23
votes
7answers
3k views

Spectral graph theory: Interpretability of eigenvalues and -vectors

I thought "Wow!" when I learned that the eigenvector of the adjacency matrix of a cycle graph $C_n$ corresponding to the second largest eigenvalue gives the coordinates of the vertices when equally ...
6
votes
1answer
1k views

why isn't the mobius band an algebraic line bundle?

When I hear the phrase "line bundle" the first thing that pops into my head is a mobius band. But this is a bad picture from an algebraic point of view since any line bundle on an affine variety is ...
17
votes
9answers
4k views

What are the reasons for considering rings without identity?

I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
73
votes
10answers
8k views

Why is the Gamma function shifted from the factorial by 1?

I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial ...
17
votes
2answers
1k views

How do you axiomatize topology via nets?

Let $X$ be a set and let ${\mathcal N}$ be a collection of nets on $X.$ I've been told by several different people that ${\mathcal N}$ is the collection of convergent nets on $X$ with respect to some ...
9
votes
1answer
451 views

Conditions for “bootstrapping” a smooth DM stack?

In the preprint "Smooth toric DM stacks", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps: 1) start ...
16
votes
7answers
1k views

Extensional theorems mostly used intensionally

Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts: $$ \int_a^b f(x)g'(x)ds = ...
14
votes
7answers
1k views

To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?

This is probably a very elementary question in symplectic geometry, a subject I've picked up by osmosis rather than ever really learning. Suppose I have a symplectic manifold $M$. I believe that a ...
9
votes
3answers
1k views

What is “restriction of scalars” for a torus?

I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am ...
104
votes
68answers
20k views

Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
14
votes
8answers
3k views

What is the “right” definition of a ring?

This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object ...
28
votes
11answers
5k views

Why is the exterior algebra so ubiquitous?

The exterior algebra of a vector space V seems to appear all over the place, such as in the definition of the cross product and determinant, the description of the Grassmannian as a variety, the ...
5
votes
2answers
1k views

What is the Theorem of the Cube?

What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?
5
votes
1answer
382 views

Is there a good version of Artin-Wedderburn for semisimple algebra objects?

Artin-Wederburn says that if you have a semisimple algebra then it is a product of matrix algebras over division rings. Suppose that $C$ is a fusion category over the complex numbers (if you want to ...
12
votes
6answers
2k views

Is a quotient of a reductive group reductive?

Is a quotient of a reductive group reductive? Edit [Pete L. Clark]: As Minhyong Kim points out below, a more precise statement of the question is: Is the quotient of a reductive linear group by a ...
0
votes
0answers
80 views

undergraduate research [on hold]

I am a undergraduate student from Sri Lanka.I'm following a maths special degree programme.I have to submit a research project.I chose the area,Linear programming/Differential equation.So can you ...
0
votes
0answers
33 views

Generating alternating cycles on a perfect matching

Given a perfect matching $M$ in a regular bipartite graph $G$, is there an efficient algorithm to randomly generate self-avoiding alternating cycles with uniform distribution? Ideally, such an ...
0
votes
0answers
54 views

Restriction of derivations on $C^\infty(X)$

In 'Kriegl, Michor - A convenient setting for global infintite-dimensional analysis', they say that for an element $x$ in a convenient (i.e. Mackey-complete locally convex) space $X$, a bounded ...
-8
votes
1answer
145 views

Proof of a cubic equation problem [on hold]

Well I was doing some questions and i found something. This equation $x^3+y^3+z^3=w^3$ has only one solution which is $x=3,y=4,z=5,w=6$. And what I have have proposed is that there is not other ...
0
votes
0answers
53 views

What does deg=0 imply for the gauss map?

I am facing the following problem. I have a Riemannian manifold $(M,g)$ with gauss curvature zero, an isometric immersion $v:M\rightarrow \mathbb{R}^3$ that is $C^{1,\alpha}$ and I consider the Gauss ...
0
votes
0answers
148 views
+50

A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows; Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...
2
votes
1answer
319 views

Do hom-sets really live in the category Set?

This isn't really a research-level question (sorry!), but I asked on math.se (link), and though the question was upvoted a few times, I didn't get any answers. So since there may well be more ...
0
votes
0answers
33 views

Integral over a point process. Asymptotic of the dispersion

I consider an integral (or a sum with random index) $$ M(t) =\int\limits_0^t f(t-u)dX(u), $$ where $$ X(u) = \sum\limits_{i=1}^{N(u)} \xi_i,\qquad N(u)=\max\{k: \tau_1+\,\dots,\,\tau_k\, <\, u\}, ...
0
votes
0answers
94 views

A linear operator on $M_{n}(\mathbb{R})$ [on hold]

In this question $O(n)$ is the orthonormal group which is equpied with a unique Haar meaure. We define a linear map $T$ on $M_{n}(\mathbb{R})$ with $$T(A)=\int_{O(n)} (g^{-1}Ag)dg$$ What is the ...
1
vote
0answers
57 views

