Newest Questions
159,064 questions
4
votes
1
answer
217
views
Model categories: "equivalence" of finite limits and finite colimits
I am needing a reference for the following statement (in case it is true): Quillen functor between stable model categories preserve finite limits iff it preserves finite colimits.
For stable $\infty$-...
- 893
2
votes
0
answers
93
views
Name for isomorphisms canonically identifying all elements in a category
Say in a category, for any two objects $A,B$, we have an isomorphism $\iota_{AB}:A\to B$ with the property that $\iota_{BC}\circ\iota_{AB}=\iota_{AC}$ and $\iota_{AA}=\mathit{id}$.
Essentially, such a ...
- 896
4
votes
0
answers
118
views
Smooth map from manifold to good orbifold
I have the following question: there seem to be several inequivalent definitions of smooth maps between orbifolds, as indicated in Borzellino and Brunsden - The Stratified Structure of Spaces of ...
- 41
5
votes
1
answer
208
views
Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
- 889
0
votes
1
answer
214
views
number of representations by sums of three squares (with coefficients)
There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for
$$
\#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\}
?$...
- 3,062
2
votes
0
answers
139
views
Distributive law of the non-empty list comonad over the non-empty list monad
Preliminaries
A monad is a triple $(M, \eta, \mu)$, with $M$ a functor, $\eta : \mathit{id} \Rightarrow M$ and $\mu : M^2 \Rightarrow M$ two natural transformations such that the following diagrams ...
- 21
26
votes
2
answers
3k
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History of right hand rule
I am not sure if this is the right place to ask, but many mathematicians are knowledgeable and interested also in history of math, so here I am.
I am curious to know when the right-hand-rule for ...
- 645
1
vote
2
answers
202
views
Classification of compact isotropy irreducible homogeneous Kaehler manifolds
Is classification of compact isotropy irreducible homogeneous Kaehler(-Einstein) manifolds known?
Here, a homogeneous space is called isotropy irreducible if the isotropy representation is irreducible....
- 21
4
votes
1
answer
446
views
Upper bounds on the Gromov–Hausdorff distance using persistent homology
In persistent homology theory, stability theorems are important to show that the topological signatures extracted are stable under small changes. A key result is the following bound on the bottleneck ...
- 183
0
votes
0
answers
63
views
Bounds for the Dispersal Problem in convex regions
We add a bit to: Bounds for minimax facility location in a convex region
Two earlier posts: Cutting convex regions into equal diameter and equal least width pieces - 2 and Facility location on ...
- 5,979
0
votes
1
answer
129
views
Sequence of functions converges pointwise to identity [closed]
Let
For $n\in \mathbb{N}$ and $k\in \{0, 1, 2, ..., 2^{n}-1 \}$ is defined
$$I_{k}^{n}=\left[\frac{k}{2^{n}}, \frac{k+1}{2^{n}}\right)$$
and $f_{n}:[0, 1) \rightarrow \mathbb{R}$ is defined by
$$f_{n}(...
- 251
6
votes
0
answers
243
views
Global well posedness of $\phi^4_1$
We consider the $\phi^4_1$ model: $\partial_t\phi=\Delta\phi-\phi^3+\xi$ on $[0,T] \times \mathbb{R},$ where $\xi$ is a space time white noise.
I know how to solve this equation locally on the torus, ...
- 573
1
vote
0
answers
150
views
Number of cluster variables associated to A type quivers
In a seminar/reading course about cluster algebras, we came across the fact that the number of cluster variables for the cluster algebra associated to mutating the quiver $A_n$ is $n(n+3)/2$ (rather, ...
- 495
2
votes
0
answers
125
views
Smoothness of locus of triples $(B_1,B_2,i)$ in Nakajima's notes
In section 1.4 of Nakajima's notes on Lectures on Hilbert Schemes, it is mentioned that $(\mathbb A^2)^{[n]}$ is identified with the space of triples $\{(B_1,B_2,i)\}/GL_n$. Here $B_1,B_2$ are $n\...
