# All Questions

**0**

votes

**0**answers

25 views

### Multiplicity of a point in positive characteristic

Let $C$ be a plane curve over a field of characteristic $p>0$ and let $x\in C$ be a point. How is the multiplicity of $C$ at $x$ calculated (in char = 0 one uses partial derivatives) ?

**2**

votes

**1**answer

21 views

### existence of a special conformal mapping

Sorry I don't know how to give an appropriate title.
In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...

**0**

votes

**0**answers

41 views

### Why is $\psi(\lambda)$ a complex linear polynomial?

Let $\phi(\lambda)$ be an entire complex function with bounded modulus. Now we know that $\phi(\lambda)$ is constant.
If, however, we know that $$\phi(\lambda) = \exp[\psi(\lambda)]$$ for some entire ...

**-4**

votes

**0**answers

21 views

### How can I turn this series notation to a formula? [on hold]

I have the summation SIGMA(0 to h) of ((h+1)(2^h)).
I need a way to convert this in to a formula f(h) = x where x is the right answer. I remember doing arithmetic and geometric series' in maths in ...

**0**

votes

**0**answers

31 views

### algebraic leaves of foliation on a product of two curves

Let $S=E\times C$ be a product of two curves, where $E$ is an elliptic curve and $C$ is a curve of genus at least two. Consider a foliation on $S$ generated by a global holomorphic 1-form ...

**2**

votes

**1**answer

89 views

### Tensor product of fields over integers

Inspired by this question we ask;
Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$?:
1.A field $K$ with the property that ...

**0**

votes

**0**answers

47 views

### Tradeoff between different forms of the partitions of unity [on hold]

This is cross-posted in MSE:
In Wikipedia, a partition of unity of a topological space $X$ is a set $R$ of continuous functions from $X$ to the unit interval $[0,1]$ such that for every point, $x\in ...

**-5**

votes

**0**answers

37 views

### Recommended Books for Advanced Undergraduate Students of Mathematics [on hold]

What are the "must read" mathematics books for advanced undergrad students? I have read the satirical book "Mathematics Made Difficult", any other recommended books?

**4**

votes

**0**answers

175 views

### What is the best way to learn about Modular Forms?

I am a senior Mathematics Major, and I am interesting in learning about Modular Forms. I have a layman's general sense of what they are but I was wondering if there is a lecture(I am willing to pay) ...

**2**

votes

**0**answers

44 views

### isomorphism of Chern character in kk-theory

Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...

**3**

votes

**0**answers

41 views

### Mirror for Flag Supermanifolds

I recently learned the following two things, and I wish to know how to make them reconciled.
(1) As far as I understand, the flag manifolds serve as a tractable class of examples for the very ...

**10**

votes

**1**answer

421 views

### Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$?

Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners ...

**8**

votes

**10**answers

983 views

### What are fun elementary subjects in probability?

I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just ...

**3**

votes

**0**answers

30 views

### Lower bound on the sum of singular values for a sum of Hermitian matrices

Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that ...

**6**

votes

**0**answers

92 views

### Cap product on Leray-Serre spectral sequences

Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and ...

**-5**

votes

**0**answers

37 views

### Probability question (urgent) [on hold]

Let Y be an absolutly continuous random variable with f (PDF):
f(y)=|2e^(-2(y-1)) if y>1
|0 otherwise
(a) Set the probability distribuction and identify the median of Y.
(b) Identify the law ...

**5**

votes

**1**answer

56 views

### Real rank zero of group $C^*$-algebras

The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...

**28**

votes

**2**answers

1k views

### Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...

**4**

votes

**1**answer

188 views

### Who said “the naive counting numbers don't exhaust $\Bbb N$”?

In the context of Robinson's framework, or more precisely its reformulation by Ed Nelson, one of the practitioners in the field expressed the sentiment something like "the naive counting numbers don't ...

**9**

votes

**1**answer

131 views

### Are these two quotients of $\omega^\omega$ isomorphic?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. For $f,g\in\omega^\omega$ we say $f\simeq_{\text{fin}} g$ if there is $n\in \omega$ such that $f(k) = g(k)$ for all $k\geq n$.
...

