0
votes
1answer
47 views

find a point in 3D space

Suppose we have $3$ fixed points $P_1, P_2, P_3$ in $3$-D space, their coordinates are $(x_i, y_i, z_i)$ for $i=1,2,3$. The problem is to find a point $P$ so that the distances from $P$ to ...
2
votes
2answers
94 views

How to prove the inequality: $\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$

Prove that: $$\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$$ subject to the constraints: $$x,y,z >0$$ and $$x+y+z=1.$$
0
votes
1answer
75 views

Possible to use exact differential to prove Euler's test?

I know that when we want to prove an equation is exact differential, we use $$\frac{d^2Q}{dxdy}=\frac{d^2Q}{dydx}$$ But I wonder why? $$dQ=\frac{dQ}{dx}dx+\frac{dQ}{dy}dy$$ Then differential again ...
0
votes
1answer
71 views

Locus in the complex plane given an equation

I have the question Let $a$ and $e$ be two positive real numbers, with $0 < e < 1$. Describe the locus of the points $z$ in the complex plane which satisfy $|z - ae| + |z + ae| = 2a$. I ...
1
vote
2answers
55 views

convergence ratio of the serie $e^{xn}$

How can I determine the values of $x$ such that the series converge: $$\sum_{n=0}^\infty e^{xn}$$ I'm really lost in this problem, please help.
5
votes
2answers
234 views

Cannot follow proof that between two reals there is a rational.

I am following a textbook and as is good practice, I am only skipping things I can do. this is self-learning I always struggle with chapter 1, the one that builds the "axioms", hate it. Anyway, I am ...
3
votes
1answer
278 views

Math Shock in graduate program

People call it Culture shock but I call it Math Shock... let me explain my Problem... First I am graduate student in a good university in USA ( I get scholarship from my country). Before I lived in ...
0
votes
6answers
91 views

Factoring $s^2+4s+13$

I was looking at an example, and it was factored as follow: $$ s^{2}+4s+13 = (s+2)^{2}+9 $$ How can we do that?
2
votes
1answer
151 views

Is Fourier transform of a $L^{1}$ integrable function is $L^{1}$ integrable?

Let $f:\mathbb R \to \mathbb R$ such that $$f(x)= \frac{\sin \pi x}{x (x^{2}-1)}$$ for $x\in \mathbb R - \{ 0, -1, 1 \}$ and $f(x):= \pi $ for $x=0$ and $f(x)=-\frac{\pi}{2}$ for $x= -1, 1$. Let ...
1
vote
2answers
448 views

harder expected value probability question

I have a question on expected value. I have the solutions for it but they havent explained exactly what they have done, and i am a bit confused whilst revising for an exam in a few days. Here is the ...
3
votes
1answer
66 views

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$.

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$. Let $N=N_G(H)$ and assume that $N/H$ is a nonabelian simple group. Prove that $N=N_G(N)$. $N=N_G(N)$. That means ...
7
votes
1answer
104 views

Show that $\max_{x\in[a,b]}|f'(x)|\geq\frac{4}{(b-a)^2}\int_a^b|f(x)|dx$ [duplicate]

Let $f:[a,b]\to\mathbb{R}$ be differentiable and $f'$ is continuous. Suppose $f(a)=f(b)=0$, show that $$\max_{x\in[a,b]}|f'(x)|\geq\frac{4}{(b-a)^2}\int_a^b|f(x)|dx$$ My approach. For any $x$, by ...
0
votes
1answer
73 views

Algebra 2-Factoring sum of cubes by grouping

Factor the sum of cubes: $81x^3+192$ After finding the prime factorization of both numbers I found that $81$ is $3^4$ and $192$ is $2^6 \cdot 3$. The problem is I tried grouping and found $3$ is ...
0
votes
1answer
78 views

How prove this $4A^5+2A^3+A=7I$ [duplicate]

Let $A_{n\times n}$ is Hermitian matrix,and $A_{n\times n}\neq I_{n\times n}$,where $I_{n\times n}$ is Identity matrix, prove or disprove $$4A^5+2A^3+A=7I$$ my try: if such this condition: then ...
4
votes
4answers
271 views

Anyone has a good recommendation of a free pdf book on group theory?

Anyone has a good recommendation of a free pdf book on group theory? I am specially interested in its application for computer science, however, I do not want it to be less mathematically rigorous ...
0
votes
2answers
75 views

Eigenvalues of $\frac{d^2}{dx^2}$ in $C^2(\mathbb{R})$

Consider the eigenvalue problem \begin{equation} \left\{ \begin{array}{l} \Phi \in C^{2}(\mathbb{R}) \ \text{and bounded }\\ -\Phi^{''}(x)=\lambda\Phi(x), \ x\in \mathbb{R}. \end{array} \right. ...
1
vote
2answers
163 views

Prove a metric space is compact, if every infinite subset in it has a limit point.

