I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help on a possible approach would be much appreciated. Thanks very much!

**Given:**

- The polynomial ring $A = \mathbb{R}[x, y]$. It can be graded according to total polynomial degree as $A = \oplus_{k\geq 0} A_k$, with the Hilbert function of the $i^{th}$ piece given by $HF_i(A)= i+1$.
- The $A$-module $M = \oplus_{i}[\alpha_i]A$ generated by formally independent elements $\alpha_i$ coming from some finite set of cardinality $m$.
- The submodule $M \supset N = \sum [\beta_i] A$, where each generator $[\beta_i] = [\alpha_j] - [\alpha_k]$ for some $j$ and $k$.
- A surjective map $f: \{[\alpha_i]\} \rightarrow \{[\gamma_j]\}$, where $\gamma_j$ are formally independent elements of another finite set of cardinality $p < m$. Assume that, $$ \forall [\beta_i] = [\alpha_j] - [\alpha_k], \quad f([\beta_i]) := f([\alpha_j]) - f([\alpha_k]) \neq 0. $$
- A map, $$ g : \sum_{i \in I} [\alpha_i]a_i \mapsto \alpha_{\max(I)} a_{\max(I)}\;, $$ where $A \ni a_{i} \neq 0$ for all $i \in I$, and $I \subseteq \{1, \dots, m\}$.
- The submodules $N_0$ and $N_1$, $$ N_0 = \sum f([\beta_i])A, \qquad N_1 = \sum f\circ g([\beta_i])A. $$

**Claim:** Assuming the natural grading on $N_0$ and $N_1$, the following inequality holds:
$$
HF_i(N_0) \geq HF_i(N_1).
$$

**EDIT**

A simplified/alternate version of the problem statement on has been posted on SE by posing the problem in terms of real vector spaces. If that helps on nailing down an approach, you can find it here: https://math.stackexchange.com/questions/2728748/vector-space-dimension-after-nonlinearly-mapping-spanning-vectors

The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the category of $G$-modules to the category of Abelian groups sending each module $M$ to its subgroup consisting of all elements of $M$ which are fixed by $G$ action.

Now we fix a natural number $n\in \mathbb{N}$ and do an obvious generalization of the above construction:

Instead of the above left exact functor $FIX$ we consider the functor $P_n$ which send a $G$-midule $M$ to the following subgroup of $M$: $$P_n(M)=\{x\in M \mid g.x= nx, \forall g \in G\}$$.

In this way what would be the corresponding derived functor?What kind of cohomology theory would appear? Is this a trivial generalization of classical "Group Cohomology"?If not, what would be a topological analogy?

The later question is motivated by the fact that the group cohomology corresponds to singular cohomology of the corresponding Eilenberg-Mclane space)

Let $E$ be an infinite-dimensional complex Hilbert space.

The spectral radius of a **commuting** multivariable operator $A = (A_1,\cdots,A_n)\in\mathcal{L}(E)^n$ (i.e. $A_iA_j=A_jA_i$ for all $i,j$) is given by
\begin{align*}
r_a(A_1,\cdots,A_n)
& =\displaystyle\lim_{m\to \infty}\left\|\displaystyle\sum_{|\alpha|=m}\frac{m!}{\alpha!}{A^*}^{\alpha}A^{\alpha}\right\|^{\frac{1}{2m}} \\
&=\sup\{\|\lambda\|_2,\;\;\lambda \in \sigma_{ap}(A)\},
\end{align*}
where
$$\sigma_{ap}(A)=\bigg\{\lambda\in \mathbb{C}^n: \;\exists\;(x_k)_k\subset E;\,\,\|x_k\|=1\;\;\hbox{such that}\;\;\\\lim_{k\longrightarrow \infty}\sum_{1\leq j\leq n}\|(A_j-\lambda_j)x_k\|=0\bigg\}.$$

If $n=1$, it is well known that $r(A)=r(A^*)$. I **claim** that if $A_iA_j=A_jA_i$ for all $i,j$, then in general
$$r_a(A_1,\cdots,A_n)\neq r_a(A_1^*,\cdots,A_n^*).$$
I hope to find an example which show that the claim is true.

It is well-known that $$H^*(ko,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(1)}\mathbb{Z}/2$$ $$H^*(tmf,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(2)}\mathbb{Z}/2$$ where $\mathcal{A}$ is the mod 2 Steenrod algebra.

$H^*(MSpin,\mathbb{Z}/2)$ and $H^*(MString,\mathbb{Z}/2)$ are closely related to the above because of the Atiyah-Bott-Shapiro orientation and Witten genus.

I find in Adams and Priddy's Uniqueness of BSO: $$H^*(ko,\mathbb{Z}/p)=\bigoplus_{s=0}^{\frac{p-3}{2}}\Sigma^{4s}\mathcal{A}_p/(\mathcal{A}_pQ_0+\mathcal{A}_pQ_1)=\bigoplus_{s=0}^{\frac{p-3}{2}}\Sigma^{4s}\mathcal{A}_p\otimes_{E(Q_0,Q_1)}\mathbb{Z}/p$$ where $\mathcal{A}_p$ is the mod $p$ Steenrod algebra for odd primes $p$ and $Q_0=\beta,Q_1=P^1\beta-\beta P^1$.

I want to know what is $H^*(MSpin,\mathbb{Z}/p)$, $H^*(tmf,\mathbb{Z}/p)$ and $H^*(MString,\mathbb{Z}/p)$ for odd primes $p$.

Any references and partial answers are appreciated.

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies what happens to (Lagrangian) tori which are invariant under the Hamiltonian flow $\phi_{H_{\epsilon}}$, associated to $$ H_{\epsilon}(q,p)=H_0(p)+\epsilon F(q,p) \quad \forall \ (q,p)\in \mathbb{T}^n\times \mathbb{R}^n,$$ when $\epsilon$ varies. If the rotation vector of the restriction of the Hamiltonian flow $\phi_{H_{0}}$ to the torus $T(p_0):=\mathbb{T}^n\times \{p_0\}$ is Diophantine, then the KAM theorem (Kolmogorov-Arnold-Moser) guarantees, for all small enough $\epsilon>0$, the existence of an invariant Lagrangian torus $L\approx \mathbb{T}^n \subset \mathbb{T}^n\times \mathbb{R}^n$ which is "close" to $T(p_0)$ and invariant under $\phi_{H_{\epsilon}}$.

Is $L$ also guaranteed to be Hamiltonian isotopic to $T(p_0)$?

Any explanations/references will be much appreciated.

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).

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.)

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.

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?

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!

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)** $B_{i,p}=0~~\forall p\in [c_B]$

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 $k^2$.

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 $\{T^ng \mid n = 0,1,\ldots\}$?*

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.

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?

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 $x$)?

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

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.

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!

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.