Each module is a sum of irreducible modules


2. It's easy to describe all irreducible modules.

In particular, when you'll be solving 2, you'll notice that there is exactly one irreducible modle of dimension 2. In fact, there is one of dimension 3, one of dimension 4, etc.

dg.differential geometry - $SU(2)$ and the three sphere

Just write $SU(2)$ in some local coordinates (some of the standard systems are the double-polar system (a.k.a Hyperspherical coordinates) or the single angle coordinates (a.k.a Hopf coordinates) and then one sees that the special unitary condition forces the $4$ parameters you will need to satisfy the $3$-sphere equation. Hence one gets an explicit local map to the $3$-sphere by mapping that matrix to the ordered tuple of $4$ numbers. Smoothness is assured since the functions are all polynomials.

These coordinate systems make what MTS said below explicit.

After all one is very likely to want to do some geometry on $SU(2)$ now that one knows it is $S^3$ and these coordinate systems are naturally adapted to do them. Like computing vielbiens on $SU(2)$ in these systems look very natural and make the symmetries of the spherical structure underneath very clear.

A related extra stuff:

One can look up a very nice analysis of this in the first chapter of Gregory Naber's book "Geometry, Topology and Gauge Fields" Volume 1.

In that section he will do what gets called the Heegard Decomposition of $S^3$ using very simple high-school maths!

Basically Naber will rationalize why these coordinate systems are in some sense natural.

You can also look up these analysis in these two write-ups I had done during my undergrad , this and this one.

pr.probability - Motivation for strong law of large numbers

I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?

I think it comes down to not having a good sense of why, practically speaking, a.s. convergence is better than convergence i.p. Sure, I can prove that one implies the other and not conversely, but the counterexamples feel contrived. I understand the advantages of a.s. convergence on a technical level, but not on the level of everyday life.

So my question: how would you explain to, say, an engineer, the significance of having a.s. convergence as opposed to i.p.? Is there a "real-life" example of bad behavior that we're ruling out?

ag.algebraic geometry - equation for abelian varieties with a given polarization

Let $A$ be an abelian variety of dimension g and a polarization $L$ of type $(d_1,.....,d_g)$ (let alone the case $d_i=d_j,$ $forall i, j$). What is the degree of the generators of the homogeneous ideal of A projectively embedded via the sections of L?

I know that Gross and Popescu gave results for surfaces with L of type $(1,d)$ - for instance if $d>10$ the ideal is generated by quadrics - but what for other polarizations and most of all for higher dimensions? Is this known?

ag.algebraic geometry - Smoothness and fullness of determinantal varieties

So, I'm in the following situation:

I have vector spaces $H,V$, and a map $A:Hlongrightarrow hom(H,V)$, sending $x$ to $A_x$ (notation); i need to consider the variety

$Sigma_1(A)={[x]in mathbb{P}(H)|\; rank(A_x)leq 1}$.

Mostly, I'm interested in wether this variety is smooth, reducible, and/or full (by full, i mean not contained in any proper projective subspace).

Are there any conditions on $A$ that allow me to know any of these properties?

I don't know what $A$ is, but here is a couple of things i know:

1. A has no kernel;




2. If I define $hat{A}:Vlongrightarrow hom(H,H)$ by $hat{A}_v(x):=A_x(v)$, then $hat{A}_v$ is skew-symmetric for all $vin V$.

Many thanks in advance!

dg.differential geometry - Integration in equivariant K-theory

It sounds like what you are after is a theory of genuine equivariant K-theory (Which exists!)

If you think of a cohomology theory as a sequence of functors to abelian groups together with some properties and suspension isomorphism, then a genuine G-equivariant cohomology theory can be similarly thought of as a sequence of functors (indexed on the representation ring of G) together with "suspension isomorphisms" where you suspend by any G-representation, i.e. take the G-sphere with is the one point compactification of W (with its G-action) and smash with it. This is only the rough picture. The real theory is somewhat technical and develops the theory of genuine G-spectra. There were some comments about this here with some references. The punchline is that yes K-theory is an example of a genuine G-spectrum.

The next step is that you need to identify your K-theory with compact supports as the reduced K-theory of the suspension. I don't think this is too hard. Then your integration map reduces to the suspension isomorphism:

$tilde{K}^*_G( Sigma^W X) cong tilde{K}^{* -W}_G(X)$

As Michael and Kevin mention, under a suitable equivariant K-theory orientation hypothesis there will be similar integration maps for more general vector bundles, but that's another story.

riemann surfaces - Weil's theorem about maps from a discrete group to a Lie group.

Let K be a group (with discrete topology), G be a Lie group. Let $operatorname{Hom}(K,G)$ be the space of all group homomorphisms from K to G. This is a closed subvariety of the space of all the maps from the generators of K to G, and as such has a topology.

Andre Weil's paper "On Discrete Subgroups of Lie Groups" proves that an important subset $U subset operatorname{Hom}(K,G)$ is open. U is defined as the set of all homomorphism $Kto G$ such that the homomorphism is injective, the image is discrete, and the quotient $G/image(K)$ is compact.

Questions:

1. What happens if you remove the condition that the quotient is compact?




2. How often/where is this taught? What kinds of books would it be in, what kind of courses would have it? This looks like a basic result that could be taught anywhere,
   but it's completely new to me (not that I know much about representation theory).
   while Weil's paper fortunately seems very readable, I couldn't easily find any other
   source that would such questions.

Motivation:

In the case where $K=pi_1 (S)$ is the fundamental group of a surface and $G=PSL_2(mathbb R)$, the space $operatorname{Hom}(K,G)$ is very closely related to the Teichmuller space of S. Every Riemann surface is a quotient of $PSL_2(mathbb R)$ by a discrete subgroup. So, for an element of $operatorname{Hom}(K,G)$, the quotient $G/image(K)$ corresponds to a Riemann surface and the data of the actual map $Kto G$ gives a marking on it.

Not every homomorphism $Kto G$ corresponds to a point of Teichmuller space. For example, the map that sends all of K to the identity is clearly no good, as the quotient $K/G$ is not topologically the same as the surface S. However, if the map is injective and the image of K in G is discrete, all will be well. So, Weil's theorem basically says that the Teichmuller space of S is an open subset of $operatorname{Hom}(pi_1(S),PSL_2(mathbb R))$.

