Math Overflow Recent Questions


An efficient way of solving this type of problem

Sun, 03/04/2018 - 00:03

My questions are:

Which section of math the problem falls under?

Is there a formula to find the number of possible solutions for either of the scenarios?

What is the most efficient way of solving this type of problem?

Thank you

The problem

There are variables X1, X2, X3, and X4. Each variable is equidistant from the subsequent one (distance between variables is unrelated to their values). I need to find Y(max) under 2 scenarios. Scenario 1: Intervals cannot overlap, intervals have to be equidistant, and an interval can only be formed from right to left by subtraction. Scenario 2: Intervals cannot overlap, intervals don’t have to be equidistant, and an interval can only be formed from right to left by subtraction?

If I am not mistakes all possible solutions for scenario 1 are

Y1 = (X2-X1) + (X3-X2) + (X4-X3)

Y1 = (X2-X1) + (X4-X3)

Y1 = (X3-X2) + (X4-X3)

Y1 = (X2-X1) + (X3-X2)

Y1 = (X3-X2) Y1 = (X3-X1) Y1 = (X4-X3) Y1 = (X4-X2) Y1 = (X4-X1) Y1 = (X2-X1)

Scenario 2 all possible solutions would be all of the above with addition of

Y1 = (X2 - X1) + (X4 - X2)
Y1 = (X3 - X1) + (X4 - X3)

So my questions are which section of match this type of problem falls under? Is there a formula to find the number of all possible solutions for either scenario? What is the most efficient way to solve these type of problems? (Excel or program etc) Thank you.

Hilbert polynomial and dimension of a multigraded module

Sat, 03/03/2018 - 21:15

Let $S$ be a $\mathbb N^s$-graded finitely generated $R$-algebra where $R$ is local, Artinian and $S'$ be a $\mathbb N^s$-graded standard subring of $S.$ Let $M$ be a finitely generated $\mathbb N^s$-graded both $S$ and $S'$

Is it true that Hilbert polynomial ( of $M$ as $S$ and $S'$-modules are same?

Is it true $\dim_SM=\dim_{S'}M?$

Geometric or topological flavored proof of Nevanlinna five valued theorem?

Sat, 03/03/2018 - 20:26

In a very early state of the development of the Nevanlinna theory, Nevanlinna proved what is now called Nevanlinna five valued theorem,

Let $f$ and $g$ be two transcendental meromorphic function. Assume $\{a_i\}_{i=1}^{5}$ be the five distinct values in the $S^2$, if $\{z:f(z)=a_i\}=\{z: g(z)=a_i\}$ for $1\leq i\leq 5$. Then $f\equiv g$.

I only know the proof which used the mechanism of Nevanlinna theory. And I am really curious whether there exists a proof with geometrical or topological flavor.

Any comments and reference will be appreciated.

Is a cubic hypersurface determined by its Fano variety of lines?

Sat, 03/03/2018 - 20:03

Consider a smooth cubic complex hypersurface $X\subset\mathbf{P}^{n+1}$ of dimension $n\geqslant 3$. The associated Fano variety of lines $F(X)$ is a smooth variety of dimension $2n-4$. Can one recover $X$ from $F(X)$? The answer is positive for $n=3$ (due to Clemens-Griffiths, Tjurin) and $n=4$ (due to Voisin, I think). Are there any conjectures for general $n$?

What do I know about a group if its representations are filtered?

Sat, 03/03/2018 - 18:39

Let $G$ be an affine group scheme over a field.

Say that, for every finite-dimensional representation of $G$, I have a $\mathbb{Z}$-grading on the underlying vector space, compatible with tensor products, with the property that every $G$-equivariant map is graded. Then by abstract Tannaka duality I have a map of group schemes $\mathbb{G}_m\to G$.

