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most recent 30 from mathoverflow.net 2018-07-15T13:14:40Z

If p is a prime number and p>3 then p | (p-1)(p-2)/2+(p-2)(p-3)/2+...+2*1/2.

Fri, 04/06/2018 - 10:50

Suppose p is a prime number and p>3 also suppose f(p)=(p-1)(p-2)/2+(p-2)(p-3)/2+...+2*1/2 then p|f(p).

Decomposition of the spectrum of an unbounded opeator

Fri, 04/06/2018 - 10:32

The Wikipedia article on spectral decomposition, see here

https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis)

says the following:

A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis ${ei}_i \in I$ consisting of eigenvectors for A.

Why is this true? What is a reference for a proof? (Also to be sure, I guess that pure point spectrum means that the spectrum of the operator is equal to its eigenvalues.)

When is a minimal immersion holomorphic?

Fri, 04/06/2018 - 10:12

Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:

$\phi\colon X\to Y$

be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am not mistaken, every holomorphic map from a Riemann surface to a Kahler manifold is a minimal immersion. I am interested in the opposite question: I would like to know the weaker set of necessary conditions currently available in the literature (such that compactness, curvature conditions etc) on $(X,g_X)$ or $(Y,g_Y)$ that guarantees that such minimal immersion is holomorphic. The literature on these beautiful topics is huge, so it is not so easy to dive in and cleanly extract a number of clear necessary conditions for the case I am interested in.

Thanks.

A generalization of Erdos-Ko_Rado theorem

Fri, 04/06/2018 - 09:29

I just wanted to know, is there any result know about the following generalization of Erdos-Ko-Rado theorem?

Let $n, k, r, s$ be positive integers. We call a family $\mathcal{F}$ of k-subsets of the set $\{1,\ldots, n\}$, $(r, s)$-intersection family if among every $r$ elements of $\mathcal{F}$ at least two of them has $s$- elements in their intersection. What is the maximum size of such a family?

Sovling classical parabolic equation by using Littlewood-Paley theory

Fri, 04/06/2018 - 09:00

Consider the following classical PDE in $R^n$: $$ \partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0. $$ Is there any references on solving the above equation by using the Littlewood-Paley theory? More precisely, I wonder whether the following result is known or not: $$ f\in L^p(R_+\times R^n),\quad b\in L^\infty(R_+;B^\alpha_{q,\infty}(R^n)) $$ with $p>1$ and some conditions on $\alpha,q$ (especially for $\alpha<0$), then there exists a unique solution $u$ to the above equation.

Many thanks for the help!

Maximum rank in a class of $0\,$-$1$ partitioned matrices satisfying combinatorial constraints

Fri, 04/06/2018 - 08:31

We are given a matrix $M \in \{0,1\}^{n\times n}$ satisfying the following property.

The rows and columns of $M$ can be partitioned into $k$ rowgroups and $k$ colgroups respectively, such that in each block $B \subseteq M$ induced by these partitions, whenever an entry $B_{i,j}$ is equal to $0$, all the entries of the $i$-th row or the $j$-th column of $B$ are equal to $0$ too.

Namely, given any such block $B\in \{0,1\}^{r_B\times c_B}$ of $M$, $B_{i,j}=0$ implies (i) $B_{i,p}=0~~\forall p\in [c_B]$ or (ii) $B_{q,j}=0~~\forall q\in [r_B]$ (hence, we may even have simultaneously both (i) and (ii)).

Note that this is equivalent to say that, given any such block $B\in \{0,1\}^{r_B\times c_B}$, we have $0$ or more rows and $0$ or more columns of $B$ containing only $0$-entries, while all the remaining entries of $B$ are equal to $1$.

Question: What is the maximum rank of $M$?

I only know the maximum rank of $M$ is upper bounded by $k^2$.

What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?

Fri, 04/06/2018 - 08:24

Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $f(x) := \int_0^x g(t) \, dt$ is in $S$ too.

A basic example is the (unital) polynomial algebra $A := \mathbb{R}[x] \hookrightarrow C([0,1],\mathbb{R})$, which according to Weierstrass's theorem is dense in the uniform (i.e., $L^{\infty}$) norm. That means that in the $L^{\infty}$ norm, the closure $\overline{A} = C([0,1],\mathbb{R})$. Generalizing this, consider more generally the closure $\overline{S}$ of $S$ in the $L^{\infty}$ norm. The manifest possibilities for that closure are the linear subspaces $$ \{f \in C([0,1],\mathbb{R}) \mid f|_{I} \equiv 0 \} $$ defined by a vanishing condition along some open ($I = [0,a)$) or closed ($I = [0,a]$) initial segment $0 \in I \subset [0,1]$, possibly empty or reduced to the point $\{0\}$.

