Questions tagged [tag-removed]
This tag is to be used only when re-tagging highly(!) off-topic questions where none of the actual tags would make sense; all actual tags the questioner has used are removed and something is needed to have some tag, which is enforced by the software, so this tag is used. However note that this tag is also used in the process of moderators deleting tags. Thus a question having this tag does most of the time not mean that somebody found it off-topic.
346
questions
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Coloring Points in the Plane
Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most ...
- 5,205
5
votes
3
answers
990
views
Is the hyperspace of the Hilbert cube homeomorphic to the Hilbert cube
Question: Is the hyperspace of the Hilbert cube $H=[0,1]^\mathbb {N}$ homeomorphic to $H$?
Remarks and definitions:
1) The Hilbert cube $H$ is a compact metric space, where the metric is given by ...
- 328
5
votes
4
answers
2k
views
Integer division: the length of the repetitive sequence after the decimal point
When dividing two integers, there may be an infinite sequence of digits after the decimal point (e.g. in the cases of 1/3, 1/7 etc).
As far as I know, if the two numbers divided are integers, this ...
- 159
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votes
2
answers
268
views
Branch locus of a 6:1 cover of the grassmannian G(1,3)
Given a general quartic surface $S$ in $\mathbf{P}^3$, there is a natural 6:1 surjective map
$\phi: Hilb^2(S) \to G(1,3)$ sending $\{P,Q\}$ to the line through them in $\mathbf{P}^3$.
Can you ...
- 453
5
votes
2
answers
2k
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Why are the Dynkin diagrams E6, E7 and E8 always drawn the way they are drawn?
The Dynkin diagrams of type ADE are ubiquitous in mathematics as solutions of various classification problems. The diagram E6 is usually drawn by five dots in a row with a sixth dot above the third, ...
- 51
5
votes
2
answers
511
views
Freeing a sphere from within a sphere
We can embed $S^2\times I$ into $\mathbb{R}^3$ by taking a compact 3-ball and removing an open 3-ball from its interior. Taking the boundary gives an embedding $i: S^2\sqcup S^2\hookrightarrow\mathbb{...
5
votes
2
answers
2k
views
Construction of Serre Spectral Sequence
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with ...
- 305
5
votes
1
answer
558
views
question about higher geometric stacks
I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Spec(A))= Hom(A,Gamma(...
5
votes
2
answers
486
views
Some weird "system" of inequalities in nonnegative integers.
Suppose I have a bunch of nonnegative integers $(a_{ijkl})_{1 \leq i \leq j \leq k \leq l \leq 17}$ such that for all 17-tuples nonnegative integers $w_t$ (for $1 \leq t \leq 17$) we have that $$\min_{...
- 5,123
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votes
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399
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An extension of Edelstein's Attraction Theorem?
Let $X$ be a metric space. Recall that a function $f: X \rightarrow X$ is contractive if there exists $C \in (0,1)$ such that for all $x,y \in X$, $d(f(x),f(y)) \leq C d(x,y)$; a function $f$ is ...
- 64.8k
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388
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Computing the limit of a certain recursively defined sequence
The following is not exactly a research question (it was originated from manufacturing of exercises for calculus), and has no other motivation than explaining a phenomenon. I apologize if it is ...
- 56.6k
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votes
2
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759
views
Can we calculate the inner product of a semicontinous function with the Dirac delta function?
Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
- 1,619
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Equation between the two branches of the lambert w function
My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?
For example the two square roots $r_1(y)$ and $r_2(y)$ of ...
- 497
5
votes
1
answer
910
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Identity involving Fresnel integrals
In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical
Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...
- 16.4k
5
votes
1
answer
749
views
Convex Analytic or linear algebraic proof that a certain psd matrix is a sum of rank 1 psd matrices
Can you prove the following using techniques from convex analysis or linear algebra? I was originally seeking an elementary proof, but I think it is better to broaden the scope for this bounty ...
user2529
5
votes
1
answer
810
views
Fiber functor of category of D-module on affine Grassmannian.
