Newest Questions
159,026 questions
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Book on analysis and algebra at the undergraduate level [closed]
I am writing this post because I would like to know what are your references concerning math book showing the interplay between analysis and algebra at an undergraduate-advanced undergraduate level.
...
1
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1
answer
96
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Number of ergodic transverse measures for geodesic laminations - bounded by the genus?
Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...
1
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0
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105
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Question about ergodic flows and periodicity
Let $X$ be a compact Haussdorf space, let $\mu$ be a Borel measure on $X$ with $\mathrm{supp}(\mu)=X$ and let $(\phi_s)_{s\in\mathbb R}$ be a one-parameter group of homeomorphisms which is
continuous ...
- 769
1
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1
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119
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weakly separated sequences in RKHS are separated by Gleason metric
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
- 151
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127
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Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
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92
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Explicit CW-complex replacement of the space of reparametrization maps
Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
- 3,901
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4
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929
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Interesting examples of systems of linear differential equations with constant coefficients
In this paper, Gian-Carlo Rota wrote:
A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal
processing, even in ...
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0
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104
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Literature for noncommutative birational invariants
Let $k$ be an algebraically closed field of zero characteristic. All fields under discussion are fields over $k$, and all division rings are division algebras over $k$.
There is rich theory of ...
- 3,318
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98
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Isomorphism and counting for tree quivers
Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of ...
- 26.5k
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184
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Question about $n$ random points in a regular polygon, and a limiting probability
Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...
- 3,567
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113
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Lengths and additive invariants which preserve positivity
The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
- 64k
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155
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Universal cover of the configuration space of points on surface
Let $S$ be a closed oriented surface and $C(S, n)$ be the configuration space of $n$ points on $S$, i.e., the space of $n$-tuples of distinct points of $S$ with the topology induced from $S^n$. Let $V ...
- 353
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94
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When can an affine functional on the dual be represented as an element of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
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105
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An open problem about simple Noetherian rings
The following is a well-known open problem in ring theory (see, for instance, Goodearl, Warfield, 'An introduction to noncommutative Noetherian rings, Appendix, Problem 19)
Question: Let $R$ be a left ...
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2
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2
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432
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Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
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281
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Does there exist a faithful exact embedding of $D^b(\dim(N)) \to D^b(\dim(N-1))$
$\DeclareMathOperator\Hom{Hom}$I am trying to show that if $X,Y$ are nice schemes with $\dim(X) > \dim(Y)$ there is no faithful FM transform $\Phi_{K}: D^b(X) \to D^b(Y)$.
Does someone have a proof ...
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229
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Abscissa of convergence of the $\tau$ Dirichlet series
Define the $\tau$ Dirichlet series $L$ by
$$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$
where $\tau$ is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$....
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1
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167
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Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
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208
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Does this pseudo-holomorphic triangle contribute to the product $\mu_2$ in Lagrangian Floer cohomology?
I'm computing the product map $$\mu_2 : CF(L_0,V)\otimes CF(V,L_1)\to CF(L_0,L_1)$$ in Seidel's exact triangle for this specific case:
This is a genus 2 surface, and I color-coded the three (...
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1
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201
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In a weak factorization system, the left class is left cancellative iff the right class is what?
Let $\mathcal K$ be a locally presentable category, and let $(\mathcal L, \mathcal R)$ be a cofibrantly-generated weak factorization system thereon. We say that $\mathcal L$ is left cancellative if $...
- 64k
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1
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253
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Prime divisors of $p^n-1$, primitive prime divisors
Let $p,q,t_1,t_2$ be distinct prime numbers and let
$$k=\frac{p^{qt_1t_2}-1}{p^q-1}.$$
Suppose that $\gcd(k,qt_1t_2)=1$. Is there any reason that $k$ is divisible by at least $7$ distinct prime ...
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75
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Optimal perfect matchings in magic squares
Question:
what is known about minimum/maximum weight perfect matchings in magic squares with or without special properties like e.g. being pandiagonal?
I am especially interested minimal/maximal ...
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126
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On 'special' points on uniform planar convex regions defined in terms of moment of inertia
The following can be easily proved using perpendicular axes theorem and intermediate value theorem:
Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...
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148
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conjecture for general form of minimax estimator
I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
- 6,541
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1
answer
142
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Does every proximal dynamical system have zero topological entropy?
A dynamical system is proximal if $$\:\forall (x,y) \in X \times X, \: \liminf_{n \rightarrow \infty} d(f^{n}(x),f^{n}(y)) = 0 $$ (where $X$ is a compact metric space with metric $d$). Is it true that ...
