**-7**

votes

**0**answers

35 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
Please answer it

**-9**

votes

**0**answers

36 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 please answer it

**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 ...

**4**

votes

**3**answers

636 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 ...

**1**

vote

**2**answers

271 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, ...

**-1**

votes

**1**answer

44 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 ...

**-1**

votes

**0**answers

61 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, ...

**3**

votes

**0**answers

34 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 ...

**-2**

votes

**0**answers

39 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 ...

**2**

votes

**0**answers

91 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 ...

**0**

votes

**0**answers

88 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 ...

**5**

votes

**1**answer

123 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 $\eta^k(m z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$ and $m=\frac{24}{gcd(k,24)}$. What I want to know is the cases ...

**-1**

votes

**0**answers

94 views

### Quantifier elimination - Existence of solution of a differential equation [on hold]

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 ...

**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:
$$
...

**6**

votes

**0**answers

85 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 ...

**-1**

votes

**0**answers

39 views

### Recognition of a transversal in finite group [on hold]

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

**1**answer

137 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 ...

**1**

vote

**0**answers

46 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 ...

**21**

votes

**5**answers

1k views

### History of Mathematical Notation

I would like to see a simple example which shows how mathematical notation were evolve in time and space.
Say, consider the formula
$$(x+2)^2=x^2+4{\cdot}x+4.$$
If I understand correctly, Franciscus ...

**5**

votes

**2**answers

234 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 $p_n$ ...

**1**

vote

**0**answers

13 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 $K > 0$ ...

**7**

votes

**0**answers

173 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 ...

**1**

vote

**0**answers

111 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

**0**answers

41 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}.
...

**2**

votes

**0**answers

61 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 ...

**9**

votes

**1**answer

151 views

### Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...

**5**

votes

**1**answer

90 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

141 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 ...

**4**

votes

**0**answers

152 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 ...

**3**

votes

**1**answer

142 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:
...

**1**

vote

**1**answer

60 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 ...

**17**

votes

**3**answers

446 views

### All compact surfaces $S\subseteq \mathbb{R}^3$ are rigid?

Recently I've come across a lecture in differential geometry by Fernando Codá (in portuguese!) in which he stated that the following problem is (at least, up to 2014) open:
Given ...

**0**

votes

**0**answers

14 views

### A weak topology generated by weakly $p$-summable sequences

Let $1\leq p<\infty$ and $X$ be a Banach space. $N_{p}(X)$ is to denote the subspace $\{x^{**}\in X^{**}:$ there exists a weakly $p$-summable sequence $(x_{n})_{n}$ in $X$ such that the sequence ...

**3**

votes

**1**answer

232 views

### Which commutative rings have irreducible (maximal) spectra?

Does there exist any term (or, maybe, a "description"?) for commutative unital noetherian rings such that their Jacobson ideals are prime (and so, their maximal spectra are irreducible)? What is the ...

**0**

votes

**0**answers

53 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 ...

**9**

votes

**1**answer

297 views

### Product-like structures on spheres

For $i=1,2$, let $j_i$ denote the inclusion of $S^n$ into the product $S^n \times S^n$ as the $i^{\text{th}}$ factor. I would very much like to know the answer to the following question, which seems ...

**-2**

votes

**1**answer

27 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

137 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 ...

**1**

vote

**0**answers

33 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 ...

**6**

votes

**1**answer

144 views

### A lift of the second Chern class

Let $X$ be a complex manifold (not necessarily Kahler or even compact).
For the first Chern class $c_1(E)\in H^2(X,Z)$ of a holomorphic vector
bundle $E\to X$ there is an obvious lift to $H^1(X, ...

**0**

votes

**0**answers

18 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 $A=QR$ ...

**2**

votes

**1**answer

64 views

### How are the real-space RG transformations defined?

I'm reading Shang-keng Ma's book Modern theory of critical phenomena, and I'm a bit confused as to how the real-space RG transformations are defined. Ma basically says that these transformations are ...

**22**

votes

**4**answers

592 views

### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...

**1**

vote

**0**answers

31 views

### Show existence maximal clique of order $s$ in an multigraph where each vertex is colored with a set of colors

You are given a multigraph $G$ with $n$ vertices as follows:
$V := (v_1, v_2, \dots ,v_n)$
$C := \{c_1, c_2, \dots\}$, be an infinite set of colors.
$f: V \rightarrow \mathbb{P}_{\le m}(C) $, a ...

**5**

votes

**1**answer

357 views

### realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a direct question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an ...

**1**

vote

**0**answers

105 views

### Homeomorphism of fibers of holomorphic maps

EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...

**33**

votes

**1**answer

536 views

### Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : number of groups of order $n$
Does $N(n)=n$ hold for some $n>1$ ?
I checked the OEIS-sequence https://oeis.org/A000001 as well as the squarefree numbers in the range ...

**9**

votes

**3**answers

200 views

### What characterizations of relative information are known?

Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as relative information, relative entropy, information gain or Kullback–Leibler divergence is defined ...

**6**

votes

**1**answer

59 views

### Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions

Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends ...

**0**

votes

**1**answer

139 views

### Rational maps between elliptic curves [on hold]

I am studying Silverman's "The Arithmetic of Elliptic Curves" and I got the following question:
In the first chapters he defines rational between projective varieties (see the first definition in ...