Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates) \begin{align} & {p_1}^2 {p_2}^2 c_1 {t_1}^2 t_2 + {p_1}^2 p_2 {c_1}^2 c_2 {t_1}^2 t_2 + {p_1}^2 p_2 c_2 {t_1}^4 + {p_1}^2 c_1 {c_2}^2 {t_2}^3 \\ & + 2 p_1 p_2 {c_1}^3 c_2 {t_2}^3 + p_1 p_2 {c_1}^2 {c_2}^2 {t_1}^2 t_2 + p_2 {c_1}^3 {c_2}^2 {t_2}^3 = a {p_1}^2 p_2 c_1 c_2 {t_1}^2 t_2. \end{align} According to the article, the genus of the curve is the number of integer lattice points in the interior of the Newton polygon of the curve.

The Newton polygon of the curve is the convex hull of the points $(1,0)$, $(2,1)$, $(0,1)$, $(1,2)$, $(3,1)$, $(2,2)$, $(3,2)$, $(1,1)$. There are two integer lattice points in the Newton polygon: $(1,1)$, $(2,1)$. Therefore the genus of this curve is $2$. Is this correct? How to check that whether or not this curve is smooth? Is this curve an elliptic curve? Thank you very much.

I have already found two definitions for a Baer group.

$G$ is a Baer group if it is generated by all cyclic subnormal subgroups.

$G$ is a Baer group if every cyclic subgroup is subnormal.

I want to prove the equivalence of the two definitions. Obviously, (2) implies (1). Please help me with the converse.

I'm wondering if it is possible to run a set of numbers ('target numbers') through a function and get out a number that when queried against in some way (with a 'target' number as all/part of the query) would return a true / false as to whether or not that number exists in the output of the function. Tough for me to verbalize so I'll show an example

say my list of numbers is ListOfNumbers = [1,5,100,46736,3]

Can I run it through some function f(ListOfNumbers) = outputNumber

So that I can then run another function f(queryNumber,outputNumber) = true/false

also, the outputNumber (or one of the functions) has to be such that if I did something like this..

f(46736,outputNumber) = true but (23368,outputNumber) = false <-- it's a factor of 46736 but should not return true, only the numbers in the list return true.

Also, if it's not possible with one function, I wonder if it is possible with a couple different functions and if you get two or 'x' trues and no falses it is in the list.

**Edit** : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one.

I want to find weak, non trivial, **continuous**, solutions of $$\Delta u - \lambda u = 0$$ for a square domain in $\mathbb{R}^N$, $N \ge 2$, under periodic boundary conditions, and under an added constraint that, the weak solutions $u$ should take given values, at a given finite set of points in the interior of the domain. $u(x_i) = d_i$, $x_i$ lie in the interior of the domain, and $d_i$ are reals.

Reference request, if someone already solved it, or partially solved it or any relevant work. I am trying to solve it and I want to know if it makes sense, and I am not re-inventing, or barking up the wrong tree.

PS : Solving, I mean, having a numerical solution that converges pointwise, to the actual solution.

I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\overline{\mathbb{F}_p}[x_1, \dots, x_n]$ for some prime $p$. One way to do this is to calculate the Gröbner basis of the ideal $$I = (f_1, \dots, f_m) \subseteq \mathbb{Z}[x_1, \dots, x_n].$$ If this basis contains some integer $k > 1$, then we can deduce that the ideal is not trivial inside $\overline{\mathbb{F}_p}[x_1, \dots, x_n]$ for any prime $p$ dividing $k$. If this basis contains a $1$, we find that $I$ is trivial in any characteristic, while if the basis does not contain an integer, we find that $I$ is non-trivial characteristic $0$. Since the computation of a Gröbner basis can be very difficult, I was wondering if there is another more direct way of computing this integer $k$ (whether or not it is $0$,$1$ or $>1$). Maybe there is another way altogether to determine whether or not a set of equations has a root in some characteristic.

I am also very interested in software that is capable of doing such a computation. So far I've used Mathematica and Sage for Gröbner basis computations, but I am not sure if these packages are the most well suited for the job.

I want to find a visual proof of the following fact:

For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.

All proofs I know require some integral estimates. I would prefer a more visual proof (even if it is more involved).

