1
vote
0answers
63 views

Ordinals as Trees

I'm trying to understand countable ordinals and their tree representation. I understand that $\omega$ is the first "non branching tree" of infinite height. I also understand that the exponent of ...
2
votes
2answers
79 views

Prove sequence $S_n$ converges

If $S_1 = \sqrt{2}$, and $S_{n+1} = \sqrt{2 + \sqrt{S_n}}$ (n = 1,2,3....), prove that $\{S_n\}$ converges, and that $S_n < 2$ for all $n \in \Bbb{N}$ This is one the questions ...
9
votes
1answer
202 views

The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
2
votes
4answers
102 views

Integer solution to multiple modular arithmetic equations

So i understand how to do this when it is just x, but now with multiples of x I am a little confused, and there's no example in my textbook of this. I just need a push in the right direction for how ...
6
votes
0answers
139 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
1
vote
1answer
168 views

The Residues of an even function or an odd function on $U$ subset open symmetric

I have to proof that for $f$ even function holomorphic with singularities isolated then $$res_{z}f=-res_{-z}f$$ an simmetric for $f$ odd, $i.e.$ $$res_{z}f=res_{-z}f$$ My hint is proof that laurent ...
2
votes
1answer
306 views

A real matrix whose eigenvalues have all negative real parts

While taking a look in some lecture notes of an ODE course, I found the following claim, which appeared in the text as an exercise: Let $A$ be a real $n\times n$ matrix whose eigenvalues have all ...
10
votes
1answer
180 views

Partial derivatives on Manifolds

Let $F : A \times B \to C$ be a map of smooth manifolds. Define the following maps ("partial derivatives"): $E_1 F: TA \times B \to TC$ $E_1 F(a,v,b) = D_a F(-,b) v $ where $v \in T_a A$ $E_2 F: A ...
0
votes
1answer
99 views

Is the induced map between quotient vector spaces automatically an inclusion?

If $T: V_1 \to V_2$ is a linear map and $W \subset V_1$ and $W \subset V_2$ with $T(W) \subset W$, then the (induced) map $T: \frac{V_1}{W} \to \frac{V_2}{W}$ given by $$T(x + W) = T(x) + T(W)$$ is ...
0
votes
1answer
89 views

$\{ x \in H: x=\sum_{k=1}^{\infty}c_{k}u_{k}$, $|c_{k}| \leq \frac{1}{k}\}$ is compact

Let $H$ be a complex inner product space that is also a complete metric space with respect to the distance induced by the inner product. Assume $\{u_{k}\}_{k=1}^{\infty}$ be an orthonormal set in ...
0
votes
2answers
42 views

How would this equation be converted to polar coordinates?

How would this equation be converted to polar coordinates? $(x^2 + y^2)^2 = 2xy$ First, I changed $(x^2 + y^2)^2$ to $(r^2)^2 = r^4$ Then, rewrote the equation to $r^4 = 2xy$ What needs to take ...
2
votes
2answers
50 views

How do I find this partial derivative

I have the following function u(x,y) defined as: $$u(x,y) = \frac {xy(x^2-y^2)}{(x^2+y^2)}$$ when x and y are both non zero, and $u(0,0)=0$ I want to compute its partial derivative $u_{xy}$ at ...
0
votes
2answers
44 views

From which equation of motion was this formula derived from in physics

When solving problems involving projectile motion I use: $\sqrt{2 * \dfrac{\text{height above ground}}{9.8}}$ Eg calculate the time it takes for a bomb to impact if it is travelling 4.9km above ...
0
votes
2answers
336 views

How to convert a big number back to decimals when you divide 1/15000 using a basic simple calculator?

This question has always stumped me since using a simple basic calculator over 20 years ago. I'm using the basic calculator on windows or it can be any for that matter. I input the following into ...
0
votes
2answers
118 views

modular arithmetic congruence

Simplify the following congruence: $$−169 \equiv \text{ ?} \mod 52 $$ (By simplify, we mean find the smallest non-negative whole number which is congruent to $-169$ modulo $52$.) Simplify the ...
1
vote
1answer
47 views

Show that there is a series in R^infinity has some term greater than or equal to 1/n but that also is arbitrary close to the zero sequence.

