# All Questions

114,973 questions
Filter by
Sorted by
Tagged with
59 views

68 views

### Characteristic polynomial of the Gcd matrix

Let $A_n$ be the $n \times n$-matrix with entries $Gcd(i,j)$ and $f_n$ the characteristic polynomial of $A_n$. Question: Is $f_n$ irreducible for all $n$ except $n=8$? This is true for $n \leq 60$.
25 views

### Upper bounding the sum with hypergeometric and binomial probabilities

Could you please help me upper bound this tricky expression: $$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$. So far I only ...
104 views

### The space of rearrangements of a plane curve

I learned of the paper "Closing curves by rearranging arcs" by L. Alese (arXiv link) by a post on Reddit and was curious about related questions and generalizations. Here a rearrangement of ...
81 views

### Holomorphic tubular neighborhood of divisors at infinity

For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. Hironaka tells us that ...
139 views

### A set of objects classically generates the full subcategory of compact objects iff it generates the whole category

Sorry in advance if my question doesn't have the level of this community. I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
40 views

### How do you find the Cholesky decomposition of the sum of two positive definite matrices without adding the matrices directly?

If you're given two positive definite matrices ($A_1,A_2$) and the Cholesky Decomposition of those two matrices ($L_1,L_2$ such that $A_1=L_1L_1^T, A_2=L_2L_2^T$). Is there a way to find the Cholesky ...
30 views

### Relation between variables (vertexes, edges, regions and faces) in three dimensional Voronoi diagram

A Voronoi diagram is a kind of tesselation that divided the medium into polygons in 2D and polyhedrons in 3D. In two dimensions, any Voronoi diagram has vertexes(V), edges(E) and regions(F) that equal ...
127 views

### $X-Y$, where $X$ and $Y$ are sums of Bernoulli random variables

Let $X = x_1 + x_2 + \ldots + x_n$ and $Y = y_1 + y_2 + \ldots + y_n$, where each $x_i$ is an independent Bernoulli random variable with success probability $p_i$ and each $y_i$ is a Bernoulli random ...
72 views

### Counting quadratic curves in $\mathbb P^1 \times \mathbb P^1$ passing through seven points in general position

Let $p_1,\dots,p_7 \in \mathbb P^1 \times \mathbb P^1$ be 7 points in general position. What is the number of maps $F=(F_0,F_1):\mathbb P^1 \to \mathbb P^1 \times \mathbb P^1$ modulo domain ...
77 views

### Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result: (C-S-T): Let $G$ be a ...
55 views

223 views
+50

148 views

### Pell equation and quadratic residues

We say an integer $k$ is Pell if there exist some integers $p,q$ such that $$p^2k-q^2=1$$ In studying a physics system we ended up with two weaker notions of Pell: We say an integer $k$ is pre-Pell ...
85 views

### What is the multiplicative order of this number

Let $q, r \in \mathbb{P}$ and $r$ is the next prime to $q$. What is the multiplicative order of $r$ modulo $\displaystyle\bigg( \prod_{\substack{p \leq q \\\text{p prime}}} p \bigg)$ ? In other word ...
29 views

### Uniquesnes of basis of the matrices with degenerate eigenvalues [closed]

How to prove the statement, "when two or more eigenvalues of a matrix, say $M$, are equal, the basis of eigenstates of $M$ is not unique"?
35 views

### Link between characters and isotypic components

I am currently studying finite complex reflection groups using the book written by Lehrer and Taylor called "Unitary Reflection Groups" and I am unsure if I understood isotypic components ...
69 views

### Commutative C*-rings

Let us consider the unital commutative $C^*$-algebra $C[0,1]$. We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions: 1- $A$ is an involutive unital subring (closed ...
41 views

100 views

### Renorming of $C[0,1]$ for a strictly convex dual

Let $C[0,1]$ be the space of all Real valued continuous functions on $[0,1]$ with the usual supremum norm. Does there exist an equivalent renorming on $C[0,1]$ such that the corresponding dual norm is ...
23 views

### Barycentric spanner for set of subsets

Let $V=\{1,...,n\}$ be the ground set, and $\mathcal{S} \subseteq 2^V$ be a set of subsets of $V$. Are there algorithms to compute barycentric spanner of $\mathcal{S}$ of size O(n)? Let us describe ...
95 views

### When is the inside of a Jordan curve open? [closed]

I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this ...
15 views

### Error in second derivative using lagrange quadratic interpolation

Quadratic lagrange interpolation is given by $P(x)=\sum _{k=0}^2 l_k(x) f_k$ with error term $E(x)=\frac{(x-x_0)(x-x_1)(x-x_2) f^{'''}(\xi )}{3!}$. To calculate, error for second derivative at point ...
83 views

### Natural candidates for sub-half-exponential which limit to half-exponential function from below

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However sub-half-exponentials (functions whose composition grows ...
142 views

### When did the main conjecture in Vinogradov's mean value theorem first appear in literature?

Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...
31 views

43 views

### Bruhat-Tits theory: how does the normalizer act on an apartment?

Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the ...
Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
For a graph $G=(V,E)$, is there a standard notation for the induced subgraph on $V \setminus \{v,w\}$ where $v,w$ are the endpoints of some edge $e$? I know $G[V \setminus \{v,w\}]$ is an option, but ...