**5**

votes

**0**answers

46 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

186 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

51 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

53 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

51 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 ...

**2**

votes

**1**answer

80 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

146 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 ...

**3**

votes

**1**answer

88 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

60 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.

**5**

votes

**1**answer

132 views

### How to determine whether a power of eta function is a eigenform? [on hold]

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 ...

**0**

votes

**0**answers

81 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

639 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 ...

**14**

votes

**1**answer

367 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

301 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 ...

**5**

votes

**2**answers

236 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

145 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

40 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

34 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

122 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 ...

**10**

votes

**1**answer

162 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

**0**answers

31 views

### Probability questionss [on hold]

In a population brain volume is distributed according to the normal distribution with a mean value of 1400 cm3 and an SD of 125 cm3. What is the probability that a randomly chosen individual will have ...

**-6**

votes

**0**answers

48 views

### Probability questions [on hold]

The probability that a new drug prevents infection by a certain flu strain is 40%. What is the probability that the drug will be effective in one out of 5 exposed persons? ( can someone please answer ...

**2**

votes

**1**answer

100 views

### Is the complement of the ends of a manifold bounded?

Let $M$ be a connected manifold with precisely $k$ ends $\epsilon_1,...,\epsilon_k$. Choose a collection $(U_i)_{i=1}^k$ of pairwise disjoint open $\epsilon_i$-neighborhoods. Then I wonder how to ...

**1**

vote

**0**answers

68 views

### Which fields have no extensions of degree divisble by a fixed prime?

Let $p$ be a prime. What are the most general examples of a field $K$ such that for any finite extension $L/K$ the degree $[L:K]$ is prime to $p$?
Certainly, there are algebraically closed examples ...

**3**

votes

**0**answers

118 views

### On the relationship between two lesser-known recurrence relations

On January 2004, in his work Integer-valued polynomials on prime numbers and
logarithm power expansion, Jean-Luc Chabert showed that
\begin{equation} \left(-\frac{\ln(1-x)}{x}\right)^m = \sum_{n = ...

**6**

votes

**1**answer

153 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, ...

**1**

vote

**0**answers

41 views

### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...

**0**

votes

**0**answers

44 views

### $\mathsf{GCD}$s of random linear form

Given $a,b\in\Bbb N_{<M}$ where $M\in\Bbb N_{>\exp(18)}$ is arbitrary with $(a,b)=1$, the probability that $\mathsf{gcd}(ax_1+by_1,ax_2+by_2)=1$ where $x_1,x_2,y_1,y_2\in\Bbb N_{>\ln M}$ is ...

**-4**

votes

**0**answers

38 views

### Proving equality of the union over a family of sets? [on hold]

Click here for problem
So I know that this probably uses a two part proof where I have to prove it both ways, but I don't see how its even true in the first place. If we let x belong to the left hand ...

**2**

votes

**0**answers

51 views

### Planar triangulations for which all distinct 4-colorings consist of exactly 6 Kempe chains

Are there any internally 6-connected planar triangulations other than the icosahedron all of whose distinct 4-colorings consist of exactly 6 Kempe chains, one for each of the 6 color-pairs?
Addendum: ...

**8**

votes

**1**answer

105 views

### Spectral properties of the Laplace operator and topological properties

Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its ...

**-2**

votes

**0**answers

34 views

### Chance of throwing dices [on hold]

Let's say we have a n-sided dice. And we throw it p-times. What's the formula that shows whats the chance to get the integer k ?

**8**

votes

**1**answer

131 views

### Boolean-Valued Models: Why is $\| x=y \| \cdot \| \phi(x) \| \leq \| \phi(y) \|$?

Let $B$ be a complete Boolean algebra. Jech defines a Boolean-valued model $\mathfrak{A}$ of the language of set theory to consist of a Boolean universe $A$ and functions of two variables with values ...

**1**

vote

**0**answers

32 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 ...