- 1,563
2
votes
0
answers
208
views
Are infinite loops possible in the game Prodway?
I'd like to know if infinitely repeating sequences of moves (i.e. cycles) are possible in the following game:
Prodway is a game for two players (Black and White) that is
played on the intersections (...
- 21
6
votes
0
answers
270
views
Torsion in the Lie algebra cohomology of gl(n,Z)
What is known about the Lie algebra cohomology $H^*(\mathfrak{gl}_n(\mathbb{Z}),\mathbb{Z})$? After passing to $\mathbb{Q}$-coefficients, the question is classical: $H^*(\mathfrak{gl}_n(\mathbb{Q}),\...
- 4,487
4
votes
3
answers
1k
views
Is there a pyramid with all four faces being right triangles? [closed]
If such a pyramid exists, could someone provide the coordinates of its vertices?
2
votes
1
answer
188
views
$(S\otimes T)^{it}= S^{it}\otimes T^{it}$ for unbounded operators
Let $S,T$ be unbounded, closed operators in Hilbert spaces $H,K$. In that case, we can form the tensor product operator $S\otimes T$ on the Hilbert space $H\otimes K$ which is the closure of the ...
- 175
5
votes
0
answers
569
views
What sets are known to have cardinality equal to $\mathbb{N}$ or $\mathbb{R}$ but open as to which?
A long time ago a similar question was asked on math.stackexchange.
There are many sets which we know to be either finite or infinitely countable but do not know which cardinality specifically.
An ...
0
votes
2
answers
329
views
A doubt regarding the extended form of the Weierstrass factorization theorem
I want to represent $\sin(x)-\dfrac{1}{\sqrt{2}}$ as a product of it's zeroes
According to the Weierstrass factorization theorem, the sine function can be represented as a product of its factors:
$$\...
- 103
2
votes
0
answers
320
views
Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
0
votes
0
answers
45
views
Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on
For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...
- 101
1
vote
0
answers
149
views
Ways of proving that a framework is locally rigid
Given a (bar-and-joint) framework/linkage, I would like to know what are possible ways of showing that the framework is locally rigid. Also, what is known about the computational complexity of ...
- 1,305
2
votes
1
answer
295
views
Are there always flat connections?
Let $G$ be a simply connected Lie group and $\Gamma$ a cocompact discrete and torsion-free subgroup. Is there a (real or complex) smooth vector bundle $E$ over the manifold $G/\Gamma$, which does not ...
user473423
-1
votes
1
answer
300
views
Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]
Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$
i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$
Now let $X$ be the restriction of its ...
- 308
7
votes
0
answers
313
views
A citation from Dieudonné about existence, classification and construction
I am looking for an exact quote from Jean Dieudonné saying more or less that
there are three kinds of important problems in mathematics:
problems of existence, classification and construction.
The ...
- 18.7k
3
votes
0
answers
93
views
Epimorphisms in Kleisli categories
I recently asked a question about characterizing epimorphisms in ${\bf Rel} = {\rm Kl}(P)$, where $P:\bf Set \to Set$ is the covariant powerset monad.
In my question I observed that a morphism $R : A \...
- 1,083
3
votes
0
answers
125
views
Extracting moments of $\max(X_1,\ldots,X_k)$ from asymptotic behavior of $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$
For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions.
Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{...
- 1,295
2
votes
1
answer
207
views
Orthogonal complements in exterior powers
I previously asked this on Mathematics Stack Exchange, to no result:
Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge ...
- 6,206
2
votes
1
answer
234
views
(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve
Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
- 1,833
0
votes
0
answers
40
views
Two particular combinations of Gauss hypergeometric functions
Browsing this site and the web I could not find a reference on the following combinations of Gauss hypergeometric functions $F={}_2F_1$, for which I have reason to believe that they can be simplified ...
- 1,942
0
votes
0
answers
92
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 563
1
vote
2
answers
178
views
Prove that the ideal of $\mathbb{C}G$ generated by a family of elements $\lbrace p_i\rbrace_{i=1}^n$ is equal to $\mathbb{C}G$
Given a finite abelian group $G$ consider the group algebra $\mathbb{C}G$ and a set $\mathcal{P}=\lbrace p_i\rbrace_{i=1}^n$ of elements of $\mathbb{C}G$. Define $I$ to be the ideal of $\mathbb{C}G$ ...
