1
vote
0answers
38 views

A possibly surprising appearance of Lucas numbers

Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, $g(1)=\{1\}$, $g(2)=\{1,x\}$, ...
1
vote
1answer
85 views

Automorphisms of ideals of $\mathbb{C}[t]$

Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$. The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following: For ...
2
votes
0answers
30 views

Asymmetric random walk on the line with barriers

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers). More generally, suppose we fix any ...
0
votes
0answers
43 views

Linearization of line bundle [on hold]

In the definition of the linearization of a line bundle in Dolgashev book, he had adding the condition that the zero section is $G$-invariant; I see that this condition is contained in the first one, ...
1
vote
0answers
23 views

“Prime” fusion rings

Surely this concept is known! (But I don't recall seeing it - maybe under another name? But "prime" is the obvious name choice.) Example. Open the Gepner/Kapustin paper at ...
5
votes
2answers
135 views

de Rham cohomology of smooth affine varieties

Let $U$ be a smooth variety over $\mathbb{C}$. We know that there exists a smooth compactification $X$ such that $X-U$ is a normal crossings divisor $D$ and that the de Rham cohomology of $U$ can be ...
2
votes
2answers
85 views

Expectation of trace of nth power of unitary matrices

I am trying to find the answer of $$\int dU \ |Tr(U^m)|^2$$ where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
1
vote
0answers
16 views

Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful. I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...
2
votes
2answers
133 views

Fano manifold admit an smooth anti-canonical divisor?

Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$
2
votes
0answers
64 views

Can a Brownian motion be fast at its extrema?

After pondering this MO question > Location of maximum of Brownian motion with rough drift <, I wonder whether a Brownian motion can be fast (i.e. beats the law of the iterated logarithm) at its ...
2
votes
2answers
33 views

An example of two cofibrant dg categories whose tensor product is not cofibrant

I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and ...
1
vote
0answers
33 views

Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok. A real finite-dimensional vector space $V$ defines the following two spaces, also real. $\mathcal{P}(V)$ generated by ...
2
votes
1answer
53 views

When the vertex covering number is smaller than the chromatic number

For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. As noted here, we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm interested in graphs $G$ ...
1
vote
0answers
29 views

A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...
0
votes
1answer
34 views

Proof of closed walk generating function identity

In 'Spectral Conditions for the Reconstrucibility of a Graph' Godsil and McKay give a short proof of an identity (Lemma 2.1) that relates the generating function for the number of closed walks ...
0
votes
0answers
31 views

Uniform $L^p-L^{p'}$ bound of a Fourier multiplier

Let $(\tau,\xi)\in\mathbb{R}\times \mathbb{R}^n$, and consider the function $$ m_{\epsilon}(\tau,\xi)=\frac{1}{\tau+|\xi|^4+\epsilon|\xi|^2+i} $$ in $\mathbb{R}^{n+1}$. My first question is that does ...
5
votes
0answers
64 views

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity Let's ...
5
votes
0answers
51 views

Do free profinite groups satisfy Howson's theorem?

Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
15
votes
2answers
290 views

What is the purpose of the flat/fppf/fpqc topologies?

There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully. As someone who originally started in ...
0
votes
0answers
76 views

On a particular Cayley Graph

Pick a prime number $P$. Pick $M$ positive numbers $g_1<\dots<g_M$ each less than $P-1$. Denote $G_{P}[g_1,\dots,g_M]$ to be Cayley graph on $P-1$ vertices generated by $g_1,\dots,g_M$ as ...
0
votes
0answers
18 views

Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$ where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is $$ \Lambda = \begin{bmatrix} x ...
-5
votes
0answers
42 views

Multi-dimensional space [on hold]

Let $x(t):{\rm I\!R} \to {{\rm I\!R}^n}$ and $f(t,x(t)):{\rm I\!R} \times {{\rm I\!R}^n} \to {{\rm I\!R}^n}$ What norm is Appropriate to show the following inequality? $$\left\| f(t,x) \right\| \le ...
2
votes
1answer
52 views

Antoine's Necklace and positive Hausdorff/Lebesgue measure

I have the following question: The usual construction of the Antoine's Necklace produces a Cantor of $1$-dimensional Hausdorff measure in $\mathbb R^3$. I would like to know whether one could adapt ...
1
vote
1answer
24 views

Sensitivity of inverse normal cdf

Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < \epsilon < p,p' < 1 - \epsilon$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful ...
1
vote
1answer
65 views

recognising weak equivalences of simplicial sets

$\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper Weak Equivalences ...
-2
votes
0answers
16 views

Can we predict next sample using the existing samples? [on hold]

Suppose that I have 18 data points and I'm sampling 3 data points each time. Suppose that I have 60 samples (each has 3 data points). Can we predict the next sample (of 3 points) from the existing ...
0
votes
0answers
25 views

Can we conclude that $\varphi(L_xf_0)\neq0$ for every $x\in G$?

Let $H$ be a compact subgroup of locally compact topological group $G$ and $A=\{f\in L^1(G); R_hf=f(a,e)\forall h\in H\}$ as a subalgebra of $L^1(G)$ by convolution of $L^1(G)$. If $\varphi \in ...
2
votes
0answers
72 views

Does this symmetrization operator have a name? Any theory?

Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define $$f_{\mathrm{symm}}(x_1,\ldots,x_n) = 2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1} ...
0
votes
0answers
39 views

Are A and A^T unitarily equivalent over a p-adic field?