Is there $C > 0$ such that the inequality $$ \prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right) $$ holds for all finitely supported sequences $(a_n)$ with $a_n\geq 0$ and $\sum_n a_n = 1$ and polynomials with nonnegative coefficients $p\in \mathbb{R}_+[X]$?

Is it even possible to take $C=1$?

It's clear that the inequality trivially holds with $C = 1$ whenever $p$ is a monomial.

If $P(z)=\sum_{k=0}^na_kz^k, (a_n=1)$ having no zeros in $|z|<1,$ I am trying to prove $$\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}\leq \frac{n-1}{2}+\frac{1}{1+|a_0|}.$$ The result is true for $n=1.$ Can I apply induction principle to prove this?

Attempt at a proof: let us try to show that the inequality holds by induction on the degree $n$ of the polynomial $P(z)$.

If $n=1$ then $P(z)=z-w$ with $|w|\geq 1$, and we have

$$\displaystyle\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}=\frac{1}{1+|w|}\leq \frac{1}{1+1},$$
which is nothing but the given inequality when $n=1$.

Let $Q(z)=(z-w)P(z)$ with $|w|\geq 1$, where $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\geq 1$. $$\displaystyle\frac{\max_{|z|=1}|Q'(z)|}{\max_{|z|=1}|Q(z)|}=\frac{\max_{|z|=1}|(z-w)P'(z)+P(z)|}{\max_{|z|=1}|(z-w)P(z)|}$$ \begin{equation}\label{p1}\displaystyle\leq\frac{\max_{|z|=1}|P'(z)|}{\max_{|z|=1}|P(z)|}+\frac{1}{\max_{|z|=1}|z-w|} (?).\end{equation}

By induction hypothesis we will have then $$\displaystyle\frac{\max_{|z|=1}|Q'(z)|}{\max_{|z|=1}|Q(z)|}\leq \frac{n-1}{2}+\frac{1}{1+|a_0|}+\frac{1}{1+|w|}\leq \frac{n}{2}+\frac{1}{1+|a_0||w|},$$ which is true follows from the fact that $$-\frac{1}{2}+\frac{1}{1+|a_0|}+\frac{1}{1+|w|}-\frac{1}{1+|a_0||w|}=\frac{(1-|a_0|)(1-|w|)(1-|a_0w|)}{2(1+|a_0|)(1+|w|)(1+|a_0w|)}\leq 0.$$

Thus the proof by induction will be complete once we establish (?) part of the above.

Let us consider the first-order logic extended with the least fixed point operator (FO+LFP). That is, together with the usual first-order formulas, we also have formulas of the form:

$$\mu X[\overline{y}] . \phi(X, \overline{y})$$

where $X$ (must occur positively in $\phi$) is a "predicate" variable of arity equal to the length of sequence of "parameters" $\overline{y}$. The semantic of this formula (in a given algebraic structure) is the least set $X^*$ such that:

$$X^*(\overline{y}) \Leftrightarrow \phi(X^*, \overline{y})$$

For example, if $R$ is a binary relation symbol, then:

$$\mu X[y_1, y_2] . R(y_1, y_2) \vee (\exists_z R(y_1, z) \wedge X(z, y_2))$$

defines the transitive closure of $R$.

If $A$ is an algebraic structure, let us write $\mathit{Th_{lfp}}(A)$ for the first-order theory of $A$ extended with the least fixed point operator (i.e. the set of all FO+LFP sentences that are true in $A$).

Does there exist an algebraic structure $A$ such that both of the following hold:

- $\mathit{Th_{lfp}}(A)$ is decidable
- FO+LFP is strictly more expressive than FO over $A$ (i.e. there is a FO+LFP formula that is not equivalent to FO formula over $A$)?

An example of a structure that satisfies the first property (but does not satisfy the second) is the structure of rational numbers with the natural ordering $\langle\mathcal{Q}, \leq\rangle$.

An example of a structure that satisfies the second property (but does not satisfy the first) is the structure of natural numbers with the natural ordering $\langle\mathcal{N}, \leq\rangle$.

One intuition is that there should be no such structure $A$ --- if $A$ defines arbitrary long well-founded orders, then the theory of $A$ should be undecidable; and if it does not define, then LFP seems to be pretty useless.

Another intuition is that there might be such a structure, because the above intuition is difficult to formalize, thus may contain essential holes.

