# All Questions

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### Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer. Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let $Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let ...
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### When two Dedekind sums are equal

The (classical) Dedekind sum $s(h,k)$ is defined as $$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\bigg(\frac{hr}{k}-\Big[\frac{hr}{k}\Big]-\frac{1}{2}\bigg)$$ for $\gcd(h,k)=1$. A natural question is, when ...
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### What is the smallest partition lattice PART(M) containing the lattice P(N) of subsets of a finite set of N elements

It's been known for 35 years that every finite lattice can be embedded in a finite partition lattice ( Pudlak and Tuma algebra universalis 1980, Volume 10, Issue 1, pp 74-95). I don't follow the ...
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### rational cohomology of finite dimensional real grassmannian

Let $G_k(R^n)$, $n>k$, be the finite dimensional real grassmannian. What is the rational cohomology algebra $H^*(G_k(R^n);Q)$? I have searched out that $H^*(BO_k;Q)=Q[p_1,p_2,...,p_[k/2]]$ is the ...
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### Prize allocation of scratch codes to ensure correct number of prizes to give away [on hold]

We would like to give away 100 prizes. We have 13.5 million codes, divided into 20 categories. To win a person must collect one code in each category. If 15 of those categories have 877,195 codes ...
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### Finiteness of geometric valuations

I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference: Let $X$ be a $\mathbb{Q}$-factorial variety with log ...
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### Mathematicians wearing hats on arbitrary total orders

I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$: Question: We have some totally-ordered set $T$ of mathematicians, each wearing a hat ...
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### How can i change 8_19 to (3,4)-torus knot K(3,4)?

In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two ...
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### Show that among any 6 non-negative integers one can find 2 integers so that their difference is divisible by 5 [on hold]

I have the following question in homework I have been assigned in Discrete Mathematics relating the pigeonhole principle: "Show that among any 6 non-negative integers one can find 2 integers so that ...
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### A certain matrix associated to graphs

I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph ...
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### Partial Sum of the Binomial Theorem [duplicate]

The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function $$\sum_{k=0}^mC_n^kr^k, \quad m<n$$ for fixed $n$ and $r$, and both $m$ and ...
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### Classes of knots that have known Bridge spectra

Bridge spectra is a knot invariant first defined by Doll, who established some basic properties. Tomova has shown that high distance knots have bridge spectra $(n,n-1,\ldots,2,1,0)$. Zupan has ...
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### Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...
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### Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
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### Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
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### Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...
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### How can I calculate the global log canonical threshold?

Let $X$ be a normal variety and let $D$ be a $\mathbb{Q}$-divisor. The log canonical threshold of a pair $(X,D)$ is $lct(X,D)=sup\{c|(X,cD)$ is log canonical$\}$. If $X$ is $\mathbb{Q}$-Fano ...
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### A strange condition on containment of special complex numbers in cyclotomic fields

In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied: $\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$ where $a\in\mathbb C^*$ and ...
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### Is there a maximal connected Hausdorff space?

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. Is there a maximal ...
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### Maximal connected topologies

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. If $(X,\tau)$ is ...
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### Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
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### LyX preamble to auto add space before and after math (CTRL+M) [migrated]

I tried everything, but nothing works. How can I make auto space bofore and after math formulas in the same line? Thank you.
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I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism. I have a question about what he calls a "class III ...
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### Irreducibility of a general fibre

Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact? If $A$ is algebraically ...
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### Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...
608 views

### (Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
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### How to estimate some combinatorial expression? [on hold]

How to estimate (or explicitly compute) the following sum $\sum_{j=1}^{k}\left|\binom{x}{j}\binom{k-1}{j-1}\right|$ from above? The most convenient estimation ...
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### Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices. Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...
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### Monoidal structure on simplicial sheaves

Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category ...
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### Deriving sequence of number added after sum of the number [on hold]

I just want to know , is there any mathematics operation by which we can achieve the below mentioned task: 1> I will sum up a sequence of number, it could be any number(For the task I am free to ...
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### Hyperellptic curve defined by a set of rational points

If we fix a field $\mathbb{F}$ of positive characteristic, and a a genus $g$ , how many rational points are enough to build a unique hyperelliptic curve of genus $g$ over $\mathbb{F}$?. The thing is ...
I have few questions about the subsets of a (for example) Hilbert Space endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? This happens for ...
### The number of $k$-free integers not exceeding $x$
We say that an integer $n\in\mathbb{N}$ is $k$-free if for each prime $p\mid n$, one has $p^k \nmid n$. Let $\mu_k(n)$ be the characteristic function of $k$-free numbers, where $k\ge 2$. Let ...