Let $(X, T)$ be a minimal subshift, i.e. $X$ is a closed $T$-invariant subset of $A^\mathbb{Z}$, where $T$ is the shift. A pair $x,y\in X$ is asymptotic if $d(T^nx, T^ny)$ goes to zero as $n\to\infty$. Always exists such a pair when $X$ is infinite: for every $n\geq1$ there exists $x^{(n)}, y^{(n)} \in X$ such that $x^{(n)}_0\not= y^{(n)}_0$ and $x^{(n)}_{[1,N]}= y^{(n)}_{[1,N]}$ (if not, for some $n$, $x_{[1,n]} = y_{[1,n]}$ implies $x_0=y_0$, i.e., $x_{[1,n]}$ determines $x_0$, and this forces $X$ to be periodic), and any pair of convergent subsequences of $x^{(n)}$ and $y^{(n)}$ will do the trick.

**My question is: do exist $k$-tuples of asymptotic points, for every $k\geq1$?**
More precisely, is it true that for every $k\geq1$ there exists $x_1,\dots,x_k\in X$ such that
$$\lim_{n\to\infty}d(T^nx_i, T^nx_j) = 0\ \forall i,j$$

Suppose $A$ is a non-unital residually finite-dimensional (RFD) $C^*$-algebra, then the multiplier algebra $M(A)$ is also RFD. I wonder whether there exists a trace on the corona algebra $M(A)/A$?

Given an automorphic representation of the general linear group over a totally real number field, there are 3 properties it might have: regular, algebraic (or "of algebraic type"?) and cuspidal. There are $2^3=8$ possible combinations of which properties are true for a given representation. Can you give explicit examples for each of the 8 combinations?

Can you give examples for the general linear group over $\mathbb{Q}$? If yes, what is the minimum $n$ (as in $\mathrm{GL}_n$) such that all combinations arise?

Let $G$ be a simple “cycle plus triangles” graph, that is, a graph with $3k$ vertices, $k>1$, the edges of which can be partitioned into a set that induces a $3k$-circuit, together with sets that induce disjoint triangles (3-circuits). Note $G$ is 4-regular.

UPDATED QUESTION (2019 may 17):

QUESTION: Is $G$ class 1, that is, can it be edge-4-colored, if it has an even number of triangles?

(SIDE) FACT : If $G$ has an odd number of triangles then it is class 2. This is because a cycle plus triangles graph with an odd number of triangles is a regular graph of odd order ... all such graphs are class 2 (suppose it is class 1 ... then each vertex is incident with an edge of each color. But the edges of a given color are independent, so they are incident with an even number of vertices, while there is an odd number of vertices, contradiction).

The hypothesis that C+T graphs which have an even number of triangles are class 1 is supported by computer experiments.

Cycle plus triangles graphs became well known when Erdös posed the “cycle plus triangles problem” (whether such graphs are vertex-3-colorable). This was solved affirmatively by Fleishner and Stiebitz using the Alon-Tarsi theorem, and later Sachs, inductively.

Perhaps the answer to my question above is known, but I am unaware of it.

In the generalization of this question to “cycle plus even $k$-cliques” (I hope it is clear what this means) the answer is that they are all class 1, as you can edge color the edges of the $k$-cliques with $k$-1 colors, and use two remaining colors for the edges of the even cycle in the ($k$+1)-regular graph. It is tantalizing to speculate that a similar answer as the answer for the “cycles plus triangles” case would hold for the general “cycle plus odd cliques” situation, of which the cycle plus triangles is a special case. It would be nice to settle at least this special case with proofs, if it is not settled already.

Imagine two beads each on parallel rods. The beads are connected to each other by a spring of stiffness $k$. The beads each exert an equal and opposite force $F$ to each other. The distance between the rods is $l$. If the beads are initially both at height zero and released they will begin to move in opposite directions before eventually coming to an equilibrium position.

The equilibrium position is the point at which the vertical component of the tension in the spring is equal to the force exerted by the beads.

There is a non-linear relationship between the force exerted by the beads and the final vertical distance between them. The same goes for the final angle.

If either the height or the angle between the beads at equilibrium was known the problem would be straight forward, but they are not known.

The Physics of Springs provides a method to solve more complex spring systems as long as the angle between the springs is assumed fixed as this allows a linear relation between distance and force. However, the angle is not fixed in this case.

This problem can be solved using numerical methods but takes time to converge and for a single node pair isn't worth it.

My question is: What analytical method can I use to find the distance between the two beads?

