0
votes
0answers
72 views

asymptotic behavior of minimum dilatations on punctured surfaces

Let $l_{g,n}$ be the logarithm of minimum dilatation for pseudo-Anosov homeomorphisms on surface of genus $g$ with $n$ punctures. Let $n$ be fixed and $g$ varies. Is the asymptotic behavior of ...
-4
votes
0answers
38 views

Proving the area of a pentagon [closed]

Suppose that a regular pentagon circumscribes a circle of radius r. Show that the area of the pentagon is 5r²tan(36°). I know that the area of a triangle is 1/2 ...
4
votes
1answer
156 views

Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page. I am asking if it ...
8
votes
1answer
190 views

Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number ...
0
votes
0answers
31 views

A C(B)-module structure on the function algebra of the total space of a vector bunlde $\pi:V \to B$

For a continuous vector bundle $\pi:V \to B$ vector bundle over a compact Hausdorff space $B$, and $C(B)$, $C(V)$ the continuous complex valued functions on $B$ and $V$ respectively, we can give ...
9
votes
0answers
141 views

Simplest example of failure of finite Galois descent in algebraic $K$-theory?

Let $E \to F$ be a $G$-Galois extension of fields. What is the simplest example where the natural map $K(E) \to K(F)^{hG}$ is not an equivalence on connective covers (i.e., where finite Galois ...
0
votes
0answers
24 views

How to incorporate novel observation into covariance matrix [closed]

I have a 3D covariance matrix with known values (but not the data it was calculated from). edit: I do have the means. |a 0 0| |0 b 0| |0 0 c| I receive a novel ...
0
votes
0answers
39 views

Software on finite field arithmetic? [closed]

is there any software, library,or toolkit that support arithmetic with normal basis on $GF(2^n)$ field? What is the best one?
-7
votes
0answers
125 views

Can Suslin's problem be decided with these axioms? [on hold]

The axiom of completion denoted +C, states: "Only theorems of the axioms are true." Let ZFC plus the axiom of completion be indicated by ZFC+C. It appears at first trivial to show that the continuum ...
18
votes
2answers
787 views

The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts. What ...
7
votes
0answers
102 views

Is there an efficient algorithm for testing isomorphism of projective planes?

Isomorphism testing is a core problem in computational complexity. Recently, Babai has shown that Graph Isomorphism problem for general graphs can be solved in quasipolynomial time. Long time before ...
3
votes
1answer
50 views

Dual of colimit in $\text{Ban}_1$

I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}_1$ of Banach spaces ...
1
vote
1answer
43 views

How to retrieve eigenvectors from shifted QR algorithm?

I understand that the key to retrieve eigenvectors in the non-shifted QR algorithm is to accumulate the transformations at each steps in the following way: $Q = \Pi_i Q_i$ Can we accumulate the ...
0
votes
0answers
17 views

Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$. Let $A \subset E$. Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator. I want to prove that $A$ is absorbing. I know that it is ...
2
votes
1answer
168 views

Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$

I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times ...
3
votes
0answers
71 views

Integer Gelfand-Kirillov dimension

Let $R$ be a (noncommutative) Noetherian affine $K$-algebra. The Gelfand-Kirillov dimension is known to be an integer for many classes of affine Noetherian algebras. I wonder, if this is true for any ...
3
votes
1answer
121 views

Irreducible monic polynomials

I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here. For instance, for the family of ...
2
votes
0answers
152 views

K theory and derived categories

Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra". As next step, I moved to ...
4
votes
1answer
199 views

Reference for: Every local field can be realized as the completion of a global field

It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...
2
votes
0answers
67 views

Is $\sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$ bounded as Im(s) goes to infinity?

Let $\rho$ denote the zeros of the Riemann zeta-function and $\delta > 0$. Is the function $f(s) = \sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$ bounded as Im(s) goes to infinity?(the real part ...
0
votes
0answers
87 views

Numbers with many prime divisors

Is there a positive $c$ such that for every $n$ there exists $m$ such that $2^m-1$ has at least $n$ distinct prime divisors and $m$ is not greater than $n^c$? I'm also interested in this question ...
9
votes
0answers
142 views

Finite dimensionality of Ext(M,N)

Let $K$ be a field of characteristic $0$ and let $R$ be a (noncommutative) Noetherian $K$-algebra. Let $M$ and $N$ be simple left $R$-modules and assume further the following conditions on $R$: ...
2
votes
0answers
34 views

Pro-V topology on a free group

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is ...
3
votes
2answers
199 views

Gcd of polynomials over a finite field [closed]

Let $\mathbb{F}_q[X]$ be the polynomial ring over the finite field with $q$ elements. Let $f$ be a polynomial of the form $x^n-a$ and let $g$ be a polynomial of the form $x^m-b$. Is it known whether ...
17
votes
2answers
815 views

What is the smallest positive integer for which the congruent number problem is unsolved?

