# All Questions

**12**

votes

**0**answers

94 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 ...

**1**

vote

**0**answers

32 views

### How close to being well-orderable does this make my powerset?

Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set $k$ satisfying $k\times k \simeq k$, and consider its powerset $X = 2^k$. I have a technical condition ...

**0**

votes

**0**answers

8 views

### Inequality for integral $\ell_p-$norms implies inequality for all $\ell_p-$norms?

Let $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ be two decreasing sequences of positive numbers such that
$$\sum_{k=1}^\infty a^p_k\leq \sum_{k=1}^\infty b^p_k<\infty$$ for all $p\in\mathbb N.$
...

**1**

vote

**0**answers

37 views

### Bockstein cohomology

There is a notion of Bockstein homomorphism $\beta$. I am interested precisly in the case of sequence $$0 \rightarrow \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p^2 \mathbb{Z} \rightarrow ...

**2**

votes

**0**answers

30 views

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

This is less of a 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 elliptic curve ...

**4**

votes

**1**answer

57 views

### non commutative polynomial which is zero for all matrix evaluation

I want to work on $K$ an algebraic closed (commutative) field of characteristic zero (even if it seems to be more general).
We can define the free K-algebra of polynomials in non commutative ...

**7**

votes

**1**answer

38 views

### On linear integer programs with infinitely many solutions

Suppose that a linear system of inequalities $Ax \le b$ with integral coefficients has an infinite number of integral solutions $x$.
Can one conclude that there is a ray containing infinitely many ...

**-3**

votes

**0**answers

19 views

### Curve with Matlab [on hold]

I have posted this question:
http://math.stackexchange.com/questions/1547373/curve-with-matlab
but I have not answers. Can you help me?

**0**

votes

**0**answers

54 views

### Proving a functional inequality

Consider two monotonically decreasing functions $f_0(x)$ and $f_1(x)$ for which the following holds:
\begin{equation}
1-t\leq f_0(t)\leq f_1(t)\leq1.
\end{equation}
Let $n,m\in\mathbb{N}$ and $m\leq ...

**0**

votes

**0**answers

54 views

### Bounded power series on the closed unit disk [on hold]

Consider the function $f(z) = \sum\frac 1{n^5+in^2-zn^5}$. Is it bounded on the closed unit disk ? (Ofcourse it is convergent on the closed unit disk.) I am looking for a function convergent on the ...

**9**

votes

**1**answer

173 views

### An inequality improvement on AMM 11145

I have asked the same question in math.stackexchange, I am reposting it here, looking for answers:
How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$:
...

**2**

votes

**0**answers

46 views

### Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$).
But what do we know about ...

**3**

votes

**0**answers

30 views

### Traces of fractional Sobolev spaces $W^{s,p}$ with $0<s<1/p$

I've stumbled upon a problem involving the trace of a function in a fractional Sobolev space of the form $W^{s,2}(H)$, where $H$ is a half-plane in $\mathbb{R}^2$. Would it be possible to define a ...

**0**

votes

**0**answers

112 views

### Who was the first to mention the nine problem? [on hold]

When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111… Later, Koenig found a way to save the diagonal argument in binary ...

**7**

votes

**2**answers

410 views

### Can all the sporadic groups be expressed as permutation groups based on a single big cycle?

Working on M11, I came up with that it can be generated using the following permutations:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
[[2, 0, 1, 7], [3, 4, 5, 6]]
[[4, 0, 6, 7], [2, 3, 1, 5]]
[[0, 7], [4, 6], ...

**-4**

votes

**0**answers

89 views

### Can the following expressions be regarded as formula of prime numbers? [on hold]

Commonly accepted opinion is that there is no any formula for prime numbers. But we propose expressions for two pairs of 2-dimensional arrays which contain indexes $p$ in the sequences $6p + 5= 5, 11, ...

**2**

votes

**0**answers

67 views

### Wreath product of an abelian group with a nilpotent group

By work of Coulbois, the wreath product of two finitely generated free abelian group is $LERF$; i.e, every finitely generated group of this wreath product is closed in the profinite topology.
Is there ...

**3**

votes

**1**answer

102 views

### When can you canonically extend an ultrafilter after forcing?

Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes.
Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...

**-6**

votes

**0**answers

62 views

### Show that $G$ a group? [on hold]

Suppose that $G$ is a semigroup , and for every $a$ in $G$ there is unique $a^*$ in $G$ that
$aa^*a=a$
Prove that $G$ is a group.

**6**

votes

**0**answers

58 views

### Inequality for the maximum of Gaussian variables

Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors
with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix
$\Gamma_Y-\Gamma_X$ is positive definite. Is it ...

