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Rings with all non-prime ideals finitely generated

Sat, 06/30/2018 - 02:49

Motivated by this question, I would like to ask:

If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?

Note that for zero-dimensional rings, the answer is yes by the linked question. So we only need to think about rings of positive dimension.

NOTE: All our rings are commutative with unity.

Is Riemann zeta function injective in some strips $a<\Re(s)<b$, or strips $c<\Im(s)<d?

Sat, 06/30/2018 - 02:04

Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, or strips $c<\Im(s)<d$?

Would you like a subject class for semigroup theory on the arXiv?

Sat, 06/30/2018 - 02:00

After contacting the arxiv recently about possibly adding semigroup theory as a subject class, they suggested I canvas the research community to establish whether such a subject class would be used and whether there's enough support available for moderation etc..

I have created an email address and website as an attempt to allow members of the semigroup community to indicate whether they would likely submit papers under the subject class were it created and/or indicate whether they are able to offer support for moderation etc.. The website is available at https://arxiv-semigroup-theory-class.gitlab.io with the email address listed on the website.

It would be excellent if people from the research community could consider their own commitments there and to possibly pass the url around with other members from the semigroup theory research community (probably better someone has this pointed out to them several times rather than not at all).

Do two inequalities as follows holds $\sum_{k=1}^n \frac{{a_k(x)}}{b_k^\alpha} >0$ and $\sum_{k=1}^n \left( \frac{{a_k(x)}}{b_k} \right)^k >0$?

Sat, 06/30/2018 - 01:27

Let series $b_1, b_2,\cdots, b_n$ and $a_1(x), a_2(x), \cdots , a_n(x)$ so that

$$b_n > b_{n-1} > \cdots > b_1 \ge 1 $$

and

$$\frac{a_1(x)}{b_1} > |\frac{a_k(x)}{b_k}| \; \text{and} \; |a_k(x) \le 1| \; \text{for all} \;k =2, 3, \cdots, A < x < B $$

and $$\sum_{k=1}^n \frac{a_k(x)}{b_k} >0 \; \text{for all} \;n =1, 2, \cdots, A < x < B$$

Then do two inequalities as follows holds?

$$\sum_{k=1}^n \frac{{a_k(x)}}{b_k^\alpha} >0 \; \text{where} \; \alpha \ge 1$$ and

$$\sum_{k=1}^n \left( \frac{{a_k(x)}}{b_k} \right)^k >0 \; \text{where} \; \alpha \ge 1$$

An variation of an assignment problem in combinatorics: Assign items to customers

Sat, 06/30/2018 - 00:40

Suppose we want to assign $n$ items to $m$ customers ($m \geq n$). Each assignment from an item $i$ to a customer $j$ will has an associated cost $c(i,j)$. Find an assignment that maximizes the total cost. Here we must assign every item, therefore a customer might receive more than one item. Moreover, we require that each customer must receive at least one item.

This problem can be modeled as an integer linear programming. However, I wonder there might be a better approach (a polynomial algorithm??). Note that if we do not require each customer must receive at least one item, then we just assign each item to the customer with the maximum cost. Does anyone have any idea?

Bounds on the Cardinality of the Intersection of Maximum and Minimum Weight Perfect Matchings

Sat, 06/30/2018 - 00:32

This question is motivated by the observation, that the intersection of the edgesets of maximum and of minimum weight perfect matchings ($PM_{max}$ and $PM_{min})$ need not be empty and the possible implications for TSP heuristics.

Consider the set of vectors $\lbrace(0,0),\ (1,0),\ (2,0),\ (3,0),\ (4,0)\rbrace\cup\lbrace(0,1),\ (1,1),\ (2,1),\ (3,1),\ (4,1)\rbrace$
and edges weighted according to the $L_2$-norm, then:

$PM_{max} = \lbrace[(0,1),(4,0)],\ [(1,1),(3,0)],\ [(2,0),(2,1)],\ [(0,0),(4,1)],\ [(1,0),(3,1)]\rbrace$ $PM_{min} = \lbrace[(0,0),(0,1)],\ [(1,0),(1,1)],\ [(2,0),(2,1)],\ [(3,0),(3,1)],\ [(4,0),(4,1)]\rbrace$,

i.e. $PM_{max}\cap PM_{min}\ =\ [(2,0),(2,1)]$


Question:

what can be said about upper bounds on $card\left(PM_{max}\cap PM_{min}\right)$, meaning

  • existence
  • constants
  • order of growth

I used the plural in the formulation of my question to account for the possibility, that the answer depends on properties of the edgeweights (e.g. metric, random, etc.)

Is $C^{\infty}(M)$ a projective Frechet $C^{\infty}(N)$-module for a smooth map $M\to N$ between compact smooth manifolds?

Fri, 06/29/2018 - 21:05

Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\alpha}||\partial^{\beta}(f)||$.

Now let $M\to N$ be a smooth map between compact smooth manifolds. Then it is clear that $C^{\infty}(M)$ is a Frechet $C^{\infty}(N)$-module.