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1
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301
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Uniqueness of the $J$ invariant
It seems that
The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that
$$J(e^{2\pi i/3})...
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2
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347
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Are there $2^{\aleph_0}$ pairwise non-isomorphic countable groups containing every finite group?
Let us call a group $(G,\cdot)$ finitarily complete if $G$ is countable, and every finite group is isomorphic to a subgroup of $(G,\cdot)$.
Is there a collection of $2^{\aleph_0}$ pairwise non-...
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2
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283
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Regularity of solution of $(-\Delta + w)f = 0$
I am studying the following Schrödinger equation:
$$(-\Delta + w)f = 0$$
which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
- 1,305
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1
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395
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Two dimensional perfect sets
Consider the following family of sets
$$ \begin{align*}
\mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
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73
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The $n$-th reproducing kernel of orthogonal polynomial
Let $N$ be a non negative integer. Define the sequence of monic orthogonal polynomials $\{P_n(x)\}_{n}$ with respect to the inner product
$$
\langle f , g\rangle =\sum^{N}_{k=0}{f(k)g(k)\rho(k)}
$$
...
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3
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1
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193
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Transitivity axiom for a Grothendieck Topology
I am currently trying to define a Grothendieck Topology on the category Prob which consists of finite probability spaces with measure preserving maps between them.
I declared the covering sieves of an ...
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235
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Injection of Catalan objects into 3-connected planar graphs
Let $C_n = \frac{1}{n+1}\binom{2n}{n}$ be the $n$-th Catalan number, counting, for example, the number of (rooted) triangulations of the $(n+2)$-gon.
Let $P_n$ be the number of three-connected planar ...
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93
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Inflation-restrction sequence for maximal $S$-ramified extension
Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension.
There is a inflation-restriction exact sequence,
$0\to H^1(Gak(L/K), ...
- 1,541
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0
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123
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Contracting a family of rational curves in a Calabi Yau threefold
Suppose we have a Calabi-Yau 3-fold $X$ (not necessarily compact, over $\mathbb{C}$) that contains a ruled surface over a smooth curve $C$ of genus $g$. I am using a strong definition of a ruled ...
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1
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198
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Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces
I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
- 6,025
23
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2
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967
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Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?
Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
- 1,170
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0
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277
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Using the von Neumann crossed product to introduce a measure on the orbit space?
Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space).
Question: is there a natural way of using the ...
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Generalization of category algebra
Let $R$ be a commutative ring. Let $\mathcal C$ be a category that has finitely many objects. The category algebra $R[\mathcal C]$ of $\mathcal C$ consists of finite sums $\sum a_i f_i$, where $f_i$ ...
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512
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Why should we study the total complex?
Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
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Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?
In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
- 492
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81
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Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$
Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
- 4,547
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348
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If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?
Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that
$C$ does not separate our surface $Q$ into two connected regions and ...
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89
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What happens if I take a doubly-free simplicial abelian group?
Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
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2
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541
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When are two elliptic curves with zero j invariant isogenous?
Consider elliptic curves of the form $E_B\colon y^2=x^3+B$ for $B\in\mathbb Q$. These are exactly the elliptic curves with zero $j$-invariant. I would like to know when are two elliptic curves $E_B$ ...
- 89
2
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1
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355
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Families of Galois representations over disks
Edit on Nov. 20, 2023. This question is answered below in the case that $0<r_i<1$. And indeed it is shown in the answers to not be an interesting question in that case. So please take all $r_i=1$...
anon
2
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2
answers
297
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Density of subsequences in Bolzano-Weierstrass
Let $(M, d)$ be a metric space and $K$ compact. It is known that $K$ is sequentially compact, so we can "run" Bolzano-Weierstrass on it.
I want to identify the set $\mathcal{F}$ of all ...
- 83
2
votes
1
answer
271
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Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
- 279
1
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1
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172
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Ratio of the constants of the Marcinkiewicz–Zygmund inequality for p=1
The Marcinkiewicz–Zygmund inequality states that
$$
{\displaystyle A_{p}E\left(\left(\sum _{i=1}^{n}\left\vert X_{i}\right\vert ^{2}\right)_{}^{p/2}\right)\leq E\left(\left\vert \sum _{i=1}^{n}X_{i}\...
- 13
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0
answers
303
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Connections in non-commutative geometry
Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
- 985
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0
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96
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Can 2 coverages generate the same Grothendieck Topology if the category is large?
I am currently analyzing a category which is not small, but locally small. I have seen that any coverage on any small category $\mathcal{C}$ generates a unique Grothendieck Topology on $\mathcal{C}$ ...
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