However, since Weil's theorem requires the quotient to be compact, this won't work if S is a non-compact Riemann surface. I wonder how much more difficult life becomes in this case.

Disclaimer/Another question:

The above has a small lie in it. To get the Teichmuller space, you actually need to look at the quotient $operatorname{Hom}(K,G)/G$ where G acts on $operatorname{Hom}(K,G)$ by conjugation of the target. In the case of compact surfaces, this is not supposed to mess up the fact that the subset is open; this seems to be a result of William Goldman but I don't have the exact reference. If you can say anything about this, I'd appreciate it too.

Thank you very much!

nt.number theory - Intrinsic construction of other Galois extensions

Let me first give a formulation of the question that makes a little bit of sense.
Let $K/F$ be a quadratic extension of a number field, and let $L/K$ be a class field
for an ideal group $D_L$ with index $2$ in the group of all idealc coprime to some
defining modulus.

1. Which properties of $D_L$ guarantee that $L/F$ is biquadratic?


2. If $L/F$ is biquadratic, how can we realize the intermediate fields
   $K_1$ and $K_2$ different from $K$ as class fields over $F$?

For base field $F = {mathbb Q}$, these questions are answered (in the classical
ideal language) in Cohn's A classical invitation to algebraic numbers and
class fields, Springer-Verlag 1978: see in particular Thm. 18.17, where
the necessary and sufficient condition for 1. to hold (assuming that $L/F$ is
normal, which happens if and only if the ideal group $D_L$ is invariant under
Gal$(K/F)$; see Cohn, Thm. 18.13) is that the ideal group contain a ring class
group modulo some conductor $f$, from which the fields $K_1$ and $K_2$ can then
be determined.

More generally, given an class field $K_1/F$ for the ideal group $D_1$, you can
use the translation theorem of class field theory to realize the compositum
$L = K_1K$ as a class field over $K$ by taking the group of ideals whose norms
land in $D_1$. This result is contained in most classical sources on class field
theory, quite likely also in Janusz.

Since there is a change of conductors involved when pushing an extension up some
tower, it is perhaps better to use the language of idèles, but I'm more
fluent in ideals.

ca.analysis and odes - Good example of a non-continuous function all of whose partial derivatives exist

I feel it's more informative to have the thought process that leads to the example rather than just be told some magic formula that works, however simple that formula might be.

Here is a way of explaining it. We'll decide that our function is going to be discontinuous at (0,0). In order to ensure that the partial derivatives there are defined we'll try the simplest thing and make the function zero on the two axes. How can we make sure that it is discontinuous at (0,0)? Well, a simple way might be to make the function equal to 1 on the line x=y. (If you object that that step was unmotivated, then read on -- it will become clear that I could have chosen pretty well any function and the argument still works.) How are we going to make sure that the function has partial derivatives everywhere? We could do it by trying to give it a nice formula everywhere. We care particularly about how the function behaves when you keep x or y constant, so let's see what happens if we try to choose the nicest possible dependence on y for each fixed x. If we do that then we'll be tempted to make the function linear. Since f(x,0)=0 and f(x,x)=1, this would tell us to choose f(x,y)=y/x.

Unfortunately, that doesn't work: we also need the function to tend to zero for fixed y as x tends to zero. But at least this has given us the idea that we would like to write f as a quotient. What properties would we need of g and h if we tried f(x,y)=g(x,y)/h(x,y)? We would want g(x,0)=g(0,y)=0, g(x,x)=h(x,x), and h(x,y) is never 0 (except that we don't mind what happens at (0,0). We also want g and h to be nice so that the partial derivatives will obviously exist. The simplest function that vanishes only if (x,y)=(0,0) is $x^2+y^2$. The simplest function that vanishes when x=0 or y=0 but not when $x=yne 0$ is $xy$. Multiply that by 2 to get 1 down the line x=y and there we are.

Now suppose we had wanted the value to be, say, $e^{1/x}$ at (x,x), so that the function is wildly unbounded near (0,0). Then we could just multiply the previous function by $exp((2/(x^2+y^2))^{1/2})$.

The main point I want to make is that we could just as easily have chosen many other functions. For example, $sin(x)sin(y)/(x^4+y^4)$ vanishes on the axes and clearly does not tend to zero down the line x=y (in fact it tends to infinity).

ct.category theory - Category Theoretic Interpretation of Matroids?

I have come to the conclusion that a lot of mathematical structure on sets (e.g. constructs) can be defined through (combinations of) relations

$
(1) quad S_X^Fsubset F(X)times X^{I}
$

for some underlying set X, some functor $F$: Rel $rightarrow$ Rel and some set $I$.

Examples:

• Magmas (monoids, groups,...) are defined through functions $Ssubset X^2times X$.


• Graphs are defined through relations $Ssubset Xtimes X$.


• Metric spaces could be defined through relations $Ssubset (mathbb{R}times X)times X$, where $((r,x),x')in SLeftrightarrow d(x,x')=r$. Or better: through $d(x,x')le r$, since the category Met have retractions as morphisms.


• Topological spaces could be defined by $Ssubset 2^Xtimes X$, where $(M,x)in SLeftrightarrow xin Mintau$ or by the closure $xin overline M$.


• Uniform spaces could be defined through relations $Ssubset 2^{Xtimes X}times X^2$, where $(U,(x,y))in S Leftrightarrow (x,y)$ is U-close. (Wikipedia)

(See Can any construct be characterized as a relation?)

This works for any construct I know and there even seems to be a general rule to generate the morphisms between the constructs, showed by the (in general not commuting, if the relations not are functions) diagram of sets and relations:
$require{AMScd}$
begin{CD}
F(X) @>F(f)>> F(Y)\
@V S_X^F V V(2) @VV S_Y^F V\
X^{I} @>>f^{I}> Y^{I}
end{CD}
$(2)quad (phi_X,phi_Y)in F(f)Rightarrow [(phi_X,(x_i)_I)in S_X^F Rightarrow (phi_Y,((f(x_i))_I)in S_Y^F]$.

