G-equivariant Whitehead's Theorem

2012-07-18T00:15:04Z

Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume that $\pi_n(X/G)\cong \pi_n(Y/G)$ for all $n\geq 0$ and are induced by the cellular inclusion $Y/G\hookrightarrow X/G$.

Whitehead's Theorem implies that there is a strong deformation retraction (SDR) from $X/G$ to $Y/G$.

In this setting, does there exist a $G$-equivariant SDR from $X$ to $Y$?

If not, what if one further assumes the existence of a SDR from $X$ to $Y$ (not assumed $G$-equivariant). Would that then imply the existence of a $G$-equivariant SDR from $X$ to $Y$?

EDIT: After Tom Goodwillie answered both questions negatively, I have decided to add another assumption; namely, assume that the fixed point set $X^G$ is contained in $Y$ (or perhaps assume that $X^G$ $G$-equivariantly retracts to a subspace of $Y$).

Importance of separability vs. second-countability

2013-05-18T17:05:31Z

For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which can be made for the importance of separability.

Let me subsume the situation: Both notions are intended to guarantee smallness known from classical spaces, from geometry and analysis. Second-countability is a stronger condition, but for metrisable spaces both conditions are equivalent—the word “separable” seems to be more popular in these cases (for example in functional analysis and descriptive set theory). However, under most weaker conditions than metrisability, second-countability is not guaranteed by separability (and in fact metrisability is implied by second-countability and regularity, that is Urysohn’s metrisation theorem): For example the space $[0,1]^{\mathbb{R}}$ with the product topology is separable, but not second-countable, although it is compact and even a product of compact Lie-groups (are there nicer spaces?). There are even locally euclidean, separable spaces which are not second-countable, as required in the usual definition of a topological manifold (see this question). In the locally compact case second-countability implies $\sigma$-compactness, which is useful for integration theory, and the space $X$ is second-countable if and only if the space $C_0(X)$ of continuous numerical functions vanishing at infinity is second-countable. For metrisability there are no analoga. For locally compact groups the second-countability is equivalent to the second-countability of $L^2$ with respect to the Haar measure (it should also hold more generally for certain non-degenerate Borel measures on general locally compact spaces).

Some classical analytic methods using sequences can be used for second-countable spaces: For second-countable spaces compactness is equivalent to countable compactness and sequential compactness. In first-countable Hausdorff spaces you can choose convergent subsequences from every convergent net. Especially in first-countable topological vector spaces (or abelian groups) the convergence of the net of all finite partial sums of a set of vectors is equivalent to the unconditional convergence of a series (the series converges independently of the order). In the separable function space $\mathbb{R}^\mathbb{R}$ this does not work.

Some more general points: Second-countability imposes a strict smallness condition (the cardinality of the topology and in the Hausdorff case the cardinality of the space must be at most the cardinality of the continuum), while separable Hausdorff spaces might consist of $\beth_2$ points. Separability has the advantage of being preserved under continuous images–however, it has the big disadvantage of not being preserved under taking subspaces.

Do you know of any important theorems/theories where separability is crucial—not second-countability? Which generalisations of important concepts from classical analysis only depend on separability? Is the popularity of the word/concept of “separability” just due to the special case of metric spaces? I have even seen some authors using the word “separable” instead of “second-countable” (which sounds reasonable, since “second-countable” sounds cumbersome).

Hyperbolic pair of pants.

2013-05-23T09:51:42Z

Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential simple arc $\sigma$ in $Y$ with end points on a same boundary component of $Y$ and $l$ denote the length of unique geodesic in the homotopy class of $\sigma$. Then, my question is weather the inequality, $l$ $\geq$ min{$l_i | i= 1, 2, 3$} hold or not for every pair of pants $Y$.

analytically isomorphic singularities

2013-05-23T12:29:12Z

Two plane curves $X,Y$, defined by polynomials $f(x,y)=0$ and $g(x,y)=0$,are analytically isomorphic(at the origin). i.e., the complete local rings $k[[x,y]]/(f)$ and $k[[x,y]]/(g)$ are isomorphic.