My question is: is $C^{\infty}(M)$ always a projective Frechet $C^{\infty}(N)$-module?

I think this question is trivial for experts. Please let me know if there is any references or it the question is not suitable for mathoverflow.

Where is a full proof of the precise Torelli theorem in the literature?

Fri, 06/29/2018 - 20:05

The precise form of Torelli's theorem is as follows (translated from Serre's appendix to Lauter - Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields):

Let $k$ be a field, and let $X_{/k}$ be a nice (= smooth, projective and geometrically integral) curve over $k$ of genus $g > 1$. Let $(\operatorname{Jac}(X),a)$ denote the Jacobian of $X$ together with the Jacobian's canonical principal polarization $a$, which is of degree 1. Let $X'_{/k}$ be another nice curve. Any isomorphism $f: X \to X'$ defines by transport of structure an isomorphism $f_J: (J, a) \to (J, a')$.

Torelli's theorem says that we get almost all of the isomorphisms: $(J, a) \to (J', a')$ in this way. More precisely:

Theorem 1. Suppose $X$ is hyperelliptic. Then, for every isomorphism of polarized abelian varieties $F: (\operatorname{Jac}(X),a) \stackrel{\sim}{\rightarrow} (\operatorname{Jac}(X'),a')$, there exists a unique isomorphism $f: X \stackrel{\sim}{\rightarrow} X'$ such that $F = \operatorname{Jac} f$.

Theorem 2. Suppose $X$ is not hyperelliptic. Then, for every isomorphism of polarized varieties $F: (\operatorname{Jac}(X),a) \stackrel{\sim}{\rightarrow} (\operatorname{Jac}(X'),a')$, there exists an isomorphism $f: X \stackrel{\sim}{\rightarrow} X'$ and $e \in \{ \pm 1\}$ such that $F = e \cdot \operatorname{Jac} f$. Moreover, the pair $(f,e)$ is uniquely determined by $F$.

So, we conclude:

$$\operatorname{Aut}(\operatorname{Jac}(X), a) \simeq \begin{cases} \operatorname{Aut}(X) & \text{if $X$ is hyperelliptic} \\ \operatorname{Aut}(X) \oplus \mathbb{Z}/2 & \text{if $X$ is not hyperelliptic} \end{cases} $$

Serre also states there that he does not know of a place where the precise Torelli theorem is proved in its full glory. In the algebraically closed field case, it is in Weil's Zum Beweis des Torellischen Satzes.

Is there a source that proves it over general fields?

Smooth morphisms to the moduli stack of elliptic curves

Fri, 06/29/2018 - 19:54

Fix a prime $p$, and let $M^\mathrm{ord}$ be the ordinary locus of the moduli stack $M$ of elliptic curves over $W(\overline{\mathbf{F}}_p)$. Is there a smooth projective scheme $X/W(\overline{\mathbf{F}}_p)$ with positive Kodaira dimension that admits a smooth map $f:X\to M^\mathrm{ord}$ (which defines a certain family of ordinary elliptic curves over $X$)? One could also ask the analogous question for the ordinary locus of the Deligne-Mumford compactification $\overline{M}$ of $M$ (which, concretely, is the open substack of the stack of cubic curves which are smooth or have a nodal singularity). I don't have much intuition for the smooth site of $M$ (hence of $M^\mathrm{ord}$ and of $\overline{M}$), so any insight into this would be much appreciated.

What are the disadvantage and advantages of the moment method and the resolvent method in Random Matrix Theory?

Fri, 06/29/2018 - 17:55

I am learning random matrix theory . I am aware that the most popular successful techniques for obtaining the limiting spectral measure of large Hermitian random matrices are the moment method and the Stieltjes transform method. The moment method is indeed very combinatorial and the Stieltjes transform method is analytical nature. The key idea is to derive, by using analytical tools, a self-consistent equation ($m(z)\approx -\frac{1}{m(z)+z}$) for the normalized trace of the resolvent $$m(z)=\frac{1}{n}\text{Tr}\, (H-z)^{-1}= \frac{1}{n}\sum_i \frac{1}{\lambda_i-z}$$ for Hermitian $n\times n$ matrices $H$ with eigenvalues $\lambda_1,\dots,\lambda_n$ and $z$ in the upper half plane. The Stieltjes transform method is also used in Wireless telecommunication for dealing with deterministc equivalents, witch are roughly speaking matrix combinations. The Stieltjes transform method is generally preferred over the moment method since it considers the eigenvalue distribution of large dimensional random matrices as the central object of study, while the moment approach is dedicated to the specific study of the successive moments of the distribution. Note in particular that not all distributions have moments of all orders, and for those that do have moments of all orders, not all are uniquely defined by the series of their moments. My questions is: What are the disadvantages of the moment method over the resolvent method in Random Matrix Theory?

$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0 ?$ 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.

Question:
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:
https://www.encyclopediaofmath.org/index.php/Elementary_functions

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 math.stackexchange.com 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?

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