Example: If $I=1$, $F$ is the (contravariant) functor defined as $2^Xoverset{2^f}longrightarrow 2^Y$, where
$ (M,M')in 2^fLeftrightarrow M=f^{-1}(M')$ and
$S_X^F$ is defined as $(M,x)in S_X^{F}Leftrightarrow xin overline{M}$.

Then due to $(2)$:

$M=f^{-1}(M')Rightarrow (xin overline{M}Rightarrow f(x)in overline{M,'})$, so $xin overline{f^{-1}(M,')}Rightarrow f(x)in overline{M,'}$. (Continuity).

In case of matroids $(X,mathcal I)$ I can see two possibilities that fits into my scheme:

1. $(A,x)in SLeftrightarrow xin Ainmathcal I$, that gives a condition for morphisms $f^{-1}(A')inmathcal IRightarrow A'inmathcal I'$;


2. $(A,x)in SLeftrightarrow xin cl(A)$, that gives the condition $r(f^{-1}(A'))=r(f^{-1}(A')cup{x})Rightarrow r(A')=r(A'cup{f(x)})$, where $cl(A)={xin X|r(A)=r(Acup{x})}$ and $r$ is the rank function.

It seems to me as the former definition of a morphism is more natural, given the scheme, since the exchange axiom doesn't have to affect the form of the morphism more than associativity affect the form of the group homomorphism. So my primary candidate is:

A function $f:Xrightarrow X'$, where $(X,mathcal I)$ and
$(X',mathcal I')$ are matroids, is a morphism if it holds for any set
$A'subseteq X'$ that $f^{-1}(A')inmathcal I Rightarrow A'inmathcal I'$.

I don't claim that this is the answer and I can't evaluate the result because of lack of experience of matroids, but this is what I got from the empirical scheme.

dg.differential geometry - Convexity and Strong convexity of subsets of Surfaces

I believe your example is correct. As you are allowing $A$ closed the following is pretty similar. Choose a real number $ N > 1.$ Take the smooth and analytic curve (write it as a level curve and check the gradient) $$ y^2 = x^3 - 3 N^2 x^2 + 3 N^4 x = left( frac{x}{4} right) left( (2 x - 3 N^2)^2 + 3 N^4 right) $$
Note that $$ 2 y y' = 3 x^2 - 6 N^2 x + 3 N^4 = 3 (x - N^2)^2.$$
Also $$ 4 y^3 y'' = 3 (x^4 - 4 N^2 x^3 + 6 N^4 x^2 - 3 N^8 ) = 3left((x-N^2)^4 + 4N^6(x-N^2)right). $$
Furthermore, when $$x = N^2, y = pm N^3.$$
Revolve this around the $x$-axis, making a simply connected surface. There is now a closed geodesic along $ x = N^2,$ of circumference $ 2 pi N^3.$ The minimizing geodesic between
$( N^2, N^3, 0)$ and $( N^2, -N^3, 0)$ is the original curve in the plane $z=0,$ of length no larger than $ 2 N^2 + 2 N^3.$ The length of half the closed geodesic is $ pi N^3,$ which is larger for large enough $N.$ So, as I did not say, we are taking $N > 2$ and
$$ A = left{(x,y,z) in mathbb R^3 : y^2 + z^2 = x^3 - 3 N^2 x^2 + 3 N^4 x ; mbox{and} ; x leq N^2.right} $$

na.numerical analysis - solving series of linear systems with diagonal perturbations

I would like to solve a series of linear systems Ax=b as quickly as quickly as possible. However, the systems are related. Specifically, each matrix A is given by:

cI + E

where E is a fixed sparse, symmetric positive definite real matrix (unchanged in all the linear systems), I is the identity matrix, and c is a varying complex number.

In other words, I am wondering how to quickly solve a series of complex linear systems which are all identical except for complex perturbations along the diagonal. I should say that the resulting matrices are not necessarily Hermitian, so currently I compute the LU decomposition. This works, but given the large number of rather closely related systems to be solved, I wonder if there is a better way to solve the problem, perhaps by using a more expensive (e.g. QR) decomposition up front.

(Edit for Jiahao: Yes, the bs are all the same.)
(Edit for J. Mangaldan: The matrices are of order n=10^5 ~ 10^6, with about 10 times that many nonzeros.)

Update:

I'd like to thank everyone here for their suggestions. My implementation is ugly, but in the end interpolation was the key to a reasonable (10x) speedup. Since the c are quite close (imagine a small region of the complex plane, small in the sense that the spectrum of the matrix E is much larger) I could get away with computing solutions for a subset of the values of c and interpolating a solution for a given value of c using the precomputed values. It isn't elegant at all but it's something.

ag.algebraic geometry - obstruction to smooth lifting of smooth schemes

Ravi Vakil's paper Murphy's Law in Algebraic Geometry … gives many references of such things: see Section 2 of

http://arxiv.org/abs/math/0411469

The first example is due to Serre:

---

Serre, Jean-Pierre
Exemples de variétés projectives en caractéristique $p$ non relevables en caractéristique zéro. (French)
Proc. Nat. Acad. Sci. U.S.A. 47 1961 108--109.

Here is the MathReview by I. Barsotti:

An example of a non-singular projective variety $X_0$, over an algebraically closed field $k$ of characteristic $p$, which is not the image, $text{mod},p$, of any variety $X$ over a complete local ring of characteristic 0 with $k$ as residue field. The variety $X_0$ is obtained by selecting, in a 5-dimensional projective space $S$, and for $p>5$, a non-singular variety $Y_0$ which has no fixed point for an abelian finite subgroup $G$ with at least 5 generators of period $p$, of the group $Pi(k)$ of projective transformations of $S$, but which is transformed into itself by $G$; then $X_0=Y_0/G$. The reason for the impossibility is that $Pi(K)$, for a $K$ of characteristic 0, does not contain a subgroup isomorphic to $G$. {Misprint: on the last line on p. 108 one should read $s(sigma)=exp(h(sigma)N)$.}

---

ADDENDUM:

After looking back at the question, my reference to Serre's paper is inappropriate: this is an example of a variety which does not lift to characteristic zero, whereas the poster asked for one which didn't lift (even) to $mathbb{Z}/p^2 mathbb{Z}$. It is still true that this sort of thing is discussed in Vakil's paper, but now the canonical primary source seems to be

---

Deligne, Pierre; Illusie, Luc
Relèvements modulo $p^2$ et décomposition du complexe de de Rham. (French) [Liftings modulo $p^2$ and decomposition of the de Rham complex]
Invent. Math. 89 (1987), no. 2, 247--270.