Now instead of a grading, say I have a $\mathbb{Z}$-indexed filtration, compatible with tensor products, such that every $G$-equivariant map $f:V\to W$ strictly preserves the filtration, meaning for all $n\in\mathbb{Z}$, $$ f\big(\mathrm{Fil}^n V\big)=f(V)\cap\mathrm{Fil}^n W. $$ What does this tell me about $G$?

homology of meridian circle of genus two handlebody

Sat, 03/03/2018 - 17:47

We know for genus one handlebody (solid torus), the meridian circle is unique up to isotopy on the torus.

Now for a genus two handlebody, the homology group of the its boundary has 4 generators, denoted by $a_1,\ b_1,\ a_2,\ b_2$ (the intersection number of $a_i,\ b_i$ is 1). Denote the homology of the meridian circle by $ma_1+nb_1+ra_2+sb_2$. it is easy to construct examples where m, r are 0, 1 or -1. Are there other possibilities?

It seems $n$ and $s$ have to be 0. $m$ and $r$ have to be coprime. ($ma_1+ra_2$ is primitive, not sure how to prove). I tried but I cannot construct a meridian disk whose boundary represent $2a_1+3a_2$.


Zero energy resonances for scaling critical Schrodinger operators

Sat, 03/03/2018 - 17:04

Given a real valued potential $V\in L^1(\mathbb{R}^3)$, we say that the Schrodinger operator $-\Delta + V$ has a zero-energy resonance if there exists $\psi\in L^2_{loc}(\mathbb{R}^3)\setminus L^2(\mathbb{R}^3)$ such that $$(-\Delta+V)\psi=0.$$ Is there a simple way to show the existence of a potential $V\in L^{1}(\mathbb{R}^3)\cap L^{3/2}(\mathbb{R}^3)$ such that the corresponding Schrodinger operator actually has a zero-energy resonance?

Are all these representations supercuspidal

Sat, 03/03/2018 - 16:25

Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries \begin{align*} \mathrm{GU}(V, q) & = \left\{ g \in GL(V) \ : \forall x, y \in V, q(gx,gy) = q(x,y) \right\} \end{align*}

These groups are known to be the inner forms of $GSp(4)$ over $F$. Moreover, they have compact automorphic quotient if and only if $D$ is ramified at a certain real place and $q$ is positive-definite or negative-definite (since anisotropy is sufficient at one place to have global anisotropy, by the local-global principle for quadratic forms).

Let take such an inner form with compact automorphic quotient. For every finite place $p$, there is a unique hermitian space giving the unique non-trivial inner form of $\mathrm{GSp(4)}$ ovet $F_p$, and it is isotropic.

What can be said about the representations of this unique non-trivial inner form? Are they all supercuspidal? (or: is this group compact modulo the center?)

I am interesting in the possibility of using matrix coefficients for selecting representations.

Show vector triple product graphically [on hold]

Sat, 03/03/2018 - 12:51

How do you show $a\times(b\times c) = (a\cdot c)b - (a\cdot b)c$ graphically?

This question came to mind while I was trying to derive Rodrigues' rotation formula.

I wonder if a similar graphical proof exists for the vector triple product. I tried but didn't succeed.

Any help would be appreciated.

On 2-actions of strict 2-groupoids?

Sat, 03/03/2018 - 09:11

I'm looking for an opinion if the following makes sense.

A linear representation of a groupoid $\mathcal{G}$ is a functor $$\nabla: \mathcal{G}\longrightarrow \mathsf{Vect}_{\mathbb K},$$ where $\mathsf{Vect}_{\mathbb K}$ is the category of vector spaces over $\mathbb K$.

I presume we could define analogously a linear 2-representation of a strict $2$-grupoid $2\textrm{-}\mathcal{G}$ as a strict 2-functor:

$$2\textrm{-}\nabla: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Vect}_{\mathbb K},$$ where $2\textrm{-}\mathsf{Vect}_{\mathbb K}$ is the $2$-category of 2-vector spaces (the objects are groupoids internal to $\mathsf{Vect}_{\mathbb K}$). I don't know if this definition appears in the literature but I believe it sounds reasonable.