This is an exercise in W. Rudin's book. Actually my question is, what is an open cover of a metric space? Since an open set is embedded in a certain metric space, how can it cover those points which ...
1
vote
1answer
73 views

Convolution of distribution and Poisson kernel

I know that for a general tempered distribution (see here) $f$ the convolution $f\star P_t$ is not meaningful. Where $P_t$ is the Poisson kernel (see here) which is given by ...
4
votes
0answers
118 views

Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
60
votes
24answers
12k views

How would you explain to a 9th grader the negative exponent rule?

Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = ...
3
votes
1answer
124 views

$B$ is a finitely generated module over $A$. Then if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring.

Let $A$ be a subring of $B$. Suppose $B$ is a finitely generated module over $A$. Prove that if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring. I am quite ...
0
votes
1answer
72 views

stationary function of an integral

Find the stationary function $y=y(x)$ of the integral $\int_o^4[xy'-(y')^2]dx$ satisfying the conditions $y(0)=0$ and $y(4)=3$. I don't know what a stationary function is. Can you anyone suggest me ...
3
votes
1answer
88 views

Number of subgroups of order 48 in $\mathbb{Z}_{8} \oplus\mathbb{Z}_{12}$

I thought it would have sufficed to show that every subgroup of G=$\mathbb{Z}_{8} \oplus\mathbb{Z}_{12}$ must be formed by couples (a,b) whose set of a's and the set b's form a subgroup of ...
0
votes
1answer
48 views

A problem about the sign of pitch angle in rotation.

The sign of pitch angle in rotation. In the Yaw-Pitch-Roll convention, for example, XYZ-Zup coordinates system. When I'm reading page 194-195 of 《Linear Algebra with Applications》 7ED, of STEVEN ...
2
votes
0answers
53 views

What (if anything) can I say about the inverse of the matrix product B'AB if B is not square?

Suppose I have: a matrix $A$ with dimension $n \times n$ a matrix $B$ with dimension $n \times m$. $C = B^{T}AB$. I'm interested in finding an expression for $C^{-1}$ when $m < n$. The ...
1
vote
2answers
306 views

Does someone know why raising the element of a group to the power of the order of the group yields the identity?

Does someone know the why raising the element of a group to the power of the order of the group yields the identity? By (finite) group I mean a tuple (G,*) that satisfies the following: closure ...
2
votes
1answer
86 views

Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$.

Let $G$ be a finite group and $p$ be a prime. Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$. Suppose $P \subseteq H^g$ for some $g \in G$. Prove ...
2
votes
1answer
64 views

Reading help, ideal of an ideal is an ideal?

While in general it is not true that an ideal of an ideal is an ideal, this proof in Humphreys confuses me. Could someone please explain to me why the underlined sentences are true?
0
votes
1answer
21 views

multi-dimensional impulse with kernel

We know that a 1d dirac function "extracts" the function of an integral, but what about for nd? $$\int \cdots \int f(x_1, \cdots, x_n) \prod_{i=1}^{n} \delta(x - x_i) dx_i$$ is this equal to ...
0
votes
3answers
70 views

Truth value of a statement?

How do I prove the following statement? For all $\forall x \in R $ there exist $\exists y\in R$ such that $y^6-xy^2=-x^2$ How do I approach this? Thank you.
2
votes
1answer
79 views

Statement about rational approximations for $\pi$

According to a result by Kurt Mahler, there is $q_0$ such that $\forall p, q \in \mathbb{Z}, q > q_0,$ $$\mid\frac{p}{q}- \pi\mid > q^{-42} $$ However, I have never been able to find a proof of ...
6
votes
1answer
179 views

How should I think about Ihara's Lemma?

I am trying to learn about Ihara's lemma because I see it mentioned in many papers in arithmetic geometry. To be more precise, I would like to know: What is the significance of this result? Why is ...
1
vote
1answer
97 views

7) Prove that $2n-3 \leq 2^{n-2}$ for all $n \geq 5$ by mathematical induction [duplicate]

Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction I have to prove by mathematical induction that: $2n-3\leq 2^{n-2}$ , for all $n \geq 5$
0
votes
3answers
95 views

Mathematical Logic Problem?

