I was randomly browsing, when I found this puff piece claiming a proof of the Lindelöf hypothesis by Fokas. Note that the Wikipedia article says that he claimed, then withdrew his claim in 2017, but the USC piece is dated June 25 2018. So, what is the truth?

Consider all knots with fixed genus $g\ge 2$ (I am considering the classical 3-genus). Do there exist infinite families of genus $g$ knots with arbitrarily large volume?

The answer seems like it should definitely be yes, but I can’t seem to find any references.

*Disclaimer. The practical execution of the algorithm in question might be illegal in certain jurisdictions, and is thus strongly discouraged by the poser of the problem.*

This recent post on the Muffin problem made me think of the following question.

Can we cut 100 banknotes into pieces of size at least $10\%$ each, and reassemble them into 101 banknotes of size $100\pm2\%$ each?

So each original banknote is cut into at most $10$ pieces of substantial size, and each new banknote also consists of at most $10$ pieces. The patterns on these newly formed banknotes should match, so we also demand, say, that no part of a banknote appears twice on a new banknote. Of course, these numbers are quite ad hoc, I'm happy to see any similar result. Note that if we don't require each piece to be at least $10\%$, then it is easy to make the trick by cutting each banknote into only two (sometimes very unequal) parts. I also wonder if non-vertical cuts might help, but I would like to keep the pieces simply connected regions bounded by Jordan curves.

Also, is there some implication between this question and the Muffin problem?

Consider a Randers space $(M,F)$ that is the solution of the zermelo's navigation problem associated to a wind $W$ which is homothety; $\mathcal{L}_Wh=\sigma h$, $\delta$ constant, on a Riemannian space $(M,h)$. Then the Randers geodesics can be found using Theorem 2 of Robles.

Now I am wondering if there is any way with which one can find the Randers geodesics of the Randers space ($\mathbb{R}^3,F$) which comes from putting the wind $W=(b+a\sin kx,0,0)$ on the Euclidean space $\mathbb{R}^3$. By using the equation of the geodesics it sounds quite difficult.

P.S.: here $a, b$ and $k$ are some constant.

This question (as the title obviously suggests) is similar to, or a continuation of, this question that was asked years ago on MO by a different user. The present question, though, is different from the old one in some ways:

The old question focussed on reference request. My question instead, while clear and readable references are welcome, is more focussed on getting some quick and dirty intuitive understanding. Most probably what I'm looking for is already buried in Tom Leinster's comprehensive monograph

*Higher operads, higher categories*, but currently I'm not planning on reading it (or going in a detailed way through other technical material on the topic).I already have an idea of how algebraic theories (also called Lawvere theories) and monads differ from each other. I would also like to understand how operads fit into this picture.

In the old question the semantic aspect was not considered (yes, there's the expression "model of theories" in that question, but just in the sense of "way of understanding the notion of mathematical algebraic theory", not in the more technical sense of semantics).

In what follows I may be missing some hypothesis: let me know or just add them if necessary.

Here is what I remember about the algebraic theories vs monads relationship. Algebraic theories correspond one-to-one to finitary monads on $\mathbf{Set}$, and the correspondence is an equivalence of categories. If one wants to recover an equivalence on the level of *all* monads on the category of sets, then one has to relax the finitary condition on the other side, so getting a generalization of the notion of Lawvere theory to some notion of "infinitary Lawvere theory". Furthermore, given a (just plain old, or possibly non-finitary) Lawvere theory $\mathcal L$ and the corresponding monad $\mathscr T$, a model of (also called algebra for) $\mathcal L$ in $\mathbf{Set}$ corresponds to a $\mathscr{T}$-algebra (necessarily in $\mathbf{Set}$), and this correspondence extends to an equivalence of the categories of algebras
$$\mathrm{Alg}^{\mathcal{L}}(\mathbf{Set}):=\mathrm{Hom}_{\times}(\mathcal{L},\mathbf{Set})\simeq \mathrm{Alg}^{\mathscr T}(\mathbf{Set})=:\mathbf{Set}^{\mathscr T}\,.$$
There is also the variant with *arities*. While usual Lawvere theories / finitary monads have finite ordinals as arities, and the corresponding non-finitary versions have sets as arities, one can introduce the notions of *Lawvere theory with arities* from a category $\mathfrak{A}$ and *monads with arities* from $\mathfrak{A}$, and again one has an equivalence between these notions, and an equivalence between the algebras in $\mathbf{Set}$.

