All Questions

9 views

Green function and translational symmetry

I have met this problem in solving the classical field theory of a scalar field with a cubic term. I am able to solve exactly each equation, given in a form of odes, but this question escapes my ...
16 views

Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
21 views

Determine line crossing corner of cuboid and POV of a camera faced towards it, based on the angles in the photo

Say you were to take a picture to a corner of a room with 90 degree angled walls and ceiling. In that picture you'd see three radial lines (the edges of the walls) starting in the same point (the ...
23 views

Basis for Witt algebra

The Witt algebra $W(n,m)$ is defined as the set of element $\{\sum f_jD_j$ such that $f_j\in A(n,m)\}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of ...
14 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ open set. Consider an $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha$$ ...
29 views

Color identical pairs and the 4-color theorem

If one could prove that every color identical pair in a 4-chromatic planar graph is separated by a cycle, would that be a proof of the 4-color theorem? Explanation: A pair of vertices $\{u, v\}$ ...
46 views

Probability of Brownian motion to have a zero in an interval

I have what should be a very simple questions for Brownian motion experts... Let $[a,b]$ be a given time interval. Let $f(x)$ be the probability that a linear Brownian motion with initial value $x$ at ...
30 views

Modal logic decidability vs first order logic

I have "simple" question . Can anyone explain me why the modal logic is decidable and first order logic is not? Whay are the difference? Many thanks Alan
39 views

Is the decomposition of kronecker product $s_\mu \otimes s_\lambda$ always unique? [on hold]

Let $s_\mu$ $s_\lambda$ be two irreducible representations of $S_n$ over $\mathbb{C}.$ Is the decomposition of kronecker product $s_\mu \otimes s_\lambda$ always unique ? That is, suppose ...
63 views

Is there a polynomial over the plane whose zero set is a half-line? [on hold]

Is there a polynomial f(x,y) whose zero set is nonnegative x's, i.e., {(x,0) | x >=0 }?
37 views

Must the coordinates of a polynomial iteration have about the same size?

The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them. Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if either it takes ...
69 views

Can infinte dimensional algebras be smooth?

Is it possible for an infinte dimensional algebra to be smooth? That is can an infinte dimensinal algebra have finite global dimension? For example is the polynomial ring $R[x_n]_{n \in \mathbb{N}}$ ...
45 views

Tensor products [on hold]

I'm currently trying to teach myself about tensors but I'm having some trouble with understanding what's going on. I've managed to come to a basic understanding of the ranks of tensors, rank n ...
103 views

Are the asymptotics of A003238 known?

Sequence A003238 of the OEIS counts rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is ...
62 views

Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...
47 views

Characterizations of the GOE/GUE family of distributions

For a random symmetric matrix of size $n\times n$, with entries drawn from a Gaussian Ensemble, the joint probability of eigenvalues can be written as: ...
107 views

I am currently an undergraduate junior. I have taken most of the standard undergraduate math courses and a few introductory graduate courses (measure theory, algebraic topology, complex analysis, ...
193 views

Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...
87 views

trick question: how to construct the centre of a given circle? [on hold]

This is a trick question I heard from a high-school teacher today: find the origin of a given circle. You can use any mathematically correct tool and method. Give me the shortest, the most elegant or ...
31 views

24 views

Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...
20 views

integral involving nth order incomplete gamma function [on hold]

\begin{eqnarray} \int_0^{\infty}\,x^{k+r+\xi-1 }\,e^{-\lambda ^{-k}\,x^k}\, \left( \Gamma\left(1+\frac{\xi }{k},\,x^k \lambda ^{-k}\right)\right)^n\,{\rm d} x\,. \end{eqnarray}
47 views

The largest size of a boolean subgraph (a hypercube) of a given graph

Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...
112 views

Chow ring of two varieties

Suppose we are given two smooth projective varieties $X$ and $Y$. Maybe this is elementary but what is the Chow ring $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$?
97 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
133 views