I'm trying to solve this mathematical logic problem, can someone please at least give me a tip on how to approach this problem? The square of any positive real number is a positive real number. ...
2
votes
1answer
68 views

Palais-Smale conditions of functional involving noncoercive differential operator

I am working on mountain-pass like theorems for the problems $$ - u_{xx} - a u = \pm |u|u+|u|^2u , \ x \in (a,b), \quad u(a)=u(b) = 0$$ where $a \in L^\infty((a,b))$ is positive (I take the one ...
1
vote
2answers
195 views

How to find the zeros of an equation of nth degree

I was working on problems in my math textbook and I saw this problem as a side note and I couldn't figure it out. The author states: ...
0
votes
2answers
60 views

Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction

Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction I have to prove by mathematical induction that: $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ Thank you for the Review.
0
votes
1answer
57 views

Find the spec of a localized ring.

I must find the following: $$\operatorname{Spec}\left(\Bbbk[x,y]/\left<xy-1\right>\right)$$ Is there a way to describe that set? I am trying to find the laurent polynomials rings prime ideals. ...
4
votes
0answers
152 views

A bounded function and lebesgue measurable sets.

Let $f : [0, 1] \to {\mathbf R}$ be bounded. (a) Show that the set where $f$ is continuous is Lebesgue measurable (even if $f$ is not Lebesgue measurable). (b) Show that if $f$ is not continuous on ...
4
votes
2answers
50 views

Continuation from here…

p.s I have no idea how to type math on this program so I just copied and pasted from a document. p.p.s I also tried doing it with log base a and that was a travesty. This was my best attempt ...
3
votes
1answer
140 views

If $f'(x)=\sin{(x)}$ and $f\left(\pi\right)=3$, then $f(x)=$?

If ${\rm f}'\left(x\right) = \sin\left(x\right)$ and ${\rm f}\left(\pi\right) = 3$, then ${\rm f}\left(x\right) =\ ?$. I understand that the derivative of $-\cos\left(x\right)$ is ...
0
votes
2answers
46 views

Definite integral question.

Use the form of the definition of the integral to evaluate the integral. $ \int _{ 2 }^{ 5 }{ (4-2x)dx } $ I got to this $ \int _{ 2 }^{ 5 }{ 4\quad dx } \quad -\quad 2\int _{ 2 }^{ 5 }{ x\quad dx ...
-1
votes
1answer
86 views

Application of convergence of Fibonacci series

'There are infinite prime numbers' is a fact that can be deduced by 'reciprocal of primes diverges' statement, so from this can we deduce the fact that --> 'there are finite Fibonacci numbers in ...
1
vote
3answers
117 views

Show that if $a_n$ is descending and $ ∑a_n$ is convergent then $n∗a_n→0$

Show that if $a_n$is descending and $\sum a_n$ is convergent then $n*a_n \to 0$. Does $\sum a_n$ has to be convergent? I see that, but I can't prove it. I think that it need not to be convergent, ...
0
votes
1answer
141 views

Problem with simple Projectile motion formula

I'm really new to all this, so please don't be rude. I have here what appears to be a simple formula. Yet I cant get my head around it: $$\begin{align} y &= y_0 + v_{0y}t + (1/2)a_y t^2 \\ 0 ...
3
votes
0answers
77 views

Clarifying Semi-Direct Products: Example

I'm working through some questions on semi-direct products, and although I can work out these problems (for the most part), I usually have trouble completing them. I have identified some of the things ...
0
votes
2answers
35 views

integral of 2-variable function in the real plane

$\displaystyle{% \iint_{D}{{\rm d}x\,{\rm d}y \over \sqrt{\vphantom{\large A}1 - x^{2} - y^{2}\,\,}\,} \quad\mbox{where}\quad D \equiv \left\lbrace\left(x,y\right)\ |\ x^{2} + 4y^{2} \leq ...
0
votes
2answers
101 views

How to solve $\frac{2}{3\sqrt{2}}=\cos\left(\frac{x}{2}\right)$?

How do you solve $\dfrac{2}{3\sqrt{2}}=\cos\left(\dfrac{x}{2}\right)$ for $x$ in the interval $0 \leq x \leq 2\pi$? This comes from a question that I asked before. I frequently get stumped when ...
1
vote
1answer
72 views

A special cubic curve

How can I transfer following cubic curve to a Weierstrass normal form? $$2x^2y+4xy^2+2y^3-2axy-ay^2+a=0,$$ where $a$ is a fixed rational number.
3
votes
2answers
263 views

Localization of an integral domain and fields of fractions

Is it true that every localization of an integral domain is isomorphic to a subring of its field of fractions? How are the localizations of an integral domain related to its field of fractions? Is ...

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