**a)** How do operads fit into all this? Are operads somehow *more general* or *less general* objects than monads/theories (with all bells and whistles)?

There is an asymmetry between the notion of semantics for Lawvere theories and for monads (In what follows I will drop the finitary assumption). Namely, every Lawvere theory $\mathcal{L}$, as remarked above, is essentially a monad $\mathscr{T}_{\mathcal{L}}$ on $\mathbf{Set}$; though it can have models in every (suitable) category $\mathcal C$: just define the category of models of $\mathcal L$ in $\mathcal C$ to be $\mathrm{Hom}_{\times}(\mathcal L,\mathcal C)$. On the other hand, a monad $\mathscr T$ on $\mathbf{Set}$, by definition, *only* has models (aka algebras) in $\mathbf{Set}$. To remedy this, one introduces $\mathcal{V}$-*enriched* monads $\mathscr T$, where $\mathcal{V}$ is a given monoidal category, and $\mathscr T$-algebras are now objects of $\mathcal{V}$ (endowed with some further structure). If I get it correct, there is now an equivalence of categories $\mathrm{Hom}_{\otimes}(\mathcal{L},\mathcal V)\simeq \mathcal{V}^{\mathscr{T}}$.

But models of $\mathcal L$ are related to each other: you can take, again if I get it correct, (nonstrict?) $\otimes$-functors $\mathcal{V}\to\mathcal{V}'$ intertwining the (strict?) tensor functors $\mathcal{L}\to\mathcal{V}$ and $\mathcal{L}\to\mathcal{V}'$.

**b)** Can we read these "inter-model" relationships in the language of (enriched) monads?

**c)** How does the above semantic aspect go for operads and how, roughly, does the translation go from there to monads/theories (and viceversa)?

**d** Is there a "natural" symultaneous generalization of all the three things (theories, monads, and operads)? Does also the notions of semantics have a "natural" common generalization?

**Edit. I had written the above (comprising questions a),...,d) ) some time ago, and just posted it now. But I forgot that I also wrote essentially the same questions in a perhaps more systematic way! I'm now going to post it below, without deleting the above paragraphs.**

**Q.1** What is the most general monads/theories equivalence to date? (possibly taking into account at the same time: arities, enrichment, and maybe sorts)

**Q.2** What is the most general monads/algebras adjunction? (again, possibly throwing arities, enrichment, and maybe sorts, simultaneously into the mix)

$$\mathrm{Free}^T:\mathcal{E}\rightleftarrows \mathrm{Alg}^T(\mathcal{E})=:\mathcal{E}^T:U^T$$

where $T$ is a monad on $\mathcal{E}$, $U^T$ is a forgetful functor and $\mathrm{Free}^T$ a "free $T$-algebra" functor.

**Q.3** Which is the relation between the different notions of semantics (for monads and theories)? Can enrichment "cure" this asymmetry?

This question was further explained a bit in question **b)** above.

**Q.4** How far are operads from being algebraic theories?

There's an article by Leinster in which it is shown that the natural functor

$$G:\mathbf{Opd}\to \mathbf{Mnd}(\mathbf{Set}),\quad P\mapsto T_P$$

where

$$T_P(X):=\amalg_{n\in \mathbb{N}}P(n)\times X^{\times n}$$

is *not* an equivalence, and it is shown that the essential image of $G$ is given by "strongly regular finitary monads" on $\mathbf{Set}$, or equivalently by those Cartesian monads $T$ admitting a Cartesian monad morphism to the "free monoid monad". In the case of symmetric operads, $G$ becomes an equivalence.

So, question Q.4 is about such a functor $G$ in the most general setting (arities, enrichment,...), and can be seen as asking: which properties does such a functor $G$ have? What is a characterization of its essential image? What is the (essential) fiber of $G$ over a $T\in G(\mathbf{Opd})$?

**Q.5** How does the notion of semantics for operads (well, it's algebras for an operad) relate to the notion of semantics for monads?

(this is similar to question Q.3, but featuring operads instead)

**Q.6** In the light of the above, what is an operad, intuitively?

An operad $P$ has, in general, "more structure" than its associated monad $T_P$ ($G$ not injective). Also, $P$ is "more rigid" an object than $T_P$ ($G$ not full). So what do these extra things amount to? This must be some sort of detail, because clearly both monads and operads "want" to be a formalization of the intuitive notion of "algebraic theory".

