# All Questions

**0**

votes

**0**answers

72 views

### No graph with eigenvalue -1/2 [on hold]

I came across an exercise of book Spectra of Graphs.
Show that there does not exist graph whose adjacency matrix eigenvalue
is -1/2.
Any thougts?

**4**

votes

**0**answers

44 views

### Archimedean $\varepsilon$-factors

Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...

**2**

votes

**0**answers

51 views

### Generalization of Ito's formula

If $f:R\to R$ is a convex function then we have Ito-Tanaka formula. Now my question is that if we are given a function $u: R\times R_+\to R$ such that $u(s,\cdot)$ is smooth for every $s\in R$ and ...

**8**

votes

**1**answer

434 views

### Is $\mathbb{R}$ a $\mathbb{C}$-module without AC?

Assuming ZFC. We can make $(\mathbb{R},+)$ into a nontrivial(scaler multiplication is not identicaly zero) $\mathbb{C}$-module.
Now my questions are?
0.Is it consistent with $ZF$ that $\mathbb{R}$ is ...

**1**

vote

**0**answers

56 views

### Does the twisted product $K^{\mathbb{C}}\times_{Z(k)^{\mathbb{C}}} X^k$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, and suppose that $(X,J,\omega)$ is a compact Kähler manifold on which the group $K$
acts holomorphically such that the group $K$ preserves the
Kähler ...

**4**

votes

**2**answers

165 views

### A Karrass-Solitar theorem for surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...

**5**

votes

**2**answers

142 views

### Contracting a rational curve in a Calabi-Yau threeolfd

Let $X$ be a Calabi-Yau threefold and $C \subset X$ be a rational curve with $N_{C/X}\cong \mathcal{O}\oplus \mathcal{O}(-2)$. Can one contract the curve $C$? Assuming the answer is yes, what kind of ...

**-1**

votes

**0**answers

29 views

### Under what conditions the limit process of a sequence of Markov processes is Markov?

Given a sequence of Markov process $X_t^n$, if $X^n_t$ convergences to $X_t$ in the sense of some topology, under which condition can we prove that the limit process $X_t$ is Markov? Please give some ...

**0**

votes

**0**answers

113 views

### Example of Graph [on hold]

First of all, I review some terms and notations. A 1-way infinite path is called a ray, a 2-way infinite path is a double ray, and the subrays of a ray or double ray are its tails. Two rays in a graph ...

**0**

votes

**0**answers

30 views

### Standard Arguments of Calculus of Variations [duplicate]

I am working on calculus of variations in solid mechanics. I did my studies in Civil Eng., so I haven't passed any courses on Math Analysis. I do have problems with main properties of Hilbert and ...

**9**

votes

**1**answer

193 views

### What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...

**0**

votes

**0**answers

61 views

### Are major arcs always around a fraction with small denominator? [on hold]

In the usual circle method we might have a trigonometric polynomial $F(\theta)=\sum_{n}a_n e(n\theta)$ and we need to estimate the integral $\int_0^1 F(\theta)d\theta$ by breaking the domain into ...

**3**

votes

**0**answers

84 views

### The Klee Trick for subsets of $\mathbb{R}^3$

I asked the question Is dimension given by the Klee trick ever sharp?
That question remains unanswered, so I thought I might ask a slightly more concrete question along those lines.
Given a metric ...

**19**

votes

**7**answers

890 views

### Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...

**5**

votes

**1**answer

171 views

### Locally Closed Orbits in Real Algebraic Geometry

Let $G$ be a real algebraic group, and let $X$ be a real affine $G$-variety. I am looking for conditions on $G$ and $X$ for which the $G$-orbits are known to be locally closed in the Zariski topology ...

**2**

votes

**0**answers

63 views

### p-adic height on CM abelian varieties

Let A be a CM abelian variety over a number field and p a prime of ordinary reduction. Is the p-adic height of a non-torsion point on A over a p-adic field non-zero?
When A is an elliptic curve, this ...