The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...
4
votes
0answers
55 views

Geometrically-explicit upper bound for on-diagonal heat kernel

Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form $$K(t;z,z) \leq ...
0
votes
0answers
36 views

Limit probability of a complete bipartite random graph $G(n,n,p)$ is connected [closed]

I need to calculate the following probability limit for a complete bipartite random graph $G(n,n,p)$ in the Erdos-Renyi model: \begin{equation} \lim_{n\rightarrow\infty}\mathbb{P}[G(n,n,p) \text{ is ...
4
votes
2answers
189 views

Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map

Say we have a number field $K$. Let $G_K = \text{Gal}(\overline{K}/K)$. Let $M$ be a discrete $G_K$-module. We know that $H^1(K, M) := H^1(G_K, M)$, i.e. profinite group cohomology. For each place $v$ ...
15
votes
1answer
390 views

Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...
2
votes
2answers
245 views

Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
3
votes
1answer
104 views

Moving from $\Re(s) = 1+\epsilon$ to $\Re(s) = \frac{1}{2}$ in the proof of the Weil-Guinand explicit formula

In all proofs of the Weil-Guinand explicit formula, there's this step (this is from Paul Garrett's notes): Now consider this: (1) $\frac{\zeta^\prime(s)}{\zeta(s)}$ has poles at $s=1$ and ...
-3
votes
0answers
29 views

Compute score for a set of data [closed]

So let's explain my problem. I have a set of items which have a score from 0 to 100. This set is dynamic which means that several values are expected to be added from moment to moment. In each item is ...
6
votes
1answer
54 views

Horizontal distribution of a totally geodesic foliation

Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner ...
9
votes
1answer
578 views

Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)?

I know that (co)ends (i.e. universal wedges) follow Fubini-like relation, i.e. $$ \int_{\langle c,d\rangle} F(c,d,c,d) \cong \int_c\int_d F(c,c,d,d) \cong \int_d\int_c F(c,c,d,d) $$ where we regard ...
0
votes
0answers
30 views

Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of ...
1
vote
0answers
27 views

Zero-dimensional $F$-space which is not strongly zero-dimensional

Does anyone know of an example of a (Tychonoff) $F$-space which is zero-dimensional but not strongly zero-dimensional? By an $F$-space we mean every cozeroset is $C^*$-embedded. By zero-dimensional ...
3
votes
0answers
71 views

Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$

I am trying to find the inverse of the following kernel in 3 dimensions $$ \nabla^2-x^2, $$ where, $$ x^2=\vec{x}.\vec{x} $$ It seems quit simple and one would think there should already be solutions ...
2
votes
0answers
94 views

Derived Categories provide a good Framework for Sheaf Cohomology?

I'm a bit new to this sheaf cohomology business. Can someone explain how derived categories provide a good setting for Sheaf Cohomology? I understand that sheaf coho arises as right derived functors, ...
3
votes
1answer
144 views

Reference request: The consistency of a tall tower in $\mathbb{N}^\mathbb{N}$

A $\kappa$-tower in $\mathbb{N}^\mathbb{N}$ is a sequence $\langle a_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$ that is $\le^*$-increasing with $\alpha$ and has no $\le^*$-upper ...
1
vote
1answer
67 views

Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say ...
12
votes
3answers
446 views

Nice things that can be proved easily with characteristic classes

A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...
5
votes
0answers
67 views

In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?

I will present my question in the context that I encountered it, although I believe it probably applies in general context. Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...
14
votes
1answer
233 views

Does the injection $\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ split?

Let $F_n$ be the free group on $n$ letters. The question is as in the title: letting $i:\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ be the natural injection, does there exist a homomorphism ...
-2
votes
0answers
16 views

If derivative of f(x) is f'(x), then what is the integral of (f'(x))^2 in terms of f(x)? [migrated]

Is there some procedure for figuring this out or am I venturing into unsolved territory here?
1
vote
0answers
101 views

Finding the Chern Class of a the pushfoward of a invertible sheaf

I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of ...
2
votes
0answers
149 views

Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...
0
votes
0answers
15 views

Sampling functions [closed]

Suppose there were two spaces $A$ and $B$, A consists of functions that computes a scalar from an array of values($x$) that have coefficients($w$); $B$ consists of functions that does some ...
1
vote
0answers
127 views

Stronger version of Bertini's theorem

In char 0, is there a generalised version of Bertini's theorem that will ensure that for a proper map $f: Y\rightarrow X$ between smooth projective varieties and for every point $x\in X$ we can find ...
-4
votes
0answers
33 views

Question on logarithm Exponentiation [closed]

I know it's not the best title but I had no idea how to be specific about it. Also sorry if I mess up the Latex syntax :/ Basically what I'm looking for is a rule that states how [log^2(a^{f(x)})] ...
0
votes
0answers
26 views

Lower bound of general bilinear form [closed]

Suppose I have a bilinear form $X^TAY$ where $X \in R^n, Y \in R^m$ and $Α \in R^{n \times m}$. All elements of $A$ are bounded, that is $\exists \bar a_{ij}>0:|a_{ij}|\le \bar a_{ij}, \forall ...

15 30 50 per page