**Q.7** Do all the above considerations naturally extend to the case of $T$-*categories* instead of $T$-*algebras*? Maybe one has to consider colored operads? Or sorts?

**Q.8** Which further structure on a monad/theory describes an *equational* theory? Same question in arities, enriched, or sorts, flavor.

This is about the distinction between e.g. the (logical formal) equational theory of groups and the *Lawvere* theory of groups (or, equivalently, the corresponding monad). So, what further structure on a theory/monad can be seen as a "presentation" of it.
Also, which properties does the functor

$$\{ \textrm{equational theories}\}\to \mathbf{Mnd}$$

have? Ess. surjective? Full? (I think not). Faithful? (again, no...).

My calculations for $x=1, 3,\dotsc, 10^7-1$ lead to the following conjecture.

Let $x$ be an odd integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{n+m}-P_n-P_m,$$ where $P_n$ is the $n^{th}$ prime.

I am looking for a comment, reference, remark, or proof.

Maybe a general of the conjecture above as follows **(but weaker)**:

Let every integer $r_0$ exist positive integer $x_0$ such that every odd number $x \ge x_0$ has the form $x=P_{n+m+r_0}-P_n-P_m$, where $m, n \ge 2$ (PS: inspired from the comment of Lev Borisov be low)

Or simpler:

*Is every odd integer of the form $P_{c}-P_a-P_b$?*

I am reading https://arxiv.org/abs/0806.4160 to understand orbifolds as stacks.

Definition : Let $D\rightarrow Man$ be a stack over category of manifolds. An atlas for $D$ is a manifold $X$ and a map $p:X\rightarrow D$ such that for any map $f:M\rightarrow D$ from a manifold the fiber product $M\times_D X$ is a manifold and the map $pr_1:M\times_D X\rightarrow M$ is a surjective submersion.

I fail to understand what this means. I know what is a stack but I am not sure what it means to say a map from a manifold to a stack.

Any clarity would be welcome.

Before giving this definition, he says,

To keep the notation from getting out of control we drop the distinction between a manifold and the associated stack $\underline{M}$. We will also drop the distinction between stacks isomorphic to manifolds and manifolds.

This only made the definition complicated and not easier (for me) :D

Help me to understand the notion of atlas.

If it helps, I am trying to read about geometric stacks which are defined to be stacks over manifolds which possesses an atlas.

Here is a naive outsiders perspective on set theory: A typical set-theoretical result involves constructing new models of set theory from given ones (typically with different theories for the original model and the resulting model). Typically, from a meta-perspective we are allowed (encouraged, or even required) to assume that the models are countable.

To the extent that this view is correct, set-theoretic constructions correspond to partial, multivalued operations $T : \subseteq \{0,1\}^\mathbb{N} \rightrightarrows \{0,1\}^\mathbb{N}$ which are defined on sequences coding a model of the original theory, and are outputting a sequence coding a model of the desired theory. These are multivalued, because the constructions may involve something like "pick a generic filter for this forcing notion".

**Question: Are these operations typically computable, and if not, how non-computable are they?**

The Weihrauch degrees (https://arxiv.org/abs/1707.03202) provide a framework for classifying non-computability of operations of these types. An answer, however, could also take forms like "For arguments like this, a resulting model is typically computable in the join of the original model and a 1-generic."

The Fejer-Jackson inequality as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$

I conjecture that the inequality as follows holds:

$$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$$

How to prove this inequality? Can You give a comment or a proof or a reference?

The last digit of 12^12 + 13^13 – 14^14×15^15 =? - By Abdullah Allad

Let $A$ be a positive definite matrix. Then $A$ is diagonalized by an orthogonal matrix $P$.

I want to know, when this matrix is also an involution. ie. $P^2 = I$.

If there is any characterization of such $A$ kindly share.

Thank you.

I asked this question on math.stackexchange.com, and getting no good answers, I offered a bounty, and still didn't get any answer that was somewhat complete (but the system automatically awarded the bounty). So maybe I can get something better here.

