# All Questions

**0**

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7 views

### Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here.
Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...

**0**

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13 views

### Explicit equations for conormal bundle to an affine toric variety

Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...

**0**

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16 views

### A Function $ g[0,\infty) \mapsto [0,1)$ Sharply Changing on Both Ends [on hold]

I need a function $ g[0,\infty) \mapsto [0,1)$ that sharply decreases near 0 and sharply increases near 1. Preferably, it wouldn't be defined in a piecewise manner.
Can anyone provide an example, ...

**3**

votes

**1**answer

17 views

### On the conformal removability of Jordan curves

We say that a compact subset $E$ of the Riemann sphere $\mathbb{C}_\infty$ is (conformally) removable if every homeomorphism of $\mathbb{C}_\infty$ conformal outside $E$ is actually conformal ...

**-2**

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**1**answer

17 views

### convex upper bound of a non-convex function [on hold]

is there any formula for finding convex upper bound of a non-convex function? Like for finding convex lower bound of a non convex function, fenchel dual exists.

**0**

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19 views

### preservation of localness among certain Krull domains

The following question essentially appeared (http://math.stackexchange.com/questions/931801/preservation-of-localness-among-certain-krull-domains) on math.SE a while ago, but nobody has done anything ...

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21 views

### On the global and local hermitian space

Let $E/F$ be a quadratic extension of number fields and $v$ a finite place of $F$.
Then I am wondering if there is a global hermition vector space over $E$ such that for all finite places $v$, local ...

**5**

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**0**answers

46 views

### Periodic group with bound on order of finite subgroups

I have asked the same question previously on stackexchange without any answer (http://math.stackexchange.com/questions/923638/periodic-group-with-bounded-subgroups):
I am looking for infinite ...

**0**

votes

**1**answer

105 views

### Are there any references for the measures without any growing tails?

I am trying to find whether there are any references to characterize measures without any growing tails. For example, the Lebesgue measure and the following measure
\begin{equation}
...

**1**

vote

**1**answer

42 views

### Strictly positive solutions of a random linear system

Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, ...

**3**

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**1**answer

62 views

### Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...

**-1**

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17 views

### What does it mean for a stochastic sequence to be “stochastically smaller” than some other stochastic sequence?

A relatively simple type of random graph is the Erdős-Rényi random graph. The graph created by means of the following process: Let $V$ consists of $n \in \mathbb{N}$ vertices, and let each edge of the ...

**21**

votes

**2**answers

287 views

### Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$.
For example $f(6)=6+3+2=11$, $f(5)=5$.
Note that $x$ is a fixed point for ...

**-4**

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**0**answers

31 views

### Help? Algebra question about ratios and proportions [on hold]

A community center is going on a trip to Philadelphia via several buses. The ratio of men to women to children is 1:2:3. If there are 150 people going on the trip, how many men are going? How many ...

**1**

vote

**1**answer

34 views

### Unbiased sample from a product

Let $X = (x_1,\ldots,x_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod_{i=1}^n x_i$
Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that on average considers only a ...

**0**

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**0**answers

63 views

### Squarefree Parts of Mersenne Numbers with prime exponent [on hold]

The $n$-th Mersenne number is $M_n=2^n−1$. Write $M_n=a_n b^2_n$ where $a_n$ is positive and squarefree. In the discussion
Squarefree Parts of Mersenne Numbers , the lower bound of $a_n$ has been ...

**-6**

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**0**answers

109 views

### Does $\pi$ encode the prime numbers? [on hold]

I have a question regarding whether or not $\pi$ encodes the sequence of primer numbers. It is common knowledge that
$$ \zeta (2) = \sum_{i = 1}^{\infty} \frac{1}{n^2} = \prod_{p \in \mathbb{P}} ...

**0**

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**1**answer

17 views

### Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue

Given the Sturm-Liouville type (time independent Schroedinger) equation
\begin{equation}
\frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R}
\end{equation}
where ...

**4**

votes

**1**answer

49 views

### Geodesic flow on infinite surfaces

The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ...

**0**

votes

**1**answer

26 views

### Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...

**1**

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**2**answers

68 views

### Non-DS circulant graphs

Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic circulant graphs on $p$ vertices.

**2**

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**1**answer

176 views

### Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?

The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation
$[X]=[Y] + ...

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42 views

### numerical method (implicit) for nonlinear pde

If $\newcommand{\lbar}{\underline{\lambda}}$
$$ \lambda(t)= \lbar+(\lambda_0-\lbar)\exp \left( (\mu-\frac{1}{2}\sigma^2)t+\sigma W^\lambda_t\right) $$
and $\mu$ , $\sigma$ , $\lbar$ , $\lambda_0$ , ...