Let $E/F$ be a quadratic extension of p-adic field. Let $U=\{u\in GL_2(E): uu^*=1\}$ be the unitary group of rank 2. My question is: given a matrix $A\in GL_2(E)$ can we find $u_1,u_2\in U$ such ...
-4
votes
0answers
46 views

Convert 1-5 Grading Scale to 1-100 Grading System [on hold]

I am creating a formula in Excel to convert 1-5 Grading Scale to 1-100 Grading System Suppose that I have the following table: 97-100 = 1.00 94 - 96 = 1.25 91-93 = 1.50 88-90 = 1.75 85-87 = 2.00 ...
0
votes
0answers
28 views

Linear elliptic estimates

i am interested in solutions of the following $$-\Delta \phi =f \; \; \; A_\lambda, \qquad \phi=0 \; \; \partial A_\lambda,$$ where $ A_\lambda=\{ x \in R^N: \lambda <|x|<1 \}$ with $ ...
0
votes
0answers
59 views

Computations of derivations, $d^2$, of a (Grothendieck) spectral sequence

The maps $d^1:E_1\rightarrow E_1$ have a nice description. Is there any text providing us with a description of the higher derivations $d^2:E_2\rightarrow E_2$ arising from a Grothendieck spectral ...
0
votes
0answers
15 views

Examples of Sigma-Adequate Links that are not simply Adequate in A and B type sense?

I am looking to see if anyone has constructed explicit examples of $\sigma$-adequate links a la Makoto Ozawa's Essential State Surfaces for Knots and Links? This technique is centered around taking ...
-3
votes
0answers
85 views

What is the interpretation of f'(x)/f(x)? [on hold]

I'm studying Fisher information and the function d/dx ln(f(x)) thus f'(x)/f(x) appears. I'm trying to interpretate what this is quantier could represent in a function. Thank you.
1
vote
2answers
275 views

Popular books written by great mathematicians [on hold]

I read: H. Poincare. Value of science F. Klein. Development of Mathematics in the 19th Century J.E. Littlewood. A Mathematicians Miscellany G.H. Hardy. A Mathematician’s Apology R. Courant, ...
3
votes
0answers
50 views

reference for higher spin - not gravitational nor stringy

Other than the papers of Berends,Burgers and van Dam, are there any papers that study the general case of deforming a free field theory with higher spin fields to be interactive?
2
votes
0answers
29 views

Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...
3
votes
0answers
249 views

Survey of Erdős' “Tricks” [on hold]

Is there a kernel of "tricks", techniques and tools that Paul Erdős was particularly fond of and therefore employed a lot in his research work? Could you point out some survey papers that deal with ...
1
vote
0answers
37 views

Free profinite products

Let $F$ be a nonabelian finitely generated free profinite group, and let $x \in F$. Must there be some $1 \neq y \in F$ such that $\langle x,y \rangle$ is isomorphic to the free profinite product of ...
1
vote
0answers
89 views

Hodge structures generated by cohomology groups of varities with dimension less than $n$

Let $X$ be a smooth projective variety over $\mathbb{C}$ with dimension $n$. Is it true that for every $i<n$, the Hodge structure on $\mathrm{H}^i(X,\mathbb{Q})$ is generated by Hodge structures of ...
2
votes
0answers
52 views

The level set of convolution

Suppose $u\in C^2(\bar \Omega)\cap C^\infty(\Omega)$ is a function which is compact supported on $\Omega$ with $u\equiv 0$ at $\partial\Omega$, and $\Omega\subset \mathbb R^3$. We also assume that ...
2
votes
2answers
206 views

cohomological obstructions and rational points

Let $X$ a (nice) scheme over $\mathbb{Q}$. Are there cohomolgical obstructions answering the following questions: 1) is $X(\mathbb{Q})$ an empty set ? 2) is $X(\mathbb{Q})$ a finite (non empty) set ...
0
votes
0answers
25 views

Laplace equation between circles [on hold]

I need to solve the simple Laplace equation $$\nabla^2f(r,\theta)=0$$ with boundary conditions: $$f(a,\theta)=g(\theta)$$ $$\lim_{A\rightarrow\infty}f(A,\theta)=1$$ where $a$ is a fixed real radius. ...
0
votes
1answer
91 views

Alexeev's projective torus embeddings

I'm reading Alexeev's "Complete moduli in the presence of semiabelian group action" and I just started to study toric geometry. In chapter 2 in order to obtain an affine toric variety he takes ...
0
votes
0answers
39 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
7
votes
1answer
105 views

Dual of Banach-valued $L^p$

Let $X$ be an infinite-dimensional Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb ...
3
votes
1answer
133 views

Proof of a Proposition regarding the reduction of N-torsion groups on elliptic curves

In Diamond-Shurman A first course in Modular forms p.334 Prop. 8.4.4. It is stated, For E elliptic curve over $\bar{\mathbb{Q}}$ with good reduction at the prime ideal $\mathfrak{p}$ the reduction ...
0
votes
0answers
21 views

Separating Two Groups of Data using Fisher's Linear Discriminant

I found an article (starting on page 8) that gives a neat method for finding the line/plane/hyperplane that maximizes the separation between two groups of data points in n-dimensions. It uses Fisher's ...
7
votes
1answer
110 views

Orders in Central Simple Algebras. Applications

It is known that orders in quaternion algebras (over a number field) are used for constructing geometric objects like hyperbolic orbifolds and Shimura curves. Moreover, if one knows embedding ...
0
votes
0answers
35 views

Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$

Let $\mathcal T = D^b(\mathbb P^1)$, the bounded derived category of $\mathbb P^1$ over a field $k$. It is well known that $\mathcal T$ admits a strong and full exceptional sequence $\{\mathcal O, ...

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