I have encountered a simple problem in probability where I would not have expected to find conditional convergence lurking about, but there it is. So I wonder:

- Is any insight about probability to be drawn from the fact that infinity minus infinity appears here?
- In particular, do the two separate terms that evaluate to infinity have some probabilistic interpretation?
- Is any insight about analysis or anything else to be found here?
- Generally in what situations does one write $\displaystyle \int_{x\,:=\,0}^{x\,:=\,1} u\,dv = \Big[ uv \Big]_{x\,:=\,0}^{x\,:=\,1} - \int_{x\,:=\,0}^{x\,:=\,1} v\,du$ and find that those last two terms are both infinite even though the first one converges absolutely?
- In particular, are any instances of this particularly notable or worth knowing about?

**The probability problem**

Suppose $X_1,\ldots,X_6$ are independent random variables each having the same exponential distribution with expected value $\mu,$ so that $$ \Pr( X_1 > x) = e^{-x/\mu} \quad\text{for } x\ge0. $$ It is desired to find this expected value $$ \operatorname E(\max\{\,X_1,\ldots,X_6\,\}) = \mu\left( 1 + \frac 1 2 + \frac 13 + \frac 1 4 + \frac 1 5 + \frac 1 6 \right) = 2.45\mu. \tag 1 $$ One fairly routine way to show this goes like this: $$ \Pr(\min\{\,X_1,\ldots,X_6\,\} > x) = \big( \Pr(X_1>x) \big)^6 = e^{-6x/\mu}\quad \text{for }x\ge0, $$ and therefore $$ \operatorname E(\min) = \frac \mu 6. $$ Let $X_{(1)}, \ldots, X_{(6)}$ be the order statistics, i.e. $X_1,\ldots,X_6$ sorted into increasing order. Then we have $$ \operatorname E(X_{(1)}) = \frac \mu 6, \quad\text{and } \operatorname E(X_{(2)} - X_{(1)}) = \frac \mu 5 $$ because that difference is the minimum of five exponentially distributed random variables. And so on through the last one.

No conditional convergence appears above.

**But suppose instead we just reduce it to evaluation of an integral.**

\begin{align} & \Pr(\max \le x) = \Pr(X_{(6)} \le x) = \Pr( \text{all of }X_1,\ldots,X_6 \text{ are} \le x) = \left( 1 - e^{-x/\mu} \right)^6 \text{ for } x\ge0. \\[10pt] & \text{Hence for measurable sets $A\subseteq[0,+\infty)$ we have } \Pr(\max\in A) = \int_A f(x)\, dx \\[10pt] & \text{where } f(x) = \frac d {dx} \left( 1 - e^{-x/\mu} \right)^6 = 6\left( 1- e^{-x/\mu} \right)^5 ( e^{-x/\mu}) \frac 1 \mu. \end{align}

So here's our integral: $$ \operatorname E(\max) = \int_0^\infty xf(x)\, dx. $$ No suggestion of conditional convergence, right?

\begin{align} \operatorname E(\max) = \int_0^\infty xf(x)\, dx & = \int_0^\infty x 6\left( 1- e^{-x/\mu} \right)^5 ( e^{-x/\mu}) \, \frac {dx} \mu \\[10pt] & = \mu \int_0^\infty s 6( 1-e^{-s})^5 e^{-s} \, ds \\[10pt] & = \mu \int s\, dt = \mu st - \mu\int t\,ds \\[10pt] & = \mu s(1-e^{-s})^6 - \mu \int (1-e^{-s})^6 \, ds. \end{align} Now a substitution: \begin{align} r & = 1-e^{-s} \\[6pt] s & = -\log(1-r) \\[6pt] ds & = \frac{dr}{1-r} \end{align} Our integral becomes \begin{align} & \mu ( - r^6 \log(1-r) ) - \mu \int \frac{r^6}{1-r} \, dr \\[10pt] = {} & \mu ( - r^6 \log(1-r) ) - \mu \int \left( -r^5 - r^4 - r^3 - r^2 - r - 1 + \frac 1 {1-r} \right) \, dr \end{align} Now the temptation is to write $$ \require{cancel} \xcancel{\left[ \mu \left( -r^6 \log_e(1-r) \right) \vphantom{\frac11} \right]_0^1} - \xcancel{\mu \int_0^1 \left( -r^5-r^4-r^3-r^2 - r -1 + \frac 1 {1-r} \right) \, dr }. $$ The problem is that this is infinity minus infinity, so we have conditional convergence. So suppose we write it like this: \begin{align} & \left[ \mu \left( -r^6 \log(1-r) \right) - \mu \int \left( -r^5-r^4-r^3-r^2 - r -1 + \frac 1 {1-r} \right) \, dr \right]_0^1 \\ & \text{(The above is not standard notation, as far as I know.)} \\[10pt] = {} & \mu \left[ (1-r^6) \log_e (1-r) + \left( \frac{r^6} 6 + \frac{r^5} 5 + \frac{r^4} 4 + \frac {r^3} 3 + \frac{r^2} 2 + r \right) \right]_0^1 \end{align} After we use L'Hopital's rule to evaluate the first term, this ends up being just what we see in $(1).$

Maybe I'll post my own answer if I am so inspired, but other answers may provide valuable alternative points of view. (I don't have an answer to post yet.)