Why $\ell(k[x,y]/(f,f_x,f_y))=\ell(k[x,y]/(g,g_x,g_y))?$ Here, $\ell(M)$=the length of $M$=the number of modules in a composition series of $M$.

Hartshorne 1.5.mensions "analytically isomorphic" and exercise 5.14. tells something about plane singularities. I can't prove and understand the above equality. How can I prove?

Is there anyway to rewrite a partial differential equation using language of differential forms, tensors,.etc

2013-05-23T11:10:11Z

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a lauguage of vector calculus in a local coordinate, is there anyway (or any algorithm) that we can use to rewrite it using language of differential forms, tensor, exterior calculus, hodge star and other operators which are coordinate independent. An example, the Grad f can be rewritten as a geometric form: (df)#, where # is a sharp operator turing a one-form into a vector. I am currently facing this problem to turn a partial differential equation into its coordinate-independent form, which involves forms, tensors, exterior calculus and other operators.

Thank you for oneany who help me about this problem!

A family of words counted by the Catalan numbers

2013-05-23T11:39:57Z

In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's Catalan exercises in EC2 and his Catalan addendum, but I don't see anything that looks to be clearly equivalent, and a bijection to Dyck paths isn't jumping out at me. So I have two questions:

Has anyone seen these words, or some equivalent objects, before?

Do you see a nice bijection between these words and any family of "classic" Catalan objects such as Dyck paths or noncrossing partitions?

Let $w$ be a word of length $n$ over the natural numbers (including $0$). Then $w$ lies in our family if it satisfies three rules:

For all $k\le n-1$, $w_{k+1}\ge w_k-1$.
If $w$ contains an $i\ge 1$, then the first $i$ is followed by some $i-1$.
Every $i\ge 2$ follows an $i-1$.

(The words "followed" and "follows" do not imply contiguity, so for the second rule, once we read the first $i\ge 1$ (reading left to right) we must eventually read an $i-1$.)

For example, the only word of length $1$ in this set is $0$, for length $2$ the set contains $00$ and $10$, for length $3$ it contains $$ 000, 010, 100, 101, 110, $$ and for length $4$ it contains $$ 0000, 0010, 0100, 0101, 0110, 1000, 1001, 1010, 1011, 1021, 1100, 1101, 1110, 1210. $$

Solution formular for Laplace equation

2013-05-23T12:07:18Z

I want to slove the Laplace equation on $R^3_+$ with Neumann boundary condition. The equation reads: $-\Delta u = f$ in $R^3_+$, $\partial_3 u|_{x_3=0}=g$ on $R^2$. If $f$, $g$ satisfy compatibility condition, can I write the explicit formular of $u$?

PDE with the Jacobian Determinant

2012-12-04T17:33:48Z

Hello,

Could you please help me in answering the following question?

Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not been able to control the support.

The question seems so natural that I'm sure it has been studied somewhere in some form but I have not been able to dig out the right reference. Few minutes in MathScinet did not come up with what I'm looking for. All references seem to handle the situation when $f$ is positive which is clearly not the case here. Any suggestion or reference in this direction is welcome.

Thank you.

QUESTION:

Let $U\subset\mathbb{R}^n$ be open, connected and let $f\in C_{0}^{\infty}(U;\mathbb{R})$ satisfy $$ \int_{U}f(x)dx=0. $$ Is it true that there exists a $u\in C_{0}^{\infty}(U;\mathbb{R}^n)$ satisfying $$ \operatorname*{det}(\nabla u)=f \text{ in }U? $$ Note that $$ C_{0}^{\infty}(U;\mathbb{R}^n)=\{u\in C^{\infty}(U;\mathbb{R}^n):\operatorname*{supp}(u)\text{ is compact and }\operatorname*{supp}(u)\subset U\}. $$

Growth of Thompson's group $F$

2013-05-22T21:27:32Z

EDIT: Mark Sapir pointed a reference (in the comments) giving a lower bound of $2^{1/4}$ for the minimal rate. Is this the state of art? The third question remains unanswered. If the answer is NO then the lower bound jumps suddenly to $\frac{\sqrt{5}+3}{2}$ by known results. END OF EDIT

What is it known about the minimal growth rate of the Thompson's group $F$? Is there an easy lower bound? Is there a generating set growing slower than the standard one?

