Trending questions
159,064 questions
0
votes
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42
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Topologizing quasi orders with regards to products
This morning I was asked by a colleague for the "right" way to construct a topology on a quasi-order (aka preorder, a reflexive and transitive relation) such that the topology on a product ...
- 1,422
-2
votes
1
answer
142
views
Solution to Erdos-Ulam problem [closed]
I have solved the Erdos-Ulam problem (see link) and can construct a set that satisfies the conditions (dense in R2 with all interpoint distances rational). I have expanded the solution from two ...
6
votes
1
answer
433
views
Asymptotic behavior of partial sums of Dirichlet series
Consider the Dirichlet series:
$$\sum_{n \geq 1} \frac{a_n}{n^s} = \frac{\zeta(s+1/3)}{\zeta(s)}$$
where $\zeta(s)$ is the Riemann zeta function.
Question: Assuming the Riemann Hypothesis (RH), how ...
- 611
2
votes
0
answers
90
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 93
2
votes
0
answers
153
views
Isoperimetric inequality for Kähler manifolds
I am interested in the following form of isoperimetric inequality for Kähler Manifolds (for example unit ball $B^n\subset \mathbb{C}^n$ with Bergman metric). It should say something like this: if $F$ ...
- 37
2
votes
0
answers
85
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Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 6,313
1
vote
1
answer
88
views
Bounds on the number of proper 3-colorings of cubic graphs
Are there known bounds on the number of proper 3-colorings of a 3-regular in terms of vertex count?
- 83
3
votes
0
answers
51
views
Hat knot Floer Homology with Z coefficients calculation
I would like to ask for recommended references which carry out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{\operatorname{HFK}}(K;\mathbb{Z})...
- 171
7
votes
2
answers
491
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
2
votes
0
answers
95
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Pullback of an ample bundle under an embedding is ample
In Example 11.8 on JP Demailly's book on Complex Analytic and Differential Geometry it is being said that
The pullback of a (very) ample line bundle by an embedding is clearly also (very) ample.
I ...
- 21
1
vote
0
answers
161
views
Efficient algorithm for A217061
Let $a(n)$ be A217061. Here
$$
a(n) = \sum\limits_{m=1}^{n}\frac{1}{(m-1)!}\sum\limits_{k=0}^{n-m}(n+k-1)!\sum\limits_{j=0}^{k}\frac{1}{(k-j)!}\sum\limits_{\ell=0}^{j}\frac{2^{\ell-j}(-1)^{\ell+j}s(n-...
- 4,948
10
votes
2
answers
404
views
Impact of the squarefreeness of the level for modular forms
I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in ...
- 249
2
votes
0
answers
95
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Using Ramanujan's cubic continued fraction $C(q)$ and $x^3+y^3=1$ to solve the Bring quintic?
The octic Ramanujan-Selberg continued fraction $S(q)$ and $x^8+y^8=1$ can solve the Bring quintic. So can the Rogers-Ramanujan continued fraction $R(q)$ and $x^5+y^5=1.\,$ It turns out Ramanujan's ...
- 12.6k
4
votes
0
answers
128
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Errata for "Foliations and Geometric Structures" by Aurel Bejancu and Hani Reda Farran
I'm reading "Foliations and Geometric Structures" (2006) by Aurel Bejancu and Hani Reda Farran and have been looking for an errata sheet. Unfortunately Prof. Bejancu has passed away. I ...
3
votes
0
answers
101
views
Historical appearance of using $\operatorname{SO}_3$-representation theory for spherical harmonics
$\DeclareMathOperator\SO{SO}$The spherical Laplace equation and the spherical harmonics are a beautiful example of a differential equation dominated by the representation theory of the Lie group of ...
- 1,846
149
votes
71
answers
21k
views
Nonequivalent definitions in Mathematics
I would like to ask if anyone could share any specific experiences of
discovering nonequivalent definitions in their field of mathematical research.
By that I mean discovering that in different ...
1
vote
0
answers
68
views
Perpendicular intersection of complex hypersurfaces
Let $(X,\omega)$ be a Kaehler manifold and $D_1,D_2$ a pair of compact smooth divisors in $X$ which intersect transversely, i.e. $D_1$ and $D_2$ are codimension 1 complex submanifolds of $X$ and for $...
