All Questions
153,436
questions
6
votes
1
answer
1k
views
Classification of compact connected abelian groups
It is known that torsion-free compact abelian groups are exactly the product of the maximal solenoid group $\Sigma_{(2,3,\cdots)}$ (which is the Pontryagin dual of the additive group $\mathbb{Q}$ of ...
- 1,576
7
votes
0
answers
475
views
The universal property of composition of morphisms
$\def\K{\mathcal K}$
Preamble.
Given a locally small category $\mathcal K$, its "composition law" is a class of maps
$$
c_{abc} : \K(a,b)\times\K(b,c)\to \K(a,c)
$$
with the universal property of an ...
- 13.2k
3
votes
1
answer
357
views
Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
1
vote
2
answers
318
views
Differential construction of mixed Hodge structure on smooth open varieties
Let $\bar{X}$ be a complete smooth variety over $\mathbb{C}$ and $D$ be a simple normal crossing divisor. Denote $X:=\bar{X}\backslash D$. Then it is known that $H^\ast(X,\mathbb{C})$ admits a ...
- 1,072
10
votes
2
answers
909
views
Expressions for the inverse function of $f(x) = \ln(x)e^x$
Can the inverse of $ \ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function ...
- 101
7
votes
0
answers
229
views
GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?
GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
- 10.3k
3
votes
3
answers
799
views
Approximation of half-integers modified Bessel function of the second kind
I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian
its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ...
- 131
6
votes
2
answers
237
views
Generalization of Bieberbach's second theorem
Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...
- 419
2
votes
1
answer
110
views
$M$ is a manifold and isometrically embedded in $X$, homotopy type of $M$ is determined by polyhedrons $P$ s.t. $M\subseteq P \subseteq X$?
This is the setting.
$M$ is a compact, connected Riemannian manifold without boundary. and it is isometrically embedded in some larger metric space $X$ ($X$ is not necessarily manifold). So, one can ...
- 449
3
votes
1
answer
320
views
Triangular conjecture (that implies the Frankl conjecture)
Let $M$ be a $n\times n$ triangular matrix, that entries are $0$ and $1$ , and such that diagonal entries are $1$.
A row or a column will be said to be small, if its number of $1$s is at most $(n+1)/...
- 469
5
votes
0
answers
255
views
When can one expect that the $\mu$-invariant of a $\mathbb{Z}_p$-extension of a number field is zero?
What is special about $\mathbb{Z}_p$-extensions which are motivic to ensure that their $\mu$ invariant is zero? Is there a simple conceptual reason.
Here are some examples.
Let $F$ be a totally real ...
user130124
11
votes
1
answer
1k
views
Maximal inequality for the average of i.i.d. random variables
Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like ...
- 485
2
votes
0
answers
132
views
Which countable discrete groups have a metrisable group compactification?
Let $G$ be a countable discrete group. A group compactification of $G$ is a compact Hausdorff topological group $H$ such that there is a group homomorphism $\iota\colon G\to H$ with dense image. For ...
- 1,588
8
votes
1
answer
491
views
Maps which are both completely positive and positive
Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...
- 42.5k
2
votes
1
answer
184
views
Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder
I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions
$$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$
$$\int_{\mathbb{R}^+} f \in \mathbb{R}^+$$
...
- 21
24
votes
3
answers
1k
views
A problem on Gauss--Bonnet formula
While teaching a course in differential geometry, I came up with the following problem, which I think is cool.
Assume $\gamma$ is a closed geodesic on a sphere $\Sigma$ with positive Gauss curvature.
...
- 43.7k
7
votes
0
answers
139
views
Algebras for the codensity monad
Given a functor $j : A \to B$, when this extension exists we call $T_j = Ran_jj$ the codensity monad of $j$.
I was wondering if there's a general rule to guess the shape of $Alg(T_j)$ given $j$.
- 13.2k
15
votes
1
answer
416
views
Does a complex Fano manifold have simplicial Mori cone when all extremal contractions are fiber type?
Let $X$ be a complex Fano manifold such that each extremal ray of $\overline{\text{NE}(X)}_{\mathbb{R}}$ is generated by a primitive class in $H_2(X;\mathbb{Z})$ of a free rational curve. Thus, the ...
- 4,091
4
votes
2
answers
479
views
Distance between primes that are quadratic residues modulo an other prime
Question: Is there an infinite sequence of primes $\{q_i\}_{i=1}^{\infty}$ that is not too sparse ( $q_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ ...
- 3,004
1
vote
1
answer
144
views
Is there a transient graph whose spectral dimension two?
Let $G = (V(G), E(G))$ be an infinite connected simple graph.
Let $((S_n)_n, (P^x)_{x \in V(G)})$ be the simple random walk on $G$.
Let $p_n (x,y) = P^x (S_n = y)$.
A spectral dimension of $G$ is ...
- 195
3
votes
0
answers
71
views
Deformations of nilpotent parts of superalgebras
I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598
After ...
- 95
6
votes
2
answers
527
views
Is the underlying vector space of the minimal model of an $A_{\infty}$-algebra canonical?
On the page 4 of these notes it is stated that an $A_{\infty}$-algebra $A$ is necessarily is quasi-isomorphic to an $A_{\infty}$-algebra $HA$ with trivial differential. Moreover, $HA$ is unique up to ...
- 305
3
votes
3
answers
210
views
If $X$ has the fixed point property, what about $\text{Cont}(X,X)$?
If $(X,\tau)$ is a topological space, we denote by $\text{Cont}(X,X)$ the collection of all continous functions $f:X\to X$. We say that $(X,\tau)$ has the fixed point property if for any $f\in\text{...
