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Non-closed range space of Laplace operators?

Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed? Sorry if this question is trivial. I am not familiar with theory of ...
Yidong Luo's user avatar
1 vote
2 answers
218 views

Sharply Estimating Pythagorean Triples [closed]

Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$ Is there a way to give a sharp estimate?
user avatar
1 vote
1 answer
215 views

Worpitzky-like identities?

Let $$r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$ Computations suggest that $$r_{2k}(x)=\sum_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r_{2k+1}(x)=\sum_{j=0}^{k^2-k}a(2k+1,j)\...
Johann Cigler's user avatar
1 vote
0 answers
120 views

Projective direct image and commutation

Let us denote by $f:X\rightarrow Y$ a projective map between two varieties, and $i:Z\rightarrow X$ an open inclusion. Is it correct to write : $$f_*(Im(i_!\mathscr{F}\rightarrow i_* \mathscr{F})=Im(...
Jean Lescure's user avatar
1 vote
0 answers
111 views

Can we construct a general counterexample to support the weak whitney embedding theorm?

The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$. ...
li ang Duan's user avatar
1 vote
1 answer
187 views

Inequality and integral

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
mathex's user avatar
  • 391
1 vote
1 answer
129 views

Two consecutive continua

Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ but there is no topological space $Z$ with $$X<Z<Y.$$ The later relation means that $Z$ ...
Ali Taghavi's user avatar
1 vote
0 answers
70 views

Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
Kei's user avatar
  • 267
1 vote
1 answer
250 views

Can we invoke "almost supermartingale" Theorem for deterministic sequences?

Perhaps stupid question. Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems? Attempt ...
user550103's user avatar
1 vote
0 answers
164 views

Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let \begin{equation} T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1. \end{equation} Prove the inequality \begin{equation*} \frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
qifeng618's user avatar
  • 836
1 vote
1 answer
263 views

Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation? $$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-...
Ali Taghavi's user avatar
1 vote
1 answer
231 views

Elliptic interface problem without conditions on the interface

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$. For a model case, consider a ball split in a smaller ball and an anulus. Consider the following elliptic ...
user avatar
1 vote
1 answer
333 views

Is the manifold of complex points of a quotient of compact groups just the tangent bundle?

In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
178 views

Measure, volume and cardinality on Minlos' book on statistical physics

The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...
MathMath's user avatar
  • 1,255
1 vote
1 answer
156 views

Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple ...
Garabed Gulbenkian's user avatar
1 vote
1 answer
148 views

'Convex' slices of proper actions

Consider a Lie group $G$ acting properly on a manifold $M$. Then by the slice theorem we can find for any point $m\in M$ a submanifold transverse to the orbit $\mathcal{O}$ through $m$ and which is (...
David Roberts's user avatar
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1 vote
1 answer
136 views

Comparison of hitting probability of two Markov chains both with only one absorbing state version 2 under stronger condition

Let $N_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}_i\in N_n\}\big)_{i=0}^\infty$ for $j\in\{1,2\}$, both of which have two absorbing states at $1$ and $n$. $\text{Pr}\...
Hans's user avatar
  • 2,169
1 vote
1 answer
316 views

On odd perfect numbers and a GCD - Part III

Let $m = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. It is known that $$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^...
Jose Arnaldo Bebita Dris's user avatar
1 vote
2 answers
214 views

Behavior of a non-linear differential equation

Let us consider the following differential equation $$ \dot{x}(t)=a - b\sin(x(t)), \quad a,b\in\mathbb{R}. $$ My question. Suppose $a>|b|$ and $x(0)=x_0\in\mathbb{R}$. Can the solution to the ...
Ludwig's user avatar
  • 2,682
1 vote
2 answers
1k views

2-class group of a quadratic imaginary extension

Let $p\equiv 5 [8]$ be a prime number, and consider $K=\mathbb{Q}(\sqrt{-p})$. I would like to check that the $2$-Sylow subgroup of the class group $C_K$ has order $2$ (I'm pretty sure it's true). ...
GreginGre's user avatar
  • 1,661
1 vote
1 answer
141 views

How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?

Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$ How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
636 views

Properties of the trace term in the Itō formula

Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where $U,H$ are separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ is nonnegative ...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
778 views

Picard group of a K3 surface generated by a curve

In Lazarsfeld's article "Brill Noether Petri without degenerations" he mentions the fact that for any integer $g \geq 2$, one may find a K3 surface $X$ and a curve $C$ of genus $g$ on $X$ such that ...
Youloush's user avatar
  • 365
1 vote
1 answer
140 views

Congruences that aren't "finite from above," take 2: semigroups

This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
Noah Schweber's user avatar
1 vote
1 answer
104 views

A claim on the concurrency of area bisectors of planar convex regions

We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia Definitions: Given a ...
Nandakumar R's user avatar
  • 5,453
1 vote
1 answer
6k views

Finding the square root modulo n, when the factors of n are known

Last month, I asked whether there is an efficient algorithm for finding the square root modulo a prime power here: Is there an efficient algorithm for finding a square root modulo a prime power? Now, ...
Craig Feinstein's user avatar
1 vote
0 answers
228 views

Does analytic isomorphism imply local isomorphism?

If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
Schemer1's user avatar
  • 789
1 vote
1 answer
126 views

About extensions between morphisms on the multiplier algebra

Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism $$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \...
user avatar
1 vote
0 answers
95 views

Zeros of modular functions and automorphisms

Let $F_N$ be the field of modular functions of level $N$ and with Fourier coefficients in $\mathbf Q(\zeta_N)$. We have $$F_N=\mathbf Q(j, f^{(r,s)}_N),$$ where the $f^{(r,s)}_N$ are the Fricke ...
Shimrod's user avatar
  • 2,335
1 vote
1 answer
377 views

Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding

The two sets are, of course, supposed infinite. This question is related to that one Commutation of tensor products with inverse limits in a specific case where it received a (partial) answer ($A$ ...
Duchamp Gérard H. E.'s user avatar
1 vote
1 answer
311 views

Symmetry in Hardy-Littlewood k-tuple conjecture

Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...
Sylvain JULIEN's user avatar
1 vote
3 answers
554 views

Surface analog of clothoid: curvatures covering $\mathbb{R}$

The clothoid $C$, a.k.a. the Euler spiral, is one among many curves with the property that its curvatures cover $\mathbb{R}$ in the sense that, for every $x \in \mathbb{R}$, there is a point $p \in C$ ...
Joseph O'Rourke's user avatar
1 vote
1 answer
658 views

Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?

Hi! I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the ...
Malte Muth's user avatar
1 vote
1 answer
127 views

Computational Geometric Aspects of Greedy Tour Expansion

Has the following problem already been investigated from the Computational Geometry point of view and what are the results regarding worst case complexity? Given a finite set $\mathcal{P}...
Manfred Weis's user avatar
  • 12.6k
1 vote
2 answers
1k views

projection formula for birational map

Let $X$ be a smooth projective variety of dimension $n$ , and $D$ be a divisor. Suppose the linear system $|D|$ induce a birational map $$f: X -\to Y, $$ and let $H$ be the very ample line bundle ...
Li Yutong's user avatar
  • 3,362
1 vote
2 answers
71 views

Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textbf{y} \|^2} \right|^2$, with complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
Felipe Augusto de Figueiredo's user avatar
1 vote
2 answers
232 views

Find $\inf_{P_{X_1,X_2}}P_{X_1,X_2}(\|X_1-X_2\| > 2\alpha)$ , where $\alpha > 0$ and inf is over couplings

Let $\mathcal X$ be a seperable Banach space with norm $\|\cdot\|$, and let $X_1$ and $X_2$ be random vectors on $\mathcal X$ with finite means. Question. Given $\alpha > 0$, what is value of, ...
dohmatob's user avatar
  • 6,716
1 vote
1 answer
456 views

