0
votes
0answers
3 views

linear algebra subspaces and polynomials

Let P3 be the vector space of all polynomials of degree less than or equal to 3: a. With p(x)=1-x^3 and q(x)=1+x-2*x^2, find the subspace of P3 spanned by p(x) and q(x) b. find the orthogonal ...
0
votes
0answers
4 views

If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
0
votes
0answers
5 views

Volume of solid of rotation about x-axis

Rotation of the region bounded by x=2y^2-1, x=y^2 and x-axis about x-axis. I draw out the graph, and found intersection is at (1,1) and (-1,-1) So is it correct to continue doing by finding volume ...
0
votes
0answers
18 views

Hardy Littlewood Circle Method

I'm working through Vaughan's book on the Hardy Littlewood circle method, which uses the following lemma: Suppose that $\alpha \geq \beta$ are positive real numbers, and that $\beta \leq 1$. Then: $ ...
2
votes
0answers
32 views

Research Paper on Math

I have to write a research paper for American history and it can be about anything so I want to write it about math. The thing is, in order to do so I have to be able to tie it back into American ...
0
votes
0answers
12 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
0
votes
1answer
14 views

limit evaluation calculus I

How do I evaluate $\lim_{x\to \infty} x-\sqrt{x^2+x}$?
0
votes
0answers
13 views

Trying to justify each step correctly in proof sequence

I am trying Justify each step in the proof sequence below for correctly with [A → (B ∨ C)] ∧ B' ∧ C' → A' So I justified my steps here but I am not sure at 1 to 3 if I did it correctly. A → (B ∨ ...
0
votes
0answers
9 views

Compactification of a straight line

Like in the case of mapping a infinite-plane to a sphere (Riemann Sphere), I can understand, that I can map the infinite line ($-\infty,\infty$) to a circle. Secondly, I can also map a finite line ...
0
votes
0answers
16 views

Prove $a^2+6a+1\perp 375$ for all $a\in \mathbb{Z}$.

Prove $A=a^2+6a+1\perp 375$ for all $a\in \mathbb{Z}$ I thought to write $375=3\cdot5^2$. So if $A$ is coprime with $3\cdot5^2$ they must share no prime factors. Then I test if $3$ or $5$ divide $A$ ...
-1
votes
0answers
10 views

Renewal equation

I have a question on renewal equation. m(t) = F(t) + integral from 0 to t (m(t − x)dF(x)) can someone tell me how to compute the integral part? it's better if you can tell me step by step until we ...
-3
votes
0answers
34 views

Is it true that If $n$ is an integer such that $n > 2$, $x$ and $y$ are positive integer such that $x > y$ , then

How to Prove or Disprove: If $n$ is an integer such that $n > 2$, $x$ and $y$ are positive integer such that $x > y$ , then $\sqrt[n]{\frac{x^{2n} - y^2}{4}}$ is not a positive integer.
0
votes
1answer
17 views

Simple combination question help

Hi, I have a question to ask regarding subquestion three in the picture. I solved it by using 10c2 since 2 of the 4 houses are fixed already, which I thought would leave me with 10 choices. However, ...
0
votes
0answers
55 views

Is Euclidean geometry really a “dead” subject? If so, why? [on hold]

It seems that Euclidean geometry is a "dead" subject nowadays. In the time of the Greeks, mathematicians and geometers were one and the same. Today, very few professional mathematicians study ...
2
votes
0answers
24 views

Question about of the polynomial $x^p -x -a$

If $F$ a field with $char(F)=p$. Prove: If $x^p -x -a$ is reducible in $F[x]$ , then this it splits in distinct factors in $F$. I know if for hypothesis $x^p -x -a = P(x)Q(x)$ with $P(x),Q(x) \in ...
0
votes
0answers
10 views

Solvable by Radicals and Polynomials

Let $f(x)$ be an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is ...
4
votes
1answer
28 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
3
votes
0answers
19 views

Relationship between cohomology and higher-homotopy

Let $M$ be a connected, compact, and orientable manifold ($H^3(M)\cong\mathbb{Z}$), and let $G$ be a simple Lie group satisfying $\pi_1(G)=\pi_2(G)=0$. Let $\pi_M(G)$ denote the set of homotopy ...
0
votes
1answer
9 views

Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
0
votes
0answers
2 views

Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...
4
votes
1answer
42 views

Need Help Understanding Notation With Functions

Original picture: LaTeX approximation: $$f\color{blue}{\substack{(x)\\x\to\infty}}=\pm\sqrt{\frac{(x^2+x)^3}{\pi}}.$$ What does the notation highlighted in blue mean? I understand that ...
1
vote
1answer
7 views

Let $R$ be a commutative ring with 1. If $R$ is a PID, then every prime ideal is either zero or maximal.

