For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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Cauchy's theorem in a disk (Proof Verification)

Consider the following proof of Cauchy's theorem in a disk. My question is pasted at the bottom of the picture. (Note that in the proof below, a reference is made to "Theorem 2". In my textbook ...
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7 views

With Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable parametrisations for ...
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0answers
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Compact and bounded if and only if $X$ is finite dimensional

I tried to prove the theorem Let $X$ be a Banach space. Then $K(X) = B(X)$ if and only if $X$ is finite-dimensional. Please could someone check my proof? Let us use the fact that a linear ...
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1answer
30 views

Convergence/Divergence of $\displaystyle \sum \frac{n^2}{2^{nr}}$

Determine whether $\displaystyle \sum \frac{n^2}{2^{nr}}$, $r \in \mathbb{R}$, diverges or converges Working: Consider the ratio test: \begin{align*} \lim \left| \frac{(n+1)^2}{2^{(n+1)r}} ...
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21 views

Compact operators is a (vector) subspace of bounded operators

Let $X,Y$ be Banach spaces. Let $B(X,Y)$ be the set of bounded linear operators and let $K(X,Y)$ be the set of compact linear operators. I want to prove that $K(X,Y)$ is a vector subspace of ...
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0answers
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Zariski closures exercise.

Compute the Zariski clousures $\overline{S} \subset \mathbb{A}^2(\mathbb{Q})$ of the following subsets: (a) $S=\{(n^2,n^3):n \in \mathbb{N}\}\subset \mathbb{A}^2(\mathbb{Q})$; (b) $S=\{(x,y): ...
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2answers
45 views

Please review my proof.

I am working on a problem from Spivak 13.7 which states: Prove: $$m_i'' + m_i' \leq m_i$$ Where: $$m_i'' = \inf \{f(x): t_{i-1} \leq x \leq t_i\}$$ $$m_i' = \inf \{g(x): t_{i-1} \leq x \leq ...
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1answer
15 views

Path components of Wedge Sum

I couldn't find this anywhere else, so I decided to post it here. I suspect that the wedge sum $⋁X_α$ of pointed spaces $X_α$ has as path components all components of the topological sum $\oplus X_α$ ...
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1answer
17 views

Norm and Matrice Proof

I'm trying to show that the following statement is true: If $||\mathbf a - \theta \mathbf b||^2 - ||\mathbf a||^2 \geq 0$ for all $\theta \in [0,1]$, then $\mathbf a^T \mathbf b \le 0$. Is this ...
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1answer
21 views

Partial differentiability with respect to $x$ and $y$ of $\int_0^x z(s,y)ds$ where $\partial_y z \in C^0$

I have the following (not accredited and not mandatory) Exercise: Problem: Let $z : \mathbb{R}^2 \to \mathbb{R}$ be continuous and in respect to its second variable partially differentiable. ...
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1answer
36 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
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1answer
28 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
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1answer
24 views

Express $a^5$ in terms of $c_0+c_1a+c_2a^2.$

Let $F=\mathbb Z_2,f(x)=x^3+x+1\in F[x].$ Suppose $a$ is a zero of $f(x)$ in some extension of $F.$ Then $F(a)\simeq F[x]/\langle f(x)\rangle$ and there is an isomorphism $$\phi:F[x]/\langle ...
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1answer
26 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
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30 views

Show by explicit calculation that $\varphi\colon\mathbb{Z}\to\mathbb{Z}_n, m\mapsto m\% n$ is a surjective ringhomomophism

Consider $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Show by explicit calculation ...
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1answer
12 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
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2answers
25 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
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69 views

$\ f \colon X \to X $ ,continuous function where X is compact,Hausdorff space.Show $\exists A$ st $f(A) =A$.