- 45.5k
1
vote
1
answer
136
views
How much can a map $R^n\to R^n$, $R$ a DVR, increase the valuation?
Let $R$ be a DVR, and $f:R^n\to R^n$ a map. Suppose $f(r_1,\dots,r_n)=(s_1,\dots,s_n)$, and write $d=\min(v(s_1),\dots,v(s_n))$, where $v$ is the valuation on $R$. Knowing $d$, what is the best bound ...
- 2,768
2
votes
0
answers
225
views
action of Weyl group element on Weyl vector
Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...
- 1,219
5
votes
1
answer
210
views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
- 51.6k
5
votes
1
answer
531
views
Bounds for the size of arrays with distinct subarray sums
Consider an array $A$ of length $n$ with $A_i \in \{1,\dots,s\}$ for some $s\geq 1$. For example take $s = 6$, $n = 5$ and $A = (2, 5, 6, 3, 1)$. Let us define $g(A)$ as the collection of sums of all ...
- 3,195
6
votes
0
answers
169
views
Cell structures of simply-connected 5-manifolds (classified by Barden's 1965 paper)
In Barden's 1965 paper: Simply-connected five manifolds, Barden gave a complete list of diffeomorphism classes of simply-connected 5-manifolds:
$$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$
...
- 1,319
6
votes
0
answers
162
views
Singular strong generator
Let $V$ be a model of $\mathrm{ZFC}$ and let $j\colon V \to M$ be an elementary embedding with a critical point $\kappa$ ($M$ is transitive). A strong generator of $j$ is an ordinal $\zeta \geq \kappa$...
- 5,192
3
votes
1
answer
282
views
Every group of totally disconnected type is locally profinite?
Let $G$ be a Hausdorff topological group in which every point has a neighborhood basis of open compact neighborhoods. Let's call this a group of totally disconnected (td)-type.
On the other hand, we ...
- 6,120
9
votes
1
answer
456
views
Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?
The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space
$X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...
- 327
4
votes
0
answers
69
views
Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains
Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
- 83
2
votes
0
answers
47
views
If $f \in \operatorname{c-Ind}_Q^P \sigma$, then $f|_N$ is compactly supported modulo $Q \cap N$?
There is a claim in a proof in Casselman's notes on representation theory which I have not been able to verify. I have asked several people and nobody knows why it is true.
The thing I can't figure ...
- 6,120
5
votes
1
answer
289
views
Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain $\Gamma(p)$?
Does every index $p$ subgroup of $SL(2,\mathbb{Z}_p)$ contain the principal congruence subgroup $\Gamma(p)$?
Equivalently, must it be the preimage of an index $p$ subgroup of $SL(2,\mathbb{Z}/p\...
- 7,161
10
votes
1
answer
662
views
Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?
Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
- 2,506
3
votes
2
answers
331
views
Exponential decay of resolvent kernel
For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
- 9,591
9
votes
0
answers
244
views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
- 195
1
vote
0
answers
60
views
A variation on Dixmier's counterexample concerning centralizers in $A_1$
This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ are commuting elements of the first Weyl algebra. Is it true that $P$ and $Q$ are polynomials in some ...
- 2,783
1
vote
0
answers
166
views
Interpretation of deformation of complex structure
Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...
- 11
2
votes
0
answers
262
views
Polarization of the Jacobian in Torelli's theorem
I'm studying an example in book Yuji Shimizu and Kenji Ueno. Advances in Moduli Theory. Translations of Mathematical Monographs, vol. 206, that shows the importance of isomorphism as principally ...
- 530
1
vote
1
answer
161
views
A proof of recontruction of Sacks generic filter from it's Sacks real (M[G] = M[f])
Given the Sacks forcing $ (\mathbb{S} = \{T \subset 2^{<\omega} : T \text{ is perfect}\},\subset) $ and $G$ generic over M, we have $f = \bigcup \bigcap G = \bigcup_{T \in G}stem(G) $ a path ...
- 11
4
votes
1
answer
337
views
Locally presentable categories
Under category
Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...
- 511
1
vote
0
answers
134
views
Construction of Feller's pseudo-poisson process
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurable space
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued time-homogeneous Markov chain on $(\...
- 161
4
votes
0
answers
277
views
Orthonormal vector fields on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. We equip the tangent bundle $TM$ with the Sasaki metric $g_s$.
Assume that $X: M \to TM$ is a vector field on $M$.
We say that $X$ is an orthonormal vector field ...
- 312
2
votes
1
answer
221
views
Localization of a model category with respect to a class of maps
I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:M\leftrightarrow N: R $$
and let $(...
- 530
4
votes
1
answer
1k
views
Spin groups in terms of matrices and/or linear operators
Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
- 329
9
votes
3
answers
385
views
Determinant of a block matrix with many $-1$'s
For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...
- 13.2k
2
votes
0
answers
132
views
Iwasawa Theoretic Interest in a certain type of result
This question is probably going to sound vague (since it does to me) and I wish I could make it more precise, but here goes. For $p\in \{107, 139, 271,379\}$ Ohtani and Blondeau (in separate papers) ...
user130124
3
votes
0
answers
142
views
Is the Mandelbrot set weakly self-similar?
A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...
- 25.9k
1
vote
0
answers
85
views
Cobordism of an annulus with a non-vanishing vector field
Let $M$ be a compact three-dimensional manifold with corners, which is a cobordism of the two-dimensional annulus. In particular, the codimension one boundary of $M$ consists of two copies of the ...
- 768