MathReview by Thomas Zink:

The degeneration of the Hodge spectral sequence $H^q(X,Omega^p_{X/k}) Rightarrow H^n_{text{DR}}(X/k)$ for a smooth and projective algebraic variety over a field of characteristic zero is a basic fact in algebraic geometry. Nevertheless, only recently has one found an algebraic proof in connection with the comparison of étale and crystalline cohomology (Faltings, Fontaine, Messing).

In this beautiful paper the authors give a short, elementary, algebraic proof for the degeneration by methods in characteristic $p>0$.

Let $k$ be a perfect field of characteristic $p>0$, and let $X$ be a smooth variety over $k$, which lifts to the Witt ring $W_2(k)$. Denote by $X'$ the variety obtained by base change via the Frobenius automorphism, and let $F:Xto X'$ be the relative Frobenius morphism …. More precisely, such splittings correspond to liftings of $X'$ to $W_2(k)$. This theorem implies the degeneration of the Hodge spectral sequence for $X$ in dimension $<p$, and, by Raynaud, a Kodaira vanishing theorem for $X$. The corresponding facts in characteristic zero may be deduced by the usual reduction process.

The authors show that their argument extends to the case where $k$ is replaced by an arbitrary base $S$ of characteristic $p$ and $Omega^cdot_{X/S}$ is replaced by differentials with logarithmic poles.

---

order theory - Filter-closed vs. chain-closed

Indeed, your conjecture is correct.

Theorem. If L is a complete lattice and S is a subset of L, then S is chain-closed iff S is filter-closed.

Proof. Clearly filter-closed implies chain-closed, since every chain is a filter base. Conversely, suppose that S is chain-closed, and that A is a filter base contained in S. Note that S is trivially filter-closed with respect to any finite filter base. So suppose by induction that S is filter-closed with respect to any filter base of size smaller than |A|. Enumerate A = { a_α | α < |A| }. Let b_β be the meet of { a_α | α < β }. This is the same as the meet of the filter sub-base of A generated by this set. This filter sub-base has size less than |A|, and hence by induction every b_β is in S. Also, the b_β are a descending chain in S, since as we take more a_α, the meet gets smaller. Thus, by the chain-closure of S, the meet b of all the b_β is in S. This meet b is the same as the meet of A, and so we have proved that S is filter-closed. QED

This argument is very similar to the following characterization of (downward) complete lattices (which I had posted as my original answer).

Theorem. The following are equivalent, for any
lattice L.

• L is complete, in the sense that every subset of L has a
  greatest lower bound.




• L is filter complete, meaning that every filter base in
  L has a greatest lower bound.




• L is chain complete, meaning that every filter base in L
  has a greatest lower bound.

Proof. It is clear that completeness implies filter
completeness, since every filter base is a subset of L, and
filter completeness implies chain completeness, since every
chain is a filter base. For the remaining implication, suppose that L is chain
complete. We want to show that every subset A of L has a
greatest lower bound in L. We can prove this by transfinite
induction on the size of A. Clearly this is true for any
finite set, since L is a lattice. Fix any infinite set A.
Enumerate A as { a_α | α < |A| }.
By the induction hypothesis, for each β < |A|, the
set { a_α | α < β } has a
greatest lower bound b_β. Note that {
b_β | β < |A| } is a chain, because
as we include more elements into the sets, the greatest
lower bound becomes smaller. Thus, there is an element b in
L that is the greatest lower bound of the
b_β's. It is easy to see that this element b
is also a lower bound of A. QED

One can describe the method as finding a linearly ordered
cofinal sequence through the filter generated by the filter
base. This proof used AC when A was enumerated, and I
believe that this cannot be omitted.

One can modify the argument to show that for every infinite
cardinal κ, then a lattice is κ-complete (every
subset of size less than κ has a glb) iff every
filter base of size less than κ has a glb iff every
chain of size less than κ has a glb.

Note that if the lattice is bounded (meaning that it has
a least and greatest element), then having greatest lower
bounds for every set is the same as having least upper
bounds for every set, since the least upper bound of a set
A is the greatest lower bound of the set of upper bounds of
A. Thus, a complete lattice is often defined as saying
that every subset has a glb and lub.

There have been a few questions here at MO concerning
complete lattices. See this one and this one.

Questions about the degree of completeness of a partial
order often arise in connection with forcing arguments, and
when one is speaking of partial completeness and partial
orders (rather than lattices), and the situation is
somewhat more subtle. For example, a partial order P is
said to be κ-closed if every linearly ordered subset
of P of size less than κ has a lower bound. It is
κ-directed closed if every filter base in P of size
less than κ has a lower bound. With these concepts,
it is no longer true that a partial order is
κ-directed closed if and only if it is
κ-closed. One example arising in forcing would be the
forcing to add a slim κ-Kurepa tree, which is
κ-closed but not κ-directed closed. The
difference between these two concepts is related to
questions of large cardinal indestructibility, for Richard
Laver proved that every supercompact cardinal κ can
become indestructible by all κ-directed closed
forcing, but no such cardinal can ever be indestructible by
all κ-closed forcing, precisely because the slim
κ-Kurepa tree forcing destroys the measurability of
κ.

soft question - Famous mathematical quotes

Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass:

"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance."

Grothendieck, of course, always pioneered this approach, and considered for example that Jean-Pierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way.

nt.number theory - Relation between Hecke Operator and Hecke Algebra

The fact that Hecke operators (double coset stuff coming from $SL_2(mathbf{Z})$ acting on modular forms) and Hecke algebras (locally constant functions on $GL_2(mathbf{Q}_p)$) are related has nothing really to do with the Satake isomorphism. The crucial observation is that instead of thinking of modular forms as functions on the upper half plane, you can think of them as functions on $GL_2(mathbf{R})$ which transform in a certain way under a subgroup of $GL_2(mathbf{Z})$, and then as functions on $GL_2(mathbf{A})$ ($mathbf{A}$ the adeles) which are left invariant under $GL_2(mathbf{Q})$ and right invariant under some compact open subgroup of $GL_2(widehat{mathbf{Z}})$.