Consider $\mathbb{R}^\infty = \{(a_n): \sum_{n = 1}^{\infty} a_n^2 < \infty\}$ with the metric $d((a_n$), ($b_n$)) = $[\sum_{n=1}^{\infty} (a_n - b_n)^2]^{1/2}$. Let $A = \{(a_n) : |a_n| < 1/n ...
1
vote
0answers
44 views

Pullback of area form of manifold by local chart map

Let $M$ be an orientable $2$-manifold in $\mathbb{R}^n$, with area form $dA$. Let $\phi: U \to M$ be a local coordinate system for $M$, with $U \subseteq \mathbb{R}_{uv}^2$. It is stated that ...
3
votes
0answers
56 views

Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
1
vote
2answers
50 views

Modular Arithmetic/Number Theory

(Not really sure about my work, so if you could tell me if I am on the right track that would be great!) Find an integer x so that: a. $x\equiv1\pmod{13}$ and $x\equiv1\pmod{36}$ Using the ...
1
vote
1answer
32 views

Confused about Random Variables question.

The question is as follows: "Let Y be a random variable with values 1,2,3. If P(Y = 1) = .3, P(Y = 2) = A, what is P(Y = 3)? What is P(Y >= 2)?" I understand the concept of random variables but I'm ...
2
votes
3answers
200 views

Prove that if A is ANY $n\times n$ matrix, then $det(adj(A)) = (det(A))^{n-1}$. (how to when A is singular?)

Prove that if $A$ is ANY $n\times n$ matrix, then $det(adj(A)) = (det(A))^{n-1}$. first of all, since I did it about 3 or 4 times before, I started off by proving the case where $A$ is an INVERTIBLE ...
0
votes
0answers
140 views

Using trilateration with unknown points to determine change in position

I have multiple fixed points to which I know the distance. I do not know the position of these points before hand. I will know my absolution position (using other means) at the start, but afterwords I ...
9
votes
1answer
173 views

Show every $f_t$ is Morse for $t$ is sufficiently small

Let $f$ be a Morse function on the compact manifold $X$. Let $f_t$ is a homotopic family function with $f_0=f$. Show every $f_t$ is Morse for $t$ is sufficiently small Here is my argument, but my ...
3
votes
2answers
164 views

Commutator of a group

A commutator in a group $G$ is an element of the form $ghg^{-1}h^{-1}$ for some $g,h\in G$. Let $G$ be a group and $H\leq G$ a subgroup that contains every commutator. $(a)$ Prove that $H$ is a ...
1
vote
2answers
36 views

How to write $x^2 + (y-2)^2 = 4$ in polar coordinates?

How can I convert this equation $x^2 + (y-2)^2 = 4$ into polar coordinates? I know that $x=r\cos(\theta)$ and $y=r\sin(\theta)$. I tried plugging those knowns in but i felt stuck.
2
votes
1answer
89 views

cut-edges, cycles, and a 2 connected graph

Read this statement in a textbook: If G - e - f is disconnected, then f is a cut-edge in G-e, whence f belongs to no edges in G-e, and thus every cycle in G containing f must contain also e. How can ...
0
votes
2answers
67 views

Quick way to find the GCD of 7602 and 7710

I've been reading through my book and I see that to find the GCD of these two numbers, I can look at the difference of these two numbers. However, how do I determine the GCD from the difference? I've ...
1
vote
1answer
78 views

Why does Proposition 1.8 in Atiyah-Macdonald imply that the smallest prime $\mathfrak{p}$ containing a primary ideal is equal to its radical?

Proposition 4.1 in Atiyah-Macdonal states that the radical of a primary ideal is the smallest prime ideal containing the primary ideal. They start the proof claiming that showing the radical is a ...
2
votes
1answer
47 views

Prove sequence converges

Let $x_n = (1+\frac{1}{2})(1+\frac{1}{4}) ... (1+\frac{1}{2^n})$, $n \in \mathbb{N}$. Prove that $\{x_n\}$ converges. I know we have to show that the sequence is increasing and bounded above. I was ...
0
votes
1answer
31 views

Fourier cosine series for a interval $[0, l]$

It is asked to find the Fourier Cosine Series for the function defined by $$f(x) = \cos \frac{\pi x}{l}, x \in [0, l/2]$$ $$f(x) = 0, (l/2, l]$$ I thought it should be $$\frac{a_o}{2} + \sum a_n ...
2
votes
3answers
302 views

Differentiate the Function $f(x) = \pi^2$

What does it mean to differentiate $\pi$? Is it similar to constant thus $0$?
0
votes
2answers
47 views

infinite limit question from Calc I

Find the limit $$\lim_{x\to\infty}\sqrt{x^2+x+1}-x$$ This limit is part of a question involving squeeze theorum, the limit is $\frac12$ but i don't know how to prove it because of the polynomial in ...
1
vote
0answers
42 views

What's the best way to define $\mathbb{H}$?