Now, how to make sense of a non-linear version of a 2-representation of a 2-groupoid?

I'm inclined to define a 2-action of a strict $2$-groupoid $2\textrm{-}\mathcal{G}$ is a strict two functor

$$\rho: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Grpd},$$ where $2\textrm{-}\mathsf{Grpd}$ is the 2-category of 2-grupoids (the objects are groupoids internal to $\mathsf{Grpd}$).

To me it sounds $2\textrm{-}\mathsf{Grpd}$ must be the non-linear version of $2\textrm{-}\mathsf{Vect}_{\mathbb K}$.

But I'm not quite certain if this is really coeherent with what an action must mean. I believe category theory must provide some prototype of what an action must be in any setting and I should follow that, but that is beyond my categorical knowledge.

To be precise, my question is, is it "right" to define a 2-action of a strict 2-groupoid as strict 2-functor $\rho: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Grpd}$?

I'm looking for arguments which could support it is "natural" to make such definition.


Can every curve be made transversal to a foliation by applying a pseudo-Anosov?

Sat, 03/03/2018 - 00:49

Let $F$ be a compact oriented surface with a foliation $\cal F$ with $k$-prong singularities only (or, if it helps, assume that $\cal F$ admits an invariant measure). Is it true then there exists a pseudo-Anosov $\phi$ such that for every loop or an arc (with endpoints in $\partial F$) $\gamma$, $\phi^n(\gamma)$ is transversal to $\cal F$ for large $n$?

Here is a rough argument why it is probably true: The $MCG(F)$ action on ${\cal PMF}(F)$ is minimal, cf. Fathi-Laudenbach-Poenaru, Thurston's Work on Surfaces, for closed $F$. (A reference for bounded $F$ would be useful although the original proof seems to work). This implies that the unstable foliations $\cal F^u_\phi$ of pseudo-Anosovs $\phi$ are dense in ${\cal PMF}(F)$. So it is reasonable to expect that there is $\phi$ with all angles between $\cal F^u_\phi$ and $\cal F$ greater than some $\alpha>0.$ (But I don't have a rigorous argument for that.) Now by the work of Thurston (cf. the book above), the angles between $\phi^n(\gamma)$ and $\cal F^u_\phi$ converge to $0$ as $n\to \infty.$ Consequently, $\phi^n(\gamma)$ is transverse to $\cal F$ for large $n$.

EDIT: To address the comment below, let us just assume that $\cal F$ admits an invariant measure of full support.

Maximizing quadratic form subject to inequality constraints

Fri, 03/02/2018 - 19:42

Given a $n \times n$ symmetric matrix $\rm S$, solve the optimization problem in $n \times k$ (where $n \geq k$) matrix $\rm X$

$$\begin{array}{ll} \text{maximize} & \mbox{tr} \left( \mathrm X^\top \mathrm S \,\mathrm X \right)\\ \text{subject to} & \mathrm X \in [0,1]^{n \times k}\end{array}$$

A strongly non-integrable distribution

Fri, 03/02/2018 - 15:26

What is an example of a three-dimensional smooth distribution $D$ of $\mathbb{R}^4$ with this property:

Not only $D$ is not integrable but also there is no a two-dimensional foliation $F$ of $\mathbb{R}^4$ such that for every $x\in \mathbb{R}^4$, the tangent space to each leaf at $x$ is contained in $D_x$.

Finitely generated group splitting non-trivially over an infinite virtually cyclic subgroup

Fri, 03/02/2018 - 06:37

Let $G$ be a finitely generated group splitting non-trivially over a infinite virtually cyclic subgroup $V$, namely an amalgamated free product

$$ A*_V B,\; V \neq A,B. $$ Question: can such a group $G$ be simple ?

Confused about the definitions of Hardy space

Fri, 03/02/2018 - 06:31

Can anyone help me to be clear about some definitions of Hardy space in literatures?!

