Recent MathOverflow Questions

Generation of peaks with EMG

Math Overflow Recent Questions - Sat, 08/11/2018 - 08:34

I need to build a function like this with the following parameters:

A - peak area

h - peak height

w - peak width in the middle

b/a - asymmetry factor

I have a function described in the file, and I'm trying to use it to get the peak with the necessary parameters, fitting the the tau and sigma, but error(the square of the sum of the error for each parameter in percent) is still to big. Does anyone know a function that can build a peak with given parameters? P.s. sorry for my "russian" english

Is any real closed extension of $\mathbb R$ characterized up to isomorphism by its ladder?

Math Overflow Recent Questions - Sat, 08/11/2018 - 08:30

Let $R$ be a real closed field. Recall that $x,y \in R$ are comparable if there are $m,n \in \mathbb Z$ such that $mx > y$ and $ny > z$. Recall that the ladder of $R$ is the linear order divisible ordered abelian group obtained by quotienting by comparability a certain equivalence relation.

Note that $R$ has trivial ladder iff $R$ is a subfield of $\mathbb R$. If $R$ has trivial ladder and $L$ is a linear order divisible ordered abelian group, let $R\langle\!\langle x^L\rangle \!\rangle$ be the real closure of the purely transcendental extension $R(L)$ with transcendence basis $L$, ordered as in $L$ field of Puiseux series. Then $R\langle \!\langle x^L\rangle \!\rangle$ has ladder $L$. Conversely, from any real closed field, we may extract a maximal subfield of $\mathbb R$ and a ladder. I'm wondering whether that's all there is to it, i.e. whether every real closed field is of the form $R\langle\! \langle x^L \rangle\! \rangle$. Let me state this more formally:

Questions: Let $R$ be a real closed field with ladder $L$, and let $R_0 \subseteq R$ be a maximal subfield with trivial ladder.

  1. Is $R$ characterized up to isomorphism by $R_0$ and $L$?

  2. Is $R$ characterized up to isomorphism over $R_0$ by $L$?

  3. The relative version: for any extension $R \to S$ of real closed fields such that $R_0 = S_0$, is $S$ characterized up to isomorphism over $R$ by the induced map on ladders?

As indicated in the title question, I'm happy to assume that $R_0 = \mathbb R$ if that simplifies things.

Hölder continuity for operators

Math Overflow Recent Questions - Sat, 08/11/2018 - 03:04

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$

we obtain $1/2$-Hölder continuity for the square-root.

I would like to know if $x,y$ are positive Hilbert-Schmidt operators. Does it follow then that for some $C>0$

$$\left\lVert \sqrt{x}-\sqrt{y} \right\rVert_{HS} \le C \left\lVert x-y\right\rVert_{HS}^{\frac{1}{2}}.$$

Sounds natural, but on the other hand, it is less obvious to me how this should follow.

One remark however is that if it would hold for finite-rank operators, then a density argument yields the claim.

Isoperimetric inequality inside a regular polygon

Math Overflow Recent Questions - Sat, 08/11/2018 - 01:12

Let $\mathcal{P}_n$ be a fixed $n$-sided regular polygon with area $A:=\vert \mathcal{P}_n\vert>0$. For any $c\in (0,A)$, I would like to find the shape of the domain $D\subset \mathcal{P}_n$ such that $\vert D \vert=c$ and the perimeter $\vert\partial D \vert$ is minimal. Do you have an idea how to tackle the problem or is there any related work you know of?

Of course, for small $c$, we can choose $D$ to be a disc because it is a global solution to the isoperimetric problem. But what happens if $c$ is too large such that the corresponding disc can not stay within the polygon?

Best wishes

Infinite difference length of integer subsets

Math Overflow Recent Questions - Sat, 08/11/2018 - 01:05

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:

We say $A$ has infinite difference length, if

(a) For every integer $n$ there exist a number $k=2^q$ (for some positive integer $q$) and $a_1,\cdots,a_k\in A$ such that $$ n=a_1+\cdots+a_{\frac{k}{2}}-(a_{\frac{k}{2}+1}+\cdots+a_k). $$

Now, denote by $k(n)$ the least $k$ obtained from (a).

(b) The set of all $k(n)$, where $n$ runs over all integers, is unbounded above.

For example, if $\gcd\{a,b\}=1$ then $A=\{a,b\}$ has infinite difference length, but not $A=\mathbb{Z}^+$ (it does not have the second condition (b)).

Now, my questions are:

(1) Does the set of all Fibonacci numbers have infinite difference length? (see https://math.stackexchange.com/questions/1989375/representation-of-integers-by-fibonacci-numbers)

(2) What about the Euler numbers?

