# All Questions

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### 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?
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### 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 ...
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 ...
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 ...
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### 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 ...
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 ...
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 ...
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### 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 ...
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 ...
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### 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 ...
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 ...
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 ...
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### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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. ...
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 ...
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 ...
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### 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 ...
160 views

### Pictures of the von Neumann polytope

Are there any graphic portrayals of von Neumann polytopes in low dimensions?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\partial ^{*}E$ ...
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 ...
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 ...