**-1**

votes

**0**answers

22 views

### Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral ($\ln(\cos x)/(1-x)$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown ...

**0**

votes

**0**answers

8 views

### Absolute value of a polynomial as the Euclidean distance from its root

Say that you have a polynomial $f(x)$ of degree 2 in one real variable. Then, if the polynomial has only one unique root $r \in \mathbb{R}$, it factorizes as $f(x) = (x - r)^2$, which expresses the ...

**0**

votes

**0**answers

2 views

### Efficient packing in Gaussian measures on Hilbert spaces

Let $B(0,1)$ be the unit ball in a separable Hilbert space $H$ with a Gaussian measure $\mu$ on it. For a small $r > 0$, can we have $x_i \in B(0,1)$, $0 < r_i \leq r$ and a constant $K > 0$ ...

**0**

votes

**0**answers

14 views

### Math Education Paper Request [migrated]

If this is the wrong forum for this post I apologize but I'm not sure of another well-suited medium for this question (and any reference to one is appreciated).
I am wondering if any research in ...

**0**

votes

**0**answers

13 views

### Uniqueness of the reduced rank QR decomposition

Let $A$ be a $n\times n$ matrix that is a real, symmetric, positive semidefinite and has rank $r$.
I read about the rank reduced $QR$ decomposition (here for example) as the QR variant where $A=QR$ ...

**1**

vote

**1**answer

53 views

### A generalized theorem of Hall's marriage theorem

We all know Hall's marriage theorem as following:
A bipartite graph $G$ with bipartition $\{ A,B \}$ contains a matching of $A$ if and only if $|N(S)|\geq |S|$ for all $S\subseteq A$.
And I am ...

**4**

votes

**0**answers

41 views

### “Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at ...

**-2**

votes

**0**answers

25 views

### Why standard deviation is preferred over mean deviation? [on hold]

I was doing my homework when I come across both these quantities which tells us dispersion in data. But, I am able to understand mean deviation as it tells on an average how much a value deviates from ...

**3**

votes

**0**answers

33 views

### geometric interpretation of derivation between two algebras

Given a smooth manifold $M,$ it is well known that any derivation of the algebra of smooth functions $C^{\infty}(M)$ can be seen (or it is associated to) a smooth vector field on $M.$ I am looking for ...

**3**

votes

**0**answers

24 views

### orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...

**-1**

votes

**0**answers

45 views

### On simple complex loops [on hold]

To apply Jordan theorem, a curve $\Gamma$ must be a simple closed continuous curve. (Its parametrization is injective)
Now consider an element $[\gamma] \in H_1(\mathbb{\Omega})$ (singular homology, ...

**2**

votes

**2**answers

25 views

### Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$.
With $nz(M) := ...

**-7**

votes

**0**answers

29 views

### Interesting Pemutation 65+ [on hold]

Ques no. 6& 7 on
http://s30.postimg.org/l6o96cfdt/Screenshot_2015_11_30_15_26_44.jpg
It is very interesting question

**-6**

votes

**0**answers

28 views

### Permutation 235 [on hold]

Ques no. 3&4 on
http://s21.postimg.org/li25onft3/Screenshot_2015_11_29_23_08_54.jpg
It is very interesting

**0**

votes

**0**answers

18 views

### Sum of N Gamma distributed random variables being N a Gamma distribution random variable [migrated]

Thanks in advance.
Let X a gamma-distributed random variable having scale θ and shape k:
$$
X \sim \Gamma(k, \theta) \equiv \textrm{Gamma}(k, \theta)
$$
with its probability density function is:
$$
...

**0**

votes

**1**answer

22 views

### Maximal induced cycles on $n$-clique graphs

For any set $X$ we set $[X]^2 = \big\{\{a,b\}: a, b\in X\text{ and } a\neq b\big\}$.
We say a simple undirected graph $G=(V,E)$ is an $n$-clique graph if there are $S_1,\ldots,S_n\subseteq V$ such ...

**3**

votes

**0**answers

23 views

### Convergence rate of Fagin's 0-1 law for first-order properties of random graphs

Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the ...