Now there's just some general algebra yoga which says that if $H$ is a subgroup of $G$ and $f$ is a function on $G/H$, and $gin G$ such that the $HgH$ is a finite union of cosets $g_iH$, then you can define a Hecke operator $T=[HgH]$ acting on the functions on $G/H$, by $Tf(g)=sum_i f(gg_i)$; the lemma is that this is still $H$-invariant.

Next you do the tedious but entirely elementary check that if you consider modular forms not as functions on the upper half plane but as functions on $GL_2(mathbf{A})$, then the classical Hecke operators have interpretations as operators $T=[HgH]$ as above, with $T_p$ corresponding to the function supported at $p$ and with $g=(p,0;0,1)$. Because the action is "all going on locally" you may as well compute the double coset space locally, that is, if $H=H^pH_p$ with $H_p$ a compact open subgroup of $GL_2(mathbf{Q}_p)$, then you can do all your coset decompositions and actions locally at $p$.

Now finally you have your link, because you can think of $T$ as being the characteristic function of the double coset space $HgH$ which is precisely the sort of Hecke operator in your Hecke algebra of locally constant functions. Furthermore the sum $f(gg_i)$ is just an explicit way of writing convolution, so everything is consistent.

I don't know a book that explains how to get from the classical to the adelic point of view in a nice low-level way, but I am sure there will be some out there by now. Oh---maybe Bump?

abstract algebra - when is A isomorphic to A^3?

Edit: In the comment below, Emil Jeřábek points out that my proof is wrong. But I'll leave this answer here for posterity.

Here's a partial answer to the Stone space question. The answer is yes for metrizable Stone spaces: if $X cong X + X + X$ then $X cong X + X$. I assume you're using $+$ to denote coproduct of topological spaces.

Proof: Write $I(X)$ for the set of isolated points of a topological set $X$. (A point is isolated if, as a singleton subset, it is open.) Then $I(X + Y) cong I(X) + I(Y)$ for all $X$ and $Y$. So, supposing that $X cong 3X$, we have $I(X) cong 3I(X)$. But $X$ is compact, so $I(X)$ is finite, so $I(X)$ is empty. Hence $X$ is a compact, metrizable, totally disconnected space with no isolated points. A classical theorem then implies that $X$ is either empty or homeomorphic to the Cantor set. In either case, $X cong X + X$.

I guess metrizability of the Stone space corresponds to countability of the corresponding Boolean ring.

The topological theory of Stone spaces is more subtle in the non-metrizable case, if I remember correctly.

ag.algebraic geometry - A closed subscheme of an open subscheme that is not an open subscheme of a closed subscheme?

Does this work?

Let $k$ be a field, and let $R$ be the subring of $prod_{i=1}^{infty} k[t]$ consisting of those sequences that stabilize. Let $I$ be the ideal comprised of those sequences that stabilize at $0$. Set $X=mathrm{Spec}(R)$, and let $U$ be the complement of $V(I)$ in $X$. (The inclusion $Uto X$ is quasiseparated, but not quasicompact; here $j_*mathcal{O}_U$ is not quasicoherent.)

If $e_i$ denotes the sequence that is $0$ except in the $i$th spot, where it is $1$, then the $e_i$'s generate $I$, and since $e_ie_j=delta_{ij}$, we deduce that $U=coprod_{i=1}^{infty}X_{e_i}=coprod_{i=1}^{infty}mathbf{A}_k^1$.

Now consider the ideal $(t^i)subset k[t]$ as an ideal sheaf on each $X_{e_i}$; these glue to give a quasicoherent ideal sheaf $I$ on $U$, which in turn defines a closed subscheme $Z$ of $U$, which is open in $X$.

I claim that there is no closed subscheme $V$ of $X$ such that $Z$ is open in $V$. If there were, it would be $V=V(J)$ for an ideal $Jsubset R$ such that $Jcdot R_{e_i}=(t^i)$. But this is impossible, since any element $f=(f_i)in J$ would have to have $f_iin (t^i)$ but would nevertheless have to stabilize. So it would have to stabilize at zero; hence $Jsubset I$, so $V(I)subset V(J)$, whence we ... EDIT: do not have a contradiction.

ct.category theory - If a category is "monadic", is it necessarily so in a unique manner?

You can certainly have non-equivalent monadic functors. Here's one example: Let $mathcal{V}_k$ be the category of $k$-vector spaces. For a vector space $V$, let $H_V: mathcal{V}_kto mathcal{V}_k$ be the functor
$$
H_V(W) = hom_k(V,W).
$$
Such a functor is always monadic, as long as $V$ is non-zero and finite dimensional. The associated monad is
$$
T_V(W) = hom_k(V, Votimes_k W) = End_k(V)otimes_k W,
$$
so this is presenting a Morita equivalence: $k$-vector spaces are equivalent to modules over the matrix ring $End_k(V)$.

You wanted functors to set; let $U_V:mathcal{V}_kto Set$ be given by the same formula as $H_V$. Then again, this will be monadic, as long as $V$ is non-zero and finite dimensional (and I'm not sure you even really need the finite dimensionality condition for either of these examples; added: you certainly don't in the first example, since $H_V$ is an exact functor, so the hypotheses of the Barr-Beck theorem certainly hold, though $T_V$ is not tensoring with an endomorphism ring if $V$ is infinite.).

gr.group theory - Construction of a proper uncountable subgroup of $mathbb{R}$ without Choice.

It is straightforward to construct proper uncountable subgroups of $mathbb{R}$. One can construst a basis for $mathbb{R}$ over $mathbb{Q}$, and then there are many possibilities (just consider the group generated by the basis or the vector subspace generated by some proper uncountable set of the basis).

However, the first step (constructing the basis) requires the axiom of choice.

So does anyone know of any proper uncountable subgroup of $mathbb{R}$ that does not require choice to construct?

or is this not possible.

Meaning are there models not involving choice where every uncountable subgroup of $mathbb{R}$ is equal to $mathbb{R}$.

homotopy theory - The Dold-Thom theorem for infinity categories?

