1
vote
1answer
19 views

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. ...
0
votes
0answers
13 views

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 $x+y\in B$ ...
1
vote
2answers
9 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is an ...
-4
votes
0answers
21 views

How to find area of a curve?

Here is the equation of a curve: $ a^2 \cdot y^2 = x^3(2a - v) $. Now I want to find the whole area of the curve . How can I find the whole area?
2
votes
1answer
9 views

inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
3
votes
0answers
13 views

Multiple integral involving product of gamma function

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
0
votes
1answer
16 views

Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
2
votes
0answers
16 views

Area of weird circumcenter triangle equals area of medial triangle

Let $X$, $Y$, $Z$ be the midpoints of sides $BC$, $AC$, $AB$ respectively in triangle $ABC$. Let $O_{A}$, $O_{B}$, and $O_{C}$ be the circumcenters of triangles $AZX$, $BXY$, and $CYZ$ respectively. ...
1
vote
0answers
11 views

Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
1
vote
0answers
13 views

Shifting integration variables

I'm not sure how to pose this question precisely, but I'll try. I'm trying to see what happens when you have an integral of the form $\int \mathrm{d}x \,f(x-g(z))$ and you try and write it as $\int ...
0
votes
2answers
13 views

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 ...
0
votes
2answers
40 views

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) ...
1
vote
0answers
47 views

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 ...
0
votes
0answers
8 views

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 ...
1
vote
0answers
30 views

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 ...
2
votes
2answers
32 views

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 ...
0
votes
0answers
4 views

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 ...
2
votes
1answer
20 views

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 ...
1
vote
1answer
15 views

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 ...
2
votes
0answers
9 views

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 ...
1
vote
3answers
45 views

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 ...
3
votes
5answers
53 views

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 ...
1
vote
1answer
12 views

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 ...
1
vote
0answers
15 views

Ability to View Answers in LaTex [migrated]

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, ...
0
votes
1answer
15 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
1
vote
2answers
32 views

Health Risk Probability

Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman ...
2
votes
1answer
20 views

Find the continuous function such that the Riemann integrable is the same

Find all functions $f$ such that $f$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t) dt$ for every x $\in (0,1)$ I can't think of any function that would satisfy this property! ...
1
vote
4answers
37 views

How to prove this equation by induction?

I am trying to prove this equation by mathematical induction $$f_{n+1}f_{n-1} = f_{n}^{2}+(-1)^n$$ is true where $f_{n} = $ the nth number in the Fibonacci sequence. I don't quite get how to do this ...
-1
votes
0answers
13 views

Differential equation to space state excercise

This is a "back of chapter" excercise which im trying to solve, my answer doesnt match the solution printed on the book, I want to write the equation in state space matrix form without using the ...
2
votes
1answer
32 views

Chance of winning a game of hearts with four players

I play hearts with a computer game program. The game is set up so that four people are playing the game. The question is: What are the mistakes, if any, with assuming that the probability of winning a ...
-4
votes
0answers
35 views

complex problem in linear algebra [on hold]

Let $A$ be an $n$ by $n$ matrix. Let $D$ be an $n$ by $n$ diagonal matrix with distinct diagonal entries, and let $u$ be an $n$ by $1$ column vector with all non-zero entries. Let $Aq=\lambda q$ with ...
2
votes
1answer
19 views

Can a low-rank matrix set have nonempty interior?

The answer to this question may be super simple, but it is very not obvious to me. Consider the space $S^n$ of symmetric $n\times n$ matrices. Consider $T\subset S$ the set of rank $n-1$ matrices. ...
0
votes
1answer
26 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
-7
votes
1answer
41 views

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).
1
vote
0answers
6 views

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 $M(t)=E[N(t)]$, ...
4
votes
0answers
32 views

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: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
-4
votes
0answers
27 views

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 $\|A-B\|_2 \geq(\|A^{-1}\|_2)^{-1}$.
4
votes
1answer
44 views

Degree of a map $S^0 \to S^0$

By definition the degree of a map $f: S^n \to S^n$ is $\alpha \in \mathbb Z$ such that $f_\ast(z) = \alpha z$ for $f_{\ast}:H_n(S^n) \to H_n(S^n)$. What is the definition of the degree of $f: S^0 \to ...
17
votes
2answers
52 views

any $2$-dimensional rep of a finite, non-abelian simple group is trivial

Let $G$ be a finite, non-abelian simple group. How would I go about proving that any $2$-dimensional representation of $G$ is trivial? If it helps, I know how to do it when we're considering ...
1
vote
0answers
27 views

How find this diophantine equation $(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$ integer solution

Find this following Diophantine equation all integer solution $$(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$$ or $$9x^2-6x+3=4y^4-12y^3+12y^2-5y$$ Maybe this equation can be solved by using Pell equation ...
0
votes
0answers
19 views

How to calculate the following conditonal expectation? am I right?

How to calculate the following conditonal expectation? Is the following calculation process right?
-4
votes
0answers
9 views

full row rank matrix and 2-norm solution [on hold]

Let $A$ be an $m$ by $n$ matrix with $m < n$ and with rank($A$)=$m$. Consider the system $Ax=b.$ (i) Find a particular solution to the row space of $A$. (ii)Find the projection onto the row space ...
3
votes
0answers
20 views

Is there any significance to complex function “monotone in norm?”

So, I was reading a question earlier where someone asked if something would be strictly monotone in the complex plane, and the comment was that this would be meaningless, since the complex numbers ...
1
vote
1answer
43 views

Let $A$, $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = O$

Let $A$ and $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = 0$. Progress I know that $ABA=0 \implies A^2B=0$. Here ...
2
votes
0answers
20 views

Geometry construction problem

Given two circles $S_1$ and $S_2$, a line $l_1$, and a length $a$ that is less than the sum of the diameters of the circles, construct a line $l$, parallel to $l_1$, so that the sum of the chords that ...
1
vote
0answers
16 views

How to calculate the following conditional expectation? Is my calculation process right?

I want to calculate the conditional person's correlation coefficient. But I don't know how to calculate the following expressions,especially the conditional expectation of ...
0
votes
3answers
57 views

Discriminant of the polynomial $f(x)=4x^3-ax-b$

Definition. The discriminant of the polynomial $f(x)=4(x-x_1)(x-x_2)(x-x_3)$ is the product $16\{(x_2-x_1)(x_3-x_2)(x_3-x_1)\}^2$. How to prove that the discriminant of $f(x)=4x^3-ax-b$ is ...
2
votes
0answers
37 views

About primitive roots and primes.

For any odd prime $p$ there exists at least one prime $q < p$ such that $q$ is a primitive root $\text{mod } p$ ; is this true?
0
votes
1answer
35 views

Ordered Field $\mathbb{F}$ Corollary Proof

I wanted to check my proof for a corollary on ordered fields $\mathbb{F}$. Here is the corollary: Corollary: Let $\mathbb{F}$ be an ordered field and $a\in\mathbb{F}.$ If $a>0$, then ...
9
votes
2answers
67 views

true story about probability? [duplicate]

A women's organization was contemplating suing a famous American university when it learned that the percentage of women who received tenure in the university was smaller than the percentage of men. ...

15 30 50 per page