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Is the axiom of choice (or equivalent) required to establish Sion's minimax theorem?
H. Komiya's elementary
[proof](http://projecteuclid.org/euclid.kmj/1138038812)
does not seem not to use any transfinite induction or choice axiom. Is this really the case, or is it implicit there in some sneaky form?
The other ... |
Does Sion minimax require choice? |
A topological space is called *irresolvable* if it is not the disjoint union of two dense subsets. So you are asking whether there is a connected, $T_2$, irresolvable space with more than one point. The answer is yes! You can find a proof, along with a few references to relevant literature, in
> D. Anderson, "On co... |
> Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
> categories. $\mathcal{A}$ reflective and closed under filtered
> colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
> $R:\mathcal{K}\to \mathcal{A}$ be the reflection. Let $G$ be a dense
> generator of $\mathcal{K}$. Can we conclude ... |
I have the following convex optimization problem:
$$\begin{array}{ll} \text{maximize}_{{q_0,q_1}} & \displaystyle\int_{\Omega} q_1^u{q_0}^{1-u}\mathrm{d}\mu\\ \text{subject to} & \displaystyle\int_{\Omega} q_0 \mathrm{d}\mu= 1,\quad \displaystyle\int_{\Omega} q_1\mathrm{d}\mu =1 \\ & f_l \leq {q_0} \leq f_u\\ & g_l \... |
I have the following problem: in the free group $F_2=\langle a,b\rangle$, we define the sequence
$\begin{cases}
w_0=a, \\
w_1=b, \\
w_{n+2}=[w_{n+1},w_{n}] & \text{for }n\ge 0.
\end{cases}$
So $w_2$ is the classical commutator $[b, a]$ (I take $[a,b]=aba^{-1}b^{-1}$ but this doesn't really matter), and then ... |
How can I prove the following?
1-x+x^2+x^5-x^7-x^12+x^15-x^22-x^26+x^35-x^40+...
= Product[(1 - x^(8 i - 7)) (1 + x^(8 i - 6)) (1 + x^(8 i - 5)) (1 +
x^(8 i - 4)) (1 + x^(8 i - 3)) (1 + x^(8 i - 2)) (1 -
x^(8 i - 1)) (1 - x^(8 i))]
Where the product is from i=1 to infinity.
It doesn't seem to fol... |
The (unsolved) [inscribed square problem][1] asks whether every closed, non-self-intersecting continuous (Jordan) curve $\Gamma$ in the plane contains the vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, every example has been like the one on the left in this figure, i... |
The (unsolved) [inscribed square problem][1] asks whether every closed, non-self-intersecting continuous (Jordan) curve $\Gamma$ in the plane contains the vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the left in this figure, ... |
How can I prove the following?
$$1-x+x^2+x^5-x^7-x^{12}+x^{15}-x^{22}-x^{26}+x^{35}-x^{40}+\dots \\= \prod_{i=1}^{\infty} [(1 - x^{8 i - 7}) (1 + x^{8 i - 6}) (1 + x^{8 i - 5}) (1 +
x^{8 i - 4}) (1 + x^{8 i - 3}) (1 + x^{8 i - 2}) (1 -
x^{8 i - 1}) (1 - x^{8 i})]$$
It doesn't seem to follow from the ... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in the plane contains the vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the left in this f... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in the plane contains the vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the left in this f... |
When considering (set-theoretic) games, we have three parameters we can adjust:
1. Definability of the payoff set
2. The set of legal moves
3. The length of the game
When working in $\textsf{ZFC}$, what are our limitations on the above three parameters? When do we reach inconsistencies? Some more specific su... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the left... |
In many contexts it is helpful to think of conditional expectations as averages of unitary conjugates, a standpoint vindicated by many standard techniques in the theory of finite von Neumann algebras. Further, this point of view is reinforced nicely in a pointwise sense (cf. Narutaka Ozawa's comments to my earlier ques... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the left... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. No proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the left... |
Let $\theta$ and $\tau$ be involutions of a semisimple complex lie algebra $\mathfrak{g}$, and suppose that they have the same Satake diagram. I want to understand why they must be conjugate to one another.
