All Questions
23,892 questions
5
votes
1
answer
151
views
Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 16.2k
2
votes
1
answer
191
views
Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
- 651
3
votes
1
answer
188
views
References for Bernstein-Zelevisnky classification
I am looking for references for the Bernstein-Zelevisnky classification of irreducible representations of GL$(n,F)$ in terms of cuspidal representations. In particular I would like to find something ...
- 111
5
votes
1
answer
261
views
Counter example for Hadamard Differentiability
I am having a hard time while trying to fully understand Hadamard differentiability.
I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
- 60.5k
4
votes
1
answer
196
views
(Lattice approximation) Does UV stability lead to continuum limit of a subsequence?
In the context of lattice approximation, the term "UV stability" seems to be used frequently. To me, it seems like
Uniform boundedness of the partition function in the limit where lattice ...
8
votes
1
answer
685
views
The state of the art on topological rings - the Jacobson topology
I was recently studying the Jacobson density theorem and I found it quite interesting.
Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
- 63.9k
2
votes
0
answers
55
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 420
3
votes
1
answer
327
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 16.2k
3
votes
1
answer
179
views
Analytic continuation to the Mittag-Leffler star using Mittag-Leffler summation
This is a reference request for a theorem I thought I had read in a book by Steven Krantz, but I can no longer find it.
Searching for Mittag-Leffler star, I can find references to the following result....
- 91.8k
0
votes
0
answers
85
views
Uniqueness of compatible cycle decomposition for Eulerian trail
Fleischner mentions in his article Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs about the uniqueness of compatible cycle decomposition that ...
1
vote
0
answers
46
views
Reference on polynomial attached to permutation group
Let $G$ be a permutation group acting on some set. Let $C(g)$ be the set of associated cycles of an element $g\in G$, and define $l(c)$ to be the length of a cycle $c$. Now set
$$T(G) = \sum_{g\in G}\...
0
votes
0
answers
28
views
Analyzing simple DDE with simple characteristic test
I'm wondering if anyone can comment on the stability of delay DE given that we can analyze its characteristic equation.
For instance, let's say we have the DDE $\frac{d}{dt}x(t) = x(t-a),$ where $a$ ...
- 11
43
votes
3
answers
7k
views
Could the Riemann zeta function be a solution for a known differential equation?
Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
- 3,738
0
votes
0
answers
72
views
Regularity estimates of Double Layer potential
Let $\Omega$ is a bounded open subset of $\mathbb{R}^n,n\ge 2$ with $C^{\infty}$ boundary. Define $$I\left[ \phi \right](x) := -\frac{1}{\omega_n}\int_{\partial \Omega} \frac{(x-y)\cdot \nu_y}{|x-y|^n}...
- 69
3
votes
0
answers
88
views
cubic twists of Mordell curve and their rank
Let $a$ be a non-zero integer. Consider the elliptic curve $E_a/\mathbb{Q}$ given by the equation
$$
E_a: y^2 = x^3 + a.
$$
For a cube-free integer $D$, define the elliptic curves $E_{aD^2}/\mathbb{Q}$...
- 1,283
20
votes
2
answers
7k
views
Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
- 6,367
0
votes
0
answers
78
views
What does analytic uniformly in $s$ mean?
Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
1
vote
0
answers
160
views
Where can I find Stefan Wewers' doctoral thesis "Construction of Hurwitz Spaces"?
I am looking for a copy of Stefan Wewers' doctoral thesis titled "Construction of Hurwitz Spaces," which was defended at the University of Essen in 1998. I have tried searching through ...
- 560
5
votes
1
answer
367
views
Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 12.9k
6
votes
1
answer
285
views
Distinguishing the Besov and Triebel-Lizorkin spaces
Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
- 301
3
votes
2
answers
349
views
Reference for proof about a result concerning Sobolev spaces and exponential growth
I'm reading an article and I saw the following affirmation without proof:
Let $u \in H^1(\mathbb{R}^2)$ and $\alpha>0$, then
$$\int_{\mathbb{R}^2}(e^{\alpha u^2}-1)dx<+\infty.$$
Is this claim ...
1
vote
0
answers
66
views
Sum of k vectors with largest possible norm
Suppose I have a family of $n$ vectors in $\mathbb{R}^d$:
$v_1,\dots,v_n.$ Is there a poly-time algorithm that computes a subset $S\subset [n]:=\{1,\dots,n\}$ of size $1\leq k\leq n$ for which the (...
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 33.8k
10
votes
2
answers
417
views
zeros on the circle of convergence
In this question some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging to the ...
- 4,713
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 111
4
votes
1
answer
253
views
Reference to formal approach to homotopy analysis method
I'm currently reading a book about the Homotopy Analysis Method (HAM), but it isn't very rigorous (it explains most things with a single example), which is bothering me.
I'm searching for papers where ...
- 1,826
2
votes
1
answer
49
views
Is any submetrizable linear topology linearly submetrizable?
Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$.
Is ...
- 16.4k
3
votes
1
answer
176
views
Are measurable maps with countably separated image in a Banach space always strongly measurable?
Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
- 285
83
votes
5
answers
14k
views
How to find ICM talks?
I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...
- 31
2
votes
1
answer
301
views
Best approximation with tensors of rank $\ge2$
Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ ...
CommunityBot
- 1
7
votes
4
answers
4k
views
Generating random curves with fixed length and endpoint distance
Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given?
The ...
- 63.9k
1
vote
0
answers
75
views
Banach lattice hull of a Banach space
I am interested in defining some Banach lattice properties for subsets of arbitrary Banach spaces. So it would be useful to have the notion of Banach lattice hull $E(X)$ of a Banach space $X$.
I ...
- 4,461
4
votes
2
answers
312
views
Injective hulls of metric spaces
In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
12
votes
0
answers
543
views
Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
- 63.1k
1
vote
1
answer
216
views
Flatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of $\...
- 12.9k
0
votes
1
answer
96
views
Existence of a complemented basic sequence
Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
- 31.5k
1
vote
0
answers
87
views
Convergence and sequential compactness for nonlinear operators
I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear.
What kind of notions of convergence does one have for such operators? I'm specifically ...
- 2,830
11
votes
0
answers
388
views
Inequality for symmetric polynomial functions of log concave variables
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...
- 505
6
votes
1
answer
241
views
Reference request: acceleration/curvature of curve in metric space
Let $(X,d)$ be a metric space. Given a continuous curve $\gamma_t : [0,1] \rightarrow X$, the metric speed is defined by $$ |\gamma_t^\prime | := \lim_{s\rightarrow t} \frac{d(\gamma_s, \gamma_t)}{|t-...
- 869
5
votes
1
answer
1k
views
Is this a new result about hexagon?
Let a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent:
Three lines $AA', BB', CC'$ are concurrent (let the point of ...
- 1,776
3
votes
1
answer
104
views
From Wightman to HK axioms for "non-neutral (charged?)" fields
Wightman axioms deal with operator-valued distributions (Wightman fields) whose values are unbounded operators in general.
On the other hand, the Haag-Kastler axioms deal with net of observables, ...
- 5,984
23
votes
0
answers
1k
views
Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
- 1,124
4
votes
1
answer
252
views
Show that $\Lambda_\varphi(x_n)\to \Lambda_\varphi(x)$ for an nsf weight $\varphi$ on a von Neumann algebra
Let $\varphi$ be an nsf weight on a von Neumann algebra $M$. Fix a square-integrable element $x\in \mathscr{N}_\varphi$. Put
$$x_n := \sqrt{\frac{n}{\pi}}\int_{-\infty}^{+\infty} \exp(-nt^2) \sigma_t^\...
- 2,471
13
votes
1
answer
1k
views
Gerhard Frey, "Links between stable elliptic curves and certain diophantine equations"
I am searching for the article by Gerhard Frey, which has indicated a connection between Fermat's Last Theorem and the Taniyama-Shimura Conjecture. The reference is give as
Gerhard Frey, Links ...
6
votes
1
answer
960
views
Original reference of six functor formalism?
What was the original reference of Grothendieck's six functor formalism?
I think it was "COHOMOLOGIE A SUPPORTS PROPRES par P. Deligne SGA IV" but maybe there was an earlier paper on the topic.
- 584
5
votes
1
answer
340
views
Equations for dual cubic curves
Suppose I have a cubic curve $C$ (over $\mathbb C$) in Weierstrass form $$y^2=x^3+ax+b.$$
I would like to find the degree $6$ equation for protectively dual curve $C^*$. Do you know any place where ...
- 14.3k
7
votes
2
answers
417
views
Catalogue of groups with short finite presentations
For various types of groups, there exist catalogues of those groups of the
particular type which are "small" in a certain sense. — For example:
The GAP Small Groups Library catalogizes ...
8
votes
3
answers
559
views
Reference for tetrahedral Coxeter group
Let $G$ be the group with 4 generators, each of order 2, such that the product of any 2, say $ab$, has order 3 (i.e., $ababab=e$).
That is, this is an infinite reflection group with Coxeter diagram a ...
- 156k
0
votes
0
answers
158
views
Understanding the Hilbert scheme of subvarieties of $\mathbb{CP}^n$
EDIT: migrated to MSE.
I am looking to get a more concrete understanding of the Hilbert scheme of projective subvarieties, specifically over $\mathbb{C}$, and to obtain good references on this subject....
- 1,763
5
votes
2
answers
218
views
Decomposition of symmetric powers of the fundamental representation of $\text{Sp}(2n,\mathbb{C})$
Let $(k,0,...,0)$ denote the highest weights vector of an irreducible representation of $\text{Sp}(2n,\mathbb{C})$. I read in Fulton-Harris, that this representation may be obtained as a direct ...
- 3,054