0
votes
0answers
2 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
3
votes
0answers
23 views

Automorphism groups for free groups with action

Let $A$ be a (finitely generated) free group and $G$ be a (finitely generated) free $A$-group - that is, a group with an action of $A$, which is free in the category of groups with an $A$-action. ...
0
votes
0answers
23 views

An equality for the dimension of the sum of subspaces (in the non-degenerate case)

This post is a sequel of An inequality for the dimension of the sum of subspaces, inspired by this famous answer on $\dim(U+V+W)$. The inequality $$\dim(\sum_{i = 1}^{n} U_i) \le \sum_{r=1}^{n} ...
0
votes
0answers
37 views

arithmetic progressions with few primes

Is this true ? Let $\beta_0$ be a positive number. One may find $\beta>\beta_0$, $0<\lambda<1$, and infinitely many $q>1$ so that there exists an arithmetic progression of step $q$, $a_1, ...
1
vote
1answer
25 views

Runge-Kutta convergence

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
2
votes
0answers
19 views

Are functions whose partial derivatives are simple functions dense in $W^{1,\infty}$?

In a 2D domain, are the functions whose partial derivatives are simple functions dense in $W^{1,\infty}$ ?
1
vote
0answers
25 views

Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$, what is the probability that given $A,B,C\in\Bbb N$ with $\mathsf{GCD}(A,B,C)=1$ such that $N^{a}(\log N)\leq A,B,C\leq N^{a}(\log ...
0
votes
0answers
18 views

A question about tensor product of algebra group

By embedding the algebraic tensor product L1(G)⊙L1(G) into L1(G×G) by mapping an elementary tensor [f]⊙[g] to [(x,y)↦f(x)g(y)] — where [⋅] denotes the taking of an equivalence class — and then ...
0
votes
0answers
9 views

Decompose a multivariate polynomial into a permutation on $F_{2^n}$ and an affine transformation

Let $S(x_1,...,x_n)=(y_1,...,y_n)$ be a secret permutation on $F_{2^n}$. $L$ is a secret $F_{2^n}\rightarrow R^{m}$ affine tranformation. $m$ can be smaller than $n$, while $n$ is ususally less than ...
0
votes
0answers
36 views

A question about tensor product

For every $f, g$ in $L^1(G)$, we know the function $[(x,y)↦f(x)g(y)]$ belongs to $L^1(G\times G)$ where here $\times$ means cartesian product. Why are these functions dense in $L^1(G\times G)$?
1
vote
0answers
64 views

Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture. We are looking for Weierstrass form and map from it of the genus one curve: $$ y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0 $$ It is ...
-3
votes
0answers
44 views

I would like to study Industrial Mathematics but needs to know it importance for project managers and the the development of third world countries [on hold]

Key Importance of Industrial Mathematics in the development of third world countries? Why industrial Mathematics important for Project Managers?
3
votes
1answer
64 views
2
votes
0answers
35 views

Separating pure states on the $2\times 2$ matrix algebra

Let $\mathcal{A}$ be the C*-algebra of $2\times 2$ complex matrices, let $\mathcal{B}$ be the C*-subalgebra of $2\times 2$ diagonal matrices, and let $v$ and $w$ be the unit vectors ...
1
vote
1answer
45 views

Elaboration of a certain section of a paper by Thanigasalam

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...
2
votes
0answers
29 views

Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...
6
votes
1answer
81 views

Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...
0
votes
0answers
22 views

Rearrangement of a spherical harmonics expansion

Referring to this article: http://i.stack.imgur.com/sfQ1C.png and http://i.stack.imgur.com/LelKb.png How is it that they get from equation 2 to equation 3? Whenever I do it, I get only cosine ...
1
vote
0answers
27 views

Calculations about the normal bundle of embedding of symmetric products

Suppose $C^{(d)}$ is the $d$-th symmetric product of a curve, embed it in $C^{(d+s)}$ by $E\to E+sP_0$ where $P_0$ is a fixed point. The normal bundle of this embedding is denoted by $N$. Suppose ...
0
votes
0answers
32 views

Crossing all boundaries on a map? [on hold]

In a variation on the traveling salesman problem, is there an algorithm (an approximate heuristic is fine) that finds a short, if not the shortest, path that crosses all boundaries between each pair ...
0
votes
1answer
86 views

Non-finitely generated, non-projective flat module, over a polynomial ring

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat ...
3
votes
0answers
88 views

Generation of cohomology of graded algebras

Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...
4
votes
1answer
87 views

Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 0}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
2
votes
2answers
105 views

What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?
0
votes
2answers
216 views

Recent progress on the busy beaver problem? [on hold]

Has there been any progress on the Busy beaver problem in the last few years? It seems like there hasn't been much work done on the problem since 2010. Is there anything amateurs can do to solve the ...
6
votes
0answers
100 views

Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$ In other words, for every finite simple nonabelian group $G$, do there exist ...
-3
votes
0answers
47 views

Numbers, multiplication and subtraction [on hold]

given a and b find c and d such that bc-ad is least and greater than zero? Also a,b,c,d are integers and all lie inside a given range i.e. [0, n]. For example if n=50, a=48 and b=49, then c=49 and ...
0
votes
0answers
65 views

Universal property of complete linear systems

Let $X$ be a projective scheme over a field $k$ and $S$ a $k$-scheme. Fix a closed immersion $i:X \to \mathbb{P}^n$ for some $n$ and denote by $\mathcal{O}_X(1):=i^*\mathcal{O}_{\mathbb{P}^n}(1)$. Let ...
6
votes
1answer
83 views

Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$. What is its $\zeta$-regularized determinant? This should be ...
8
votes
1answer
289 views

Intuition behind the Morse inequalities?

Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?
0
votes
0answers
12 views

Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients. ...
2
votes
0answers
44 views

Permutation equivalence classes with kendall-tau distance

I asked this question on Stack Exchange a few days ago with no help. I am not sure if the question seems too trivial or of not enough general interest to get any attention. Anyway, any sort of ...
4
votes
1answer
390 views

Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
5
votes
1answer
215 views

Relation between moduli spaces and classifying spaces

I hope this question is suitable to be posted here on MO. I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider ...
4
votes
1answer
108 views

Number of bases of a matroid

I would like to know the minimum number of bases of a matroid of rank $k$ and $n$ elements, knowing that each singleton is independent. At least for small ranks.
1
vote
1answer
39 views

Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed. In particular it is said that the poles form a deformed lattice that ...
2
votes
1answer
47 views

Maximal opening angle of a polygon from a point [on hold]

I'm looking for an algorithm that given a 2D convex polygon and point outside it, returns the two points of the polygon which are the two extremities of the polygon when viewed from that point. One ...
0
votes
0answers
36 views

Norm inequalities between difference operators

Assume $(v_i)$ to be a sequence in $\ell_\infty(\mathbb{R})$ for $i=1,\dotsc,N$. Define the difference operator as $\Delta v_i:=v_{i+1}-v_i$ and $\Delta^n v_i:=\Delta(\Delta^{n-1} v_i)$. Then, how can ...
3
votes
0answers
46 views

Distribution of $\sin(X) *\cos(Y)$ where $X,Y$ are iid r.v., uniformly distributed on $[0, 2 \pi]$ [migrated]

What is the probability density of $R = \sin(X) * \cos(Y)$ where $X,Y$ are independent random variables, uniformly distributed on $[0, 2 \pi]$? I am stuck with complicated integrals, not sure if ...
1
vote
0answers
72 views

Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads: If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...
0
votes
0answers
26 views

convergence of empirical distribution of random vectors

Given (a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal entries $A_{ij}\sim \mathcal N(0, 1/n)$; and (b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...
-4
votes
0answers
33 views

When does equality occur in the triangle inequality in metric space? [on hold]

When I think of R^n , n<=3 ; it is very easy given the usual metric. But what if the metric is not usual?
4
votes
0answers
125 views

Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...
0
votes
2answers
44 views

Pairwise distance distribution for point clouds (normal distribution) [on hold]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$). My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...
1
vote
0answers
93 views

Flat cohomology of an ordinary liftable Calabi-Yau threefold

Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and ...
1
vote
0answers
34 views

biharmonic equation with L^1 data and Navier Condition

I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
2
votes
1answer
162 views

Counting function for prime pair with bounded gaps between them

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible. Yitang Zhang breakthrough result established that ...
3
votes
0answers
83 views

How many different sums of parts of a vector

I hope this question isn't too basic for MO. I also asked it on math.se previously. This mathematical puzzle was given to me by a friend a while ago and I can't work out how to solve it. Does anyone ...
2
votes
0answers
84 views

Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$

I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...
-2
votes
0answers
81 views

A Game theory problem about two killers and two citizens? [on hold]

Two killers and two citizens, killers know the identities of others, citizens don't know the identities of others. Each guy will vote for a guy to be the killer in order (randomly predefined). If all ...

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