Let $R$ be a commutative ring with 1. If $R$ is a PID, then every prime ideal is either zero or maximal. My proof: Let $I = (p)$ be a non-zero prime ideal of $R$. Note that $p$ is prime. Since $R$ is ...
0
votes
0answers
7 views

Tail field of random variables in $\mathbb{Z}$

Let $X_1, X_2, \ldots$ i. i. d. with values in $\mathbb{Z}$, define $S_0 := 0$, $S_n := X_1 + \cdots + X_n$ and $R_n := \{S_n = 0\}$ for $n \in \mathbb{N}$. Show that ...
0
votes
1answer
24 views

Prove that the output of the function equals the determinant

Let $δ$ : $M_{2×2}$($F$) $→$ $F$ be a function with the following three properties. ($i$) $δ$ is a linear function of each row of the matrix when the other row is held fixed. ($ii$) If the two rows ...
2
votes
1answer
27 views

Proving that the unit cube cannot be tripled (with straight edge and compass)

I would like to show the unit cube cannot be tripled using a straightedge and compass. I note that the side of a cube that has been tripled would have a side length of $\alpha=\sqrt[3]{3}$ But ...
-2
votes
0answers
26 views

Algebra fraction simplification? [on hold]

I have the following and is solution but I don't understand how they arrived at the solution.Can someone direct me to how they arrived at this please ? $$\frac{z(z-1)(z+1)}{(z-3)^3}$$ The solution : ...
0
votes
1answer
21 views

Prove this function is uniformly continuous by verifying the $\epsilon$-$\delta$ property?

$f(x) = \frac{5x}{2x-1}$ on $[1,\infty)$ Here's what I've worked through so far: $$|f(x) - f(y)| = \left|\frac{5x}{2x-1} - \frac{5y}{2y-1}\right| = \left|\frac{5y-5x}{(2x-1)(2y-1)}\right| ...
2
votes
2answers
41 views

Logic behind finding a $ (2 \times 2) $-matrix $ A $ such that $ A^{2} = - \mathsf{I} $.

I know the following matrix "$A$" results in the negative identity matrix when you take $A*A$ (same for $B*B$, where $B=-A$): $$A=\begin{pmatrix}0 & -1\cr 1 & 0\end{pmatrix}$$ However, I am ...
2
votes
1answer
29 views

Do there exists continuous functions on compact sets with infinite length?

Is it possible to construct a continuous function from $[0,1] \to \mathbb{R}$ whose length is infinite?
0
votes
0answers
22 views

What is 'free algebra'?

I've been googling the definition of it, and it seems like somehow it's related to a polynomial ring. But I still quite don't get it. Is a free algebra just a free group with additional operation ...
2
votes
0answers
12 views

Composition of Polynomials and Galois Theory

Let $f(x)$ be a polynomial of degree $n$ over $\mathbb{Q}$, with Galois group isomorphic to the symmetric group $S_n$. How do I show that $f$ cannot be expressed as a composition $g(h(x))$ of two ...
0
votes
0answers
8 views

Infinite matrix defining a bounded operator on $l^2$

I think I only need some help to clear up the terminology and make sure I understand correctly: I've shown that for a sequence $\{f_{k}\}_{k=1}^{\infty}$ in a Hilbert space $H$, there exists $C>0$ ...
-1
votes
1answer
22 views

Gauss Method to show

Could you please give me the way to solve this problem Using Gauss method to show if $x ≠ y + 1$ then $$ \sum_{i=0}^n (x-y)^i = \frac{(x-y)^{n+1}-1}{x-y-1}. $$
2
votes
1answer
29 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $Q$ but $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not