- 911
1
vote
0
answers
169
views
Optimal transport-like problem where the objective depends on conditional probability distribution
$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data.
Consider two sets $\...
1
vote
0
answers
89
views
Bounds for minimax facility location in a convex region
An earlier question: Facility location on manifolds
A possibly related earlier post: Cutting convex regions into equal diameter and equal least width pieces - 2
The minimax facility location problem ...
- 5,979
2
votes
0
answers
331
views
Conditions for an existence of smooth solution to a parabolic PDE
I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset):
\begin{equation*}
u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
- 425
3
votes
0
answers
304
views
What are the unsolved problems in Formal groups and $L$-functions?
In the 1st page of the introduction of Hazewinkel's Formal Groups and Applications book, there are two ways of constructing formal groups (law):
$\bullet$ Given a Lie group $G$, one can define a ...
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3
votes
1
answer
321
views
Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $
Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.
Consider the natural map
$$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{...
- 1,541
2
votes
0
answers
157
views
Why do von Neumann algebras possess identity?
My starting point is that a von Neumann algebra is a $C^*$-algebra with a predual. The usual approaches to showing the existence of identity involve spectral theory (for approximate identity), but ...
- 433
3
votes
2
answers
581
views
Approximation of partial sum over prime omega function
I asked the question in Math StackExchange. Link: https://math.stackexchange.com/questions/4765476/approximation-of-partial-sum-over-prime-omega-function
I haven't got any response yet. Here are the ...
- 221
-1
votes
1
answer
177
views
Can we use linear map to approximate lipschitz continuous function $f$ in a compact domain after some linear transform?
Suppose $f : \mathbb{R}\to \mathbb{R}$ is lipschitz continuous function , $K$ is a compact domain, for any $\varepsilon>0$, can we find $d,a\neq 0,c,w,b \in \mathbb{R}$ such that $\|df(ax+c)-(wx+b)\...
- 109
0
votes
2
answers
69
views
Is the right-hand term of the dynamic system equivalent to the original system after being multiplied by a constant?
Given two dynamical systems where $f$ is lipschitz for $x$ : $\begin{cases} x'(t)=af(x,t),\\ x(0)=x_0,\end{cases} t\in[0,\tau]$ and $\begin{cases} z'(t)=f(z,t),\\ z(0)=x_0,\end{cases} t\in[0,\tau']$, ...
- 109
1
vote
1
answer
108
views
Understand Riemannian cross-derivative on product manifolds
Suppose we have a smooth function $f:\mathcal{M}\times\mathcal{N}\rightarrow\mathbb{R}$ where the domain is a product of two Riemannian manifolds. The Riemannian cross-derivative ([1], section 2) is ...
- 125
2
votes
0
answers
128
views
Go variant: cyclic or not?
I would like to know if a cycle of moves is possible in the Go variant, Savage Go. That is, you capture my stones, I capture your stones, you capture my stones... The game never ends. A position is ...
- 121
1
vote
1
answer
142
views
For a pure-injective module $M$ does the property "$\operatorname{Hom}(-,M)$ is surjective" commute with certain limits?
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a pure-injective module. Then $\Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\...
- 696
1
vote
1
answer
262
views
What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
- 305
5
votes
0
answers
245
views
Duality and compactness for pro vector spaces
I have a somewhat basic question which I haven't been able to piece together from the literature.
Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
- 51
3
votes
0
answers
228
views
Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap
I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...
- 31
2
votes
0
answers
79
views
Proof of the convergence of the Rayleigh-Ritz Method?
In this article The convergence of the Rayleigh-Ritz Method in quantum chemistry by Bruno Klahn & Werner A. Bingel they have at page 11
Let $H_B$ be that Hilbert space which can be obtained as the ...
2
votes
1
answer
298
views
Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
- 557