**Postscript:**

Where I've seen something similar before is in attempts to prove that if $\Pr(X\ge0) = 1$ and $f$ is the p.d.f. and $F$ the c.d.f. of $X$, then

$$ \int_0^\infty xf(x)\, dx = \int_0^\infty (1-F(x))\,dx. $$

If you write

$$ \int(1-F(x))\,dx = \int u\,dx = xu - \int x\, du = \text{etc.,} $$

then you get infinity minus infinity. But you can do this:

\begin{align} & \int_0^\infty xf(x)\, dx = \int_0^\infty \left( \int_0^x f(x)\,dy \right) \, dx \\[10pt] = {} & \int_0^\infty \left( \int_y^\infty f(x) \,dx\right) \, dy \\[10pt] = {} & \int_0^\infty (1-F(y))\,dy. \end{align}

Tonelli's theorem is applicable since the function being integrated is everywhere non-negative, so that justifies the change in the order of integration.

This is the "real-life" (but slightly more technical) version of a question I have asked recently.

For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of all those lines in $\mathbb F_p^2$ parallel to the lines $$ X:=\{(x,0)\colon x\in\mathbb F_p \}, \ Y:=\{(0,y)\colon y\in\mathbb F_p \}, \ Z:=\{(z,z)\colon z\in\mathbb F_p \}, $$ respectively; thus, $|\mathcal L_X|=|\mathcal L_Y|=|\mathcal L_Z|=p$. Write $$ \chi(x,y) := \omega^x,\quad (x,y)\in\mathbb F_p^2, $$ where $\omega$ is a fixed primitive root of unity of degree $p$. Given a set $S\subseteq\mathbb F_p^2$, with every element $s\in S$ associate a formal variable $x_s$, and consider the system of homogeneous linear equations \begin{gather*} \sum_{s\in S\cap\ell} x_s = 0,\quad \ell\in\mathcal L_X\cup\mathcal L_Y, \\ \sum_{s\in S\cap\ell} \chi(s)\,x_s=0, \quad \ell \in \mathcal L_Z; \end{gather*} notice that there are $3p$ equations and $|S|$ variables. Does there exist a set $S\subseteq\mathbb F_p^2$ of size $|S|<3p$ for which this system has a solution such that the set $\{s\in S\colon x_s\ne 0\}$ meets every line in $\mathbb F_p^2$?

Let $X$ be a Riemann surface and $O_{X}(D)$ be the line bundle associated with $D$. Let the metric on $O_{X}(D)$ be given by $$ |1_{O_{X}(D)}(P)|=G(P,D)^2 $$ where $G(P,D)^2$ is the Green function associated with $D$ and $P$ which vanishes up to first order on the diagonal.

My question is whether $O_{X}(D)$ equipped with this metric is a reflective Banach space over $\mathbb{C}$.

Definition : Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. A **bibundle** from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with two maps $a_L:P\rightarrow \mathcal{G}_0,a_R:P\rightarrow \mathcal{H}_0$ such that

there is a left action of $\mathcal{G}$ on $P$ with respect to an anchor $a_L$ and a right action of $\mathcal{H}$ on $P$ with respect to an anchor $a_R$.

$a_L:P\rightarrow \mathcal{G}_0$ is a principal $H$-bundle.

$a_R$ is $\mathcal{G}$ invariant.

the actions of $\mathcal{G}$ and $\mathcal{H}$ commutes.

I am trying to understand in what sense these are called generalized morphisms between Lie groupoids.

There is already a notion of generalized morphsim between Lie groupoids from Ieke Moerdijk's article Orbifolds as groupoids.

Definition : A generalized morphism from a Lie groupoid $\mathcal{G}$ to a Lie groupoid $\mathcal{H}$ is a morphism of Lie groupoids $\mathcal{G}'\rightarrow \mathcal{H}$ where $\mathcal{G}'$ is a Lie groupoid morita equivalent to $\mathcal{G}$.