Let $\mathfrak{t}_-$ be a maximal torus in the $(-1)$-eigenspace of $\theta$, and let $\mathfrak{h}$ be a Car... |
I want to classify (up to isomorphism) all transitive subgroups of symmetric group $S_n$ which are generated by double transpositions (product of two transpositions). Is there a characterization for such subgroups of $S_n$? |
Let $X$ be a smooth complex projective variety. Let $M_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\time... |
Let $X$ be a smooth complex projective variety. Let $M_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\time... |
What kind of upper bound can one get for
$$
\sum_{d|n}\frac{1}{d^{\sigma}},
$$
where $0<\sigma<1$? The best I can do is $\ll n^{(1-\sigma)/2}$ by breaking the sum at $\sqrt{n}$, using the symmetry of the divisors, and replacing the sum on $d|n$, $d<\sqrt{n}$, with the sum over all $d<\sqrt{n}$. Since the hyperbol... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
As far as I've seen, most examples have been like the one on the... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have been like the one on the left, in which the n... |
Let $X$ be a smooth complex projective variety. Let $M_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over $X\time... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have been like the one on the left, in which the n... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have been like the one on the left, in which the n... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture asks whether every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have been like the one on the left, in which the n... |
Let $X$, $Y$ be reflexive Banach spaces, and let $\imath:X\hookrightarrow Y$ be a bounded inclusion with dense image. Then for any domain $\Omega\subset\Bbb R^n$ we may define $C_{\mathrm c}^\infty(\Omega;X)$ to be the (Fréchet)-smooth functions $\Omega\to X$ with compact support, with the usual inductive limit topolog... |
Let $X\subset Y$ be a dense inclusion of reflexive Banach spaces. Then is $C_c^\infty(\Omega;X)\subset C_c^\infty(\Omega;Y)$ dense? |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve $\Gamma$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have resembled the one on the left, in which the na... |
The stable real cohomology of $SL_n(\mathbb Z)$ was computed by Borel: it is given by $\mathbb R[z_i\mid i=5,9,13,\cdots]$ with $z_i$ in degree $i$. One may wonder whether the pull back of the stable class $z_i$ on $SL(\mathbb Z)$ to $SL_n (\mathbb Z)$ for some finite $n$ is non-zero. That is, whether $\iota_n^* z_i = ... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have resembled the one on the left, in which ... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have resembled the one on the left, in which ... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all four vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have resembled the one on the left, in which ... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. Neither proof nor counter-example has ever been given.
Most examples have resembled the one on the left, in which the n... |
Given the 2D wave equation in polar coordinates:
$$u_{\rho\rho}+\dfrac{1}{\rho}u_{\rho}+\dfrac{1}{\rho^2}u_{\theta\theta}=\dfrac{1}{a^2}u_{tt}$$
with $u=u(\rho,\theta,t),(\rho,\theta,t)\in [0,c]\times(-\pi,+\pi]\times [0,+\infty)$
and boundary conditions:
$$u(0,\theta,t)=\nu(t),\forall \theta\in(-\pi,+\pi], t\in [0... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. Neither proof nor counter-example has ever been given.
Most examples resemble the one on the left, in which the natural... |
I'd like to know whether a certain unbounded operator on a Hilbert space is the generator of a strongly continuous contraction semigroup.
(Such operators are known as [maximally dissipative operators][1].)
• Let $A$ be an unbounded <b>skew-adjoint</b> operator on $H$.
• Let $P$ be an unbo... |
Combining both observations, one has the trivial bounds
$$d(n)n^{-\sigma/2}<=S<=\frac{d(n)}{2}(1+n^{-\sigma/2})$$
Can one do better ?
|
Maybe this is the sort of direct proof you're looking for?
Suppose the action of $G$ is not $2$-transitive. Then $G$ preserves some nontrivial directed graph $\mathcal{G}$ with vertex set $X$ (the $G$-orbit of your favourite edge). Therefore $G$ preserves the eigenspaces of this graph, i.e., the eigenspaces of the a... |
Combining both observations, one has the trivial bounds
$$d(n)n^{-\sigma/2}\le S\le \frac{d(n)}{2}(1+n^{-\sigma/2})$$
Can one do better ?
|
What kind of upper bound can one get for
$$
\sum_{d|n}\frac{1}{d^{\sigma}},
$$
where $0<\sigma<1$? The best I can do is $\ll n^{(1-\sigma)/2}$ by breaking the sum at $\sqrt{n}$, using the symmetry of the divisors, and replacing the sum on $d|n$, $d<\sqrt{n}$, with the sum over all $d<\sqrt{n}$. Since the hyperbol... |
Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c \ln \ln n}{\ln^\beta n}$ holds for infinitely many $n$.
Statement 2 : if the R.H. is false then there exist constants $0<\be... |
Does Statement 1 imply Statement 2? |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. Neither proof nor counter-example has ever been given.