Geometric Satake correspondence allows us to construct Langlands dual group in a canonical way. In Mirkovic-Vilonen's paper, they prove that category of spherical perverse sheaves is an commutative ...
- 1,447
5
votes
1
answer
1k
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Possible to find a set of log-concave functions with log-concave sums?
While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
- 51
5
votes
2
answers
980
views
Intersection forms of 4-manifolds with boundary
Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
- 53
5
votes
1
answer
358
views
Can one "soup-up" the LAW OF THE MEAN in the following way?
Let J be a closed interval of real numbers whose length is finite and positive. Let f be a real valued
function defined on J which has a continuous second derivative at all points of J.
QUESTION: If ...
- 6,105
5
votes
2
answers
350
views
Distributing points with respect to a concave function
Suppose I have a concave function defined on the unit interval such that $f(0) = f(1) = 0$ and $\int_0^1 f(t) dt = \alpha$, where $\alpha$ is "small" (say $0.01$ or thereabouts). Say I distribute $n$ ...
- 635
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466
views
On the multidimensional generalisation of Gamma function
Gamma function is defined as
$$
\Gamma(z) = \int\limits_{0}^{+\infty} x^{z-1} e^{-x} \; dx
$$
I'm looking for multidimensional generalisation of this definition. I consider the class $Q$ of positive,...
- 1,269
5
votes
1
answer
540
views
Localizability of differential operators a la Grothendieck
Hello,
Maybe this question is trivial, so sorry
Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).
Then we can define the module of differential operators $D^{\leq n} (A)$,...
- 5,492
5
votes
2
answers
494
views
How to specify a finite group up to inner automorphism?
I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...
- 18.3k
5
votes
0
answers
396
views
Azimuthal and polar integration of a 3D Gaussian
Numerical evaluation of the following integral of a 3D gaussian $G$ seems to result in a 1D Gaussian $g$:
$$\int_{0}^{2\pi}\int_{0}^{\pi}G(R,\phi,\theta)\sin\theta\ \text{d}\theta \ \text{d}\phi= g(R)$...
- 347
5
votes
0
answers
228
views
Does every stack with a connection admit an atlas with a connection?
Dear all,
Let $S$ be a scheme in characteristic $0$,
and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme $...
- 2,230
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2
answers
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Is the notation $f(x)$ "deprecated by professional mathematicians" (as claimed by Wolfram)? [closed]
Wolfram's MathWorld website, at the page on functions, makes the following claim about the notation $f(x)$ for a function:
While this notation is deprecated by professional mathematicians, it is ...
- 167
4
votes
2
answers
874
views
Is it true that all the "irrational power" functions are almost polynomial ?
Hello all, the $\Delta$ operator on functions $\mathcal{N} \to \mathbb{R}$
(where $\mathcal N$ denotes $\lbrace 1,2, \ldots , \rbrace$ )defined by
$\Delta(f)(n)=f(n+1)-f(n)$ is well-known and
it is ...
- 3,565
4
votes
1
answer
585
views
Definition of Pontrjagin Classes
I am studying characteristic classes recently and find some quesions about Pontrjagin classes.
Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes ...
- 165
4
votes
2
answers
502
views
Vanishing of Self-Ext groups of vector bundles
Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)?
Edit: ...
- 2,444
4
votes
1
answer
447
views
Homotopy groups of K3
Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...
4
votes
1
answer
568
views
The number of simply connected 4-dimension manifold
For a simply connected four-dimension manifold, we know the Freedmen's work.
My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is ...
- 693
4
votes
1
answer
974
views
Harmonic functions on the plane
I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it ...
- 103
4
votes
3
answers
407
views
Implement intersection products
I am doing a counting problem, and it comes to compute intersection products ( Chow ring ) on some varieties. Is there any computer algebra that deals with this?