Is this a full rank matrix? [closed]

According to the answer of znt to the previous version, I revise the question as follows: Is there a real $(n-1)\times n$ matrix $A$ such that $A$ is not a full rank matrix and satisfy $a_{ii}&...
Ali Taghavi's user avatar
1 vote
1 answer
2k views

About isomorphism of $PGL(2)$ and $SO(3)$ [closed]

I need to prove that $PGL_2(\mathbb{R})\cong SO_3(\mathbb{R})$. Abstract considerations show that both can be identified with the group of projective motions of a conic curve. But maybe there is more ...
Tim's user avatar
  • 125
1 vote
1 answer
465 views

Fibration when central fibre is a Calabi-Yau variety with canonical singularities

Let $f\colon X\to Y$ be a surjective proper holomorphic fibre space such that $X$ and $Y$ are projective varieties and central fibre $X_0$ is Calabi-Yau variety with canonical singularities, then can ...
user avatar
1 vote
0 answers
247 views

partially commutative monoid [closed]

Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
GA316's user avatar
  • 1,219
1 vote
0 answers
150 views

How many critical points can you have below a Fibonacci term in an algebra of elementary embeddings?

In this question, I asked about how slowly the Fibonacci terms in a algebra of elementary embeddings could grow, but I did not inquire if there was a limit to how quickly the Fibonacci terms could ...
Joseph Van Name's user avatar
1 vote
2 answers
549 views

Prove Liouville theorem without using mean value property

How can I prove the following Liouville theorem without using the mean value property? If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $...
Lao's user avatar
  • 217
1 vote
1 answer
394 views

joining or coupling

given two shift invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$, is there a way to construct joinings of them? It's very diffcult, in general, to find exactly the minimal joining i.e, ...
Bruno Brogni Uggioni's user avatar
1 vote
1 answer
87 views

On the marginal distributions of an absorbed diffusion

This question can be seen as a variant of the post Bounded density for diffusions with diffusion coefficients bounded away from $0$ by Iosif Pinelis. Namely, consider the diffusion $$X_t=\int_0^t a(s,...
GJC20's user avatar
  • 1,220
1 vote
1 answer
291 views

Truncated sums of symmetric polynomials; reference request for an algebraic derivation

EDIT: This is a case of being too wrapped up in a formulation ($e_j,p_i,$ and the like) to try something simple. It did not occur to me to pull exp to the outside in the weeks I have stared at this. ...
Gerhard Paseman's user avatar
1 vote
0 answers
169 views

A series of conjectures on $\sum_{x=0}^{(p-1)/2}(\frac{x^5+cx^3+dx}p)$ (I)

Let $p$ be an odd prime. Here I introduce the sum $$S_p(c,d):=\sum_{x=0}^{(p-1)/2}\left(\frac{x^5+cx^3+dx}p\right)$$ with $c,d\in\mathbb Z$, where $(\frac{\cdot}p)$ is the Legendre symbol. I have a ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
1 answer
185 views

Is a polytope with vertices on a sphere and all edges of same length already rigid?

Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties: all vertices are on a common sphere. all edges are of the same length. I suspect that such a polytope is ...
M. Winter's user avatar
  • 12.5k
1 vote
0 answers
142 views

Non-trivial lower bound on the number of "Graph Diagonals"

The definition of Graph Diagonals, that are the subject of this question, is based on the notions of crossing edges and on connected graphs: Two edges $AC$ and $BD$ of a complete, symmetric and ...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
106 views

Hard instances for this graph isomorphism algorithm based on powers of weighted adjacency matrices?

In short, I found an algorithm for GI and the only hard instances I found so far are non-isomorphic strongly regular graphs with large automorphism groups. Q1 What are hard instances for the ...
joro's user avatar
  • 24.2k

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