Permutation and Combination question....

2013-05-23T11:28:59Z

Hello, I am currently studying Extension 1 Mathematics. I missed two classes and I figured out that tomorrow I will have a quiz. Can you help me to solve this permutation and combination question:

In how many ways 3 cards be selected from a pack of 52 playing cards if:

(i) at least one of them is an ace;

(ii) not more than one is an ace.
In how many ways can 9 books be distributed amongst a man, a woman and a child, if the man receives 4, the woman 3, and the child 2?

Thanks in advance!

A question about $L^p$ integral of an entire function on $\mathbb{C}$

2012-12-30T19:47:27Z

Question: Suppose that $f$ is an entire function (i.e. analytic in $\mathbb{C}$), and satisfies the condition $\iint_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1)$. I guess that $f\equiv 0$ but I do not know how to prove it.

Note. If $p\in[1,\infty)$, it is easy to prove that $f\equiv 0$. In the settting $p\in (0,1)$, one should deal with the integral of an entire function near the essential singularity point $\infty$ carefully.

EDIT. Thank Alexandre Eremenko for his answer. I also want to know the solution to the following harmonic version of question.

Question (H): Suppose that $f$ is a harmonic function ($i.e. \Delta f=0$, $f$ may be complex-harmonic in $\mathbb{C}$), and satisfies the condition $\iint_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1).$ I believe that $f\equiv 0$.

Is there a good reason why a^{2b} + b^{2a} <= 1 when a+b=1?

2010-03-05T16:18:14Z

The following problem is not from me, yet I find it a big challenge to give a nice (in contrast to 'heavy computation') proof. The motivation for me to post it lies in its concise content.

If $a$ and $b$ are nonnegative real numbers such that $a+b=1$, show that $a^{2b} + b^{2a}\le 1$.

Equivariant $K$-theory, singular vectors, and flag manifolds

2013-05-22T19:02:13Z

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ of $B$, and the differential operators on $E$ are closely linked to the representation theory of $G$.

For the special case of a flag manifold, which is to say, when $B$ is a Borel subgroup of $G$, differential operators from $E$ to itself correspond to homomorphisms of the Verma module $U({\frak g})\otimes_{U({\frak b})} V_{\lambda}$. These homomorphisms are in turn classified by the so-called singular vectors of $V_{\lambda}$, which is to say the vectors killed by the action of the positive niradical. Moreover again, these singular vectors correspond to solutions of certain hyper-geometric functions.

What I would like to know is how all this relates to equivariant K-theory. Is there some characterization of the singular vectors correspond to a Fredholm operator. Also, can the defining equivalence relation of the equivariant K-theory group $K^0$ be nicely reformulated in terms of representation theory and singular vectors?

Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

2013-05-22T08:46:19Z

Hi.

Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this convergence is that for any continuous function $\varphi\in\mathscr{C}^0([-1,1])$,

\begin{align*} \int_{-1}^1 f_n \varphi \operatorname*{\longrightarrow}_{n\rightarrow +\infty} \int^{1}_{-1} f \varphi. \qquad (1) \end{align*}

It is easy to see that $f$ is necessarily also non-negative.

Question : do we also have $(f_n)_n \rightharpoonup f$ in $L^1([-1,1])-w$, i.e. can we replace in the previous convergence the continuous function $\varphi$ by any element of $L^\infty([-1,1])$ ?

I am quite sure that the answer is no (because there is no density of regular functions in $L^\infty$), but I did not manage to find a counterexample.

Two remarks :

1) Without the assumption of non-negativeness one may consider $f_n:= n\mathbf{1}_{[0,1/n]} - n\mathbf{1}_{[-1/n,0]}$ , which, in some sense , approximates $ " \delta_{0^+}-\delta_{0-} "$, and hence $(f_n)_n \operatorname*{\rightharpoonup}^{\mathscr{M}-\star} 0$, but it may be checked easily that $(f_n)_n$ do not converge to $0$ weakly in $L^1$ : the sequence "concentrates" the eventual discontinuity in $0$.

