Math Overflow Recent Questions


$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0 ?$ with $\alpha \ge 1$ and $n=1, 2,\cdots$

Fri, 06/29/2018 - 17:06

Could You give a poof, comment or reference for the inequality as follows:

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

  • See also:

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

Conjecture on tilting modules for an Auslander algebra

Fri, 06/29/2018 - 15:29

On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism classes of modules, occurring as the $i$-th summand of some tilting module $T_x$ for the finite Auslander algebra $Λ_n$ are enumerated by the number triangle OEIS A046802, which also enumerates the positroids of the totally non-negative Grassmannians and contains the h-vectors of the stellahedra / stellohedra.

Can anyone prove this or provide supporting evidence?

Can continuity always be shown by using ε-δ? [on hold]

Fri, 06/29/2018 - 15:06

When we learn calculus we usually:
1. Prove that polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions are continuous on its domain.
2. Prove that continuity of any two functions at a point will be preserved, in the resultant function, after taking finitely many times of the four arithmetic operations and composition.
∴ Elementary functions are continuous on its domain.

For every elementary function $f$, and every point $p$ in the domain of $f$, and every $\epsilon > 0$,
can we always find a $\delta > 0$ explicitly as a closed-form expression (including "the maximum function") in terms of $f$, $p$ and $\epsilon$ such that $| x - p | < δ \implies | f(x) - f(p) | < ε$ ?
(In other words, can we always construct at least one valid $\epsilon-\delta$ style argument showing $f$ satisfies the definition of being continuous at $p$?)

After all, elementary functions are infinitely many, but human beings only have finite amount and time.

At the moment the definition of "elementary functions" follows this webpage:

Is there an integrable function which Fourier terms are all the same? [on hold]

Fri, 06/29/2018 - 14:31

I am reading a paper and it made me raise the following question: is there $f \in L_1(\mathbb{T})$ such that \begin{equation} \frac{1}{2\pi} \int_0^{2\pi} f(e^{it}) e^{int} dt = 1 \end{equation} for every $n \in \mathbb{Z}$?

Is this generalization of the Hopf map for quadratic field extensions surjective?

Fri, 06/29/2018 - 12:18

Let $k$ be a field, and let $L$ be a quadratic extension of $k$. Denote by $\sigma$ the non-trivial element of $\operatorname{Gal}(L/k)$. Let $M_2(L)$ be the vector space over $L$ of two-by-two matrices with entries in $L$. Let $$H^0_2(L/k) = \bigl\{ y \in L(2); \sigma(y)^T = y \text{ and } \operatorname{tr}(y) = 0 \bigr\}.$$ and define the map $j: L^2 \to L^2$ by $$j \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} -\sigma(v) \\ \sigma(u) \end{pmatrix}.$$ We also let $(-,-): L^2 \times L^2 \to L$ be defined by $$(\mathbf{u_1},\mathbf{u_2}) = u_1 \sigma(u_2) + v_1 \sigma(v_2)$$ where $\mathbf{u_i} = (u_i,v_i)^T$, for $i=1,2$. Define $$M^j_2(L) = \{x \in M_2(L); x j = j x \bigr\}.$$ We now define the map:

$$h: M^j_2(L) \to H^0_2(L/k),\qquad h(x) = \sigma(x)^T \begin{pmatrix} 1&0\\0&-1\\\end{pmatrix}x.$$

While it is clear that the image of $h$ lies in $H_2(L/k)$, it remains to check that $\operatorname{tr}(h(x)) = 0$, for any $x \in M^j_2(L)$. Let $x \in M^j_2(L)$. We know that $$\begin{align}\operatorname{tr}(h(x)) &= \operatorname{tr}\left(x\sigma(x)^T \begin{pmatrix} 1&0\\0&-1\\\end{pmatrix}\right) \\ &= (x\sigma(x)^T e_1, e_1)-(x\sigma(x)^T e_2, e_2) \\ &= (x\sigma(x)^T e_1, e_1)-(x\sigma(x)^T je_1, je_1) \\ &= (\sigma(x)^T e_1, \sigma(x)^Te_1)-(\sigma(x)^T je_1, \sigma(x)^T je_1) \\ &= (\sigma(x)^T e_1, \sigma(x)^Te_1)-(j\sigma(x)^T e_1, j\sigma(x)^Te_1) \\ &= (\sigma(x)^T e_1, \sigma(x)^Te_1)-\sigma(\sigma(x)^T e_1, \sigma(x)^Te_1) \\ &= 0 \end{align}.$$

Question 1: for which pairs $(k,L)$ is the corresponding map $h$ surjective? It is surjective if $k=\mathbb{R}$ and $L=\mathbb{C}$, since $h$ is essentially the Hopf map in this case. What about in general?

Question 2: what can be said about the fibers of $h$?

Edit 1: I realized that the target space should be the space of hermitian tracefree matrices, otherwise the statement would not even be true for $k=\mathbb{R}$ and $L=\mathbb{C}$. This led me to modify my original post accordingly, and introduce the "quaternionic structure" $j$ on $L^2$, and also restrict the domain to the "real slice" of $M_2(L)$ which is $j$-equivariant.

Edit 2: I further assume that $\operatorname{char}(k) \neq 2$, otherwise, the corresponding generalized Hopf map is not surjective for trivial reasons, by working out the formula for the generalized Hopf map explicitly (a small calculation).