- 183
1
vote
0
answers
40
views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
0
votes
1
answer
82
views
Dimension of a manifold derived from a dense $G_{\delta}$ subspace
Let $X,Y$ be (compact connected) topological manifolds of dimensions $n,m$, respectively. Assume that a dense $G_{\delta}$ subspace $A$ of $X$ is homeomorphic to a dense $G_{\delta}$ subspace $B$ of $...
8
votes
1
answer
224
views
Can increasing the winding number of a 2-cell make a CW complex embeddable?
Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$.
For a natural number $n\ge 2$ consider the operation of ...
- 13.6k
1
vote
1
answer
112
views
Bounding a Riemann sum by its integral limit?
Let $M_{n}(\mathbb{C})$ denote the space of complex $n \times n$ matrices and, for $a>0$, $a \in \mathbb{R}$ fixed, let $A: [0,a) \to M_{n}(\mathbb{C})$ be a given function. I will write $A(t) = (...
1
vote
1
answer
102
views
Cohomology of "symplectically self-dual" chain complex
Let $G,H$ be abelian groups (denoted additively) with their Pontryagin duals denoted as $G^*$ and $H^*$. (The cases I'm interested in are products of $\mathbb Z_n$, $\mathbb Z$, $\mathbb R$, and $\...
- 3,001
6
votes
3
answers
554
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 700
0
votes
0
answers
55
views
reference request: conditions for pointwise and operator-norm convergence of kernel projections
At a very high level, I’m interested in the following question. Suppose $X$ is a (separable) Hilbert space, and $T_n : X \rightarrow X$ is a sequence of finite rank self-adjoint maps that converges (...
- 101
6
votes
1
answer
162
views
Can there exist a set of all transitive sets in a model of NF or NFU?
Is it consistent with $\sf NF$ or $\sf NFU$ to have a set of all transitive sets? Formally:
$\exists t \forall x (x \in t \leftrightarrow x \text { is transitive})$
Where "$x$ is transitive" ...
- 11.3k
18
votes
1
answer
555
views
When can we add choice to a model of ZF
For countable transitive models of ZF, is existence of a ZFC extension with the same height a first order property?
In other words, is there a statement $τ$ (in the language of set theory) such that ...
- 7,545
1
vote
1
answer
145
views
An application of min-max characterization of eigenvalues
Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let
$$
\mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{...
- 563
0
votes
0
answers
127
views
Relative minimal models of pencils of surfaces
I would like to ask for recomendation for literature on theory relative minimal models of surfaces, where "relative" in sense of that the study objects are not surfaces alone (="absolue ...
- 5,966
2
votes
1
answer
203
views
Can a Feynman graph be an empty set?
I was reading the section about Feynman graphs from the book Renormalization - An Introduction and this question arose. To set the notations, let $p \in \mathbb{N}_{+}$ and, for each $k \in \{1,...,p\}...
- 1,305
4
votes
0
answers
47
views
Are W-spaces with countable pseudocharacter first countable?
Cross-post of a question originally asked by Almanzoris on Mathematics Stack Exchange.
A topological space $X$ is called W-space if P1 has a winning strategy at each point $x \in X$ for the following ...
- 1,422
9
votes
1
answer
429
views
A curious norm related to the L¹ norm
If $f \in C^0([0,1])$, one can define
$$\Vert f \Vert_? = \sup_{J \subset [0,1]} \left\lvert \int_J f \right\rvert,$$
where $J$ runs among all subintervals of $[0,1]$.
This is a norm on $C^0([0,1])$ (...
- 575
11
votes
1
answer
416
views
Examples of natural algebraic irreflexive relations
To motivate the question, consider the theory of rings.
Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit".
Then $\parallel$ ...
- 15.9k
1
vote
1
answer
154
views
Looking for Fitch-style Natural Deduction system that allows for open formula
I find most Natural Deduction proof systems only allow for close formulas, which are not convenient for FOLs without a constant. Most Sequent Calculus systems instead allow for open formulas, but it ...
- 127
12
votes
3
answers
801
views
The orders of the exceptional Weyl groups
Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
- 465
2
votes
2
answers
176
views
Great literature on discrete dynamical systems and/or qualitative theory of difference equations
I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
1
vote
0
answers
46
views
An internal functor in $\mathbf{Fib(B)}$
I have been thinking about this notion of an internal functor in the category $\mathbf{Fib(B)}$ of fibrations over the same base $\mathbf{B}$. Say $f \colon P \Rightarrow Q$ is an internal functor ...
