# All Questions

**1**

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**1**answer

21 views

### Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact:
If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...

**3**

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**0**answers

44 views

### Adjunction algebra - is there anything similar to this in abstract algebra?

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...

**4**

votes

**1**answer

31 views

### Subgroups of Nilpotent groups with prescribed center

Let $G$ be a torsion-free, finitely-generated, nilpotent group of nilpotency class at least 3. Does there exist a normal subgroup
$N\leq G$ such that $G/N\cong \mathbb{Z}$ and $Z(G)=Z(N)$? (By ...

**0**

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**0**answers

12 views

### Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...

**0**

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**0**answers

40 views

### Curvature and the area

Assume that $\Omega$ is a Jordan domain with $C^2$ boundary $\gamma$ with $2\pi$ length and let $\kappa_z$ be the curvature of $\gamma$ at a point $z$. Is this formula known:
$$Area(\Omega)=2\pi ...

**0**

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**0**answers

26 views

### Unbounded difference on multiples of any irrational number [on hold]

For some irrational r consider the sequence of integral multiples as r, 2r, ..., ...

**5**

votes

**1**answer

64 views

### continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...

**2**

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**0**answers

37 views

### Topology for bounded operators quotiented by Schatten ideal

I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here.
Given the $C^{\ast}$-algebra of bounded ...

**0**

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**0**answers

24 views

### The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...

**5**

votes

**1**answer

103 views

### Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find?
In the adjoint representation, one can express the Killing form as a matrix and consider it as ...

**2**

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**1**answer

84 views

### Cyclotomic character in class field theory

Let $K$ be an extension of $\mathbb{Q}_p$.
By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}_K \rightarrow \mathbb{Z}_p^\times$ corresponds to a character $\chi : K^\times ...

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**0**answers

33 views

### Re-expressing the Integral $\int h\left(t-\tau\right)x\left(t-\tau\right)x\left(\tau\right)^2\,\text{d}\tau$

Given the integral:
$$
\int^{t}_{0} h(t-\tau)x(t-\tau)x(\tau)^2\,\text{d}\tau = \int^{t}_{0} h(t-\tau)F(\tau)\,\text{d}\tau + K
$$
Can you find $F(\tau)$ so it is not a function of $t$ and find $K$ ...

**1**

vote

**1**answer

41 views

### Explicitly relating two functions containing exponential terms [on hold]

This is an extremely basic question for a forum like this, but I am unable to think of any workable approaches myself.
I have two functions related to the distribution of administered drugs in the ...

**0**

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**0**answers

66 views

### Help estimating a number

I m lost estimating a number. I m looking for a tool to apply in this situation.
Given $A_0=\begin{pmatrix}1 & 1 \\ 1 &0 \end{pmatrix}$ and $A_1=\begin{pmatrix}1 & 1 \\ 0 &1 ...

**0**

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**0**answers

17 views

### Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...

**7**

votes

**0**answers

84 views

### Easiest example where field of definition is not field of moduli

There are many examples of varieties over $\overline{\mathbb Q}$ whose field of moduli is $\mathbb Q$ but which can't be defined over $\mathbb Q$. What is the easiest such example? It should be a ...

**1**

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**0**answers

42 views

### Question on the consistency of Zermelo set theory minus specification and extensionality

Let $W=Z^{-}-Specification$ where $Z^{-}=Z-Extensionality$ and Z is Zermelo set theory. What is known about models of $W$ or $W^{+}=W+Extensionality$?

**3**

votes

**2**answers

182 views

### (reference request) Chaitin's constant is incompressible

I've been looking for a full, detailed proof that Chaitin's constant is incompressible, i.e. there is a universal constant $c$ such that every program writing first $n$ digits of $\Omega$ has length ...

**0**

votes

**1**answer

40 views

### Integral over Kronecker product

Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that
$$ \int_{0}^T A(t) dt $$ is invertible.
Does it then follow that the matrix
$$ \int_{0}^T A(t) \otimes ...

**1**

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**0**answers

63 views

### Comonads and the category of Sets

In Vicary's paper, after eq 15, he talks about how the category of internal comonoids $C_\times$ has many properties of the category of sets. We know that a comonad on a category has the same axioms ...

**2**

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**1**answer

168 views

### Can a closure make the index finite?

Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of ...

**3**

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**0**answers

30 views

### Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...

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**0**answers

29 views

### Notation for rolling median [on hold]

I am writing a paper where a stack of images is processed using, among others, a rolling median. This takes ten images and calculates, pixel by pixel, the median value. What notation do you suggest to ...

**0**

votes

**0**answers

10 views

### Characterization of the optimal solution in relative entropy minimization

The following optimization problem is related to relative entropy and to the limit of the iterative proportional fitting procedure.
For $1 \leq i,j \leq n$ and fixed $w_{ij} \geq 0$, and fixed $a_i, ...

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**0**answers

60 views

### Hilbert function of points in $\mathrm{P}^2$

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If
$$
...

**-2**

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**0**answers

96 views

### What's the second cohomology group of the tangent bundle of $CP^n$ [on hold]

I want to know what is the second cohomology of tangent bundle of $CP^n$. Especially, for $CP^1$. And is there any general method to compute cohomology for tangent bundle?