2) Keeping the non-negativeness but working on the open set $(-1,1)$ instead of $[-1,1]$ simplifies also the problem, since the mass may then concentrate to the boundary : $f_n:= n \mathbf{1}_{[1-1/n,1[}$ tends to $0$ in $\mathscr{M}^1(-1,1)-w\star$ (test functions in $\mathscr{C}^0_c(-1,1)$) but clearly not in $L^1-w$.

Thanks in advance for any advice !

Ayman

EDIT

Okay, I think I manage to get a proof in the case where $f \in\mathscr{C}^0([-1,1])$ (or almost everywhere continuous). Maybe this wil help to get an answer in the general case !

First Step : we may assume (even in the general case) that each $f_n$ is continuous. Indeed, consider a sequence of non-negative and pair mollifiers $(\rho_n)_n$ and replace $f_n$ by $f_n\star \rho_n$. Also, using a classical argument of uniqueness of accumulation points, it is enough to show that $(f_n)_n$ is relatively weakly sequentially compact in $L^1$.
Second Step : The set $\{x\,:\, \liminf\,f_n \geq f+1\}$ has an empty interior. Indeed, if it contained a non-empty open interval $I$, one could find a non-negative and non-zero continuous fonction $\varphi$ with support in $I$ and the convergence $(1)$ would lead to the following, using Fatou's lemma \begin{align*} \int_{-1}^1 (f+1) \varphi \leq \int_{-1}^1 \liminf \,f_n\varphi \leq \liminf \int_{-1}^1 f_n \varphi = \int_{-1}^1 f \varphi, \end{align*} whence a contradiction.
Third Step : $\{x\,:\, \liminf\,f_n \geq f+1\}$ is hence of empty interior and therefore so is \begin{align*} A := \bigcup_{n\in\mathbb{N}} \bigcap_{p \geq n} \{x\,:\,f_p \geq f+1\}. \end{align*} The complement of $A$ in $[-1,1]$ is therefore dense and is precisely given by the formula \begin{align*} [-1,1]\backslash A := \bigcap_{n\in\mathbb{N}} \bigcup_{p\geq n} \{x\,:\,f_p < f+1\}. \end{align*} Because of the previous formula, it is easy to see that for each $x\in [-1,1]\backslash A$, there exists an subsequence $(f_{\sigma_x(n)})_n$ for which $f_{\sigma_x(n)} < f(x)+1$ for all $n$. Now pick a coutable dense subset $D\subset [-1,1]\backslash A$, and extract diagonally along $D$ an extraction (still labeled) $(f_n)_n$ such as, for all $x\in D$, $f_n(x) < f(x) +1$.
Fourth Step : The inequality $f_n(x) < f(x)+1$ is true on a dense subset and may be hence extended (replacing $<$ by $\leq$ ) to the whole intervall for the chosen subsequence. One may hence conclude using Dunford-Pettis theorem.

In fact the same proof applies when $f$ is only assumed to be in $L^\infty$ (replace $f+1$ by $\|f\|_\infty+1$). I tried to use Luzin's theorem but it failed : the set $D$ is dense but not necessarily dense in the subsets in which $f$ is continuous.

Any advice is welcome, thanks again !

Ayman

An operation on binary strings

2013-05-23T09:28:58Z

Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ by $\alpha$ and each 0 in $\beta$ by $\overline{\alpha}$, with $\overline{\alpha}$ being the negation of $\alpha$, which one gets by replacing every 1 in $\alpha$ by 0 and vice versa.

Formally:

$$(\alpha \times \beta)[k] = \begin{cases} 1 & \text{if}\ \ \alpha[k\ \text{mod}\ a] = \beta[k\ \text{div}\ a] \\ 0 & \text{otherwise} \end{cases}$$

for $a=|\alpha|, b=|\beta|, k = 0,\dots,ab$.