It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a generic extension. Indeed, the standard proof for preservation of AC is based on the fact that well-orderability is preserved under surjection, a fact that doesn't seem to have any nice generalization for weaker choice principles at all. So I wonder if we can get any results in the opposite direction.

Are there any known results of the form "If all generic extensions satisfy [some weak choice principle] then [some stronger choice principle] holds in $V$"?

I take choice principles to include e.g. AC, DC, AC$_{\omega}$, the selection principle, "all infinite sets are Dedekind-infinite," and "(strongly) amorphous sets don't exist." Two conjectures I want to focus on are:

Plausible conjecture: AC$_{\omega}$ in all generic extensions implies AC in $V$ (the idea here is that if there's a set in $V$ without a choice function, maybe there's a way to collapse its cardinality to $\omega$ without adding a choice function),

and

Ridiculous conjecture: If every generic extension has no strongly amorphous sets, then AC holds in $V$ (I can't believe this is true, but I also have no idea what property $V$ can have to prevent forcing amorphous sets).

Let $E\neq \{0\}$ be a Banach space. For each $p\in[1,\infty), $ we define $$E\oplus_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$ Let $F$ be another Banach space. By $E\cong F,$ I mean that $E$ and $F$ are isometrically isomorphic.

Question: Suppose that $p,q\in [1,\infty).$ If $$E\oplus_p E \cong E\oplus_q E\,$$ then is it true that $p=q$?

If $E$ is of finite-dimensional, then the question is affirmative. However, I do not know what will happen if $E$ is of infinite-dimensional. I would be glad to see a proof if it is true or a counterexample if it is false.

We say that a norm $\phi:\mathbb{R}^2\to\mathbb{R}$ is *normalized* if
$$\phi(0,1) = \phi(1,0) = 1.$$

Also, $\phi$ is *monotone* if for $|a_1|\leq |b_1|$ and $|a_2|\leq |b_2|,$ then
$$\phi(a_1,a_2) \leq \phi(b_1,b_2).$$

We define $$E\oplus_\phi E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \phi(\|x\|, \|y\|) \}.$$

A more general question:

Suppose that $\phi,\psi:\mathbb{R}^2\to \mathbb{R}$ are norms that satisfy normalization and monotonicity. Assume that $\phi$ and $\psi$ are not $\ell^\infty$ norm. If $$E\oplus_\phi E \cong E\oplus_\psi E,$$ then is it true that $\phi = \psi?$?

I'm reading William Cherry and Zhuan Ye's book 'Nevanlinna's theory of value distribution, the second main theorem and its error terms'. In Section 1.12, they explains why $N$ and $T$ is used in Nevanlinna theory instead of $n$ and $A$, where $A(f,r)=\int_{D(t)}f^\ast\omega$. Then they gave some results on the comparison of $n(f,a,r)$ and $A(f,r)$. For example,

Gol'dberg in 1978 constructed an entire function $f$ such that for every $a\in \mathbb{C}$ such that $$\limsup_{r\to \infty}\frac{n(f,a,r)}{A(f,r)}=\infty.$$

In another direction, Hayman and Stewart in 1954 formulate a theorem

Let $f$ be a non-constant meromorphic function on $\mathbb{C}$ and set $n(f,r)=\sup_{a\in \mathbb{P}^1}n(f,a,r)$. Then $$1\le\liminf_{r\to \infty}\frac{n(f,r)}{A(f,r)}\le e.$$

**My question is: what can we say about $\frac{N(f,a,r)}{T(f,r)}$?**

Firstly, by FMT we know $\frac{N(f,a,r)}{T(f,a,r)}\le 1$.

I also checked some elementaty functions. For examples, exponential function $f(z)=e^z$. By a simple calculation, $N(f,\infty,r)=0,N(f,a,r)=\frac{r}{\pi}+O(\log r),T(f,r)=\frac{r}{\pi}$. So
$$\frac{N(f,\infty,r)}{T(f,r)}=0,\frac{N(f,0,r)}{T(f,r)}=1.$$
**When does this ratio nonzero for a general meromorphic function?**

Any reply or reference is appreciated.

I am looking for the following reference:

Krukenberg, Claire Emil,
*COVERING SETS OF THE INTEGERS*.
Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. 1971. 104 pp.