Embedding a finite morphism into a finite morphism of smooth varieties

Fri, 06/29/2018 - 11:50

Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (at least locally on $Y$)?

There are embeddings $i$ and $j$ of $X$ and $Y$, resp., into smooth quasiprojective varieties $X’$ and $Y’$, resp., along with a finite morphism $f’$ such that $f’\circ i = j\circ f$.

Cycle Structure of a Permutation Based on the Binary Representation

Fri, 06/29/2018 - 11:33

This is a question I posted on before but never got an answer. I am cross-posting it here.

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, c\text{ is an odd natural number}\big)$. For each $x\in X$, $x=a2^k$, where $a$ is an odd natural number and $k$ a non-negative integer. Define $$f(x) = \sum_{i=0}^{k-1}s(i)+\frac{a+1}{2},$$ and $$\sigma(x)=n+1-f(x).$$

This permutation is equivalent to the following playing card shuffling process. Given a stack of cards, counting from the top first card, take all the odd numbered cards, one by one put them with the latter one on top of the previous one and form another stack. Repeat the previous procedure on the leftover first stack with the current withdrawn cards placed on top of the second stack. Repeat this procedure until the first stack is exhausted. The final second stack is the original first stack permuted described in the first paragraph.

What can we say about the cycle structure of this permutation? What is the least common multiplier of all the cycle lengths? Is the least common multiplier of all the cycle length the maximal cycle length?

Perhaps we may start with $n=2^j$ for any natural number $j$ and recurs on $j$.

As an example, for $n=16$, the inverse of $f$ or $f^{-1}(X)$ \begin{pmatrix} 1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16 \\ 1&3&5&7&9&11&13&15&2&6&10&14&4&12&8&16 \end{pmatrix}

and the permutation $\sigma(X)$ is

\begin{pmatrix} 1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16 \\ 16&8&15&4&14&7&13&2&12&6&11&3&10&5&9&1 \end{pmatrix}

The cycle structure is $$(4)(11)(1\ 16)(2\ 8)(5\ 14)(3\ 15\ 9\ 12)(6\ 7\ 13\ 10).$$ Here the maximal cycle length is $4$ and is the least common multiplier of all the cycle lengths.

concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$

Fri, 06/29/2018 - 10:01

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be

$$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \left(1+\frac{x_2}{1+x_2+\frac{x_0}{2}+\frac{x_1}{4}}\right) \right]. $$

My guess is that $f$ is a concave function. The standard approach to prove multivariate concavity is to find the Hessian matrix and prove that it is non-positive definite. However, it seems to be an overwhelming approach for this function. Can we somehow use the structure of $f$ to prove or disprove the concavity?

Edit1: The general form is $f:~ [0,1]^n \rightarrow \mathbb{R}$ $$ f(x_0,x_1,\ldots,x_{n-1})= \log \left[ \prod_{k=0}^{n-1}\left(1+\frac{x_k}{1+\sum_{j=0}^{n-1}q^jx_{(k+j)~\text{mod}~n}}\right) \right], 0<q<1. $$

Edit2: I changed the notation and started the indices from zero to make the general case accurate.

A question concerning a short exact sequence with an action

Fri, 06/29/2018 - 08:24

Let $A$ and $D$ be two non-trivial abelian groups and $B,C$ be two non-abelian groups. Also, let $C$ is a free group and acts on $A,B,D$.

Let $0\to A \xrightarrow{f}B\xrightarrow{g}C\to 0$ be a short exact sequence of groups in which the action of $C$ commutes with maps $f$ and $g$ ($C$ acts on itself trivially). Also, let there exist homomorphisms $h_1 :D\to B$ and $h_2 :B\to D$ (commuting with the action of $C$) so that $h_2 \circ h_1 =1_D$.

If $A$ is a free $\mathbb{Z}C$-module, then is $D$ a projective $\mathbb{Z}C$-module?

Why did Euler consider the zeta function?

Fri, 06/29/2018 - 00:57

Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD conjecture, Langlands program, Riemann hypothesis,...).

Euler was perhaps the first person to consider the zeta function $\zeta(s)$ ($1\leq s$). Why did Euler study such a function? What was his aim?

Further, though we know their importance well, should we consider that the Riemann zeta function and its generalizations happen to play key roles in modern number theory?

Lindelöf hypothesis claim

Thu, 06/28/2018 - 19:15

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?

Knots of fixed genus with arbitrarily large volume

Thu, 06/28/2018 - 16:09

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.

Can we make 101 almost perfect banknotes from 100?

Thu, 06/28/2018 - 14:15

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?

How to find geodesics in a Randers spaces?

Thu, 06/28/2018 - 13:20

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.

Relation between monads, operads and algebraic theories (Again)

Thu, 06/28/2018 - 07:19

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

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


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

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...).

Is every odd integer of the form $P_{n+m}-P_n-P_m$?

Thu, 06/28/2018 - 04:58

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 $P_{c}-P_a-P_b$?

Understanding the definition of atlas of a stack

Thu, 06/28/2018 - 01:11

I am reading 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.

How (non-)computable is set theory?

Wed, 06/27/2018 - 12:53

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 ( 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."

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

Wed, 06/27/2018 - 05:17

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?

Solve this question within 10 seconds [on hold]

Wed, 06/27/2018 - 00:57

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