A reasonable framework to study properties of operator $A \mapsto KAK$ on Banach space

Let $K$ be a continuous linear operator on $C[0,1]$ (more, precisely, it is a linear integral operator). Then $K$ defines a continous linear operator $\widehat K$ on $\mathcal L(C[0,1])$ by the rule ...
-3
votes
0answers
48 views

Radius of convergence of power series [on hold]

Suppose that $f(z)=\sum_{k=0}^{\infty}c_{k}z^{k}$ is a power series with complex coefficients, such that there exists $M>0$ and $A\in (0, 1)$ with the property that $|c_{k}|\le M\cdot ...
1
vote
0answers
66 views

continuous homomorphism with open image in a product topological space

I originally posted the problem below in MathSE, but I deleted it since it is not receiving any attention. So I decided to transfer the question here instead. Let $G$ be a profinite group and $I$ ...
1
vote
0answers
50 views

Invariance of the Noether charge

The paper http://epubs.siam.org/doi/abs/10.1137/1023098 (Generalizations of Noether’s Theorem in Classical Mechanics, by Willy Sarlet and Frans Cantrijn) mentions "an interesting property of the ...
2
votes
1answer
86 views

Matching power series to infinity

As pointed out by Makoto, on this question about power series rings and the axiom of choice, an idea I had needed the axiom of dependent choice to work. However, the construction raises another ...
4
votes
0answers
275 views

Consequences of ZF+“all subsets of reals are Lebesgue measurable”

(I'm not sure if this is entirely suitable here so feel free to close it if it's not.) The statement "there is a Lebesgue measure on $\mathbb{R}$($2^\omega$)" means: there is a total $\sigma$-additive ...
-3
votes
0answers
48 views

Increasing order [on hold]

It's been a long time (20 years) since I took any math class. In High School I made it as far as Algebra II with Basic Trig I'm currently taking an online algorithm course. I have reasoned that the ...
-4
votes
0answers
50 views

Notation in groups/rings/fields [on hold]

I'm reading about Imaginary Quadratic Fields, and I came across this notation Z/nZ, but i don't know what it means. What does \mathbb{Z}/n\mathbb{Z} mean?
3
votes
0answers
100 views

a game with generic filters

The present question is a follow-up to this one. Assume GCH holds in $V$ and suppose $G \subseteq Add(\omega_1,\omega_2)$ is generic over $V$. For any $g \subseteq Add(\omega_1,\alpha)$ in $V[G]$ ...
12
votes
7answers
612 views

Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...
1
vote
1answer
277 views

Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
0
votes
1answer
82 views

Graph lifts and representation theory

Is there any connection known between the two? One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...
2
votes
1answer
30 views

Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem. The description of my problem is ...
0
votes
0answers
16 views

Column Subset Selection implementations

Are there readily available implementations of algorithms for the CSSP - Column Subset Selection Problem?
3
votes
0answers
40 views

Highest (short) roots and commutation relations in (twisted) DAHA

I am trying to understand, explicitly, the commutation relation between $X_\vartheta$ and $Y_\vartheta = T_0T_{s_\vartheta}$ in the (twisted) DAHA for a root system $R$, where $\vartheta$ is the ...
2
votes
1answer
77 views

Proper domain for operators

in this paper on arxiv in equation 27, two operators $$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$ and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + ...
0
votes
0answers
20 views

Moments in the Quantile Process

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
0
votes
0answers
68 views

Discrete measures and discrete kernels

This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence ...
1
vote
0answers
56 views

About Theorem $3.1.3$ in Kubota's book: Elementary theory of eisenstein series

My question is about the proof of Theorem $3.1.3$ given in kubota's book, which shows how the function $\varphi(s)$ appearing in the Fourier expansion of eisenstein series can be continued ...
3
votes
1answer
139 views

Verlinde Formula and Theta Function Identities

The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just ...
1
vote
1answer
136 views

rational point of a curve [on hold]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...
1
vote
0answers
113 views

Integration in C^* algebra

Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a continuous family of its automorphisms. Is it true that $$ \int d s \, f(s)\, \alpha_s(A) $$ is well defined as a ...
6
votes
2answers
168 views

Decidability of diophantine equation in a theory

Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows: Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether ...
2
votes
2answers
189 views

Rigid curves, and the “richness” of their homology class

Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H_2(X,\mathbb Z).$$ I have a very vague question about this situation: Q. ...
0
votes
1answer
123 views

A question on the Lebesgue differentiation theorem

In the paper [Jessen, B., Marcinkiewicz, J., and Zygmund, A. Note on the differentiability of multiple integrals. Fundamenta Mathematicae 25.1 (1935): 217-234] it is considered the limit $$ ...
9
votes
2answers
201 views

Is there a compendium of the consistency strength between the most important formal theories?

Preliminar Notions: A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...

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