Question. Are there any other possibilities for the closure $\overline{S}$ besides these?

Equivalently, and in the contrapositive formulation: If $g \in C([0,1],\mathbb{R})$ has $g(0) \neq 0$, must the constant function $1 = \chi_{[0,1]}$ be a uniform limit of linear combinations from $\{T^ng \mid n = 0,1,\ldots\}$?

Asymptotics of an integral by two methods

Fri, 04/06/2018 - 08:21

This was asked in MSE, here, but the answer was not satisfactory.

I want to compute the asymptotic behavior of the integral $$ f(K,a)=\int_0^1 (1-x)^Ke^{iKa\frac{x}{1-x}}x^2dx$$ when $K$ is large and $0<a<1$. I tried two different approaches.

1) My first idea was that the exponential, a fast-oscillating function around $x=1$, is killed by the $(1-x)^K$, and the integral should be dominated by the vicinity of $x=0$. Therefore, I put $x=y/K$ and approximate $(1-y/K)^K\approx e^{-y}$ and $\frac{x}{1-x}\approx \frac{y}{K}$ to get

$$f(K,a)\approx \frac{1}{K^3}\int_0^\infty e^{-y+iay}y^2dy=\frac{2}{K^3(1-ia)^3}.$$

2) On the other hand, the stationary phase approximation should be valid. If I write $$f(K,a)=\int_0^1 e^{KS(x)}x^2dx,$$ with $S(x)=\log(1-x)+iax/(1-x)$, the equation $S'(x_0)=0$ gives $x_0=1-ia$. Second derivative is $S''(x_0)=-1/a^2$. Hence, this idea leads to $$f(K,a)\approx e^{KS(x_0)}x_0^2\sqrt{\frac{\pi a^2}{K}}=(ia)^Ke^{K(1-ia)}(1-ia)^2a\sqrt{\frac{\pi}{K}}.$$

These two results are completely different! I need help understanding this.

Simpleness of the socle of polynomial rings

Fri, 04/06/2018 - 08:18

Is there any equvalent condition under which the intersection of non zero ideals of the polynomial ring $R [x] $ over a commutative ring $R $ is non zero?

Ultrafilters and diagonal arguments

Fri, 04/06/2018 - 07:11

Is there a diagonal argument to show that if $x$ is infinite then ${\cal P}(x)$ (the power set of $x$) is smaller than $\beta x$ (the set of ultrafilters on $x$)?

An approach to calculus by abstract differential geometry

Fri, 04/06/2018 - 06:41

By my knowledge, synthetic differential geometry provides us a foundation of calculus using nilpotent infinitesimal which calls smooth infinitesimal analysis. Abstract differential geometry is similar to synthetic differential geometry (Anastasios Mallios claims this). Can we construct a foundation of calculus bases on abstract differential geometry likes synthetic differential geometry and smooth infinitesimal analysis? Thank you

what is a single squiggle mean in this equation [on hold]

Fri, 04/06/2018 - 06:34

http://cdn2.hubspot.net/hubfs/2450960/InterviewQuestion_10PAGES.pdf

In the above pdf, in number 6, bullet 3, there is a formula that has what looks like a single horizontal squiggle after the first term in it. What does this symbol mean?

When the classes are unbalanced, the baseline is not 50% but the proportion of the bigger class. You could add a weight on each class to balance the error. Let $W_y$ be the weight of the class $y$. Set the weights such that $\frac1{W_y}\sim\frac1n\sum_{i\le n} 1_{y_i} = y$ and define the weighted empirical error.

The composition of weakly compact operators and completely continuous operators

Fri, 04/06/2018 - 06:03

Let $T:X\rightarrow Y$ be weakly compact and $S:Y\rightarrow Z$ be completely continuous. Clearly, the operator $ST$ is compact. My question is how to quantify this elementary fact.