Let $mathcal{M}$ denote the category of finite sets and monomorphisms, and let $mathcal T$ denote the category of based spaces. For a based space $X in mathcal T$, one has a canonical funtor $S_X : mathcal M rightarrow mathcal T$ defined by ${n} mapsto X^n$. The definition on morphisms is to insert basepoints on the factors which are not in the image of a given monomorphism.

As is well know, the homotopy groups of $mathrm{colim} S_X = SP^infty X$ give the homology of $X$ (this is the Dold-Thom theorem), and the homotopy groups of $mathrm{hocolim} S_X = SP^infty_h X$ given the stable homotopy of $X$.

Is there a model for $SP^infty X$, the ordinary infinite symmetric product, as a homotopy colimit as opposed to a categorical colimit?

The motivation for this question comes from thinking about $infty$-categories. In an $infty$-category, one does not really have a good notion (at least not one that I am aware of) of strict categorical colimits. So I'm wondering if there is, nonetheless, some easily defined functor on the $infty$-category of spaces which will let us calculate ordinary homology. In short, is there any $infty$-categorical analog of the Dold-Thom theorem?

Update: Following up on André's remark it seems using the orbit category is heading in the right direction, at least for the $n$-th approximations. I'll just quickly sketch what I have so far:

Let $mathcal O(Sigma_n)$ denote the orbit category. The objects are the homogeneous (discrete) spaces $Sigma_n/H$ (with left actions) as $H$ runs over all the subgroups of $Sigma_n$, and the morphisms are the $Sigma_n$-equivariant maps. There is a canonical functor $$Sigma_n rightarrow mathcal O(Sigma_n)^{op}$$ where we regard $Sigma_n$ as a category with one object as usual.

Given a $Sigma_n$ space $X$, right Kan extension along this inclusion produces a $mathcal O(Sigma_n)^{op}$ diagram $tilde X$ defined by $$tilde X(Sigma_n/H) = X^H$$ It turns out that the above inclusion is final so that it induces an isomorphism of colimits. Hence $mathrm{colim}_{mathcal O(Sigma_n)} tilde X cong X_{Sigma_n}$, i.e., the coinvariants. It's also not hard to see that the undercategories are copies of $BSigma_n$, hence not contractible, so we don't expect an equivalence of homotopy colimits, which is good.

On the other hand, I can now show that when $X$ is discrete, the canonical map $$mathrm{hocolim} tilde X rightarrow mathrm{colim} tilde X$$ is an equivalence. My methods here do not generalize to all spaces, so if someone has a reference for why this is true in general, that would be much appreciated. (I think something like this must appear in May's book on equivariant homotopy theory if it's true, but I did not have it available this weekend.)

The remaining part would be to let $n rightarrow infty$, but somehow this seems like it should not be too bad. (Something like: make a functor $mathcal M rightarrow mathcal Cat$ by $n mapsto mathcal O(Sigma_n)$. Take the Grothendieck construction. Some natural diagram on this category might give the right answer.)

nt.number theory - What should Spec Z[sqrt{D}] x_{F_1} Spec bar{F_1} be?

What should be $text{Spec } mathbb{Z}[sqrt{D}] times_{mathbb{F}_1} text{Spec } overline{mathbb{F}}_{1}$?

Sure, there's more than one definition.
I'm looking for any answer that uses at least one definition of scheme over $mathbb{F}_1$.

This really is more a question of opinion.
What do you think this should be?
Some monoid that has something to do with $text{Spec }mathbb{Z}[sqrt{D}][mathbb{Q}/mathbb{Z}]$ would be my guess (where the second brackets mean group ring).

This interests me from the point of view that, say, hyperelliptic curves over a finite field come (geometrically) from the group scheme of a quadratic extension of $overline{mathbb{F}}_p [t]$. In this case the frobenius acts on ideal classes, and satisfies a quadratic equation.
But, from what I understand, the natural analogue of frobenius in the arithmetic case, is like taking any positive power, and taking limits to 0 (or something of the sort).
Would this satisfy some kind of equation on, say, $text{Pic(Spec } mathbb{Z}[sqrt{D}] times_{mathbb{F}_1} text{Spec } overline{mathbb{F}}_{1}text{)}$?
(for whatever definition of Pic that should be natural here)

I've searched for information on $mathbb{F}_1$, but most just talk about making $text{Spec }mathbb{Z}$ into a curve, getting zeta functions to be Riemann's, etc.
Instead, I want to ask questions that are not just about proving the Riemann hypothesis, like the one above.

soft question - Best online mathematics videos?

My good friend Professor Elvis Zap has the "Calculus Rap," the "Quantum Gravity Topological Quantum Field Theory Blues," a vid on constructing "Boy's Surface," "Drawing the hypercube (yes he knows there is a line missing in part 1)," A few things on quandles, and a bunch of precalculus and calculus videos. In order to embarrass all involved, he posted the series "Dehn's Dilemma" that was recorded in Italy last summer.

soft question - Best online mathematics videos?

Richard Feynman gave the 1964 Messenger Lectures at Cornell University --- this is an endowed lecture series to which a number of famous scholars have been invited, including several physicists. His lectures were recorded, and Bill Gates bought the rights to them and has provided them to the public for free.

http://research.microsoft.com/apps/tools/tuva/index.html

The content is mostly designed for a general audience, so if you have never learned physics you will learn something. And if you have studied plenty of physics already, you will be pleased to see the master at work in his prime. I very much enjoyed watching it.

algebraic number theory - In what sense (if any) does the cohomology of profinite groups commute with projective limits?

Background:

Let $G$ be a profinite group. If $M$ is a discrete $G$-module, then $M=varinjlim_U M^U$, where the direct limit is taken with respect to inclusions over all open normal subgroups of $G$, and one naturally has $H^n(G,M)simeqvarinjlim H^n(G/U,M^U)$, where the cohomology groups on the right can be regarded as the usual abstract cohomology groups of the finite groups $G/U$ (this is sometimes, as in Serre's Local Fields, taken as the definition of $H^n(G,M)$).

More generally if one has a projective system of profinite groups $(G_i,varphi_{ij})$ and a direct system of abelian groups $(M_i,psi_{ij})$ such that $M_i$ is a discrete $G_i$-module and the pair $(varphi_{ij},psi_{ij})$ is compatible in the sense of group cohomology for all $i,j$, then $varinjlim M_i$ is canonically a discrete $varprojlim G_i$-module, the groups $H^n(G_i,M_i)$ form a direct system, and one has $H^n(varprojlim G_i,varinjlim M_i)simeqvarinjlim H^n(G_i,M_i)$. The statement and straightforward proof of this more general result can be found, for instance, in Shatz' book on profinite groups.