- 238
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votes
2
answers
251
views
Steinberg Group as a Lattice in a lie group
Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators
$e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$,
$p\neq q$, $1\leq p,q \leq n$
Subject to ...
- 41
4
votes
1
answer
6k
views
Inverse of a function defined by an integral
Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...
- 167
4
votes
1
answer
354
views
Convex polyhedral decomposition of spheres
Is there a decomposition of $S^2$ into $k$ (geodesically) convex polyhedra that are congruent to each other? What about $S^n$ for $n>1$?
Remarks:
A polyhedron is defined as an area enclosed by a ...
- 115
4
votes
2
answers
2k
views
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
For a image denoising problem (below):
http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf
the author has a functional E defined
$E(u) = \int\int_\Omega F \\ d\Omega$
which he wants to ...
- 103
4
votes
1
answer
291
views
Invariant lattice of algebraic surface.
Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or $H^...
- 41
4
votes
1
answer
746
views
Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?
I've been trying to find a closed form expression/series expansion for the following integral without success:
$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$$...
- 81
4
votes
1
answer
395
views
(Non)-exoticness of a diffeomorphism of a sphere
Suppose you have a standard sphere $S^n$ and a "standard" $S^{n-2}\subset S^n$. I am really thinking about $S^{n}\subset \mathbb{R}^{n+1}$ the usual sphere, and $S^{n-2}=S^n\cap \{x_0=x_1=0\}$. Let $S^...
- 1,420
4
votes
2
answers
349
views
Minimal right ideals in finite semigroup
Let $E$ be a finite semigroup. According to N. Bourbaki (Algèbre I p. 121 exerc. 14 c), if $M$ and $M'$ are minimal right ideals in $E$, then they are isomorphic. I spent some time browsing through ...
- 43
4
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1
answer
3k
views
complex gradient of a function
Let $M$ be a complex $n-$dim manifold and $u : M \rightarrow \mathbb{R}$ be some smooth function. On $M$ assume that we have a Kaehler metric $h$. How is the complex gradient vectorfield defined with ...
- 41
4
votes
1
answer
933
views
Interpolation by a function whose second derivative is bounded
I don't know if this is an easy question for specialists in the field. Consider
the following interpolation problem : let $\varepsilon >0$, $X$ be a finite
set of real numbers and $g$ be a real-...
- 3,565
4
votes
1
answer
1k
views
Exact solutions to nonlinear Klein-Gordon equation
The nonlinear pde
$$
\partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0
$$
has the exact solution
$$
\phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i)
$$
...
- 1,659
4
votes
2
answers
4k
views
Minimizing a function containing an integral
I am trying to optimize a function of the following form:
$L = \int_{t=0}^{T}(AR-x)dt$, where A is a system parameter
i.e. I am trying to find the optimum x(t) that minimizes L over all admissible x(...
- 429
4
votes
2
answers
845
views
Ansätze for solving PDEs with wavelets
It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features.
My question: Which Ansätze do you know ...
- 5,875
4
votes
0
answers
386
views
A question on an intuitive way to look at stacks
I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
- 805
4
votes
0
answers
282
views
Easy to find roots
Is there a smooth function $f:\mathbb{R} \to \mathbb{R}_{\geq 0}$ such that:
1) $\lim_{x \to \infty} = \lim_{x \to -\infty} = 0$
2) $\forall x > 0$, $f'(x) < 0$
3) $\forall x < 0$, $f'(x) &...
- 3,125
3
votes
3
answers
2k
views
Fourier transform of fractional differential operator and Plancherel formula equivalent for fractional norms
I would like to know if the the following exist or are defined
The Fourier transform $\mathcal{F}\left(\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}\right)$ of a fractional differential operator such as $\...
- 661
3
votes
2
answers
2k
views
Complexity of computing derivatives
Sorry if this is too simple. This is my first question here.
Suppose $f : R^n \to R$ is a differentiable function. Say that we can compute in $T$ arithmetic operations the value $f(x)$ at any point $...
- 33