Maybe it comes as a surprise - at least for me it did - and it's a little bit cumbersome to prove, that the operation $\times$ - even though it is not commutative - is associative, i.e. $\alpha \times (\beta \times \gamma) = (\alpha \times \beta) \times \gamma$.

Is there an elegant argument to see that $\times$ is associative? (I had to go through a couple of case discriminations and some equivalencies of modulo arithmetic like $(k\ \text{mod}\ ab)\ \text{div}\ a = (k\ \text{div}\ a)\ \text{mod}\ b$) to get to the result.)

Since $1$ is a neutral element ($\alpha \times 1 = 1 \times \alpha = \alpha$), the tuple $(\lbrace 0,1\rbrace^+,\times,1)$ is a monoid.

In this monoid it seems that each $\sigma$ has a unique “factorization” into “primes” (upto associativity). If the length of $\sigma$ is prime, $\sigma$ itself is necessarily “prime”. If $\sigma$ has length $2^n$ it can have up to $n$ prime factors, e.g. $10010110 = 10 \times 10 \times 10$. But it can also be prime, e.g. $1110$.

In which contexts and under which name has this monoid been investigated?

Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$

2013-05-23T09:54:53Z

Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given by

$V \otimes W:= \oplus_{j\in \mathbb{Z}}\oplus_{p+q=j}V_p\otimes V_q$ and for the graded vector space $\mathbb{F}[j]$, which is $\mathbb{F}$ in degree $j$ and te zero vector space ${0}$ otherwise, the shift $V[j]$ is given by

$V[j]:=\mathbb{F}[1]\otimes V$

We then define a monoidal structure on the category of graded vector space, (more or less) given by the rule on homogeneous elements

$v\otimes w= (-1)^{deg(v)deg(w)}w\otimes v$

Then there is the decalage isomorphism

$ dec: V_1[1]\otimes \cdots \otimes V_n[1] \to (V_1 \otimes \cdots \otimes V_n)[n] $

given by $dec(v_1[1]\otimes \cdots \otimes v_n[1])= (-1)^{\sum_{j=1}^n(n-j)deg(v_j)}(v_1\otimes \cdots \otimes v_n)[n]$.

Now in work on graded (stuff), it is frequently said, that this isomorphism defines a natural isomorphism of the symmetric graded tensor-algebra of $V[1]$ and the antisymmetric graded tensor algebra, that is

$S(V[1])\simeq (\bigwedge V)[n]$

*The question is: How does the decalage induces such an algebra isomorphism? Or What is the natural isomorphism? *

If $dec$ itself would be the isomorphism, then

$dec(v[1] \vee w[1])= (dec(v_1)\wedge dec(w))[2]$ should hold, but this isn't true in general.

Group action on the real line

2013-05-23T01:41:26Z

Hi,

I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct a new action such that a point p in R has the trivial stabilizer in G. Is it possible to make (possibly by a completely new action) the map G -> R defined by g -> g(p) is a group homomorphism always (when R is regareded as a group with addition)? If not, when could it be done?

Any comment and/or advice would be greatly appreciated.

Matrix Inverse with Same Principal Minors

2013-05-14T09:42:47Z

Given an invertible matrix $A \in \mathbb{R}^{n \times n}$, and index set $\langle n\rangle = \{ 1, \dots, n \}$, and the submatrix $A(\alpha)$ with the columns and rows of $A$ with indices $\alpha \subset \langle n \rangle$.

Is there are characterisation of all matrices $A$ with

$\textrm{det } A(\alpha) = \textrm{det } A^{-1}(\alpha) \qquad \forall \alpha \subset \langle n \rangle \qquad ?$

Example: Sufficient conditions are involutory ($A^{-1} = A$) or orthogonal ($A^{-1} = A^T$) matrices.