How can I show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $Q$ but that $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not. I am kind of lost with Galois Theory Thanks
0
votes
0answers
6 views

The relation of “be characteristic subgroup” is transitive? [duplicate]

If $K$ is a characteristic subgroup of $H$ and $H$ is a characteristic subgroup of $G$, then $K$, is characteristic in G? This is, the relation of "be characteristic" is transitive?
0
votes
2answers
12 views

The number of $q$-Sylow subgroups cannot be $p$ for prime $p<q$

Since $q>p $, we cannot have $n_q=p $. Here $n_q $ is the number of $q $ Sylow subgroups. Why is the above statement true? This is a statement from Dummit and Foote.
1
vote
1answer
20 views

Contour integration

Consider the real-valued function $$u(t) = \frac{1}{13-12\cos(t)}$$ By converting it to a contour integral along the unit circle in $\mathbb{C}$, evaluate $$\int_0^{2\pi} u(t)\;dt$$ I have ...
-1
votes
1answer
13 views

Probability Binomial Distribution Question

The quality control unit in a medical device company inspects 20 pacemakers each hour. Let X represent the number of pacemakers in the sample of 20 that require rework. Pacemakers are assumed to be ...
1
vote
1answer
20 views

Symbol clarification

Okay, so I've read a few different meanings for the exclamation point in a statement. For example: $$!\exists x \in O \ni 2x < 5$$ The only question I have is about the Exclamation point in front ...
0
votes
2answers
41 views

how to convert log(x) into linear form? [on hold]

I have simple function which is non-linear like log(x) I want to convert it into linear function. Anyone could help out? Thanks
1
vote
3answers
44 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
0
votes
2answers
26 views

Convergence test of an integral

$$\int_1^\infty \left( 1- \frac{2x+1}{2(x^2+x)^{1/2}} \right) \ dx$$ Anyone can suggest hint for how to determine whether it is convergent or not? Thank you
4
votes
0answers
61 views

A transfinite epistemic logic puzzle: what numbers did Cheryl give to Albert and Bernard?

I expect that nearly everyone here at stackexchange is by now familiar with Cheryl's birthday problem, which spawned many variant problems, including a transfinite version due to Timothy Gowers. In ...
0
votes
0answers
4 views

Closed subgroup of a locally compact Hausdorff group whose Haar measure is non-zero.

Let $G$ be a locally compact Hausdorff group, $H$ its closed subgroup. To avoid pathologies, we assume the underlying topological space of $G$ has a countable base. Let $\mu$ be a Haar measure on $G$. ...
0
votes
1answer
25 views

Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
-8
votes
4answers
77 views

Can you check my proof of Fermat's Last Theorem? [on hold]

I've come up with a proof of Fermat's Last Theorem and my teacher would not look at it so i was wondering if you could check. I know it's supposed to be hard to prove, but I use a "trick" from calc ...
0
votes
0answers
8 views

p-primary component of a group

I have been asked to find the $3$-primary component of the group: $$\mathbb{Z_3}\oplus\mathbb{Z_5}\oplus\mathbb{Z_9}\oplus\mathbb{Z_{153}}$$ Now, I know that we define the $p$-primary component of a ...
0
votes
1answer
34 views

Creating a language

I am given a list languages, say L, over alphabet {a,b}. A function f is defined such that f(i) = L for i ∈ N. I am trying to a construct a language D which is not in the list (aka. D =/= f(n)). I am ...
3
votes
1answer
51 views

$x^p-x-1$ is irreducible over $\mathbb{Q}$[x]

For any prime p, prove that $x^p-x-1$ is irreducible over $\mathbb{Q}$[x]. (In a field of characteristic p this is true). I asummed exist root in $\mathbb{Q}$, let's call $\frac{\alpha}{\beta} ...
1
vote
0answers
7 views

Can independence of a system and a vector be establish if there is only cross-indepedence?

Say that I have the following linear system: $$[A a'] \begin{bmatrix} x \\ x' \\ \end{bmatrix} =Ax + a'x' $$ I want to know when this system is zero if and only if ...

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