**0**

votes

**0**answers

44 views

### Hodge metric on pair

I am looking for the definition of Hodge metric, like definition 2.2 here
https://hal.archives-ouvertes.fr/hal-00322845/document
But instead of Vector bundle if we have divisor $D$ with conic ...

**1**

vote

**0**answers

31 views

### Consecutive integers divisible by consecutive small numbers

Given $n$, what is the largest set of consecutive integers in $[n,2n]$ can we have so that each integer is divisible by a distinct element from $[\log n,2\log n]$ (no partiular order)? So apriori I am ...

**-1**

votes

**0**answers

36 views

### Recognition of a transversal in finite group

Given a subset $T$ of a finite abelian group $G$ with $|T|/ |G|$, how can we determine if $T$ is a transversal of some subgroup of $G$?

**2**

votes

**0**answers

70 views

### mod 2 Bockstein and the Steenrod square

Let $M$ be a manifold, $n$ be a positive integer and $x\in H^1(M;\mathbb{Z}_2)$. I want to find some checkable sufficient conditions imposed on $M$ such that $x^n\neq 0$ can imply $x^{2n}=Sq^n ...

**2**

votes

**0**answers

51 views

### Generalizing von Staudt's synthetic construction of the complex numbers

Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...

**5**

votes

**0**answers

57 views

### Adding minimal subsets to $\aleph_\omega$

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all
$\alpha < \kappa.$
Question. Is it consistent that there ...

**0**

votes

**0**answers

80 views

### On the Frobenius coin problem [on hold]

Is there an $n_0\in\Bbb N$ such that for every $n\in\Bbb N_{>n_0}$ there are $a,b\in\Bbb N$ with $n<a,b<2n$ with $\mathsf{gcd}(a,b)=1$ such that
1. if $ax+by=rt$ for some $x,y>0$ with ...

**0**

votes

**0**answers

26 views

### On a conjecture about Riemannian metric with positive sectional curvature [duplicate]

What is the last status of the following conjecture? Is it still open? What partial or similar results are known up to now?
Conjecture: $S^2\times S^2$ admits a Riemannian metric with positive ...

**1**

vote

**0**answers

88 views

### Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...

**1**

vote

**2**answers

193 views

### Polynomials which always assume perfect power values

Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, ...

**2**

votes

**0**answers

16 views

### Fractional parts of two multiples [duplicate]

There is a theorem (I can't remember its name) saying that for any irrational number $x$ and any $0<a<b<1$, there exists a positive integer $n$ such that $\{nx\}\in (a,b)$, where $\{\cdot\}$ ...

**-2**

votes

**1**answer

19 views

### Systems with zero output for periodic inputs [on hold]

I need help with the following question:
Suppose you have an LTI system which produces the 0 output in response to any periodic input with period T. Show that the impulse response h(t) of the system ...

**2**

votes

**0**answers

35 views

### Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$

Consider the transform (see e.g., (5.1) in this paper):
\begin{equation*}
\Lambda_\mu(q)(z) := \int_{\Delta_n} ...

**1**

vote

**1**answer

121 views

### Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...

**0**

votes

**0**answers

29 views

### Complexity of $\mathsf{gcd}(a,b)\bmod N$

Given $a,b\in\Bbb N$ where each $a,b$ is $n$-bits, we can compute $\mathsf{gcd}(a,b)$ in $cn^{1+\epsilon}$ bit operations for some fixed $c\geq1$.
My query is given $N,a,b$ where $a,b$ is $n$-bits ...

**4**

votes

**0**answers

132 views

### Reference request: category of sheaves of O-modules with coherent cohomology

Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of ...

**0**

votes

**0**answers

85 views

### Quantifier elimination - Existence of solution of a differential equation

We consider the ring $\mathbb{C}[x]$ and the language $\{+, \frac{d}{dx}, 0, 1\}$.
I want to eliminate the quantifier from the formula $\exists y \ Ly=f$.
The elements of the ring are of the form ...

