# All Questions

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### Problems whose first solutions had been using Calculus but later was shown to be done by n0n-Calculus methods

I was wondering about mathematical problems whose first published solutions was obtained by using methods of Calculus but later was shown (or known) to be solvable by using non-Calculus methods. ...
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### How find the smallest $m$ such this $|A|=n,|B|=m,A\subseteq B$

let $n\ge 5$ Find the smallest $m$, there exist two set $A,B$(with element is postive integer )such following two condition: $|A|=n,|B|=m,A\subseteq B$ $\forall x,y(x\neq y)\in B$, have $x+y\in B$ ...
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### Show that the number of subsets of $S_1 \cup \dots \cup S_t$ that contain at most one element from each $S_i$ is $(a_1 + 1)(a_2 + 1) \dots (a_t + 1)$.

I found this problems on Aigner's: A course in enumeration: 1.1 We are given $t$ disjoint sets $S_i$ with $|Si| = a_i$. Show that the number of subsets of $S_1 \cup \dots \cup S_t$ that contain ...
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### Computing the limit of an alternating series,

I am looking at the series $$\sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
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### Why are quaternions useful?

What I mean is why are they used basically where they are used? Listing some advantages of using them would be better. I am taking a mechanics course where a teacher mentioned them in a discussion ...
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### Matroid Isomorphism Definition

I'm working though Welsh's Matroid Theory work, and he very casually mentions matroid isomorphisms in the first chapter but I don't think I like his statement. He says that two matroids ...
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### How prove this complex inequality with same as (2014 china CMO) Cauchy-Schwarz inequality

let $r$ is give numbers,let $z_{1},z_{2},\cdots,z_{n}$ such $|z_{i}-1|\le r,i=1,2,\cdots,n,r\in(0,1)$ show that ...
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### What is an affine space?

I am having trouble understanding what an affine space is. I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say: "The upshot is that, even in the affine plane, one can ...
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### Left and Right Discrete Maximal Functions

Define the uncentered maximal function $$\widetilde{M}f(n)=\sup_{s,r\in\mathbb{Z}^{+}}\dfrac{1}{s+r+1}\sum_{k=-r}^{k=s}\left|f(n+k)\right|,$$ where $\mathbb{Z}^{+}=\left\{0,1,2,\ldots\right\}$. Define ...
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### probability circle determined by chord determined by two random points is enclosed in bigger circle

Two points $A$ and $B$ are chosen uniformly at random from the interior of a circle $X_1$. Let $X_2$ be the circle whose diameter is the segment $AB$. What is the probability that $X_2$ is contained ...
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### Triangles formed by line segments in a square

There is a square, denoted by points A, B, C, and D. There are 30 distinct points located inside the square (call these $A_2, A_3, A_4, ... A_{31}$. Non-intersecting segments $A_iA_j$ vertices are ...
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### Distinct integers with $a=\text{lcm}(|a-b|,|a-c|)$ and permutations

Do there exist three pairwise different integers $a,b,c$ such that $$a=\text{lcm}(|a-b|,|a-c|), b=\text{lcm}(|b-a|,|b-c|), c=\text{lcm}(|c-a|,|c-b|)?$$ None of the integers can be $0$, because the ...
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### Seemingly Simple Integration: $x/(x-1)$

I am currently working on some advanced engineering math but this seemingly simple integral has me stuck. Someone please show me how to derive it. It is part of a far bigger more complex problem in ...
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### Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
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### How to calculate 2-d plane from 3 4-d points?

I want to compute 3-d cross-sections of a pentatope (4-dimensional tetrahedron). The 3-d cross-sections will be calculated as: x+y+z+w=c C is a constant that I will vary to get different ...
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Is there an option or can one be implemented so that new users like myself can view the "source code" of others answers. Obviously there are many tutorials in which we can find the correct commands, ...
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### How to describe the Cartesian product $\mathbb{R} × \mathbb{R}$? [on hold]

Let $\mathbb{R}$ denote the set of all real numbers. Describe $\mathbb{R} × \mathbb{R}$. (This is a self-answered question).
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### Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
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### Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
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### Let A,B be nxn matrices such that detA Not equal to 0, but detB = 0: Show [on hold]

Let $A$, $B$ be $n\times n$ matrices such that $\det A \neq 0$, but $\det B = 0$. Show $\|A-B\|_2 \geq(\|A^{-1}\|_2)^{-1}$.