$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$

2013-05-23T08:42:11Z

Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the sheaf $f^\ast \mathcal I$ is not a sheaf of ideals on $X$ (since it can be not contained in $\mathcal O_X$). Nevertheless, there is a natural morphism $$ f^\ast \mathcal I \to \mathcal O_X $$ and we can set $f^{-1}\mathcal I \cdot \mathcal O_X$ as the image of this morphism. In this wat $f^{-1}\mathcal I \cdot \mathcal O_X$ is a sheaf of ideals on $X$.

So, my question is the following:

Is there any reasonable condition under which we have $f^{-1}\mathcal I \cdot \mathcal O_X = f^\ast \mathcal I$?

I think this is an interesting question in general, but the real reason why I ask this question is that have a problem with the book of Faltings and Chai "Degeneration of abelian varieties". In chapter V, Section 5 of that book, they prove that the toroidal compactification of the Siegel modular variety is isomorphic to the normalization of the blowup of the minimal compactification with respect to some sheaf of ideals $\mathcal I$. The proof is quite complicated, but at page 178 they use the universal property of blow-ups to define the morphism. The point is that the show that $f^\ast \mathcal I$ is invertible, while the universal property is about $f^{-1}\mathcal I \cdot \mathcal O_X$. I suspect I am missing something quite obvious.

Any help is appreciated!

Sheaves on Contractible Analytic Spaces

2013-05-22T13:50:46Z

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to conclude, in some cases, that $\mathcal{F}$ is isomorphic to $\mathcal{O}_X^{\oplus n}$ for some $n$? If you like, you may take $X$ to be the analytic space associated to a complex affine variety.

I ask because contractibility is often a useful condition when attempting to prove a fibre bundle is trivial.

can we say that $(p^2+1)/2\ne p_0^2$ where $p$ is a Mersenne prime

2013-05-23T08:12:32Z

Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Can we say that $(p^2+1)/2$ is not equal to the square of a prime number? Many thanks for your help BHZ

probability measures with entropy equal to nonnegative number

2012-11-12T21:08:38Z

Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this number?

Eigenvalues of Symmetric Tridiagonal Matrices

2013-05-22T22:57:15Z

Suppose I have the symmetric tridiagonal matrix:

$ \begin{pmatrix} a & b_{1} & 0 & ... & 0 \\ b_{1} & a & b_{2} & & ... \\ 0 & b_{2} & a & ... & 0 \\ ... & & ... & & b_{n-1} \\ 0 & ... & 0 & b_{n-1} & a \end{pmatrix} $

All of the entries can be taken to be positive real numbers and all of the $a_{i}$ are equal. I know that when the $b_{i}$'s are equal (the matrix is uniform), there are closed-form expressions for the eigenvalues and eigenvectors in terms of cosine and sine functions. Additionally, I know of the recurrence relation:

$det(A_{n}) = a\cdot det(A_{n-1}) - b_{n-1}^{2}\cdot det(A_{n-2})$

Additionally, since my matrix is real-symmetric, I know that its eigenvalues are real.

Is there anything else I can determine about the eigenvalues? Furthermore, is there a closed-form expression for them?

Help me on proof of an equation.

2013-05-23T07:45:57Z

I wanna prove following equation $ \sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i) $

I have verified several numbers such as $n=2,3,4$, and try to prove it using mathematical induction, however I can't extend the 2 $\sum \sum$. Some guru told me to try resultant, but it seems to be sums of resultants, and can't be simply used.

Can anyone help me on this? thanks a lot.

On Perelman's paper

2013-05-23T07:26:28Z

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written:

Fix a closed manifold $M$ with a probability measure $m$, and suppose that our system is described by a metric $g_{ij}(\tau)$, which depends on the temperature $\tau$ according to equation $\frac{\partial}{\partial \tau}g_{ij}=2(R_{ij}+\nabla_i \nabla_j f)$ where $dm = udV$, $u =(4\pi \tau)^{-\frac{n}{2}}e^{-f} $, and the partition function is given by $log Z = \int (−f + \frac{n}{2})dm$.

Question 1: We know $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$ is modified Ricci flow, where $t$ is time. Why the changes of the metric respect to $\tau$ is exactly backward of the changes of the metric respect to $t$?