Question:

In general, I'm wondering if there are, under appropriate hypotheses, any similar formulae for projective limits of discrete $G$-modules. Now, given a projective system of discrete $G$-modules $(M_i,psi_{ij})$, it isn't even obvious to me that the limit will again be a discrete $G$-module, and at any rate, while each $M_i$ is discrete, the limit (in its natural topology) will be discrete if and only if it is finite. So, for the sake of specificity, I'll give a particular situation in which I'm interested. If $R$ is a complete, Noetherian local ring with maximal ideal $mathfrak{m}$ and finite residue field and $M$ is a finite, free $R$-module as well as a discrete $G$-module such that the $G$-action is $R$-linear, then the canonical isomorphism of $R$-modules $Msimeqvarprojlim M/mathfrak{m}^iM$ is also a $G$-module isomorphism (each $M/mathfrak{m}^iM$ is a discrete $G$-module with action induced from that of $M$). Moreover, in this case, one can see that the limit is a discrete $G$-module (because it is isomorphic to one as an abstract $G$-module!). There is a natural homomorphism $C^n(G,M)rightarrowvarprojlim C^n(G,M/mathfrak{m}^iM)$ where the projective limit is taken with respect to the maps induced by the projections $M/mathfrak{m}^jMrightarrow M/mathfrak{m}^iM$, and this induces similar map on cohomology. I initially thought the map at the level of cochains was trivially surjective, just because of the universal property of projective limits. However, given a ``coherent sequence" of cochains $f_i:Grightarrow M/mathfrak{m}^iM$, the property gives me a map $f:Grightarrow M$ that is continuous when $M$ is regarded in its natural profinite topology, which is, as I noted above, most likely coarser than the discrete topology, so this might not be a cochain. So, what I'd really like to know is whether or not the map on cohomology is an isomorphism.

Why I Care: The reason I'd like to know that the map described above is an isomorphism is to apply it to the particular case of $G=hat{mathbb{Z}}$. It is well known (and can be found, for instance, in Serre's Local Fields) that $H^2(hat{mathbb{Z}},A)=0$ for $A$ a torsion abelian group. In particular the higher cohomology of a finite $hat{mathbb{Z}}$-module vanishes, and I'd like to be able to conclude that the same is true for my $M$ above, being a projective limit of finite abelian groups.

Thanks!

ho.history overview - Did Apollonius invent co-ordinate geometry?

Let V be the vertex of a parabola, F its focus, X a point on its symmetry axis, and A a point on the parabola such that AX is orthogonal to VX. It was well within the power of the Greeks to prove relations such as $VX:XA = XA:4VF$. If you introduce coordinate axes, set $x = VX$, $y = XA$ and $p = VF$, you get $y^2 = 4px$, the modern form of the equation of a parabola.

Everything now depends on what "invention of coordinate geometry" means to you. I do not think that the Greeks' work on conics should be confused with coordinate geometry since they did not regard the lengths occurring above as coordinates. It's just that parts of their results are very easily translated into modern language.

In a similar vein, Eudoxos and Archimedes already were close to modern ideas behind integration, but they did not invent calculus. Euclid, despite Heath's claim to the contrary, did not state and prove unique factorization. And Euler, although he knew the product formula for sums of four squares, did not invent quaternions (Blaschke once claimed he did).
In any case, we are much more careful now with sweeping claims such as "Appolonius knew coordinate geometry" than historians were, say, 100 years ago.

ag.algebraic geometry - Topologically distinct Calabi-Yau threefolds

This is a very good question, and I would really love to know the answer since its current state seems to be quite obscure. Below is just a collection of remarks, surely not the full answer by any means. I would like to argue that for the moment there is no any deep mathematical reason to think that the Euler number of CY 3-folds is bounded. I don't believe either that there is any physical intuition on this matter. But there is some empirical information, and I will describe it now, starting by speaking about "how many topological types of CY manifolds we know for the moment".

As far as I understand for today the construction of Calabi-Yau 3-folds, that brought by far the largest amount of examples is the construction of Batyrev. He starts with a reflexive polytope in dimension 4, takes the corresponding toric 4-fold, takes a generic anti-canonical section and obtains this way a Calabi-Yau orbifold. There is always a crepant resolution. So you get a smooth Calabi-Yau. Reflexive polytopes in dimension 4 are classified the number is 473,800,776. I guess, this number let Miles Reid to say in his article "Updates on 3-folds" in 2002 http://arxiv.org/PS_cache/math/pdf/0206/0206157v3.pdf , page 519

"This gives some 500,000,000 families of CY 3-folds, so much more impressive than a mere infinity (see the website http://tph16.tuwien.ac.at/~kreuzer/CY/). There are certainly many more; I believe there are infinitely many families, but the contrary opinion is widespread"

A problem with the number 500,000,000 in this phrase is that it seems more related to the number of CY orbifolds, rather than to the number of CY manifolds obtained by resolving them. Namely, the singularities that appear in these CY orbifolds can be quite involved and they have a lot of resolutions (I guess at least thousands sometimes), so the meaning of 500,000,000 is not very clear here.

This summer I asked Maximillian Kreuzer (one of the persons who actually got this number 473,800,776 of polytopes), a question similar to what you ask here. And he said that he can guarantee that there exist at least 30108 topological types of CY 3-folds. Why? Because for all these examples you can calculate Hodge numbers $h^{1,1}$ and $h^{2,1}$, and you get 30108 different values. Much less that 473,800,776. As for more refined topological invariants (like multiplication in cohomology) according to him, this was not really studied, so unfortunately 30108 seems to be the maximal number guarantied for today. But I would really love to know that I am making a mistake here, and there is some other information.