I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that ...
2
votes
3answers
47 views

How to find the second derivative of this function: $x^2 |x|$ at $x = 0$

newbie in Calculus here. I was going to plug it into the definition of derivative function, the exact way my professor showed us to do it, except when he plugged it in, he used the $\sqrt{x}$ as ...
1
vote
1answer
59 views

how to represent the predicate member/2 in first order logic?

Lets say we have a predicate that returns true if there is a member in a list. False if not. In prolog I ended up with a code like below ...
2
votes
1answer
34 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
0
votes
1answer
36 views

find the smallest m

Let m be smallest positive integer, which complies with $$n<\sqrt[3]{m}=n+r $$ where n is an positive integer and $0<r<\frac{1}{1000}$. Find m. In other words do I have to find the smallest ...
0
votes
0answers
128 views

Monotonic function $f$ - Riemann Stieltjes Integral

If $f \in R(\alpha)$ ( where $\alpha$ is some function ) on $[a,b]$ and if for every monotonic function $f : $ $\int_a ^b f~ d \alpha = 0 $ then, prove that $\alpha$ must be constant on $[0,1]$ ...
0
votes
2answers
39 views

Find integers $r$, $s$, and $t$ such that $12r + 30s + 18t = 2$

Could someone please explain if such integers exist and how to find them? If not, could someone please explain how to prove that they don't exist? Thank you!
1
vote
1answer
137 views

Menger's theorem and how many pairwise-edge disjoint paths?

In a proof, I came across this statement: ...
2
votes
2answers
1k views

Proof that if $f$ is integrable then also $f^2$ is integrable

Prove this: Let $f :[a,b] \to \mathbb{R}$ be a bounded and integrable function. Show that $f^2$ is integrable too. I'm in trouble with this. Can anyone show how to do it?
0
votes
1answer
27 views

No simultaneous solutions (Chinese Remainder Theorem)

a. Show that $x\equiv2\pmod6$ and $x\equiv3\pmod4$ have no simultaneous solutions. If $x\equiv2\pmod6$ then x is even but if $x\equiv3\pmod4$ then x is odd. This is a contradiction, so ...
1
vote
1answer
132 views

Are 2 dependent probabilities always disjunct?

If $2$ probabilities are disjunct then they are not independent. Are any $2$ dependent probabilities also disjunct? Modus ponens for instance are $2$ dependent and mutually exclusive events ($A$ and ...
-1
votes
3answers
65 views

Find integers $s$ and $t$ such that $15s + 11t = 1$

Could someone please explain this question to me? I know that such integers do NOT exist, but I could not prove it. "Either solve it or give a brief explanation as to why it is impossible." Thank you! ...
1
vote
1answer
33 views

Possible error in Lipschutz

The problem asks to show that $f:(0,\infty)\to [-1,1]$ given by $f(x)=\sin (\frac {1}{x})$ is neither open nor closed. Where $(0,\infty)$ and $[-1,1]$ are subspaces of $\Bbb{R}$ with the usual ...
2
votes
1answer
58 views

Propositional logic and distributive law

I am having trouble trying to understand how this question passes from this point $$ ( ( p\vee q )\wedge (p \vee \neg r ) \wedge (\neg q \vee \neg r ) ) \vee ( \neg p \vee r ) $$ to this $$ ...
1
vote
1answer
496 views

Counting principles and probability question

A batch of 15 DVD players contains 4 that are defective. The DVD players are selected at random, one by one, and examined. The ones that are checked are not replaced. a. What is the probability that ...
3
votes
1answer
156 views

Another theorem of Principal value

Let $f$ holomorphic function with isolated singularities in neighborhood of $ \overline{\mathbb{H}}^+ = \{ z\in \mathbb{C} : \operatorname{Im} z \geqslant 0\}$ and suppose that f only have one ...
4
votes
2answers
68 views

Find the sum $\sum_{n=1}^\infty \frac{n}{(1+x)^{2n+1}}$

Find the sum $$\sum_{n=1}^\infty \frac{n}{(1+x)^{2n+1}}.$$ Indicating the interval of convergence for $x$. My attempt: Let $ t=\frac{1}{x+1}$. Then, applying the root test, $$\lim_{n\to \infty} ...
0
votes
1answer
192 views

Disjoint union of two affine schemes

Say I have two commutative rings with unity, $R$ and $S$. What does the sheaf of disjoint union of $\DeclareMathOperator{Spec}{Spec}(\Spec(R), \mathscr O_{\Spec(R)})$ and $(\Spec(S), \mathscr ...

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