**Note:** This question was asked on Physics overflow but was closed and deleted as being an off-topic/homework. It is also similar to this question, which was also closed. I would argue however that on the basis that for more than two nodes the system may not have an analytical solution, the analytical solution to the special case of 2 nodes is of general interest and non-trivial due to the non-linear relationship. The analytical solution of the two-node case speeds up finding the equilibria position of many network configurations.

can i get some help with solving those questions? i am not good at math and got an exam coming up so i am solving older exams as a reference but i don't have the answers for it to be sure that i solved it correctly. thanks in advance!

questions are;

mathematical induction

binomial expansion

partial fraction

Taylor's expansion

integration

differentiation

newton forward formula

matrix

Fourier series expansion

examples

I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. I have looked in the book of Ronveaux, as well as the NIST Handbook of mathematical functions and could not find any information on this problem. Is there anything known in that direction? I could find some information on where this happens for a hypergeometric function.

Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$. Suppose that $\mu \neq 0$ and $\mu \in \text{Sp}f(\lambda_0)$ for some $\lambda_0 \in D$.

Consider the set $E = \{ \lambda \in D: \mu \in \text{Sp}f(\lambda) \}$.

B. Aupetit mentions in a proof he writes for Theorem 3.4.26 in his book *A Primer on Spectral Theory*, that this set $E$ is closed by the upper semicontinuity of the spectrum. I am struggling to see why this is true.

Can anyone please point me in the right direction as to how I can show that $E$ is closed by the upper semicontinuity of the spectrum?

Consider the problem $$ \min p(x) \text{ subject to } g_j(x)\le 0 \quad p,g_j\in\text{SOS}, \qquad (*) $$

i.e. $p,g_j$ ($j=1,\ldots,m$) are *sum of squares* (SOS) polynomials. Can this problem be solved efficiently?

The paper [1] shows that unconstrained minimization of SOS polynomials can be reduced to a convex program. Surprisingly, the paper goes onto consider general (non-SOS) constrained polynomial optimization to derive the Lasserre hierarchy, but never explicitly discusses the special case $(*)$ above.

[1] *Lasserre, Jean B.*, **Global optimization with polynomials and the problem of moments**, SIAM J. Optim. 11, No. 3, 796-817 (2001). ZBL1010.90061.

In every field of pure mathematics, there are several layers of abstraction a person must absorb before they can contribute anything of note.

To absorb each layer a person needs to spend some effort (and thus some time). Say, for quite a few people, the first time they see the definition of a topological manifold it feels "weird" or somehow unintuitive. After they do some exercises, they typically understand that the definition is actually fairly natural and captures what it is supposed to capture reasonably well.

For example, in arithmetic geometry you need to understand the definition of a scheme (takes quite some time, doesn't it), a site & a topos, also the standard package of homological algebra and some other stuff too. I think an average person who decides to go into this field learns this stuff by the time they are 24-25 years old.

The life expectancy of a human being is not infinite, however. I think it is not significantly above 90 years in any given country, as of May 2019. From what I know, after we max out the advancements in medicine, the life expectancy would be 120 years (assuming we do not use replacement organs en masse).

Was there any research making an educated guess about how much time is left before a sufficient number of layers of abstraction accumulate so that a person would have to spend their entire life in the graduate school to learn all of them (and then die, having no time left to apply them to anything)?

Is there any counterexamples known to the following statements? ($A$ a commutative algebra)

If $K_1(A)$ is finitely generated then $K_n(A)$ is finitely generated for any $n\geq 2$

If $K_1(A)$ is torsion then $K_n(A)$ is torsion for any $n\geq 2$

Every matrix $A\in M_4(\mathbb{R})$ can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$.

What is the least uniform upper bound $M$ for such $n(A)$?In the other words, what is the least integer $M$ such that every $A$ admit such a representation with $n(A)\leq M$?

Is this least upper bound equal to the corresponding least upper bound for all matrices $A$ which are a matrix representation of quaternions $a+bi+cj+dk$?

As another question about tensor product representation: What is a sufficients condition for a $4\times 4$ matrix $A$ to be represented in the form of $A=B\otimes C -C\otimes B$?

Is it possible to calculate the norm associated to fractional Laplacian of $u$ and $s\in (0, 1).$

$$\|(-\Delta)^{s/2} u\|_2^2=\int_{\mathbb R^N}|(-\Delta)^{s/2} u|^2dx=C_{N,s}\int_{\mathbb R^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}dxdy$$

For example, $u(x)=(1+ |x|^2)^{-\frac{N-2s}{2}},$ can we calculate $\|(-\Delta)^{s/2} u\|_2$ just by using the above formula. I am aware $u$ is the some type of solution to the fractional Yamabe problem but I want a result independent proof. Any reference is welcome.