**2**

votes

**0**answers

36 views

### Mixed tensor index position significance

What is the significance of tensor index position?
For example the fourth order Riemann curvature tensor
\begin{align}
R^m_{ijk}
\end{align}
or
\begin{align}
R^{\phantom{i}m}_{i\phantom{m}jk}.
...

**-3**

votes

**0**answers

47 views

### Random process & probability problem

A random process r obeys the following distribution:
p(r,ṙ)=$\frac{r}{b_0}\exp{(-\frac{r^2}{b})}\frac{1}{\sqrt{2\pi b_2}} \exp{(-\frac{\dot{r}^2}{2b_2})}$, where $\dot{r}$ is the derivative of r in ...

**-4**

votes

**0**answers

25 views

### how to select a submatrix from a matrix [on hold]

suppose I have matrix, M is as such [A,B;C,D]; and I want to expand the M matrix as such,
M=P1*M*P1'+P2*M*P2'+P3*M*P3'+P4*M*P4';
where,
P1*M*P1'=[A,0;0,0];
P2*M*P2'=[0,B;0,0];
then other two.
It ...

**1**

vote

**1**answer

88 views

### Fourier transform localisation

I was just reading through these notes here on p. 19 and in the last paragraph the authors claim something (by saying "standard Fourier theory shows...") that sounds to me like:
If you have $f \in ...

**5**

votes

**1**answer

82 views

### Asymptotic enumeration of magic squares

An order-$n$ magic square is an $n \times n$ matrix over the numbers $\{1, ... ,n^2\}$, each appearing exactly once, whose row and column sums are all equal. Sometimes the sums of the diagonals are ...

**-4**

votes

**0**answers

22 views

### Parallel Optimization (fmincon) using for loops and if-then statement [on hold]

I am trying to do parallel processing optimization (fmincon) using for loops for each a,b=0:.01:1 includes if then statement because I have a normalized condition which is a^2+b^2+c^2=1 then ...

**-4**

votes

**0**answers

135 views

### does Gorenstein imply reduced? [on hold]

Let X be a projective scheme over a field, if X is Gorenstein then must X be reduced?
The definition of Gorenstein I know is that all local rings have finite injective dimension as modules over ...

**0**

votes

**0**answers

41 views

### Extending grading of subring to entire ring

Let $R$ be a (commutative) subring of $S$, and assume that $R$ is graded by an abelian group $G$. Is there anything known, possibly under less general circumstances, about the existence/uniqueness of ...

**5**

votes

**0**answers

42 views

### Chromatic numbers for coloring-constrained graphs

I am interested in any and all articles about chromatic numbers applying to constrained colorings of a graph. For example, if a graph must be (properly) colored so that there is a 2-color path ...

**7**

votes

**0**answers

150 views

### Finding non convex functions satisfying a weak form of convexity, without the axiom of choice

If a real-valued function $f$ over reals satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other ...

**-6**

votes

**0**answers

50 views

### Precalculus math question natural logs [on hold]

How do I go about expanding this expression using the law of logs
http://i.stack.imgur.com/Bo9HA.png

**3**

votes

**0**answers

45 views

### Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...

**-5**

votes

**0**answers

50 views

### General formula for composite numbers [on hold]

I propose general formula of composite numbers, except divisible by 2 and 3:
Positive integers contained in two 2-dimensional arrays:$P1(i,j)=6i^2-1+(6i-1)(j-1)$ and $P2(i,j)=6i^2-1+(6i+1)(j-1)$ are ...

**0**

votes

**0**answers

54 views

### Conic sections in high dimensions

Can every $n$-dimensional ellipsoid be obtained as a (spherical) conic section?
This is false for generic quadrics but seems true for ellipsoid.
Does anybody have any references?

**2**

votes

**0**answers

129 views

### Some examples of non trivial principal bundles

1.Is there a nontrivial pricipal bundle $P(M,G)$, with $G$ connected, such that the total space $P$ admit a foliation such that each leaf is diffeomorphic to $M$(Not necessarily via projection ...

**0**

votes

**0**answers

33 views

### concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...