Now, it seems to me that the reason, that some people say, that the Euler characteristics of CY 3-folds could be bounded is purely empirical. Namely, the search for CY 3-folds is going for 20 years already. Since then a lot of new families were found. We know that mirror symmetry started with this symmetric table of numbers "($h^{1,1}, h^{2,1})$", and the curious fact is that, according to Maximillian, what happened to this table in 20 years -- it has not got any wider in 20 years, it just got denser. The famous picture can be found on page 9 of the following notes of Dominic Joyce http://people.maths.ox.ac.uk/~joyce/SympGeom2009/SGlect13+14.pdf . So, this means that we do find new families of CY manifolds, all the time. But the values of their Hodge numbers for some reason stay in the same region. Of course this could easily mean that we are just lacking a good construction.

Final remark is that in the first version of this question it was proposed to consider complex analytic manifolds with $c_1=0$. If we don't impose condition of been Kahler, then already in complex dimension 2 there is infinite number of topological types, given by Kodaira surfaces, they are elliptic bundles over elliptic curve. In complex dimension 3 Tian have shown that for every $n>1$ there is a holomorphic structure on the connected sum of n copies of $S^3times S^3$, with a non-vanishing holomorphic form. Surely these manifolds are non-Kahler. So if you want to speak about any finiteness, you need to discuss say, Kahler 3-folds with non-vanishing holomorphic volume form, but not all complex analytic ones.

fa.functional analysis - Convergence of operators to the identity on Banach spaces

Let $U_infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_infty$. That is, $U_r subseteq U_{r'}$ if $r le r'$ and $U_infty = overline{bigcup U_r}$.

For $r in [1,infty]$, let $Y_r = C(U_r,mathbb R)$ be the Banach space of real-valued continuous functions over $U_r$ with the supremum norm. For $r le r'$, let $phi_{r,r'} : Y_{r'} to Y_r$ be the restriction maps, so that $Y_infty$ is the inverse limit of the spaces $Y_r$. Write $phi_r : Y_infty to Y_r$ for the restriction map $phi_{r,infty}$.

Suppose there exists a family of continuous linear operators $m_r : Y_r to Y_infty$ such that $|m_r| le M$ for all $r$, and $phi_r circ m_r$ is the identity map on $Y_r$.

Question: Suppose $Gamma subseteq Y_infty$ is compact. Does $m_r circ phi_r$ converge strongly to the identity operator on $Gamma$? That is, for all $epsilon > 0$, does there exist $R > 0$ such that if $r ge R$, then $$sup_{y in Gamma} left| (m_r circ phi_r)(y) - y right|_{Y_infty} < epsilon?$$

ag.algebraic geometry - Can isomorphisms of schemes be constructed on formal neighborhoods?

No. Let A be k[[t]]. Let X be A^1 setminus {-1,0,1} and Y be A^1 setminus {1,t,-1}. In explicit equations, X = Spec k[[t]][x, y]/y(x-1)x(x+1)-1 and Y = Spec k[[t]][x, y]/y(x-1)(x-t)(x+1)-1.

Over k[[t]]/t^{n+1}, the reductions of X and Y are isomorphic because all infinitesimal deformations of a smooth affine scheme are trivial. (See corollary 4.7 in Hartshorne's notes on deformation theory.)

However, X and Y are not isomorphic because the two fibers over the general point are not: For any field K, if we have P_K^1 setminus {a,b,c,d} for {a,b,c,d} in P^1(K), then the cross ratio of a,b,c and d is a well defined element of K. In particular, this applies when K=k((t)).

homological algebra - Colimit of intersections

Let $B_i^p$ be a family of sets, where $pin mathbb{N}$ and $i in I$, $I$ being a directed set, and such that, for every $i$, we have a descending chain of inclusions

$$
dots supset B_i^{p-1} supset B_i^p supset B_i^{p+1} supset dots
$$

Question: is the following implication true?

$$
bigcap_p B_i^p = emptyset, text{for all} i quad Longrightarrow quad bigcap_p varinjlim_i B_i^p = emptyset .
$$

Since $bigcap_{}$ is a limit, this seems a problem of an interchange of limits and filtered colimits and indeed there is a universal map

$$
varphi: varinjlim_i bigcap_p B_i^p longrightarrow bigcap_p varinjlim_i B_i^p
$$

If $varphi$ was a bijection, then my implication would be true with no doubts,
but, since the intersection is not finite, I can not say that $varphi$ is a bijection. Nevertheless, could my implication still be true, without $varphi$ being a bijection?

The reason behind my question is the following: let $(A_i, F_i)$ be a directed family of filtered sets (or abelian groups, or modules; in fact, in my problem they are cochain complexes). Since filtered colimits (direct limits) are exact, you can define a filtration on the colimit like this:

$$
F^pvarinjlim_i A_i = varinjlim_iF_i^pA_i .
$$

Now assume all the filtrations $F_i$ are Hausdorff; that is, $bigcap_p F_i^pA_i = 0$ for all $i$. Is it then necessarily true that the filtration $F$ on $varinjlim_iA_i$ is Hausdorff too?

This question is a sequel to my previous question Convergence of right half-plane spectral sequence bounded on the right . Despite Tilman's counterexemple to my guess there, I think I've managed almost to prove it because my spectral sequences are right half-plane and this is the final detail I need.

st.statistics - construct random variable with a fixed level of Spearman Coefficient to another

You don't even need $sigma^2$ to construct such a variable.

Let $Z$ be $+1$ with probability $q$, and $-1$ with probability $1-q$, and independent of $X$.

Let $Y = e^XZ$. This squashes X to the positive reals preserving order, and then may change the sign.

$y_1$ and $y_2$ have the same ordering as $x_1$ and $x_2$ when the greater value isn't negated. That is, if $x_1 gt x_2$, then $sign((x_1 - x_2)(y_1-y_2)) = sign(y_1)$. If $x_1lt x_2$, then $sign((x_1 - x_2)(y_1-y_2)) = sign(y_2)$. So, $E[sign((x_1 - x_2)(y_1-y_2))] = E[Z].$

Choose $Z$ to have average value $p$ (set $q=frac{(p+1)}2$), and then $X$ and $Y$ have Spearman Rank Correlation Coefficient $p$.

Actually, there is a little ambiguity (to me) about whether you allow $x_1 = x_2$, which I ignored above. If under your definition, the rank correlation of $X$ with itself is $alpha$, then the rank correlation of $X$ with $Y$ is $alpha p$, and you can get any value in $[-alpha,alpha]$.