- 615
15
votes
3
answers
1k
views
Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 261
0
votes
0
answers
52
views
The order of product of two reflections in a Coxeter group
Let $(W,S)$ be a Coxeter system. For any $s,t\in S$, denote $m_{st}$ be the order of $st$. By a reflection, we mean the element in $W$ which conjugates to some simple reflection.
My question is: Let $\...
- 71
1
vote
0
answers
159
views
Does a bijection between well orders of two sets imply a bijection between the sets? [closed]
We know that whether $|P(x)|=|P(y)|$ implies $|X|=|Y|$ is dependent on CH. Let $W(X)$ be the set of all well orders over $X$. Does $|W(X)|=|W(Y)|$ imply $|X|=|Y|$? Is the answer dependent on CH? More ...
- 77
-1
votes
0
answers
27
views
Number variance of random points (and deviations for empirical processes)
Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider
$$
V(N,x) = \...
- 2,227
5
votes
2
answers
557
views
A race to the bottom
Nate has a biased coin that comes up heads $\frac{1}{2} + \delta$ proportion of the time, where $0 < \delta \leq \frac{1}{2}$. He is competing against a large number $N$ people who each have fair ...
- 6,313
6
votes
1
answer
305
views
New Mersenne prime and compute time [closed]
GIMPS has just announced that $2^{136,279,841}-1$ is prime. Does anyone have a sense of the scale of the computational resources involved in finding this? (And maybe how it compares to, say, ...
- 133
2
votes
0
answers
189
views
Semantic equivalence between mathematical proofs
Sometimes, we recognize two proofs of the same claim to be the "same" proof. In some cases, this sameness is obvious -- for example, the proofs that $\sqrt{2}$ and $\sqrt{3}$ are irrational ...
- 225
2
votes
0
answers
117
views
Tilting complexes arising from homotopy equivalences
Let $k$ be a field and let $A$ and $B$ be finite-dimensional selfinjective $k$-algebras. Suppose we have an isomorphism of homotopy categories $F: K^b(A-mod) \cong K^b(B-mod)$ that descends to a ...
- 175
5
votes
1
answer
568
views
Dualizing sheaf of nodal curve
Let $C$ be a connected, nodal (I'm working with definition from Alper's notes on Stacks & Moduli, see p 210), projective curve over an alg closed field $k$, beeing everywhere smooth except at a ...
- 5,966
1
vote
0
answers
114
views
Solving the Bring quintic using the Ramanujan $g$- and $G$-functions?
Ramanujan defined two functions now called the Ramanujan g- and G-functions. One of the more well-known values is,
$$g_{58} = \sqrt{\tfrac{5+\sqrt{29}}2}$$
If we let,
$$2^6\big(g_{58}^{12}+g_{58}^{-12}...
- 12.6k
1
vote
0
answers
73
views
Krull dimension of affinoid algebra
Let $K$ be a complete field w.r.t. a valuation, with residue field $k$. Let $A$ be an affinoid algebra over $K$ with respect to a valuation $V$ (in the sense of Tate; in the terminology of Berkovich, $...
- 237
4
votes
1
answer
215
views
Characterisation of Sobolev spaces using their Lipschitz approximations
Let $f \in W^{1, p} (\mathbb R^n)$. A classical approximation theorem (see for instance, the book by Evans and Gariepy) says that we can approximate $f$ by Lipschitz functions, in the sense that for ...
- 6,313
0
votes
0
answers
99
views
Quotients of K3 surfaces vs cyclic covers
Let $X$ be an algebraic K3 surface (for sake of simplicity, with base field of char $\neq 2$) and $f: X \to X$ a non-symplectic morphism (i.e. non-symplectic in sense of that that the induced action ...
- 5,966
0
votes
0
answers
168
views
Combine two types of permutations in a Young diagram?
Given a Young diagram $Y$, for each row $R$ choose a permutation $\sigma_R$ of $\{1,\dots, |R|\}$, where $|R|$ is the size of row $R$. Let $\sigma_R(i)$ be the “row coordinate” of the $i$th cell in ...
- 281