This is my first question here - so if i'm out of format - let me know :)

The problem is this:

$m$ identical urns

$k$ identical cells in each urn

$n$ identical balls

The balls are distributed among the urns, with no restriction other than the maximal capacity of each urn (up to $k$ balls - one in each cell). The question at hand is: What is the probability that exactly $t$ urns are occupied by $r$ balls.

I thought of approaching it the same way Fang does (here: https://www.researchgate.net/publication/305977113_A_restricted_occupancy_problem): $P(M_t=r)=N(m,k,n,r,t)/N(m,k,n)$

Where $N(m,k,n,r,t)$ is the number of ways of distributing the $n$ balls into the $m$ urns, such that exactly $r$ of them have exactly $t$ balls.

And where $N(m,k,n)$ is the number of ways of distributing the balls into the urns.

The latter is simple to calculatr (if I'm not mistaken: $N(m,k,n)=mk(mk-1)\cdot...\cdot(mk-n+1)$), but the former seems quite hard. I have trouble using the generating functions to solve $N(m,k,n,r,t)$, because I can't find the independent variables to use (so that the generating functions can be multiplied).

I'd love it if someone could explain clearly how he would count $N(m,k,n,r,t)$. Thanks in advance:)

I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a direct attack on the question stated in the title.

**Notational conventions:** $L_p$ is short-hand for $L_p([0,1])$ with the Lebesgue measure on $[0,1]$. If $E$ is a Banach space then $L_\infty(E)$ denotes the space of essentially bounded, strongly measurable functions $[0,1]\to E$ (modulo equivalence a.e.).

In the case where $E=L_1$, we shall regard elements of the space $A=L_\infty(L_1)$ as functions of two variables which are $L_\infty$ in the *second* variable and $L_1$ in the *first* variable. So here the norm would be given by
$$
\Vert f \Vert = \operatorname{ess.sup}_{t\in [0,1]} \Vert f( \cdot, t)\Vert_1
$$
This convention has the advantage that given $f\in A$ and $\xi \in L_1$ we can define $T_f(\xi) \in L_1$ by
$$
T_f(\xi)(s) = \int_0^1 f(s,y)\xi(y)\,dy
$$
The map $f\mapsto T_f$ is an isometric embedding of $A$ as a closed subalgebra of $B(L_1)$. For given $f,g\in A$, we may define
$$
(f\bullet g)(s,t) = \int_0^1 f(s,x) g(x,t)\,dt
$$
Then $f\bullet g\in A$ and $T_fT_g = T_{f\bullet g}$. The image of $A$ in $B(L_1)$ under this embedding, which we will denote by $J$, turns out to be a much studied object in the theory of operators on $L_1$: it is the set of *representable* operators from $L_1$ to itself.

QUESTION: does $J$ (or equivalently $(A,\bullet)$) have a bounded right approximate identity? What if we drop the requirement of boundedness?

Note that the naive attempt of taking simple functions in $L_\infty(L_1)$ that approximate the "Dirac" measure concentrated on the diagonal in $[0,1]^2$ won't work, because such functions get sent by our embedding to elements of $K(L_1)$, which is properly contained in $J$ (see below).

*Some remarks for background context, which might be relevant to a solution.*

It can be shown that $J$ has no left approximate identity (bounded or otherwise). I would like to thank W. B. Johnson for indicating why this is the case; an expanded and paraphrased version of his explanation is given below.

From some vector measure theory, we know that $J$ contains the ideal $W(L_1)$ of weakly compact operators. Moreover, $J$ is contained in the ideal $CC(L_1)$ of completely continuous operators; "completely continuous" means that weakly convergent sequences are mapped to norm convergent sequences).

(The containment $J\subseteq CC(L_1)$ follows from a theorem of Lewis and Stegall, which characterizes operators in $J$ as those which factor through $\ell_1$. This also shows that $J$ is a *$2$-sided ideal* in $B(L_1)$, not just a subalgebra.)

It follows that if $S\in W(L_1)$ and $T\in J$, then $TS\in K(L_1)$. Since there exist weakly compact operators $S$ on $L_1$ which are not compact (e.g. take any non-compact map $L_1\to \ell_2$ and then compose with an isometric embedding $\ell_2\to L_1$), it follows that there is no net $(T_\alpha)$ in $J$ such that $\Vert T_\alpha S - S \Vert \to 0$.

Given the center coordinates $\{x_i,y_i\}$ and radii ($r_i$) of three circles ($i=1,2,3$) (the areas of which are certain probabilities), the three pairwise intersections being all non-empty, what is the condition that their three-way intersection is, on the other hand, empty?