**9**

votes

**1**answer

1k views

### Go I Know Not Whither and Fetch I Know Not What

Next day: apparently my original question is harder, by far, than the other bits. So: it is a finite check, I was able to confirm by computer that, if the polynomial below satisfies $$ f(a,b,c,d) ...

**3**

votes

**2**answers

91 views

### Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...

**0**

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94 views

### Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...

**4**

votes

**2**answers

159 views

### Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...

**2**

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31 views

### Weighted graph similarity

I have the following problem. Consider an undirected biconnected graph with $n$ vertices and $m$ edges ($n \leq m \leq n(n-1)/2$). The $m$ edges of the graph are then "populated" by integer weights ...

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36 views

### square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces

I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant:
...

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**1**answer

136 views

### Name for class of matrix determinants

Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $\alpha_i(x)$. Now, ...

**-8**

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46 views

### New formulas for Generating sin(nx) & cos(nx) [on hold]

I have proved formulas for generating $\sin (nx)$ and $\cos(nx)$ in term of only $\sin x$ and integer $n$, or $\cos x$ and integer $n$, such as:
$$\sin(9x) = 9\sin x -120\sin^3 x + 432\sin^5 x – ...

**-5**

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**0**answers

67 views

### Rookie looking for help with basic mathematical problem [on hold]

This should probably be very simple, but I'm just not very skilled in math :S.
I want a function that takes one variable, x, ranging from 0-1. As the input approaches 0 so should the output. As the ...

**1**

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**0**answers

41 views

### Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...

**-1**

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59 views

### Isogeny of abelian varieties over general fields [on hold]

We know that given an abelian variety $X$ over an algebraically closed field $K$ of characteristic $0$ and any integer $n$ the induced map $[n]:X \to X$ is an isogeny. As far as I understand this ...

**4**

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173 views

### $x^2+1$ attaining almost prime values

Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...

**1**

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24 views

### Jacobi Polynomial asymptotics via saddle-point methods

I'm looking at asymptotics of a Jacobi polynomial: $P_{n-2}^{\alpha_n,\beta_n}(0)$, with $\alpha_n=(n-2)-N, \ \beta_n=[cn^{3/2}]-(n-2)$, where $c>0$ is a constant, $N=\binom{n}{2}$ and $[\cdot]$ is ...

**1**

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79 views

### Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...

**1**

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**2**answers

103 views

### The zeros of alternating sign, binomial coefficient polynomials

I have a question regarding the zeros of the following polynomial, based on partial rows of pascal's triangle,
$$f(x)=\sum_{k=a}^n\binom{n}{k}(-1)^{k}x^k,$$
where $a,n\in \mathbb{Z}^+,n>a.$
...

**-2**

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50 views

### Expected probability [on hold]

Suppose, given an integer N, N divides by its divisors(1, to N) and gain new number, repeats the procedure until it becomes 1.
what is the expected number of divisions need number N become 1?
Can ...

**5**

votes

**2**answers

152 views

### Eigenstates of Fourier transformation

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by
$$
(\mathcal F u)(\xi)=\int e^{-2iπ ...

**1**

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**2**answers

94 views

### When is the induced representation factored through the initial one?

Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ ...

**6**

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**1**answer

139 views

### Generalizing “variation of parameters”

I'm stuck on generalizing an ODE formula and could use your help!
One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here ...

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108 views

### Accessible subgroups of free groups

Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...

**1**

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**1**answer

73 views

### GOE convergence

As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see this question for more background). The question is: ...

**0**

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38 views

### Relationship between ODE and PDE(Laplacian on $S^2$)

Assuming that I solved the Sturm-Liouville problem with some periodic/antiperiodic BC.'s:
$$-y''(x)+f(x)y(x) = \lambda y(x),$$
where $\lambda $ is a free parameter on $[0,2\pi]$ and with $f \in ...

**1**

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58 views

### On the universal property of certain representable functors and rational sections

Let $P_1,P_2$ be two Hilbert polynomials of subschemes in $\mathbb{P}^n$. Denote by $H_{P_1,P_2}$ the corresponding flag Hilbert scheme (parametrizing pairs $(X\subset Y)$ where $X$ has Hilbert ...

**0**

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31 views

### Continuity of Functional Represented by Surface Integral

Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $S \subset \Omega$ be a hypersurface in $\mathbb{R}^n$. Let further be $C_0(\Omega)$ the space of all continuous functions with compact ...

**-3**

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**0**answers

82 views

### Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$? [on hold]

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...

**8**

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**0**answers

86 views

### Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...

**1**

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**0**answers

67 views

### Does this integral have a closed form? [migrated]

$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$
I saw this question here. It is really hard to find a closed form. Or is there no closed form? Please give me a hand. Thanks!