0
votes
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
10answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
107answers
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
0answers
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
81answers
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
0answers
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
2answers
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
0answers
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
1answer
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
17answers
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
51answers
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
2answers
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
1answer
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
1answer
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
2answers
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
3answers
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
10answers
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
4answers
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
2answers
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
3answers
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 $*\leftarrow X \to *$, ...

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