The name generalized morphisms seems reasonable for this but I do not see in what sense a bibundle is said to be a generalized morphism.

Any comments that helps to understand in what sense bibundles are called as generalized morphisms are welcome.

Hello please help with the geometry problem and if you can check it on the electronic descriptive application is given a point inside the acute angle and you need to select 2 points on the sides of the corner so that the perimeter of the triangle formed is the smallest and check please for this condition, please 2 of the following options: if we drop perpendiculars from a given point to the sides of the angle and connect the intersection points of these perpendiculars with the sides of the angle among themselves or if the same perpendicular s on the sides of the angle to extend by the same distance and the ends of these distances join together from where and the supposedly sought-for points are determined at the intersection of the data connecting the ends of the line to the sides of the corner. Thank you very much!

Recall that the Fibonacci numbers $F_0,F_1,\ldots$ are defined by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots),$$ and the Lucas numbers $L_0,L_1,\ldots$ are given by $$L_0=2,\ L_1=1,\ \text{and}\ L_{n+1}=L_n+L_{n-1}\ (n=1,2,3,\ldots).$$ It is well known that $$F_n=\frac{1}{\sqrt5}\bigg(\left(\frac{1+\sqrt{5}}2\right)^n-\left(\frac{1-\sqrt{5}}2\right)^n\bigg)$$ and $$L_n=\left(\frac{1+\sqrt{5}}2\right)^n+\left(\frac{1-\sqrt{5}}2\right)^n$$ for all $n=0,1,2,\ldots$.

Here I ask a question on a new kind of representations involving primes, Fibonacci numbers and Lucas numbers.

QUESTION: Can we write each integer $n>3$ as $p+F_kL_m$ with $p$ an odd prime and $k$ and $m$ positive integers? Is this supported by heuristic arguments?

I conjecture that any integer $n>3$ can be written as $p+F_kL_m$, where $p$ is an odd prime, and $k$ and $m$ are positive integers. I have verified this for all $n=4,\ldots,3\times10^9$. For the number of ways to write a positive integer $n$ as $p+F_kL_m$ with $p$ an odd prime, $k>1$ and $m\ge1$, see http://oeis.org/A316141. For example, $$5=3+F_3L_1\ \text{and}\ 17=3+F_3L_4=5+F_4L_3=11+F_3L_2=13+F_2L_3.$$

Do heuristic arguments support the above conjecture?

$f:\mathbb{C}-{0}\rightarrow\mathbb{C}$be a holomorphic function and$$|f(z)|\leq|z|^2+\frac{1}{|z|^{1/2}}$$ for z around 0,determine all such function.

i just prove that $f$ is also holomorphic at 0 by taylor and cannot get something more. any idea is helpful.thanks

Let $n$, $m$ are two positive integer numbers for $n \ge 2$, $m \ge 1$; $P_n$ is $n$-$th$ prime number. How I can prove that:

$$P_{n+m} \ge P_n+P_m$$

Can you give for me a hint, a reference, or a comment, or a proof?

My friend, mister Paul Yiu has sent to me a very new nice equilateral triangle associated with two Fermat points and Kiepert hyperbola of a reference triangle via email since 4 years ago. But I have no proof. This problem is very nice, so I post at here, I hope that have a solution.

- Let $ABC$ be a triangle with two Fermat points $F_1$, $F_2$. Circles with center $F_1$ radius $F_1F_2$ meets the Kiepert hyperbola again at three points $M$, $N$, $P$.

Then triangle $MNP$ is equilateral and perspective to $ABC$ at triplet points. This mean:

- $MA$, $NB$, $PC$ are concurrent
- $MB$, $NC$, $PA$ are concurrent
- $MC$, $NA$, $PB$ are concurrent

Three points above collinear.

- Let $ABC$ be a triangle with two Fermat points $F_1$, $F_2$. Circles with center $F_2$ radius $F_2F_1$ meets the Kiepert hyperbola again at three points $M$, $N$, $P$.

Then triangle $MNP$ is equilateral and perspective to $ABC$ at triplet points. This mean:

- $MA$, $NB$, $PC$ are concurrent
- $MB$, $NC$, $PA$ are concurrent
- $MC$, $NA$, $PB$ are concurrent

Three points above collinear

**See also:**