Question 2: $\frac{\partial}{\partial \tau}g_{ij}=2(R_{ij}+\nabla_i \nabla_j f)$ is backward to heat equation PDE, Is there a solution to this?

Question 3: Why is the partition function $log Z =\int (−f + \frac{n}{2})dm$?

Thanks

Effective Chebotarev without Artin's conjecture

2013-05-23T02:11:22Z

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's $L$-function and Artin's conjecture (that the Artin $L$-function have no pole except maybe at $s=1$). But they also add, with no argument nor reference, that the same theorem is true assuming only GRH, not the Artin's conjecture. I'd like to know how this can be proved.

Let me explicit my question for a reader that would not want to open Iwaniec and Kowalski's book. Let $L/\mathbb Q$ be a finite extension Galois group $G$, and let $\rho$ be an irreducible complex representation of $G$, that one can as well suppose non-trivial (as Artin's conjecture is known for the trivial representation). For $x>0$ a real, let $\psi (\rho,x) = \sum_{p^n < x} tr\ \rho(Frob_p^n) \log p$, where the sum is on all prime power of the form $p^n$, where the prime $p$ is unramified for $\rho$. Then the crucial point in proving Iwaniec-Kowalski's effective chebotarev is the following estimate $$(1) \ \ \ \ \ \psi(\rho,x) = O(x^{1/2} \log x \log x^d q(\rho)),$$ where $q(\rho)$ is the Artin's conductor of $\rho$ and the implied constant is absolute. Estimate (1) is proved under GRH and Artin in Iwaniec-Kowalski (Theorem 5.15)

How to prove (1) under GRH alone, as Iwaniec and Kowalski suggest is possible ?

I have already given some thoughts to the question, but I am not able to solve it. Surely one of the ideas involved should be the following: even assuming only GRH for the Artin's $L$-function, not Artins conjecture, we know that $L(\rho,s)$ has no pole except maybe on the critical line, for by the theorem of Brauer, $L(\rho,s)$ is a quotient of products of Hecke $L$-functions, and those functions have no poles on the critical strip except parhaps at $s=1$ by a deep result of Hecke, and no zeros either, under GRH, except maybe on the critical line. Moreover, one also knows using the same argument of Brauer that $L(\rho,s)$ is meromorphic function of order $1$, that is quotient of two entire functions of order $1$, and therefore that $L(\rho,s)$ has a nice Weierstrass product formula (like (5.23)). Then the natural way to go is to try to follow the proof given by Iwaniec and Kowalski under GRH and Artin, using the above remarks to avoid using Artin. I see several issues with that method, the most important being the following: a crucial step in the method is the estimate of the number of zeros $N(T)$ (with their positive multiplicty) of $L(\rho,s)$ on the critical segment between $s = 1/2 -iT $ and $s=1/2 + iT$ - see Theorem 5.8. This estimate is obtained by integrating $L'/L$ on a suitable rectangle intersecting the critical line on that segment. Yet in the presence of poles this method will not count the number $N(T)$ of zeros, but the difference $N(T)-P(T)$ where $P(T)$ is the number of poles (with their multiplicty) on the same segment. So even a good estimate for $L'/L$ hence of $N(T)-P(T)$ will not prevent $N(T)$ and $P(T)$ to be arbitrary large. Then, when one computes $\psi(\rho,x)$ by some explicit formula (such as (5.53)), both zeros and poles contribute and if these are too many, that is if $N(T)+P(T)$ is too large, the precise estimate (1) will be completely ruined.

The Area of Spherical Polygons

2012-05-23T02:19:31Z

I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can assume a radius of $R=1$).

For what I am researching (I will not go into the background) I need the following conditions to be satisfied:

The edge length of all spherical polygons in $\mathbb{S}^2$ is $\pi / 3$.
The spherical polygons I want to consider may or may not be convex, in fact it is necessary that I be able to compute the area of a non-convex polygon in $\mathbb{S}^2$.