(3) Does anybody know some important well-known integer sequences with infinite difference lengths?

(4) Did anyone see something like the above property (definition) yet?

Thanks in advance

Higher roots modulo prime complexity best algorithm

Math Overflow Recent Questions - Sat, 08/11/2018 - 00:05

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$.

  1. What is the best method to find all such $x$?

  2. What is the complexity (is it $O(poly(\ell\log p)$?)?

Random numbers between 0 and 1 [on hold]

Math Overflow Recent Questions - Fri, 08/10/2018 - 23:54

John Derbyshire in his book PRIME OBSESSION says on page 366: CHAPTER 3 10.

"Here is an example of e turning up unexpectedly. Select a random number between 0 and 1. Now select another and add it to the first. Keep doing this, piling on random numbers.
How many random numbers, on average, do you need to make the total greater than 1?
Answer: 2.71828…."

My question: Can you provide a proof of the above statement? Can you indicate an experimental verification method (code and data) with the computer?

Counting promenades on graphs

Math Overflow Recent Questions - Fri, 08/10/2018 - 23:18

Let a "promenade" on a tree be a walk going through every edge of the tree at least once, and such that the starting point and endpoint of the walk are distinct. What we mean by isomorphic promenades should be clear (in particular, the underlying trees have to be isomorphic). What is the total number $N(l,r)$ of promenades (up to isomorphism) of length $l$ with $r$ distinct edges (letting the underlying tree vary)?

gaussian upper bound on spherical heat kernel

Math Overflow Recent Questions - Fri, 08/10/2018 - 21:23

It is known that the heat kernel on n-sphere satisfies $p_t(x,y)\leq Ct^{-n/2}e^{-d(x,y)^2/5t}$ for all $t\in (0,T)$. Can something be said about how big C needs to be?

Finding a metric in $ \Bbb R^2 $ depending on $s$ such that $x^s+y^s=1$ is a geodesic wrt. the metric

Math Overflow Recent Questions - Fri, 08/10/2018 - 20:25

Looking for a complete regular Riemannian metric in $ \Bbb R^2 $ depending on $s$ such that $x^s+y^s=1$ is a geodesic wrt. the metric, $x,y\in(0,1), s\in \Bbb R(1, \infty). $

Good papers on stochastic differential equations with applications in finance

Math Overflow Recent Questions - Fri, 08/10/2018 - 20:07

I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) SDEs and its applications in finance and I would like some help finding some recent papers related to or useful when doing research. I have already looked at some papers on MathSciNet by the same author but I would much appreciate if anyone can suggest some journals or papers/ articles that are relevant and useful in the current times. Thank you in advance!

calculate how much a check needs to be to get 1500 dollars biweekly [on hold]

Math Overflow Recent Questions - Fri, 08/10/2018 - 19:59

Can u calculate 12 dollars and hour Monday through Friday for 8 hours and 14.00 per hour on Saturday paid weekly or biweekly at 14 per hour for an 8 hour shift on Saturday and how many hours of overtime I’d need to come up with about 1500 dollars for a 1200 dollar car and then adding insurance and registration and title

Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice?

Math Overflow Recent Questions - Fri, 08/10/2018 - 13:05

The question of existence of sets $x,y$ such that

$$|x|<|y| \wedge |P(x)|=|P(y)|$$

is known to be independent of $\text{ZFC}$!

But are there known examples of sets fulfilling the above condition that necessitates violation of choice?

Coverage of balls on random points in Euclidean space

Math Overflow Recent Questions - Fri, 08/10/2018 - 09:50

We have n points randomly distributed in a d-dimensional unit hypercube. We randomly sample k of those points and center a ball with radius r on each of those k points. Does there exist an estimate of the radius r such that r is the smallest radius expected to cover all the points with those k possibly overlapping balls?

Reference request: the dual Coleman family

Math Overflow Recent Questions - Fri, 08/10/2018 - 09:03

Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an eigenform of weight k, level N (not necessary new) with nebentype $\chi$ and f be (one of the) p-stabilization of $f^{\circ}$, we can construct several modular forms from f: assume the $q$-expansion of f is $\sum_{n\geq1}a_nq^n$,

$(i)$ The conjugate form $f^*$ defined by $f^*(\tau)=\overline{f(-\bar{\tau})}$, the q-expansion of $f^*$ is $\sum_{n\geq1}\bar{a}_nq^n$;

$(ii)$ The twisted form $f_{\chi^{-1}}$ by $\chi^{-1}$ whose q-expansion is $\sum_{n\geq1}\chi^{-1}(n)a_nq^n$;

$(iii)$ and moreover as in Def.2.4 of the paper a note on p-adic Rankin-Selberg L-functions, the form $f^c$ which is the unique form which has level $Np^r$ for some r and whose Hecke eigenvalue away from N is same as $f_{\chi^{-1}}$.