Let $\Omega$ be a smoothly bounded domain in $\mathbb{C}^{n}$ and $p>0$.

Definition 1: $f\in H^{p}\left(\Omega\right)$ iff $f$ is holomorphic in $\Omega$ and $$ \limsup_{\varepsilon\rightarrow0^{+}}\intop_{\delta=\varepsilon}\left|f\right|^{p}dS<\infty.(1) $$

Definition 2: $f\in H^{p}\left(\Omega\right)$ iff $f$ is holomorphic in $\Omega$ and there is $\varepsilon_{0}>0$ such that

$$ \sup_{0<\varepsilon<\varepsilon_{0}}\intop_{\delta=\varepsilon}\left|f\right|^{p}dS<\infty.(2) $$

The terms in these definitions will be defined as $\left\Vert f\right\Vert _{H^{p}\left(\Omega\right)}^{p}$. Here $\delta$ is the distance function to the boundary.

My question: Is it true that if $f\in H^{p}\left(\Omega\right)$ in the Definition 1 then $f\in H^{p}\left(\Omega\right)$ in the Definition 2, and vice versa? Moreover, is it true that $$ \limsup_{\varepsilon\rightarrow0^{+}}\intop_{\delta=\varepsilon}\left|f\right|^{p}dS=\sup_{0<\varepsilon<\varepsilon_{0}}\intop_{\delta=\varepsilon}\left|f\right|^{p}dS $$ for any $f\in H^{p}\left(\Omega\right)$?

Number of solutions to polynomial congruences

Fri, 03/02/2018 - 06:17

Suppose I have $R$ homogeneous polynomials $F_1, ..., F_R$ with integer coefficients. Let $V$ be the affine variety defined by these polynomials over $\mathbb{C}$. I was wondering if some bound that looks like the following was known? $$ \#\{ \mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n: F_{i}(\mathbf{x}) \equiv 0 (\text{mod }q) \text{ for each } 1 \leq i \leq R \} \leq C q^{\dim V + \delta} $$ for some $\delta > 0$. Here $C>0$ is independent of $q$. Of course if $\delta = n - dim V$ then it's trivial so I was hoping for something smaller. I would greatly appreciate any comments on this. Thank you very much.

Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter

Fri, 03/02/2018 - 04:54

First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a topological disk.

Question. Let $(\Sigma,x)$ is a compact hyperbolic surface (without boundary) of genus $g$ and diameter $d$ with a marked point $x$. How to estimate from above in terms of $g$ and $d$ the number $A$ such that there exists a standard collection of geodesic loops on $\Sigma$ based at $x$, all shorter than $A$?

I would be especially grateful for a reference.

Buy a Dinner Set

Thu, 03/01/2018 - 04:13

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Supremum norm of certain quantity II

Thu, 03/01/2018 - 04:12

Can anyone solve the maximization problem...$\max_{|z_i|=1}\Big|\sum_{i,j=1}^nz_iz_j+\sum_{i,j=1}^n|z_i-z_j|\Big|$?

When the global section functor is a Cartesian fibration?

Thu, 03/01/2018 - 04:03

Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to this question gives some sufficient conditions for $\mathbf{E}$ to be an $\infty$-topos. I'd like to know when the converse holds. That is, if we have a functor between toposes $p : \mathbf{E} \to \mathbf{B}$, when it is a Cartesian fibration? Since this question is probably too general, it may be assumed that $p$ is the direct image of a geometric morphism and even more that it is the global section functor $\mathrm{Hom}(1,-) : \mathbf{E} \to \mathrm{\infty Grpd}$.

More generally, if $\mathbf{C}$ is an arbitrary $\infty$-category with a terminal object, when the functor $\mathrm{Hom}(1,-) : \mathbf{C} \to \mathrm{\infty Grpd}$ is a Cartesian fibration? Since this question is also probably too general, it may be assumed, if it helps, that $\mathbf{C}$ is locally presentable and that $\mathrm{Hom}(1,-)$ has either left or right adjoint.