Let us fix some notations. If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $$d(A,B)=\inf\{\|a-b\|:a\in A,b\in B\},$$$$\widehat{d}(A,B)=\sup\{d(a,B):a\in A\}.$$

Let $A$ be a bounded subset of a Banach space $X$. The Hausdorff measure of non-compactness of $A$ is defined by $\chi(A)=\inf\{\widehat{d}(A,F):F\subset X$ finite$\}$. Then $\chi(A)=0$ if and only if $A$ is relatively norm compact. The de Blasi measure of weak non-compactness of $A$ is defined by $\omega(A)=\inf\{\widehat{d}(A,K): K\subset X$ is weakly compact $\}.$ Then $\omega(A)=0$ if and only if $A$ is relatively weakly compact. For an operator $T: X\rightarrow Y$, $\omega(T), \chi(T)$ will denote $\omega(TB_{X}),\chi(TB_{X})$ respectively. For a bounded sequence $(x_{n})_{n}$ in $X$, we set $ca((x_{n})_{n})=\inf\limits_{n}\sup\limits_{k,l\geq n}\|x_{k}-x_{l}\|.$ Then $(x_{n})_{n}$ is norm Cauchy if and only if $ca((x_{n})_{n})=0$. For an operator $T: X\rightarrow Y$, we set $cc(T)=\sup\{ca((Tx_{n})_{n}):(x_{n})_{n}$ weakly Cauchy in $B_{X}\}$. Clearly, $T$ is completely continuous if and only if $cc(T)=0$.

Question. Let $T:X\rightarrow Y$ be an operator and $S:Y\rightarrow Z$ be an operator. Then $\chi(ST)\leq C\cdot\max(\omega(T),cc(S)),$ where $C$ is a universal constant?

Thank you!

What is your favourite wrong proof of RH?

Fri, 04/06/2018 - 05:00

Some of the users here receive claimed proofs of the Riemann hypotheses on a regular bases. As fas as we know all of them have been wrong. But sometimes failure is also interesting.

So for all cases of proven wrong claimed proofs:

Have there been examples where the first obvious error appeared only at a late stage in the manuscript?

Are there examples which contain/imply some interesting*/remarkable/entertaining idea?

Maybe an idea which even was exploited otherwise?

Obviously the cases also can be categorised into cases from professionals and from laymen, and maybe in cases from both (e.g. from "outside" Mathematics, like Physics).

Are there interesting cases from both sides?

Are there repeating patterns which are simple to understand / communicate?

I think some answers which could be given also for this question, are found here ("Examples of interesting false proofs") on MO and indeed at least one answer contains a link to Peter Woit's blog named "not even wrong" where a few examples of wrong RH proofs are given and discussed.

This question focusses on RH in particular and since there should be many more claimed false proofs than these few examples it seems likely that there are also many more interesting cases than those mentioned.

Examples for related questions of a second class are:

Are you aware of interesting claimed proofs of the negation of RH?

Are any particularly interesting cases among them?

Are there interesting examples of claimed $\Re{\rho}\ne\frac{1}{2}$ cases?

Are there other claimed proofs of existence?

Again also here I am only interested in proven wrong proofs, so were someone has already pinpointed the mistake(s).

Edit

*)"interesting" is in this post supposed to mean interesting in terms of mathematics or mathematical considerations arising from.

Expected value of determinant of simple infinite random matrix

Fri, 04/06/2018 - 04:13

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

I would be very happy if someone knows the result or a good reference where I could look it up.

Do algebraic elements form a subring [duplicate]

Fri, 04/06/2018 - 03:49

This question already has an answer here:

It is known that the following statements are true:

(i) If $R,S$ are commutative rings and $R$ is a subring of $S$. Then all of the elements of $S$ integral over $R$ form a subring of $S$.

(ii) If $K/F$ is a field extension then all of the elements of $K$ algebraic over $F$ form a subfield of $K$.

Can these proposition be generalized to this statement?

If $R,S$ are commutative rings and $R$ is a subring of $S$. Then all of the elements of $S$ algebraic over $R$ form a subring of $S$.

At first I want to prove it using the strategy of (ii), however I found that this may work only when the element is integral instead of being algebraic. Now I am not sure if it is right. Can anybody give a proof or counterexample?

Sets of points avoiding small angles

Thu, 04/05/2018 - 15:58

(1) $\mathbb{R}^2$.

I'd like to place $n$ points in the plane so that the smallest angle they determine is as large as possible. In a sense, such a point set is in very general position, not only avoiding three points collinear, but also avoiding near collinearities.

Define the smallest angle of a set $S$ of points to the be smallest angle of any triangle formed by three points in $S$. So the $n=4$ and $n=5$ point sets shown below have smallest angles $45^\circ$ and $36^\circ$ respectively.          

Q1. What is the maximum of the smallest angle determined by any set $S$ of $n$ points, the maximum over all $S$? Is $S$ the vertices of a regular $n$-gon?