Number of representations of an integer as an (arbitrary) sum of products

If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but ...
74 views

endomorphisms of the Jacobian of a curve

Let $C$ be a smooth, projective curve over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, ...
12 views

Saturation of a subalgebra over the Tate-algebra inside the power series ring

Let $A$ be a discrete valuation ring and $\pi$ a uniformizer. Over $A$ we consider the Tate-algebra A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert ...
40 views

States and extremal states of quantum SU(2) and the Podleś sphere

Is there any description (preferably somehow related to the original generators) for the state space (as in C*-algebras) of quantum SU(2) and the Podleś sphere? If so (this is pushing my luck) are the ...
228 views

Ground Axiom and behaviors of continuum function

The Ground Axiom ($GA$) is the assertion that the universe of sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing $P\in W$. Is $GA$ consistent with any possible ...
57 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
48 views

What is Q2 as field extension [on hold]

I am reading the article: Stabilizers of quadratic points on the Bruhat-Tits tree of SL(2) over finite extensions of Q2. By: Terence Joseph Kivran-Swaine which can be found here: ...
47 views

$U(1)$ quotient of sphere $S^{2N-1}$ is projective space [on hold]

I am physical background,and don't have much algebra geometry knowledge. There is a statement in the textbook that '$\mathbb{C}P^{N-1}$ is the $U(1)$ quotient of sphere $S^{2N-1}$'. Could someone ...
55 views

The standard topology of a module over a noetherian local ring

Let $A$ be a noetherian local ring with maximal ideal $m$. One says that an $A$-module $X$ is discrete if for every $x\in X$, there is a natural number $n$ such that $m^n.x=0$. My question is: Given ...
140 views

Sources of Theorem drafts by the original author

When I look at first time to a theorem and I try to understand it or when I try to memorise a useful theorem I always have difficulties (I am not the only one. For example: I read a question: I always ...
104 views

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
293 views

Microlocalizing Hochschild homology

A recent paper of Ben-Zvi and Nadler gives a general formalism for understanding "dimensions" in sheaf theories. Without getting too far into details, amongst other things, this formalism allows us ...
121 views

Name and references for a “twisted” addition in a ring

This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...
56 views

is the structure sheaf of a variety with quotient singularities reflexive? [on hold]

The question is kind of contained in the title. Let me say a few more words. I am interested in varieties $X$, say over the complex numbers, with only quotient singularities. Instead of taking the de ...
41 views

The Book for ergodic theory on SFT in dimension $D>1.$

I have been unable to find a good reference for a book that study in details ergodic theory on sub shifts of finite type in dimension $D>1.$ The only reference that I got was actually a book by ...
131 views

blow up and derived category

Consider the blowup $X$ of $\mathbb{P}^2$ at a single point $p$. Then, Orlov showed that there is a semiorthogonal decomposition $D^b(X)=\langle e,O_X,O_X(1),O_X(2)\rangle$, where $O_X(i)$ is the ...
31 views

Differentiation of improper integral [on hold]

Differentiate $L$ with respect to $t$, where $L= a(t)\int _{t}^{\infty}\frac{dt}{a(t)}$, and $a$ is a function of $t$. I have seen in a paper that $\frac{d}{dt}L=\frac{\dot{a}}{a}L+1$, but I don't ...
149 views

Normal polytopes - counterexample?

An integral polytope $P$ is normal if all lattice points inside the integer dilation $kP$ can be expressed as $p_1+p_2+\dots+p_k$, where $p_i \in P$ are lattice points. I am looking for an example ...
Let $n$ be an odd composite number $(>1)$ and $p$ be an odd prime such that: $p\nmid n$, $\phi(n)\nmid (n-1)$, For some even part of $\phi(n)$ (say, $\phi(n_0)$), $\phi(n_0)|(n-1)$, ...
It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and \$n ...