**0**

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**0**answers

53 views

### What is a “normal crossings divisor relative to S”?

I'm reading SGA 1 Expose XIII, as given here:
http://arxiv.org/pdf/math/0206203v2.pdf
and I'm trying to understand the given definition of a "diviseur a croisements normaux relativement a S" for some ...

**7**

votes

**1**answer

102 views

### Existence of solutions of a polynomial system

Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional real vector, and define
$$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} ...

**4**

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30 views

### Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to ...

**5**

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**0**answers

46 views

### Factor a sum of products of cofactors

Let $M$ be an $n\times n$ matrix whose first column consists entirely of 1s.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row ...

**1**

vote

**1**answer

86 views

### Weak convergence of a sequence

I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : ...

**1**

vote

**1**answer

269 views

### Are all linear transformations measurable?

Let $V$ and $W$ be topological vector spaces over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$), and let $T:V \to W$ be a linear transformation. It is well-known that $T$ is not ...

**7**

votes

**2**answers

77 views

### Convexity of a certain sublevel set

Consider the polynomial of degree $4$ in the variable $r$ :
$$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$
The discriminant of this polynomial in $r$ is the following expression
(obtained using ...

**0**

votes

**1**answer

103 views

### Function that dominates everything in little o

I have a function $f(n)$ that satisfies the following property: for any function $g(n) = o(n^{-2})$, we have $f(n) = \Omega(g(n))$ (the implied proportionality constant in the $\Omega$ expression ...

**8**

votes

**3**answers

255 views

### Circles avoiding rational points of height $\le h$

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$)
of radius $r < 1$ avoid all rational points
of height $\le h$?
A rational point is a point all of whose coordinates ...

**1**

vote

**1**answer

84 views

### Equivalent Killing vector fields via an isometry

Suppose that $(M,g)$ is a complete semi-Riemannian manifold.
We say that two Killing vector fields $V$ and $W$ are equivalent if there is $\Phi:M\rightarrow M$ an isometry such that $\Phi_*(V)=W$.
...

**0**

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**0**answers

18 views

### transfer function, Laplace transform of second order equation [on hold]

I have met such problem,I have the second order equation, as
Au(t) + By(t) = C*y''(t).where A,B,C are paramters, and u(t) is input, the output is y(t).
what is the transfer function of Y'(s)/U(s)? ...

**-2**

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**0**answers

47 views

### Volume of revolving y=sin(x) about a line y=c [on hold]

Consider the surface formed by revolving $y=sin(x)$ about the line $y=c$ from some $0\le{c}\le{1}$ along the interval $0\le{x}\le{\pi}$.
[![graph][1]][1]
Set up and evaluate an integral to calculate ...

**0**

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**0**answers

19 views

### Compatability of depth for elements in p-adic groups under base change

Suppose $F$ is a non-archimedean local field and $E$ is a tame extension of it. Let $G$ be any connected reductive group over $F$. For any $r>0$, let $G(E)_r$ be the set of elements with depth $\ge ...

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**0**answers

37 views

### extending to bimeromorphic maps

A meromorphic map of complex spaces (in the sense of Remmert) f:X→Y is a multivalued map such that its graph Γ is an analytic subset of X×Y and off some analytic subset Z⊂Γ, the projection on the ...

**4**

votes

**1**answer

73 views

### Reference or proof for the fact that $J(X_0(N))$ splits inabelian varieties with real multiplication

It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ...

**1**

vote

**2**answers

194 views

### decomposition of Hilbert space into tensor product $L^2([0,\tfrac{1}{2}]) \otimes L^2([\tfrac{1}{2},1]) \simeq L^2([0,1])$

The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$.
As an example, is it ...

**6**

votes

**1**answer

102 views

### Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of ...

**0**

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**0**answers

30 views

### When does a stochastic process have its sample paths a.s. in the reproducing kernel hilbert space (RKHS) induced by its covariance function?

Let $T$ be a compact metrizable space. Consider a centered second order measurable process $(X_t\colon t\in T)$ with continuous covariance function $c(t,s):= \mathbb{E}X_t X_s$.
Are there any known ...

**3**

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**0**answers

49 views

### Non Borel Spaces: Gauge Integral

Question
Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology?
I'm wondering since it seems as the gauge crucially uses ...

**1**

vote

**1**answer

51 views

### softening probability distribution function

I am working on ECG signals and I want to fit it's probability distribution function with gaussian mixture model (sum of 2 or 3 gaussians) to extract features but it has a very sharp pdf around zero. ...

**3**

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91 views

### mod $p$ Jacquet-Langlands correspondence

Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and ...

**0**

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**0**answers

130 views

### Structure theorem for linear algebraic groups

I was wondering whether there is a structure theorem for linear algebraic groups over a number field $k$, i.e. something which tells us that any linear algebraic $k$-group is made up of, say, ...

**1**

vote

**1**answer

89 views

### Empty real conic containing two pairs of conjugate points in the projective plane?

Given two conjugate pairs of points in general position in $\mathbb{CP}^2$, there is a pencil of real conics containing these four points. Is there a real empty conic in this pencil?

**0**

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**1**answer

148 views

### Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...