**-2**

votes

**0**answers

24 views

### Generate Gamma random number using scale or rate parameters [closed]

I am wondering if I can generate Random number from gamma distribution using the shape and the rate parameters and then take the reciprocal of this number to be like it was generated from the shape ...

**-4**

votes

**0**answers

34 views

### Find the volume generated by revolving [closed]

Can anyone tell me how to solve this particular type of problem? A reference will be wonderful.
Find the volume generated by revolving the region bounded by y= -cosx and y=0, from x=-2pi to x=-pi ...

**0**

votes

**1**answer

153 views

### Coin graph is 4-colorable

How can we prove that a coin graph is 4-colorable???Also, can we find any example of an non-3-colorable coin graph.

**3**

votes

**1**answer

106 views

### Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...

**-1**

votes

**0**answers

60 views

### Fixed point combinator and functions with no fixed point [closed]

In lambda calculus the fixed point combinator is defined as:
$$Y=\lambda f.(\lambda x.f (xx))(\lambda x.f (xx)) $$
It is very easy to see how $Yg =g(Yg)$ for any $g$ by using $\beta$-reduction.
...

**2**

votes

**1**answer

119 views

### deformations of vector bundles on curves

Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of ...

**2**

votes

**1**answer

51 views

### Solution to generalized Sylvester equation

I am interested in solving generalized Sylvester equations (for $X$) of the form:
$$ \sum_{j=1}^k A_j X B_j^T = F, $$
where $A_j,B_j,X,F\in\mathbb{C}^{n\times n}$ and $k$, $n$ are integers. I will ...

**0**

votes

**0**answers

54 views

### F-splitting and F-purity from commutative algebra viewpoint

First I define two terms:
Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...

**6**

votes

**2**answers

160 views

### Pictures of the von Neumann polytope

Are there any graphic portrayals of von Neumann polytopes in low dimensions?

**0**

votes

**0**answers

88 views

### Golod Shafarevich Inequality and Inequalities among higher Cohomology groups

As a consequence of Golod- Shafarevich, we get an inequality between second cohomology group of a $p$-group with coefficients in $F_p$ and the first cohomology group of a $p$-group with coefficients ...

**8**

votes

**0**answers

106 views

### Wrapped Fukaya categories of Stein manifolds

By the work of Abouzaid, we know that the wrapped Fukaya category of $T^\ast Q$ with $Q$ a closed smooth manifold is generated by a cotangent fiber. Basically, this is an application of Abouzaid's ...

**-3**

votes

**0**answers

73 views

### Question about continuity of Holder functions [closed]

Let the definition of the oscillation about a point be given as
$
\mathrm{osc}[f] (x) : = \lim\limits_{\epsilon \rightarrow 0} \sup_{[x -\epsilon, x + \epsilon]} f - \inf_{[x -\epsilon, x + ...

**0**

votes

**1**answer

52 views

### Eigenvalues of product of diagonal positive matrix and symmetric matrix [closed]

Assume that we have two real symmetric matrices A and B, where A is a positive diagonal matrix, and B is a symmetric matrix with one eigenvalue λ = 0. Assume that H= AB;
is it possible to proof that ...

**2**

votes

**0**answers

94 views

### Perf($\mathscr{A}$) and perfect chain complexes

Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ ...

**6**

votes

**2**answers

212 views

### Holomorphic Hoffstein-Lockhart

In the article Hoffstein, Jeffrey; Lockhart, Paul "Coefficients of Maass forms and the Siegel zero." Ann. of Math. (2) 140 (1994), no. 1, 161–181, it is stablished a good bound for the Petersson norm ...

**1**

vote

**1**answer

73 views

### Is it true that $ H^{n-1} (spt \mu _E - \partial ^{*}E)=0 ?$

In Federer's Theorem, $ H^{n-1} (\partial ^{m}E - \partial ^{*}E)=0 $, where $E$ is a set of finite perimeter in $ \mathbb R^n $, $\partial ^{e}E$ is the essential boundary of E, and $\partial ^{*}E$ ...