Most examples resemble the one on the left, in which the natural... |
Combining both observations, one has the trivial bounds
$$d(n)n^{-\sigma/2}\le S\le \frac{d(n)}{2}(1+n^{-\sigma})$$
The upper bound is optimal when $n$ is prime.
Can one do better ?
|
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where $f^*\omega$ is the usual pullback of forms, and the $d$'s denote the appropriate exterior derivatives. They then prove that... |
Let $A$ be a finite dimensional connected algebra over a field and $X$ the set of all $A$-right modules $M$ with $End_A(M) \cong A$.
Say $A$ has property $(*)$ in case the following is true:
Given $M, N \in X$, also $M \otimes_A N$ is in $X$.
In this case $X$ is a monoid.
Are there interesting algebras $A$ ... |
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where $f^*\omega$ is the usual pullback of forms, and the $d$'s denote the appropriate exterior derivatives. They then prove that... |
The following is a well-known result for elliptic operators.
> **Theorem.** Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$
> (c... |
The (unsolved) [inscribed square problem][1] or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. Neither proof nor counter-example has ever been given.
Most examples resemble the one on the left, in which the natural... |
Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where $f^*\omega$ is the usual pullback of forms, and the $d$'s denote the appropriate exterior derivatives. They then prove that... |
1. When does $\det(A+B)=\det(A)+\det(B)$ hold?
2. I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
|
Yesterday I needed to do some calculations with circles and "ventured" to calculate the arc length via the $\int{\sqrt{1+\left(f'(x)\right)^2}}$ formula and was baffled to see that in the case of unit half-circles that amounts to $\ ds := \sqrt{1+\left(f'(x)\right)^2} = \frac{1}{f(x)}$
Stuffing that differential e... |
Has this Peculiar Property of Unit Circles Already been Noticed? |
Let $X=\{x_1,\dots,x_N\}$ and $F=F(X)$ be a free group generated by $X$. Let $\phi\colon F\to F$ be an automorphism of $F$. Define a growth function of $\phi$ as:
$$
\operatorname{gr}_{\phi,X}(n)=\operatorname{max}_{1\le i\le N}\{\|\phi^n(x_i)\|_X\},
$$
where $\|.\|_X$ denotes the word length with respect to $X$. W... |
growth of a free group automorphism is same for finite index subgroups? |
This is a version of the [question I asked recently][1], but the assumptions got now strengthened substantially.
> Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\pm1\}$. How small can the rank of $A$ be, given that
>
(1) $a_{ii}=1$ for all $i\in[1,n]$;
>
(2) $\ a_{ij}a... |
> Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
> categories. $\mathcal{A}$ reflective and closed under filtered
> colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
> $R:\mathcal{K}\to \mathcal{A}$ be the reflection. Let $G$ be a dense
> generator of $\mathcal{K}$ consisting of obj... |
I had the privilege of discussing similar concerns with regard to theoretical physics
with the late Richard Feynman. He told me the following, which has always served me in
good stead: "You keep on learning and learning, and pretty soon you learn something
no one has learned before." That was his "advice"; my adv... |
You are asking whether there are functions $f$ such that $(f')^2=h(f)$. (I denote $h=g^2-1$). The answer is yes: these functions are solutions of the differential equation
$f'=\sqrt{h(f)}$. The general solution of this equation is: $$\int\frac{df}{\sqrt{h(f)}}=x+c.$$
In your case, $h$ is a quadratic rational function... |
Let $X$ be a smooth complex projective variety. Let $M_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns to any scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over ... |
Let $X=\{x_1,\dots,x_N\}$ and $F=F(X)$ be a free group generated by $X$. Let $\phi\colon F\to F$ be an automorphism of $F$. Define a growth function of $\phi$ as:
$$
\operatorname{gr}_{\phi,X}(n)=\operatorname{max}_{1\le i\le N}\{\|\phi^n(x_i)\|_X\},
$$
where $\|.\|_X$ denotes the word length with respect to $X$. W... |
I am reading a paper (the 2015 paper by A. Falkowski and L. Slominski **Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation**, page 353, proof of the Lemma 3.1) for my master degree thesis.
The framework of the problem is the following: we fix
$$
f:\Bbb R^d\to\Bbb R^... |
I like the following perspective:
> Up to scalar, trace is the only linear operator $\text{M}(n,k) \stackrel{t}{\to} k $ such that $t(AB) = t(BA)$.