What is the relation of the three forms? Would anyone provide some reference? Thanks

If F is a Coleman family of tame level N and nebentype $\chi$, could we define the corresponding family $F^*$, $F_{\chi^{-1}}$ or $F^c$ satisfying similar description on $q$-expansion? The paper A p-adic Gross-Zagier formula for diagoanl cycles(page 41) uses $F^*$ (for Hida family), but no details provided there and Lem.3.4 of 1 claims the existence of $F^c$ (at least for F is new at N) but I could not spell out the details. Would anyone please provide me some reference containing more details? Thanks

Zeros of Multivariate Complex Functions [need reference]

Math Overflow Recent Questions - Fri, 08/10/2018 - 07:05

I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions.

For my purpose, it would be interesting to see a discussion on the following topics:

  1. Zeros of single variable analytic function are discrete (isolated)
  2. Zeros of multivariable analytic functions are not isolated
  3. The set of zeros of an analytic function of $n$ variables roughly speaking lives in $n-1$ dimensional space.

I have been reading "Functional Theorem of Several Complex Variables" by Krantz. While I found the book interesting to read, I don't think that it is very accessible to a non-mathematician. Right now looking for a reference that would be more accessible to a senior Ph.D. student in engineering with a good background in single variable complex analysis. In any case, any reference on this subject that you can provide would be of great help to me.

Strongly reducible but not effectively interpretable

Math Overflow Recent Questions - Fri, 08/10/2018 - 06:50

A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A.

A is said to be effectively interpretable in B if A is strongly reducible to B via a computable functor from the category of copies of B to the category of copies of A.

  • See this paper of Harrison-Trainor/Melnikov/Miller/Montalban for a precise definition (and note that this isn't the starting definition; that definition is given on page $3$, and its characterization by functors is Theorem 5). Roughly speaking, let $\mathcal{A}$ and $\mathcal{B}$ be respectively the categories whose objects are copies of $A$ and $B$ and whose morphisms are isomorphisms (in the usual sense); then a computable functor reducing $\mathcal{A}$ to $\mathcal{B}$ is a functor from $\mathcal{B}$ to $\mathcal{A}$ given by a pair of Turing functionals, one sending objects in $\mathcal{B}$ to objects in $\mathcal{A}$ and the other sending morphisms in $\mathcal{B}$ to morphisms in $\mathcal{A}$. The claim that this yields a strict strengthening of strong (= Medvedev) reducibility is made without proof on page $5$ of the linked paper.

What is an example of two countable structures A,B such that A is strongly reducible to B but A is not effectively interpretable in B?

Is there an 11-term arithmetic progression of primes beginning with 11?

Math Overflow Recent Questions - Thu, 08/09/2018 - 18:24

i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?

Schwartz space on $\cup_{n=1}^CR^n$

Math Overflow Recent Questions - Wed, 08/08/2018 - 10:18

I have an application where I need to work with the following idea. I need to work with the shwartz space on $\cup_{n=1}^CR^n$ and its dual by defining an inner product as

$\langle\nu,\phi\rangle=\sum_{n=1}^C\int_{R^n}\phi(x_1,\cdots,x_n)\,d\nu(x_1,\cdots,x_n)$. Does all the properties of schwartz space and its dual on just $R^n$ holds even for the case of the space $\cup_{n=1}^CR^n$? If it is true, can we extend this to the case when $C=\infty$? Please give some suggestions and references for this.

What kind of computer tools topologists/geometrists use to visualize the objects they deal with?

Math Overflow Recent Questions - Wed, 08/08/2018 - 10:05

I have recently started to read a bit about geometry and topology. Hopf fibration, Lense spaces, CW complexes, stuff that are discussed in Hatcher's Algebraic Geometry and other things that require good visualization. What is apparent to me is that the further I go, the less I understand what is going on. I have searched on YouTube and found some really nice animations for some of these topics but good animations are rare like gems.

Advanced stuff in mathematics are less discussed and available on the internet. I have realized that if I want to understand math one day, at some point I should be able to create my own animations. Now, my question is rather directed at people with experience in teaching advanced mathematics or currently doing research in mathematics in areas where geometric intuition is absolutely necessary. What kind of tools do you use? Do you develop them on your own in your research team/group? Can an independent person have access to them? Is it possible for an independent person to develop this kind of tools on their own?

Can you think of a situation where you couldn't understand a geometric concept visually but you created an animation that demystified it for you?

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