# All Questions

**0**

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4 views

### When an elliptic curve is a quotient of $\mathbb{G}_a$?

I want to know when an elliptic curve $E \rightarrow S$ is a quotient of $\mathbb{G}$_a.
When $S$ is an analytic space, there is an exact sequence $$0 \rightarrow R^1 \mathbb{Z}^{\vee} \rightarrow ...

**0**

votes

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2 views

### What is the reason the eigenvalues of GUE and CUE matrices tend locally to the same distribution?

It's well known in random matrix theory that locally the eigenvalues of a random matrix from the Gaussian unitary ensemble tend to a sine-kernel determinantal point process. Likewise, locally the ...

**0**

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4 views

### Lower bound on the tail of the hypergeometric distribution

Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...

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13 views

### How to find integer solution for bilinear transformation?

Let y = (ax + b)/(cx + d), where a, b, c, d integer constants, is there any technique to find integer solution of x & y?

**2**

votes

**1**answer

17 views

### Countable chain condition in $\text{BP}(X)$

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$.
Assume $X$ is second countable Baire ...

**0**

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10 views

### A special approximation of BV functions

Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., ...

**1**

vote

**1**answer

15 views

### Nodal sets under the heat flow

Let $u(t,X)$ be a bounded smooth solution of the heat equation on $R^2$
$u_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $u(t,x,y)$ behaves like $e^{-\alpha t} xy$ for large $t$, meaning that ...

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26 views

### permutation action on cohomology of configuration space

Let $F(M,n)$ be the $n$-th configuration (ordered) of manifold $M$.
In the paper The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres, ...

**1**

vote

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20 views

### permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...

**3**

votes

**0**answers

17 views

### Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...

**-3**

votes

**0**answers

25 views

### Derivative Problem [on hold]

This is an derivatives Problem.
find the tangent to y=sqrt(x^2-x+5) at x=5.
y = ?
What I did was first find y by plugin x into the equation. The answer is y = 5.
Then i found the derivative ...

**-2**

votes

**0**answers

14 views

### function bounds and comparison [on hold]

ok so basically i need to understand how i can compare functions so that i can find big O big theta and big omega for algorithms of a program
my mathematics background is not very strong but i have ...

**-4**

votes

**0**answers

30 views

### Mathematic Theory of Computation [on hold]

Let Sigma be an alphabet. Use the principle of induction on Sigma^*, to prove that |w v| = |w| + |v| for all w, v in Sigma^.

**3**

votes

**0**answers

34 views

### Asymptotics of the multipartition function

Recall that the multipartition function $p_k(n)$ counts the number of $k$-tuples of partitions $\lambda^1,\ldots,\lambda^k$ of numbers $a_1,\ldots,a_k$ with $a_1+\cdots+a_k=n$. It has a generating ...

**3**

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**0**answers

44 views

### Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"?
Or is there something else that states ...

**0**

votes

**0**answers

22 views

### A 2D random walk on a lattice of equilateral triangles [on hold]

Calculate the characteristic ratio C∞ for a random chain on a two-dimensional lattice made
of equilateral triangles with side a. At each step, a walk has five choices (it cannot double back).

**1**

vote

**0**answers

28 views

### How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...

**-1**

votes

**0**answers

45 views

### Lawvere algebraic theory as presentation-invariant [on hold]

We can read in a lot of papers, included Lawvere's PhD thesis, that algebraic theories are "an invariant notion of which the usual formalism with operations and equations may be regarded as a ...

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51 views

### In search of a preprint by Litherland

I've seen the following citation a lot: "R. LITHERLAND: A formula for the Casson-Gordon invariants of a knot, preprint." I can't seem to find a corresponding publication. [Added in edit: apparently ...

**0**

votes

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21 views

### double Sum with a function of 3 variables inside. How to solve it? [on hold]

I am studying time scheduling problem solving with linear programming. I am reading paragraph 2.5 from this book (pages 32-35) and I am trying to solve it with Java.
I have some questions about ...

**4**

votes

**1**answer

112 views

### Establishing Duality in Tannakian Categories

I sometimes need to prove a category is Tannakian. Part of the definition of a Tannakian category is that it is rigid.
However, I find the definition of rigid categories somewhat difficult. I don't ...

**3**

votes

**0**answers

41 views

### Replacing functors by topologically or simplicially enriched functors

I believe it is true that if a functor $F:Top\to Top$ preserves weak homotopy equivalences then it is related by a natural weak equivalence to some other such functor that is in some sense continuous, ...

**-2**

votes

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43 views

### C^*-algebras and bidual [on hold]

Let $A$ be a C$^*$-algebra, $x \in A$ a non-zero positive element of $A$, $r(x)$ the range projection of $x$ in $A^{**}$ and $e = 1 - r(x)$.
How do I show that exist a projection $f$ in $A^{**}$ such ...

**0**

votes

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55 views

### Motivation of $a_p$ for non-CM elliptic curves [on hold]

For an elliptic curve $E$ without CM let $\overline{E}$ be the good reduction of $E$ modulo $p$ prime. The value $a_p = p+ 1 - \mathbb{F}_p$ is referenced by DDT on p.19 and Ribet on p.5 . However ...