If one likes vector field theory, this is the only linear operator that vanishes on commutator of vector fields. I do prefer to characterize it as the nullifier of hy... |
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2^d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I would like to know: Let $\lambda \notin (-\infty,0]$ is it true that
$\operatorname{ran}(P-\lambda)$ is dense in $... |
This is a generalization of an [older question][1].
If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its *nowhere dense covering number* $\nu(X)$ by the smallest cardinality that a [partition][2] of $X$ into [nowhere dense subsets][3] of $X$ can have. (The [older question][1] can be reformu... |
1. Who/Which paper was the first to prove or conjecture that the number of tilting modules equals the Catalan number for type $\mathcal{A}$-hereditary algebras?
2. Let $B$ be a tilted algebra of a representation finite hereditary algebra $A$ of Dynkin type $\mathcal{A}$. What is the number of tilting modules in $A$ ... |
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete [geodesic metric spaces][3] such that:
$X_{n}$ is a regular$^1$ [CW-complex][4] of constant local dimension$^3$ $n$, it is of finite type$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is [n-connected][5]... |
Is such an infinite dimensional metric space, weakly contractible? |
> **Definition 1** A subset $B$ of a metric space $(M,d)$ is called a **metric basis** for $M$ if and only if $$\forall b \in B,\,d(x,b)=d(y,b) \implies x = y \,.$$
> **Definition 2** A metric space $(M,d)$ has "**metric dimension**" $n \in \mathbb{N}$ if there exists a _minimal_ (in terms of cardinality) metric bas... |
When does a metric space have "infinite metric dimension"? (Definition of metric dimension) |
Consider the function $f$ on the prime numbers defined by $$ f(p):= \text{ the greatest prime factor of } 2p+1.$$ The iteration of $f$ from any prime $p<10^8$ converges to the cycle $$(3,7,5,11,23,47,19,13)$$
**Question**: Is it true for any prime $p$?
Let $\ell(p)$ be the number of iterations needed to join th... |
In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon.
I am wondering if there is a similar constraint for the Euclidean TSP tour
of points in $\mathbb{R}^3$.
To be specific:
> ***Q1***. Might it be true that the TSP tour of general-position points in $\mathbb{R}^3$ is neve... |
Knotted TSP tours in 3D? |
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2^d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I would like to know: Let $\lambda \notin [-1,1]$ is it true that
$\operatorname{ran}(P-\lambda)$ is dense in $\ell^... |
In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon.
I am wondering if there is a similar constraint for the Euclidean TSP tour
of points in $\mathbb{R}^3$.
To be specific:
> ***Q1***. Might it be true that the TSP tour of general-position points in $\mathbb{R}^3$ is neve... |
On $L^2(\mathbb{R}^n)$ it is true that $\Delta$ has $\sigma(\Delta)=(-\infty,0].$ Also, there are no eigenfunction. Yet, even if one would not know this, negativity $\langle -\Delta u,u \rangle \le 0$ does immediately imply that there could only be such functions satisfying $\Delta u = \lambda u$ for $\lambda \le 0.$
... |
Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces." [She died July 15, 2017](http://www.bbc.com/news/science-e... |
Let $B$ be a $C^{*}$-subalgebra of a $C^{*}$-algebra $A$ with a faithful conditional expectation $P: A\rightarrow B$. [Kumjian][1] suggests on page 15 that since $P$ is faithful we have
$\left\Vert a\right\Vert =\text{sup}\left\{ \left\Vert P\left(c^{*}a^{*}ac\right)\right\Vert ^{1/2}\mid c\in A\text{, }P\left(c^{*}... |
In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon.
I am wondering if there is a similar constraint for the Euclidean TSP tour
of points in $\mathbb{R}^3$.
To be specific:
> ***Q1***. Might it be true that the TSP tour of general-position points in $\mathbb{R}^3$ is neve... |
Kunen came up with the idea of, starting with a large cardinal notion A, kill the weak compactness of A and then resurrect some property without resurrecting weak compactness. Here we can for instance start with a weakly compact and find a generic extension in which the cardinal reflects stationary subsets but isn't we... |
Background of this question is that I had been asked for advice in clustering a very big set ($10^6$ to $10^8$) set of points in Euclidean 3D-space; these points in turn lie on 2D manifolds.
I remembered, that the combination of point clusters and optimal tours is beneficially been utilized in [TSP Art](http://www... |
Are Optimal Tours Sensitive to Clusters? |
**EDIT:**
Thanks to David's comments that help me a lot, the problem I really concern is just the following.