Update. Answered Yes by fedja with a nice proof in the comments.

(2) $\mathbb{R}^3$ (Added).

In 3D, the optimal arrangement seems to be akin to packing points on a sphere, e.g., the Tammes problem or the Thompson problem. Below shows the smallest angle realized by the $12$ vertices of the icosahedron.          
          Smallest angle $\approx 31.7^\circ$.

Q2. The same question in $\mathbb{R}^3$, and in $\mathbb{R}^d$, $d>3$.

Likely this question has been studied, in which case pointers to the literature would be appreciated.

Affiliation when invited professor

Thu, 04/05/2018 - 12:10

I am a PhD student in one university and an invited professor in another, that is I do not have a permanent position in the second one. Now I need to indicate an affiliation in a journal paper but I do not know whether to indicate or not to indicate a university where I am an invited professor.

What’s the etiquette for the affiliation to indicate in such a case - to indicate both or only the first one?

$\det(I-K(z)+\varepsilon(z,x)) $ versus $\det(I-K(z))$

Thu, 04/05/2018 - 10:58

First let me ask the general question that might interest others dealing with determinantal formulas. We are trying to compare the following two quantities

$$C_{\varepsilon} := \oint \det(I-K(z)+\varepsilon(z)) \frac{dz}{z}$$

and

$$C := \oint \det(I-K(z)) \frac{dz}{z}$$

where the contour must contain the origin, $I, K(z), \varepsilon(z)$ are $n\times n$ matrices. For $K(z)$ has entries poles wrt to z whereas $\varepsilon(z)$ has entries that are analytic.

A third condition is the following. By residue theorem we $\oint F(\varepsilon(z)) \frac{dz}{z}=F(\varepsilon(0))$ for analytic F. We assume that $\varepsilon(0)$ is the zero matrix. So if for example $F=\det$ we obtain

$$\oint \det(\varepsilon(z)) \frac{dz}{z}=0.$$

Then given this condition we ask:

Q: Ideally $C_{\varepsilon(z)}$ is close to $C$.

This is asking too much since the determinant will have all its terms coupled. So maybe we can say something interesting as $(\varepsilon)_{i,j}\to 0$.

Our particular case

For example, a typical entry for K is of the form

$$\left(\oint_{|w|=R}-\oint_{|w|=\delta}\right) e^{t(w-1)}\left(\frac{1-w}{w}\right)^{q} \frac{1}{w^{n}(1-w)^{l-n}-z^{l}} dw,$$

for some constants $l>0,q>l-n>0$, large R and small $\delta$. The $R,\delta$ are picked so that the poles of $w^{n}(1-w)^{l-n}-z^{l} $ are contained in the annulus $A(0,\delta,R)$. Whereas $\varepsilon(z,x)$ has similar entries

$$\left(\oint_{|w|=R}-\oint_{|w|=\delta}\right) e^{t(w-1)} \left(\frac{1-w}{w^{-x}}\right)^{q} \frac{1}{w^{n}(1-w)^{l-n}-z^{l}} dw,$$

with the exception of a $w^{-x}$ and so as $x\to \infty$ (even just for large enough x) the contour $\oint_{|w|=\delta}$ disappears and we are left with a quantity that is analytic in z. And so we obtain the third condition above:

$$\varepsilon(0) = \left(\oint_{|w|=R} e^{t(w-1)} \left(\frac{1-w}{w^{-x}}\right)^{q_{i,j}} \frac{1}{w^{n}(1-w)^{l-n}} dw \right)_{i,j}=0.$$

Q: So ideally we have $C_{x}\to C$ for large enough x. But again that might be asking too much.

Attempts

  1. expanding the determinant (using Jacobis formula) gives terms where $\varepsilon(z)$ and $I-K$ are coupled

  2. For both integrals expanding in z as a geometric series:

$$\frac{1}{1-\frac{z^{l}}{(w^{n}(1-w)^{l-n})}}$$

by picking the z-contour small enough so that $\left|\frac{z^{l}}{(w^{n}(1-w)^{l-n})}\right|<1$. This gave for the first integral

$$\sum_{k_{1}+ \dots +k_{n}=0} \det[F(x,k)]$$

where $F(x,k_{i})$ is of the form

$$\left(\oint_{|w|=R}-\oint_{|w|=\delta}\right) e^{t(w-1)}\frac{(1-w)^{q+(n-l)k_{i}} }{w^{-x+q+nk_{i}}} dw.$$

Then I will take its difference with that of the other integral.

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