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40 views

### Simple example game with 3 players [on hold]

I am currently writing an algorithm to compute different things as nash equilibria, dominated strategies etc for normal-form games. Since I am now trying to extend it to an infinite amount of players, ...

**2**

votes

**1**answer

40 views

### Limits at infinity of fellow-travelling sequences in Teichmuller space,

I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.
Suppose that we have closed surface of genus $g\geq 2$, and ...

**1**

vote

**0**answers

68 views

### local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal).
Is there a sufficient condition for $A$ to be noetherian?
For example, we know that the completion ...

**1**

vote

**1**answer

59 views

### Non-closability of an operator

Let $a$ be a positive continuous function nowhere differentiable on $[0,1]$. The operator $T$ in $H:=L^2(0,1)\oplus L^2(0,1)$ defined by
$$T(u_1,u_2) := (u_1' + au_2',0)$$
on $\textrm{Dom} \,T := ...

**3**

votes

**2**answers

84 views

### Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$.
Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...

**5**

votes

**2**answers

125 views

### Is every connected reductive group over a local field already defined over a global field?

Let $K$ be a local field, e.g. $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$. Let $G$ be a connected reductive group over $K$. Is it true that $G$ is already defined over a global field? More precisely, does ...

**2**

votes

**0**answers

40 views

### Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...

**-1**

votes

**0**answers

39 views

### Largest subsequence with average $\geq \delta$ [on hold]

I found a solution in Largest subarray with average $\geq$ k.
However, I fail to understand the algorithm correctly.
Let us consider the following sequence:
12, -10, 10, -10, 12, -9
Let us pick ...

**1**

vote

**0**answers

40 views

### Density of smooth functions in Sobolev space, respecting nonnegative traces

I am looking for a reference for the following result (if it is true):
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma_0\subset\partial\Omega$ be sufficiently regular.
Let ...

**0**

votes

**0**answers

47 views

### Reducedness of scheme theoretic fibers of toric morphisms

Let's consider $X=X(\Delta_X)$ and $Y=Y(\Delta_Y)$ two complete $\mathbb{Q}$-factorial toric varieties over an algebraically closed field of characteristic zero, and let $f:X\to Y$ be a flat toric ...

**0**

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33 views

### Extending and contracting an ideal by a faithfully flat homomorphism [migrated]

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $.
This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry ...

**2**

votes

**1**answer

42 views

### Largeness, generic, random points

As presented in Oxtoby's book ( http://link.springer.com/book/10.1007%2F978-1-4615-9964-7 ), there are two notions of largeness for subspace $Y$ of a given space $X$:
Topology: $X$ is a topological ...

**3**

votes

**0**answers

30 views

### Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...

**0**

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99 views

### Fermat's Theorem on p = a^2 + b^2 [migrated]

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4.
I heard that such a criterion could be possible for a given integer $n$ like
$p = a^2 + n b^2$
...

**1**

vote

**0**answers

25 views

### Constructing pointwise-or independent binary matrices

Let $M$ be an $m \times n$ matrix such that every $M_{ij} \in \{0,1\}$.
We say that $M$ is `$\lor$-irreducible' if (i) no row is a pointwise-or of other rows, and (ii) no column is a pointwise-or of ...

**0**

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**0**answers

63 views

### Proof for this expression for Dottie number [on hold]

Neither Wolfram's mathworld nor Wikipedia mentions any series expanion for the cosine's fixed point.
Therefore I am asking for a proof for this Kapteyn series representation of Dottie number:
...

**2**

votes

**0**answers

26 views

### Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved?
$$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {
...

**4**

votes

**1**answer

91 views

### Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case.
Question:
Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...

**0**

votes

**1**answer

44 views

### Fixed point property for the projectivization of manifold of fixed rank matrices

Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. The projectivization of $M$ is denoted by $PM$.
Does $PM$ satisfy fixed point property?

**-1**

votes

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42 views

### $\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev
Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...

**2**

votes

**1**answer

123 views

### Manifold with corners [on hold]

Iam looking at the following situation of a manifold $Z$ with corners.
More specifically a product of a smooth manifold X with a standard $k$-simplex $\Delta^k$.
I wish to study certain formulas for ...

**0**

votes

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56 views

### Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...

**-1**

votes

**0**answers

8 views

### Partition of function into pieces for interpolation needs [migrated]

I've got some experimental data obtained from my mate's research. There are two sets of (x,y) points for each curve. He asked me to interpolate function values between this points, so for each curve I ...

**1**

vote

**1**answer

83 views

### Weak topology on a pre-Hilbert Space

Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state.
Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product ...

**-2**

votes

**0**answers

20 views

### Is Markov Chain Sampled at stopping times a Markov chain? [on hold]

Given a Markov hain $\{X_n\}$ and $T_n$ be an increasing sequence of stopping times, is $\{X_{T_n}\}$ Markov chain ?

**0**

votes

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32 views

### Boundedness of a singular integral operator on $L^p(\mathbb{R})$, $1<p<\infty$

My singular integral operator is defined by
\begin{align}
Sf(x)=-\int_{-\infty}^{\infty}f(t-x) \frac{dt}{2\sinh\frac{\pi}{2}t},
\end{align}
that is, a convolution $-\frac{1 }{2\sinh\frac{\pi}2x}\ast ...