**0**

votes

**1**answer

85 views

### about the horizontal lift in a principal bundle

I'm currently studying Fibre Bundle by Nakahara's book, and I'm a bit confused about the following:
Imagine we have a Principal Bundle $P(M,G)$ with open chart {$U_i$} and a local section ...

**2**

votes

**0**answers

105 views

### Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?

Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons:
i) it gives the numerals |, ||, |||,.... an ersatz ...

**0**

votes

**0**answers

29 views

### Compact factors of Lie groups; possibly varying definitions [migrated]

Let $G$ be a real connected semisimple Lie group. Are the following equivalent?:
(1) $G$ has no proper cocompact Normal subgroups.
(2) $G$ has no proper cocompact connected Normal subgroups.
In ...

**5**

votes

**1**answer

286 views

### Grothendieck's paper on principal bundles, reduction to a torus step

In Grothendieck's paper "Sur la Classification des Fibres Holomorphes sur la Sphere de Riemann", there is a step I don't understand in section 4, where he proves reduction to a torus. He states (lemma ...

**-1**

votes

**1**answer

28 views

### Convex Optimization in an Ellipsoid

Suppose we want to minimize a linear objective inside an ellipsoid that is,
$\min _x l^Tx$
such that $(x - \mu)^TA(x - \mu) \leq \beta ^2$.
Here, A is PSD and $\mu$ is a fixed vector. Can this be ...

**7**

votes

**1**answer

105 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

**11**

votes

**2**answers

213 views

### Factorization of a matrix as a product of a symmetric and a skew-symmetric matrix

When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?
For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy ...

**5**

votes

**1**answer

109 views

### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...

**1**

vote

**1**answer

141 views

### Jensen formula in $\mathbb{C}^n$?

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads
$$
\log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = ...

**0**

votes

**0**answers

89 views

### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the ...

**3**

votes

**1**answer

241 views

### K theory long exact sequence

(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset ...

**13**

votes

**0**answers

147 views

### What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...

**1**

vote

**0**answers

81 views

+50

### A path optimisation problem

Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...

**1**

vote

**0**answers

33 views

### Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).
Let $ \Omega$ denote a cube in $ R^n$ and consider ...

**3**

votes

**2**answers

207 views

### Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?

**0**

votes

**1**answer

95 views

### An increasing sequence of real numbers [closed]

This was first posted to SE, but now I think its better to be posted here.
For what positive real numbers $\alpha$, the sequence $a_n = \frac{\lfloor n\alpha\rfloor}n $ is (not necessary strictly) ...

**17**

votes

**1**answer

502 views

### Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...

**3**

votes

**2**answers

269 views

### Natural operators in differential geometry - why are they natural?

I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ...

**3**

votes

**1**answer

159 views

### Exterior derivative as only (up to multiple) natural operator $\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$

In Kolar, Michor, & Slovak's book Natural Operations in Differential Geometry, it is proved the exterior derivative is universal in the following sense.
Proposition 25.4. For $k>0$ all natural ...

**8**

votes

**1**answer

94 views

### Smooth manifolds as idempotent splitting completion

The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps.
...

**119**

votes

**26**answers

28k views

### Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...

**21**

votes

**0**answers

243 views

+50

### Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the ...

**2**

votes

**1**answer

194 views

### Conductor of a CM elliptic curve and its Grössencharacter

For a CM elliptic curve $E$ and its Grössencharakter, their conductors are both supported on bad primes of $E$. Moreover, by comparing their functional equation, there should be some obvious ...

**2**

votes

**0**answers

42 views

### Action of a Drinfeld modular group on a Drinfeld symmetric space

Let $\Bbb C_\infty$ be functional field case analog of $\Bbb C$, i.e. $\Bbb C_\infty$ is the completion of the algebraic closure of the field of Laurent series $\Bbb F_q((\theta^{-1}))$, where $q$ is ...

**111**

votes

**81**answers

84k views

### Do good math jokes exist? [closed]

Have a good joke? Share.
I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)

**151**

votes

**64**answers

26k views

### Proofs that require fundamentally new ways of thinking [closed]

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...

**13**

votes

**2**answers

400 views

### What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?

Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...

**5**

votes

**1**answer

113 views

### For a simplicial set $X$, is the category of non-degenerate simplices of $X$ a full subcategory of the category of simplices of $X$?

I'm slightly confused, but I think you can help me. Let $X$ be a simplicial set. The category of simplices of $X$ and its subcategory of non-degenerate simplices are defined at ...

**101**

votes

**40**answers

11k views

### Generalizing a problem to make it easier

One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you ...

**16**

votes

**3**answers

2k views

### How did Ramanujan discover this identity?

Let $$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and
$ad=bc$, then
$$64*F_6*F_{10}=45*F_8^2$$
This fascinating identity is due to Ramanujan and can be found in ...