# All Questions

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6 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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**0**answers

30 views

### The Riemann Hypothesis and the RSA Cryptosystem

Follow up to the new RSA factorization technique below, how could we derive to an explicit proof to the Riemann Hypothesis?
For every odd number on the form $N=pq$,
where $p$ and $q$ are primes, ...

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**0**answers

7 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 ...

**3**

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**0**answers

14 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**

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**0**answers

23 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

105 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 ...

**3**

votes

**1**answer

27 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

79 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 ...

**7**

votes

**2**answers

125 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

60 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

14 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

36 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

13 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 ...

**1**

vote

**0**answers

28 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

63 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

139 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 ...

**5**

votes

**1**answer

85 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

27 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

37 views

### Non Borel Spaces: Gauge Integral

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 neighborhoods...
So far ...

**1**

vote

**1**answer

43 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. ...

**2**

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**0**answers

80 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

111 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

**0**answers

47 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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**0**answers

82 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 ...

**-3**

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**0**answers

63 views

### Order of the generators of a finitely presented group [on hold]

I want to ask a question... if a group $G =\langle S\mid R\rangle$ has infinite order and we do not know the order of its generators, can we convert this presentation in such a presentation having ...

**14**

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**0**answers

109 views

### Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra ...

**8**

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**1**answer

148 views

### Geometry description of the GSR riffle shuffle model

In 1992 Diaconis and Bayer announced their famous result which is now a well-known folklore: Seven shuffles is enough to randomize a deck of cards.
One of the key ingredients in their proof is that ...

**0**

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**0**answers

64 views

### Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...

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28 views

### Get archimedean spiral length by cartesian coordinates [on hold]

I am currently busy with writing a generator for a Sacks spiral, and the formulas I currently have are these:
...

**1**

vote

**1**answer

56 views

### Connected representant of a framed cobordism class (reference needed)

Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected.
I believe it could be proved by directly ...

**2**

votes

**1**answer

103 views

### Which polynomials define extensions of $k(t)$ unramified at the finite places

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $L$ be the extension
of $k(t)$ obtained by attaching a root of an irreducible polynomial $f\in k(t)[x]$.
Is there a way to tell ...

**5**

votes

**1**answer

117 views

### Linear systems on bielliptic surfaces

A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$.
...

**2**

votes

**1**answer

163 views

### Splitting the Hopf map in two

Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $h\circ i:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of ...

**1**

vote

**0**answers

43 views

### Existence of Colimits in the Definition of Locally Presentable Categories

Basically, my question is simple: why does the definition of a locally presentable category require all colimits exist?
The motivation for this is that I was learning about algebraic posets, and had ...

**3**

votes

**1**answer

71 views

### What is the probability of a given induced ordering of a random permutation?

I ran into the following problem in a calculation involving permutations.
Let $[n] = \{1,...,n\}$, and assume that $[n]$ is partitioned into equivalency classes. That is, $[n]$ is the disjoint union ...

**5**

votes

**1**answer

60 views

### Origins of the Jacobi matrix

I have several questions concerning history of Jacobi matrices.
Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...

**6**

votes

**3**answers

219 views

### Diagonalization via the Toda flow

According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...

**9**

votes

**1**answer

145 views

### Existence of a family of elliptic curves with large torsion subgroup

Andrej Dujella's excellent web-site "High rank elliptic curves with prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/tors.html
gives example of the (current) largest known rank of an ...

**3**

votes

**0**answers

67 views

### Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (in the sense of Rieffel) also Morita equivalent in the algebraic sense (as rings)?

Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts:
Morita equivalence for $C^*$-algebras: Equivalence ...

**1**

vote

**1**answer

76 views

### Does the Fourier transform of a non-strictly positive real kernel $\Phi(t)$ always generate entire function with complex zeros?

Question (1) Does the Fourier transform of a non-strictly positive real kernel
$f(t)$ always generate an entire function $g(z)$ with complex zeros?
$$g(z)=\int_{-\infty}^{\infty}f(t) ...

**3**

votes

**1**answer

71 views

### An orthogonal factorization system on 1-Cob?

Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W_{dom}:A\to W$ and $i^W_{cod}:B\to ...

**5**

votes

**0**answers

179 views

### Is there an infinite J-group?

For a group $G$ let $\operatorname{Sub}(G)$ be the lattice of all its subgroups.
A subgroup interval is an interval in the lattice $\operatorname{Sub}(G)$.
A group $G$ is called a J-group iff for ...

**13**

votes

**2**answers

845 views

### Is any particular algebraic number known to have unbounded continued fraction coefficients?

The continued fraction
$$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be ...

**2**

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**0**answers

97 views

### Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...

**6**

votes

**1**answer

234 views

### A divisor sum congruence for 8n+6

Letting $d(m)$ be the number of divisors of $m$, is it the case that for $m=8n+6$,
$$ d(m) \equiv \sum_{k=1}^{m-1} d(k) d(m-k) \pmod{8}\ ?$$
It's easy to show that both sides are 0 mod 4: the left ...

**0**

votes

**0**answers

35 views

### Quantile as solution to minimization problem

I posted this on Math Stack Exchange, but since I got no response, I'm trying my luck here. I'm studying basics of quantile regression now and I have trouble proving that $\tau-$th quantile of ...

**-3**

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**0**answers

46 views

### BCL algebra define a partial order. Answer [on hold]

BCL algebra define a partial order. Answer
Proof:
(i) Reflexivity: If x*x=0, then x⩽x.
(ii) Anti-symmety: If x⩽y and y⩽x, then x*y=0 and y*x=0, by axiom (2),we have x=y.
(iii) Transitivity: If ...

**4**

votes

**0**answers

51 views

### A conditional form of Holder's Inequality on Type II-1 von Neumann algebras

Suppose $V$ is a von-Neumann algebra with a trace $\tau$ so that $\tau(I) = 1$. Suppose $W$ is a sub-von Neumann algebra of $V$. Then we can define a conditional expection to be a projection ...

**0**

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**0**answers

163 views

### Why does Neeman avoid t-structures?

I have a simple question: the book "Triangulated Categories" by A. Neeman aims to be an exhaustive reference about the whole (basic) theory of triangulated categories. So why there is only a single ...

**-2**

votes

**0**answers

32 views

### multiplication of a projection matrix and PSD matrix is a PSD? [on hold]

I have a projection matrix P and X^TAX where A is a diagonal matrix with all strictly positive entries can I tell that PX^TAX is PSD?