I searched the internet but no success. I would appreciate the link for downloading this reference.

Thanks.

I am currently studying parameter dependent symbols, $s(t,x,\xi)$, where $t\in [0,1],x\in \Omega, \xi \in \mathbb{R^n}$. I wanted to know how the low regularity (for example, $s$ is just continuous w.r.t. $t$) of symbol w.r.t. parameter affects further study of symbols and the corresponding operators.

To be specific, I need this to study hyperbolic operators which are low regular in time. All the books which I have referred till now assume smoothness w.r.t. parameter.

Thanking you in advance.

Let

- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $T>0$
- $I:=(0,T]$
- $d\in\mathbb N$
- $M:\Omega\times\overline I\times\mathbb R^d\to\mathbb R$ such that $M(\;\cdot\;,\;\cdot\;,x)$ is $\mathcal A\otimes\mathcal B(\overline I)$-measurable for all $x\in\mathbb R^d$

Now, let $i\in\left\{1,\ldots,d\right\}$ and $$N(\omega,t,x,\theta):=\frac{M(\omega,t,x+\theta e_i)-M(\omega,t,x)}\theta$$ for $(\omega,t,x,\theta)\in\Omega\times\overline I\times\mathbb R^d\times\left(\mathbb R\setminus\left\{0\right\}\right)$.

Let $\delta\in(0,1]$. Assuming that for all $p\ge2$, there is a $C>0$ such that $$\int\sup_{t\in\overline I}\left|N(\omega,t,x,\theta)-N(\omega,t,y,\vartheta)\right|^p\operatorname P\left[{\rm d}\omega\right]\le C\left(|x-y|^{\delta p}+|\theta-\vartheta|^{\delta p}\right)\tag1$$ for all $(x,\theta),(y,\vartheta)\in\mathbb R^d\times\left(\mathbb R\setminus\left\{0\right\}\right)$, how can we conclude that $M(\omega,t,\;\cdot\;)$ is partially differentiable with respect to the $i$th variable for $\operatorname P$-almost all $\omega\in\Omega$ for all $t\in\overline I$?

This should be an application of a Kolmogorov-type theorem, but which version of that theorem do we need and how do we need to apply it exactly?

A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\mod n_i)$. The known examples are:

$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 5 (\mod 6),\ 7 (\mod 12)$

$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 3 (\mod 8),\ 7 (\mod 12),\ 23 (\mod 24)$

My question is that is it possible to construct a covering system of congruences for ** non perfect square odd** integers $>1$ such that

$$ \left(\frac{a_i}{n_i}\right)=-1,\qquad \text{for}\ \ 1\leq i\leq r $$

where the parenthesis is Legendre (or Jacobi ) symbol.

A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a_i (\mod n_i)$. The known examples are:

$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 5 (\mod 6),\ 7 (\mod 12)$

$0 (\mod 2),\ 0 (\mod 3),\ 1 (\mod 4),\ 3 (\mod 8),\ 7 (\mod 12),\ 23 (\mod 24)$

The proof of that the above families are each a covering system of integers are not difficult.

My question is the other side, i.e., that how can we construct a covering for integers from the given numbers for example $2,3$ with their multipliers as moduli?

**The question in a nutshell:**

Let $f:\mathbb{R}^d \to \mathbb{R}^d$ be a differentiable map. Under some natural conditions, the minors of degree $k$ of $df$ uniquely determine $df$, in a smooth way. Now suppose these minors have some "good" regularity/integrability property. Does $df$ have this property as well?

*The details:*

Let $\Omega \subseteq \mathbb{R}^d$ be an open bounded domain. Fix some integer $1<k<d$ and some $k<p<d$. Suppose that $k,d$ are not both even.

Let $f \in W^{1,p}(\Omega;\mathbb{R}^d)$ be a continuous map with $\det df > 0$ a.e. Denote by $f^i$ the $i$-th component of $f$.

Suppose that all the $k$-minors $$df^I:=df^{i_1} \wedge df^{i_2} \wedge \dots df^{i_k} \in W^{1,p}_{loc}(\Omega,\bigwedge^k (\mathbb{R}^d)^*),$$

for every increasing multi-index $I=(i_1,\dots,i_k)$.