**1**

vote

**0**answers

28 views

### VC dimension of infinite cones

What is the VC dimension of infinite $d$-dimensional cones? ( single cones not double).
I would say $2d + 1$ or $O(d^2)$
Does anybody have any reference or ideas?

**0**

votes

**0**answers

65 views

### Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where ...

**3**

votes

**1**answer

116 views

### An interpretation for filters of subspaces in Banach spaces

Let $X$ be a separable infinite-dimensional (real or complex) Banach space.
Call a collection $\mathcal{F}$ of closed subspaces of $X$ a filter if it is nonempty, does not contain $\{0\}$, is closed ...

**4**

votes

**1**answer

270 views

### A property of Mersenne primes

Consider the effect of $f(x)=\frac12(x-x^{-1})$ on the residues mod $p$ (plus $\infty$) of a Mersenne prime $p$. You get the following tree (example $p=7$):
$$
\begin{array}{ccccccc}
4\\
...

**8**

votes

**3**answers

311 views

### polynomials and symmetric functions

Suppose I have a polynomial function $f\in \mathbb{Z}[x_1, \dotsc, x_k],$ such that whenever $r_1, \dotsc, r_k$ are roots of a monic polynomial of degree $k$ with integer coefficients, we have $f(r_1, ...

**3**

votes

**1**answer

214 views

### The SGA1 version of Riemann's Existence Theorem is about analytic spaces. How does one relate it to topological covering spaces?

In SGA1, Theoreme 5.1 (Riemann's Existence Theorem) states:
Let $X$ be a $\mathbb{C}$-scheme locally of finite type, $X^{\operatorname{an}}$ the associated analytical space. Then the functor which ...

**4**

votes

**0**answers

43 views

### Fourier series of a Wightman field

From a proof that 2D Wightman CFT leads to a vertex algebra [1]:
Let
$$
Y(a,z):=\frac{1}{(1+z)^{2\Delta_a}}\Phi_a\left(i\frac{1-z}{1+z}\right),\quad\text{with}\quad |z|<1.
$$
Here $\Delta_a\ge 0$ ...

**10**

votes

**1**answer

306 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 ...

**7**

votes

**1**answer

269 views

### Arithmetically equivalent number fields and Langlands Program

Two (number) fields are arithmetically equivalent if their Dedekind zeta functions are the same. It is known that any two arithmetically equivalent fields are not necessarily isomorphic; Prasad ...

**1**

vote

**1**answer

358 views

### Is there a shorter proof of Fermat's Last Theorem for $n=4$ than that of infinite descent? [on hold]

Out of curiosity, i'm wondering whether there exists a shorter proof of FLT for $n=4$ with respect to the one of infinite descent ?
The Wikipedia article on this subject states that more proofs were ...

**2**

votes

**0**answers

113 views

### Smooth algebraic curves through smooth points

Does there always exist a smooth algebraic curve through any point of a smooth, projective algebraic variety (over $\mathbb{C}$)? (and through any smooth point of an arbitrary projective variety?)

**2**

votes

**1**answer

86 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 ...

**6**

votes

**1**answer

201 views

### What is known about Lie groups with positive(strictly) curvature?

If we consider $G$ a Lie group with left invariant riemannian metric its sectional curvature is nonnegative, when this metric is positive?
I thought a little about and only found $SU(2)=S³$.
In ...

**1**

vote

**0**answers

28 views

### Infinitesimally small elements in extensions of models of model-complete theories

Suppose that we have a first order language $\mathcal{L}$ that extends the language of rings. Let $T$ a be a topological $\mathcal{L}$-theory of fields in the sense of Pillay.. this means that not ...

**1**

vote

**1**answer

46 views

### Spaces of Killing spinors for different orientation

Simply put, I want to understand how a change of orientation on a Riemannian spin manifold can change the space of Killing spinors.
To be more precise:
Let $M$ be a spin manifold (i.e. the first and ...

**3**

votes

**1**answer

235 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 ...