**7**

votes

**1**answer

198 views

### What is the level of a positive energy loop group representation?

I am trying to learn a bit about loop group representation theory to understand its role in string geometry.
Let $G$ be a Lie group. I am thinking of $\text{Spin}(n)$, so you may assume $G$ to be ...

**4**

votes

**1**answer

157 views

### A Question Regarding Weak Diamond

In Assaf Rinot's survey article "Jenson's diamond principle and its relatives", he proves the following fact:
Fact 2.5:For every stationary set S, $\Phi_{S}$...entails that no ladder system ...

**1**

vote

**0**answers

76 views

### The convolution on the finite dimensional weak Hopf $C^*$-algebras

Let $\mathbb{A}$ be a finite dimensional weak Hopf $C^*$-algebra, and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{A}}$ and the convolution $a * b ...

**0**

votes

**1**answer

80 views

### Is the support of two odd theta characteristics on a generic curve disjoint?

Concise version of the question
On a generic curve of genus $g$, the odd theta characteristics will have exactly one global section. Therefore each odd theta characteristic will correspond to a ...

**0**

votes

**0**answers

43 views

### Localization on orbit type submanifolds (generalization of Atiyah-Bott-Berline-Vergne)

In equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem says roughly speaking that the integral of an equivariant cohomology class on the $G$-manifold $M$ has only contributions ...

**5**

votes

**2**answers

115 views

### Inverse of partial differential operator as a smooth tame map

Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an ...

**1**

vote

**0**answers

58 views

### A question about the existence of a solution for an ODE

Let $\gamma_+$, $\gamma_-:\mathbb{R}_+\to\mathbb{R}$ be two given functions. Assume that $\gamma_+$ ($\gamma_-$) is smooth, strictly increasing (decreasing) and $\gamma_{+}(+\infty)=+\infty$ ...

**4**

votes

**0**answers

135 views

### Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...

**3**

votes

**1**answer

72 views

### Independent Sets in random geometric graphs

I was wondering if a lot is known about independent sets in Random Geometric graphs? Most google searches don't bring up much. In addition I'm interested in any algorithms used for finding independent ...

**18**

votes

**1**answer

349 views

### Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant

Let $N$ be a positive integer; for simplicity I'm happy to assume it's an odd prime but I'm interested in the general case too.
Let $f \colon \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let ...

**1**

vote

**1**answer

43 views

### How to extend Dirichlet distribution to Dirichlet process

For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as
$$(X(B_1),\cdots,X(B_n))\sim Dir(\alpha H(B_1),\cdots,\alpha H(B_n))$$
...

**2**

votes

**0**answers

26 views

### The distribution of maximum of fraction Brownian motion over finite time interval

Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of
$(\sup_{0\leq t\leq ...

**0**

votes

**0**answers

107 views

### path integral and index theorem

I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form)
$\int ...

**1**

vote

**0**answers

37 views

### Connection between Strebel differentials, ribbon graphs, and Belyi maps

In this paper, a nice story is woven regarding the connection between quadratic differentials on Riemann surfaces, so-called 'ribbon graphs' drawn on those surfaces, and Belyi maps. However, I am ...

**-5**

votes

**0**answers

62 views

### Please help me to check the correctness of a paper. Thanks! [closed]

The following paper claimed to develop a differential and integral calculus for dealing with the fractional dimensional manifold:
Yong Tao, “The Validity of Dimensional Regularization Method on ...

**5**

votes

**1**answer

153 views

### Bases of surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...

**-1**

votes

**0**answers

47 views

### An isogeny from a split algebraic torus [migrated]

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...

**0**

votes

**0**answers

18 views

### Merging regions of function with similar mean and deviation using statistical test

I have got a question related to statistical tests that I would like to use in a new algorithm I am developing. Given an action space $x$, the algorithm would identify the regions in the function ...