Is it true that $f \in W^{2,q}(\Omega;\mathbb{R}^d)$ for some $q>1$?

A-priori the fact $f \in W^{1,p}$ only ensures $df^i \in L^p$, so $df^I \in L^{\frac{p}{k}}$. (You can think on $df^I$ as $\binom{d}{k}$ scalar functions). However, we are now told that in fact the minors $df^I$ have an improved regularity- they are weakly differentiable with (locally) controlled integrability.

The question is whether we can push this regularity back to $f$.

The "$k$-minors map" $\psi: A \to \bigwedge^k A$, considered as a map $\text{GL}(\mathbb{R}^d) \to \text{GL}(\bigwedge \mathbb{R}^d)$ is **smoothly invertible** from its image (which is a closed embedded Lie subgroup).

The injectivity of $\psi$ uses the assumption $k,d$ are not both even; in general $\bigwedge^k A=\bigwedge^k B$ implies (for invertible elements $A,B$) $A=\pm B$. If $k$ is odd, then of course $A=B$. If $d$ is odd, then assuming $\det A>0,\det B>0$ we again deduce $A=B$.

So, since $\det df>0$ a.e. we can "almost everywhere" invert the collection of minors $$\{df^I\}_I \stackrel{\psi^{-1}}{\to} df.$$

So, $df$ is obtained from the $(\binom{d}{k})^2 \, \,$ $k$-degree minors, which are in $W^{1,p}_{loc}$, via a composition with a smooth operator. Thus, I expect there is a chance that this "composition" (with $\psi^{-1}$) won't reduce the regularity too much- so we will have $df \in W^{1,q}_{loc}$.

It's known (due to Perelman) that in class of Alexandrov spaces of fixed dimension and bounded from below curvature Gromov-Hausdorff distance separates homeomorphism types — every $\epsilon$-close to $X$ space will be homeomorphic to $X$ for some $\epsilon$.

Well, if we have some finite metric space $X_{\delta}$ which is $\epsilon/2$-close to $X$, then $(X_{\delta}, n, C)$ define homeomorphism type of, say, compact Riemannian manifold, where $n$ is dimension and $C$ is lower curvature bound.

Now let's fix $C$ once for all (take $-1$, for example) and call finite metric space $X_{\delta}$ *a model* of a manifold $X$ if for some $\epsilon$ the only manifold $\epsilon$-close to $X_{\delta}$ is $X$ with some metric with curvature bounded below by $-1$. We can define two functions on homeomorphism (diffeo, if dim > 4, thanks to Grove-Peterson-Wu) classes of $n$-dimensional manifolds: $min \, |X_{\delta}|$ and $min \, k: X_{\delta} \to \Bbb R^k$ for isometric embedding into real space with some norm, where minimum is taken over all models. It seems appropriate to me to call first one metric complexity $mCom(X)$ and second one — essential dimension $edim(X)$.

Can $edim(X)$ be strictly less than dimension of $X$?

Are there some bounds on $mCom$ in terms of something like LS category or topological complexity (i. e. minimal cardinality of open cover over which $eval: X^I \to X \times X$ has local sections?

What is, for example, $mCom(S^1 \times S^1)$ — or something else $\geq 2$-dimensional — and what is the model? (I guess that for all surfaces answer should be derivable from known results about triangulations et cetera).

(I'm totally not an expert in this area, so maybe those questions are either very easy or hopelessly hard; if it's so, I'll gladly accept as an answer putting them into one of these two categories.)

For any (finite) group $G$ its *length* $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define *width* function $w_G(k)$ in such fashion: $w_G(k) := \#\{H<G: l(H) = k\}$. Then we can do some adjustments — make it a function $W_G:[0, 1] \to \Bbb R$ by setting $W_G(k/l(G)) = w_G(k)$, interpolating linearly and then maybe normalizing by setting integral over $[0, 1]$ to $1$. For example, $W_{\Bbb Z/n}$ is constant and $W_{\Bbb (Z/p)^n}$ is $p$-binomial distribution.

So, my question is

What is limit of $W_{S_n}$ for large $n$ — is there some "central limit theorem"? Is it dominated by $W_{Syl_2(S_n)}$?

(Exact length of $S_n$ is known (Cameron-Solomon-Turull, 1989) and asymptotically equal to length of 2-Sylow.)