# All Questions

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Let s1,s2...,s101 be 101 bit strings of length at most 9. Prove that there exist two strings, si and sj , where i≠j, that contain the same number of 0’s and the same number of 1’s. (For example, ...
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### Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies. [on hold]

Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies. Suppose dividing 1 people named A and other 9 ...
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### Does the centroid depend continuously on the curve?

Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the centroid of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on ...
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### The trace of a wedge product of matrices

I'm trying understand a computation on page 371 of Besse's book on Einstein Manifolds. I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form ...
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### Bound on measure of set difference by means of Laplacian

I am considering the following setting: Let $f,g$ be sufficient 'regular' probability density function, and $A=\{\Delta f<0\}$ and $B=\{\Delta g<0\}$, what I want is that if $||f-g||<\delta$ ...
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I am working on a problem which involves proving that a particular graph is a forbidden minor of the class of graphs that i am working on. Now i read kuratowskis theorem for planarity but i still ...
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### Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration

Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let ...
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### Boundary of fibers of submersions

Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result: ...
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### Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture

Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
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### local systems with cyclic monodromy

In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows: Let $X$ be a smooth projective variety over some field $k$ of ...
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### Derivative of Set Functions

For any closed set $I\subseteq (0,1]$, I have a set function in the form of $$\theta(I)=\int_{q(I)}YdF(Y)$$ Where $q(I)$ if the image of I under the mapping $u \mapsto q(u)$. And $F(\cdot)$ is the ...
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### Derivative of diag (A) with respect to A [on hold]

Assume matrix A has no special structure, how to calculate ? Where means getting the element from the diagonal. I'm not sure whether $$\delta(A)$$ is a column vector consisting the diagonal element ...
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### Properties of singularities that are preserved by categorical quotients

Let $G$ be a reductive group acting on an affine singular variety $X$, and let $X/G$ be the categorical quotient. I know that if $X$ has rational singularities, then so does $X/G$ ...
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### Hessian in local coordinates [migrated]

Let $f\colon M\to \mathbb{R}$ be a smooth function on a manifold $M$ with a critical point $p$. We define its Hessian at $p$ via $H(u, v)=(UVf)(p)$ where $u, v\in T_pM$ and $U$ and $V$ are vector ...
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### coends of stable infinity categories

Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote ...
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### Given an even function how to obtain the most close odd function and vise versa?

Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$? By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference ...
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### Positivity question on K3 surfaces

Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$. (Q1). do we have $L\cdot D\geq0$ ? If either one has positive self-intersection, the ...
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### Fourier Analysis in Kahane and Zelasko's characterization of maximal ideals in commutative Banach algebra

I've been told that Kahane and Zelasko (in A characterization of maximal ideals in commutative Banach algebra's Studia Math 29 1968) use a "non-trivial result" from Fourier analysis in their proof of ...
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### Exact Prime Counting Function [on hold]

There have been dozens of prime counting functions $\pi(n)$ that have been created over the years and yet the most common ones I see are either estimates (prime number theorem) or they require the ...
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### Which hyperbolic tilings are Cayley graphs?

I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here: Given a regular tiling of the hyperbolic plane is ...
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### Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS. It doesn't appear at this time. I can't find any reference to it either. Has anyone seen it? ...
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### Is this integration by parts legitimate?

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,$$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
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### A question about the inverse of a matrix [on hold]

and it's obvious that det A =w=1 or -1. My question is simple, but it seems to be chaos. I will be very happy if someone can replay.
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### reference request: Riesz/Newton potential and HLS inequality in L1.logL1

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$ $$f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,$$ which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...
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### Bivariate Function Approximation

I am working on a nonlinear control design and having difficulty in finding approximation of bivariate functions. Are there papers or methods discussing the following question: For any bivariate ...
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### What is a good poster for a math conference?

I'm going to participate to a conference and they ask me to do a poster on my research. I've never made a poster for a conference/seen a poster session in a conference. So what is important? What do ...
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