I know there are multiple ways to compute the area of a spherical triangle using the spherical law of cosines, L'Huilier's theorem, or other formulas, but I want to be more general than this. The type of information I know about the internal angles of the spherical polygons is a bound in terms of the degree of the vertex considered in a spherical simplicial $2$-complex $\mathcal{K}$ in $\mathbb{S}^2$. That is, labeling $\gamma_{i}$ as an internal angle of the spherical polygon that $$ \sum\limits_{1 \leq i \leq b} \gamma_{i} = \sum\limits_{1 \leq i \leq b} (i-1)\arccos(\frac{1}{3})b_{i}$$ where $b_{i}$ denotes the number of vertices of degree $i$ in $\mathcal{K}$. The last comment about the internal angles may or may not be confusing, but I just wanted to mention that I know something about the internal angles of the spherical polygons. For an example of how difficult this problem may be, there was a large discussion here about determining the area of a spherical $4$-gon with given side length (and the answer was quite messy), so I'm hoping that some of you have ideas!

To summarize exactly what my question is, and what information I know:

You are given a number $E$ which tells you how many edges a spherical polygon $C$ in $\mathbb{S}^2$ has (all edges have length $\pi /3$ and $C$ is not necessarily convex). Determine the area of $C$ (or a function for the area of $C$).

That is, I want to find the analogue in spherical geometry to the equations in Euclidean geometry which tell you the area of a regular polygon of a given number of sides. If such a general expression does not exist, I would be interested in the case for $E=5,E=6,...,E\approx20$.

a measurable cardinal & a real-valued measurable cardinal in the same model?

2012-12-26T09:30:20Z

Although I know that "ZFC & there exists a measurable cardinal" and "ZFC & there exists a real-valued measurable cardinal" are equiconsistent with one another, I am not sure whether "ZFC & there exists a measurable cardinal k & there exists a real-valued measurable cardinal b" is equiconsistent with ZFC. (Obviously k is not equal to b.) I would be grateful for an answer.

Order type of the smallest set containing the identity function and closed under exponentiation

2013-05-22T21:18:34Z

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$, i.e. $E=\{n \mapsto n, n \mapsto n^n, n \mapsto n^{n^n}, n \mapsto (n^n)^n, n \mapsto (n^n)^{n^n},\ \dots\}$. Let $E$ be ordered by eventual domination.

Is $E$ well-ordered? What is the least ordinal that cannot be embedded in $E$?

Do operations generate well-ordered sets only?

2013-05-23T01:22:38Z

I've read @TauMu's question about the set of functions $\mathbb N\rightarrow\mathbb N$ generated from the identity map by repeatedly applying exponentiation of two already accepted functions. @TauMu asks if such a set is well-ordered with respect to eventual domination.

It seems to me that it would be hard, perhaps impossible, to obtain a set which is not well-ordered regardless of the choice of a binary operation $\mathbb N\times\mathbb N\rightarrow\mathbb N$. Here is a patient formulation of my more general question:

Given an operation $\tau : \mathbb N^2\rightarrow\mathbb N$, let $\bigcirc^{\tau}\subseteq\mathbb N^{\mathbb N}$ be the smallest set such that it has the identity map $I_{\mathbb N}$ as its element, $I_\mathbb N\in\bigcirc^\tau$, and $h:=\tau\circ(f\triangle g)\in\bigcirc^{\tau}$ for every $f\ g\in\bigcirc^{\tau}$.

REMARK (an explanation of the notation above) $$\forall_{n\in\mathbb N}\quad h(n) := \tau(f(n)\ g(n))$$

Finally,

QUESTION (edited twice after the 1st and 2nd comment by @Joseph Van Name) Does there exists an operation $\tau:\mathbb N^2\rightarrow \mathbb N$ dominating the identity in each variable (see below), and such that $\bigcirc^{\tau}$ contains an infinite strictly decreasing sequence with respect to the relation of eventual domination?

By $\tau$ dominating the identity in each variable I mean that:

$$\forall_{k\ n\in\mathbb N}\quad \tau(k\ \ n)\ \ \ge\ \ \max(k\ \ \ n)$$

Top Questions - MathOverflow