**0**

votes

**0**answers

5 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) := ...

**-6**

votes

**0**answers

17 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

**-5**

votes

**0**answers

18 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

8 views

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

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:
$$
...

**2**

votes

**0**answers

13 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

28 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 ...

**-4**

votes

**0**answers

23 views

### Guess the number of the prize & win the prize problem

There is prize in a box. The prize has a value of a positive integer between 1 and N and you are given N. To win the prize, you have to guess its value. Your goal ...

**0**

votes

**0**answers

18 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**

vote

**0**answers

18 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 $G$?

**1**

vote

**0**answers

36 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

43 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 ...

**3**

votes

**0**answers

39 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

64 views

### On the Frobenius coin problem

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

20 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

70 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 ...

**0**

votes

**2**answers

114 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

15 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

15 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 ...

**1**

vote

**0**answers

25 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

104 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

26 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

117 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

64 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 at the formula $\exists y \ Ly=f$.
The elements of the ring are of the form ...

**2**

votes

**0**answers

24 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

44 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

24 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 ...

**0**

votes

**0**answers

63 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

78 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

20 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 ...

**-5**

votes

**0**answers

127 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

40 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

36 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

132 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

47 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

43 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

48 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

122 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

30 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 ...

**1**

vote

**0**answers

47 views

### Category of equicontinuous sets of mappings

Does this category have a name? Does it have any literature?
Objects are topological vector spaces. A morphism from A to B is any equicontinuous set of linear mappings from A to B.

**4**

votes

**0**answers

75 views

### How to determine whether a power of eta function is a eigenform?

I find that it is complicated to do this from the definition. In fact, I know that $\eta^k(z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$. What I want to know is the cases that $k=5,7,11$. In addition, ...

**0**

votes

**0**answers

63 views

### The groups with nilpotent Hall $p'$ subgroup

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements.
Theorem $2$: A group of order $p^nq^m$ is solvable.
Theorem $1$ depends on ...

**4**

votes

**3**answers

590 views

### Euler's constant: irrationality and proof theory [on hold]

Let γ represent Euler's constant. Is there a real number x such that there is a proof within Zermelo-Fraenkel set theory (ZF) that x is irrational and there is also a proof within ZF that γ + x is ...

**12**

votes

**0**answers

138 views

### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction),
as defined by Grayson in Higher algebraic K-theory: II (page 219),
takes as an ...

**11**

votes

**1**answer

274 views

### Examples and Counterexamples in Commutative Algebra

There are Counterexamples in Analysis and Counterexamples in Topology. Is there any similar book for commutative algebra? I want to see some more (counter)examples for Atiyah and MacDonald's book. Let ...

**4**

votes

**2**answers

218 views

### Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that:
$$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$
with $p_n$ ...

**3**

votes

**1**answer

136 views

### Existence of nonlinear equation

How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation:
...

**-2**

votes

**1**answer

36 views

### Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [on hold]

Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative ...

**1**

vote

**0**answers

18 views

### Construction of Stein's exchangeable pair for certain dependent random variables

Given a sequence of exchangeable random variables $X_1,\ldots,X_n$, and a measurable function $g: \mathbb{R}^n \to \mathbb{R}$. Let $S_n=g(X_1,\ldots,X_n)$. Then what is a natural construction of a ...

**3**

votes

**1**answer

116 views

### Can a length n distributive lattice be embedded into Bn?

Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in this note).
Let $n$ be the length of ...