In particular, I have in mind three circles $A,B,C$ of radii $\frac{8 \pi}{27 \sqrt{3}}$ and $\frac{1}{6}$ and $\frac{1}{6}$. The intersection of $B$ and $C$ is $\frac{1}{9}$ and $A \land B$ and $A \land C$ are both $-\frac{4}{9}+\frac{4 \pi }{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$, while as indicated, $A \land B \land C$ is $\varnothing$.

This problem pertains to my efforts (https://mathematica.stackexchange.com/questions/198019/create-a-venn-diagram-showing-the-relations-of-three-sets-of-quantum-states/198091#198091) to represent--via a Venn or related diagram--the Hilbert-Schmidt probabilities of certain quantum ("positive-partial-transpose" and "bound-entangled") "two-qutrit" states (representable by $9 \times 9$ density matrices).

So, how to locate the three centers--subject to the three circles all being contained in a circle (arbitrarily centered at the origin) of area/probability 1--is also in question (https://math.stackexchange.com/questions/3224845/given-a-circle-a-of-area-1-centered-at-0-0-give-conditions-that-another).

I also how further information as to the probabilities assigned to various unions and intersections of $A,B,C$ and their negations--which might be helpful.

Suppose I have a coalgebra $\mathcal{C}$ in the $\infty$-category of presentable stable $\infty$-categories (with continuous functors). If the structure functors for $\mathcal{C}$ all have continuous right adjoints, then passing to right adjoints I get an algebra structure on $\mathcal{C}$. However, in the general situation, I only have a full stable subcategory $\mathcal{D}$ of $\mathcal{C}$ such that the right adjoints preserve $\mathcal{D}$, i.e. they induce $\mathcal{D}\otimes \mathcal{D}\rightarrow\mathcal{D}$, and restricting to $\mathcal{D}$ they are continuous. Note that $\mathcal{D}$ is not necessarily a sub-coalgebra of $\mathcal{C}$.

Is there a clean way to show that in the above situation, $\mathcal{D}$ naturally induces an algebra structure? All I can say is that all the homotopical data needed are contained in the coalgebra structure of $\mathcal{C}$, but this seems not so clean.

Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$. I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$, then there is global solution (defined for all $t\in\mathbb{R})$ of $$ u_t = i (\Delta u + \vert u \vert ^{\alpha -1} u) $$ $$ u(\cdot,0)=u_0 $$

I have proved the existence of solution (local). Now I want to prove that is in fact global. In order to do that, I thought to use the blow-up alternative, that is, I just have to show that $\Vert u \Vert _{H^1(\mathbb{R}^N)}$ is bounded, where $u$ is a $H^1-$solution. It is clear that $ \Vert u \Vert _{H^1(\mathbb{R}^N)} = \Vert u \Vert _{L^2(\mathbb{R}^N)} + \Vert \nabla u \Vert _{L^2\mathbb{R}^N)}$.

By the mass conservation, $\Vert u \Vert _{L^2(\mathbb{R}^N)} = \Vert u _0\Vert _{L^2(\mathbb{R}^N)} $. By the energy conservation, $ \frac{1}{2}\Vert \nabla u\Vert_{L^2} ^2 - \frac{1}{\alpha + 1} \Vert u \Vert_{L^{\alpha+1}} ^{\alpha +1} \equiv E\in\mathbb{R} $ therefore, by Sobolev embedding, $$ \Vert \nabla u\Vert_{L^2} ^2\leq 2E + \frac{2}{\alpha + 1} \Vert u \Vert_{L^{\alpha+1}} ^{\alpha +1}\leq 2E + \frac{2C}{\alpha + 1} \Vert u \Vert_{H^1} ^{\alpha +1}$$

If I show that $\Vert \nabla u\Vert_{L^2} ^2 \leq 2E + \frac{2C}{\alpha + 1} \Vert u \Vert_{H^1} ^{\alpha +1} \leq \text{constant}$ I finish.
I have the assumption that **$\Vert u _ 0 \Vert _{H^1}$ is small, does it implies that $\Vert u \Vert _{H^1}$ is also small?**
As I am reading on Internet, I found something called Lyapunov stability.
Maybe is useful here, can anyone explain how could be used in this situation? Thanks in advance. Any idea is welcome!

What are the existing studies and future directions of work to be done with constructing Mobius transformations with sterogrpahic projections from Riemann Sphere to the complex plane, rigid motions taking one Riemann sphere to another, and mapping that second sphere back to the complex plane ?

Even if not, what are some possible directions of research that can be done using Mobius transformations in the plane and rigid motions (or order preserving isometries) in the space ?

Thank you for you time and suggestions.

I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new idea, or that are counterexamples to a statement that might look reasonable to a beginner but is actually false? Inspired by this question.