**13**

votes

**1**answer

528 views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

**3**

votes

**2**answers

193 views

### Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...

**10**

votes

**0**answers

160 views

### Reference Request for Hilbert Schemes

I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...

**1**

vote

**0**answers

77 views

### Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$?
Is it enough to consider the injection via Hochschild:
$Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$?
In ...

**6**

votes

**1**answer

156 views

### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...

**1**

vote

**0**answers

104 views

### On power residues

Given $n\in\Bbb N$ is there an $a_n\in\Bbb N$ such that for every $a>a_n$ there are two distinct integers $0<b<c<a$ such that $b^i\bmod a,c^i\bmod a\in(\sqrt a,\sqrt a\log a)$ for every ...

**25**

votes

**0**answers

260 views

### What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...

**1**

vote

**0**answers

118 views

### universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind.
Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...

**3**

votes

**1**answer

174 views

### $K$-Theory of finite dimensional Banach algebras

Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group?
I asked this question in MSE but I received no answer
...

**2**

votes

**2**answers

174 views

### odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: ...

**0**

votes

**1**answer

916 views

### What is the modern consensus on the difficulty of infinitesimals?

At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...

**8**

votes

**1**answer

124 views

### Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?

How can I efficiently compute
$\mathrm{trace}(A(B^{-1}))$
where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of ...

**100**

votes

**107**answers

34k views

### Most memorable titles [closed]

Apparently, for a large number of readers, the choice whether they select to read a paper or not is often strongly influenced by the title.
I was wondering if the MO-users would be willing to share ...

**0**

votes

**0**answers

48 views

### Locally nilpotent derivations on noncommutative rings [on hold]

I am interested on LNDs on noncommutative rings, specially noncommutative polynomial rings.
I have found only some stuff related to the Weyl algebras and Lie algebras.
Anyone can tell me a ...

**112**

votes

**81**answers

88k 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.)

**1**

vote

**0**answers

60 views

### Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...

**3**

votes

**2**answers

550 views

+100

### Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the classical Borsuk Ulam theorem
for the Euclidean n-sphere, $n>2$ that is based only on the theory
of Banach algebras. I checked on MR but had no ...

**2**

votes

**0**answers

79 views

### Conjugacy classes of involutions in Kac-Moody groups

Let $A=(a_{s,s'})_{s,s'\in S}$ be a generalized Cartan matrix.
Let $G=G(A)$ be the corresponding simply connected complex Kac-Moody group with Cartan subgroup $H$ and Weyl group $W$ acting on $H$.
...

**3**

votes

**1**answer

94 views

### Solve SDE $dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$

I am trying to solve the following SDE
$$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$
$c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in ...

**80**

votes

**17**answers

21k views

### What recent discoveries have amateur mathematicians made?

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...

**51**

votes

**51**answers

17k views

### Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as ...

**18**

votes

**2**answers

630 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

**11**

votes

**1**answer

764 views

### Two to the power of a triangular number: bijections

The numbers $2^{n(n+1)/2}$ come up in various enumerative contexts. In addition to the trivial example (bit-strings of length $n(n+1)/2$) and the old example of domino tilings of Aztec diamonds ...

**5**

votes

**1**answer

185 views

### Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$.
In Macaulay's words, the ...

**5**

votes

**2**answers

1k views

### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...

**9**

votes

**3**answers

269 views

### Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement:
Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs
\begin{align}
P_i(\partial / \partial x_1, \dots, ...

**19**

votes

**10**answers

4k views

### Are there any good computer programs for drawing (algebraic) curves?

I realise that I lack some intuition into how a curve (or surface, or whatever) looks geometrically, from just looking at the equation. Thus, I sometimes resort to some computer program (such as ...

**11**

votes

**4**answers

4k views

### “You can't push a rope” [on hold]

"You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students. Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code ...

**70**

votes

**2**answers

6k views

### When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...

**21**

votes

**3**answers

958 views

### Homotopy groups of spheres in a $(\infty, 1)$-topos

Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces).
You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...