**Question:**
1. Now that we know if $M$ is non-compact then $C^\infty(M,N)$ is
modeled on spaces $C^\infty_c(M,N)$. Since we know
$W^{k,p}_0(\Omega)$ is the closure of $C^\infty_c(\Omega) $ ... |
**EDIT:** Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \times [0,1]$.
Thanks to David's comments that help me a lot, the problem I really concern is just the following.
... |
Let $\theta$ and $\tau$ be involutions of a semisimple complex lie algebra $\mathfrak{g}$, and suppose that they have the same Satake diagram. I want to understand why they must be conjugate to one another.
Let $\mathfrak{t}_-$ be a maximal torus in the $(-1)$-eigenspace of $\theta$, and let $\mathfrak{h}$ be a Car... |
On $L^2(\mathbb{R}^n)$ it is true that $\Delta$ has $\sigma(\Delta)=(-\infty,0].$ Also, there are no eigenfunction. Yet, even if one would not know this, negativity $\langle \Delta u,u \rangle \le 0$ does immediately imply that there could only be such functions satisfying $\Delta u = \lambda u$ for $\lambda \le 0.$
... |
Let $A$ be a representation finite hereditary algebra.
Define the Euler form by $f(M,N):=dim(Ext^1(M,N))-dim(Hom(M,N))$ for two modules $M,N$.
Now if $A_1$ and $A_2$ are two representation-finite hereditary algebra with the same underlying quiver, there is a bijection between the basic tilting modules of these algebr... |
Schur (1916) has proven that, given $n$, when the field is sufficient large, this equation,
$$x^n+y^n= z^n$$
always has a nontrivial solution.
What conditions do the number of the solutions satisfy?
- I. Schur, Uber die Kongruenz ¨xm + ym ≡ zm (mod p)," Jahresber. Deutschen Math. Verein. 25 (1916), 114–11... |
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete [geodesic metric spaces][3] such that:
$X_{n}$ is a regular$^1$ [CW-complex][4] of constant local dimension$^3$ $n$, it is of finite type$^4$, boundaryless$^2$, unbounded, uniform$^5$, and it is the $n$-skeleton of $X_{n+1}$, which is [n-connected][5]... |
The following combinatorial question came up while trying to prove a lemma for my research. Let $[N]$ denote the $N\times N$ grid inside the integer lattice $\mathbb{Z}\times\mathbb{Z}.$ The square corresponding to $(z_1,z_2)\in\mathbb{Z}\times\mathbb{Z}$ is $[z_1-\frac{1}{2},z_1+\frac{1}{2}]\times [z_2-\frac{1}{2},z_2... |
Let $E$ be $n$-plane bundle over CW complex $X$. Then $E$ is a pullback of tautological bundle $\gamma_n$ over $BO(n)$ i.e. $E=f^*(\gamma_n)$. This $f$ is called classyfing map. One can show that the universal covering of $BO(n)$ is $BSO(n)$ and $f$ can be lifted to $BSO(n)$ (meaning that there is $\tilde{f}:X \to BSO(... |
Let $X$ be a smooth projective variety, $Z$ a smooth subvariety of $X$, and let $f:\widetilde{X}\to X$ be the blow-up of $X$ along $Z$. Then for a subvariety $V\subset X$, we have two cohomology classes on $\widetilde{X}$: The pullback $f^*[V]$ and $[\widetilde{V}]$, the class of the proper transform of $V$. My questio... |
Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum_{y \tilde \ x} \frac{1}{2d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I would like to know: Let $\lambda \notin [-1,1]$ is it true that
$\operatorname{ran}(P-\lambda)$ is dense in $\ell^1... |
Let $X$ be a smooth projective variety, $Z$ a smooth subvariety of $X$, and let $f:\widetilde{X}\to X$ be the blow-up of $X$ along $Z$. Then for a subvariety $V\subset X$, we have two cohomology classes on $\widetilde{X}$: The pullback $f^*[V]$ and $[\widetilde{V}]$, the class of the proper transform of $V$. My questio... |
Let $A$ be a (finite-dimensional graded cocommutative) Hopf algebra over a field $k$, $E$ be a Hopf subalgebra, and $R=A \otimes_E k$. Then the comultiplication on $A$ induces a coalgebra structure on $R$. Furthermore, $R$ is a coalgebra in the monoidal category of $A$-modules, with $A$ acting on $R \otimes R$ diagon... |
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