Suppose $\ f \colon X \to X $ is a continuous function from a compact,Hausdorff space to itself. Prove that there exists a subspace $A$ such that $f(A) =A$. I came up with an answer based on nets ...
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2answers
96 views

Proof of equicontinuous and pointwise bounded implies compact

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
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1answer
31 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
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3answers
42 views

Proof by induction of a recursive sequence

I am studying CIE A levels Further Maths and I am stuck at a question from June 2002: Q The sequence of positive numbers $u_1,u_2,u_3,...$ is such that $u_1<4$ and ...
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2answers
41 views

If T(S) is linearly independent, show S is linearly independent

Let $T: V \to W$ be a linear transformation. Let $S = \{v_1,...,v_k\}$ and assume $T(S)$ is linearly independent. Show S is also linearly independent. I think I just have to prove that if $a_1 v_1 + ...
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2answers
34 views

Proving corollary to Euler's formula by induction

I'm currently looking at two proofs to the following corollary to Euler's formula and I'm not quite seeing how the authors can make a specific assumption in their proof. One proof comes from my ...
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1answer
29 views

$\epsilon - N$ proof confirmation.

These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it. My limit and required assumptions: ...
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1answer
36 views

Prove that if graph $G$ is a 3-connected planar graph then its dual must be simple.

I'm trying to study for a quiz. I think I'm on the right track with this problem, however, I'm having a difficult time formalizing it. Prove that if graph $G$ is a 3-connected planar graph then its ...
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0answers
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$p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$

I need to show that $p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$ Here's what I've done: Please tell me if it's correct Over $\mathbb C,$ $x^4-2x^2-4\\=(x^2-1)^2-5\\=(x^2-1+\sqrt ...
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9answers
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Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
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4answers
348 views

Is the reasoning/algebra for my proof correct? (musical tuning theory proof)

This isn't for a class, I was just wondering if I would be able to work out a proof for something like this myself for fun, and wanted to verify that my methods are correct. Basically, what I'm trying ...
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1answer
23 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
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1answer
16 views

Dividing summations that have existing properties in each element

If $a_i/c_i > B$ for all $1 \le i \le k$, is it fair to assume that $(a_1 + a_2 + \cdots + a_k)/(c_1 + c_2 + \cdots + c_k) > B$ ? Is there a way to prove this? Thanks!
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0answers
41 views

topological equivalence on interior of $D^2$ that is not continously extendable to $D^2$

As said in the title, I'm trying to find a topological equivalence on the interior of $D^2$ that is not continously extendable to $D^2$. I have an idea about this, so here it goes: Let ...
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1answer
18 views

Proving a relation is a total order relation

Consider question #21 part a: Here is the solution: However, consider the definition of a total order relation: The solution didn't prove that the relation is a partial order relation. This ...
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2answers
21 views

Prove that this function is injective

I need to prove that this function is injective: $$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$ $$f: (x, y) \to (2y-1)(2^{x-1})$$ Sadly, I'm stumbling over the algebra. Here is what I have so ...
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0answers
19 views

$x_n,x_ny_n$ convergence implies $y_n$ converges

Assume that $x_n$ converges to a nonzero number $x$ and that the sum $x_ny_n$ converges to a limit $L$. Prove that the series $y_n$ converges. The natural guess is that $y_n$ will converge to $L/x$. ...
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1answer
78 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...
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0answers
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Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
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53 views

Construct the truth table?

Any body help me .. How to solve this? (i) $(p\land q)\to (p \leftrightarrow (q \lor r))$ (ii) $(p \leftrightarrow q) \leftrightarrow ((p\land q) \lor (\neg q \land \neg p))$ (iii) ...
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1answer
18 views

Sylvester Gallai Problem

Recently I came through a book of Arthur Engel which mentioned a problem called Sylvester Problem which states that- A finite set $S$ of $n$ points in the Euclidean Plane has the property that any ...
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23 views

A nowhere zero point in a linear mapping and Research Resources

Conjecture: If $\mathbb{F}$ is a finite field with at least 4 elements and $A$ is an invertible $n\times n$ matrix with entries in $\mathbb{F}$, then there are column vectors $x,y \in \mathbb{F^n}$ ...
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1answer
36 views

Proof of $\displaystyle \lim_{x \to p+}f(x) = l \land \lim_{x \to p-}f(x) = l \implies \lim_{x \to p}f(x) = l$

Let $f:(a,b) \to \mathbb{R}$ and $p \in (a,b)$. In proving the following implication, I am unsure about one step $\displaystyle \lim_{x \to p+}f(x) = l \land \lim_{x \to p-}f(x) = l \implies \lim_{x ...
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0answers
20 views

Question about integration and sets of measure zero

Let $Q \subseteq \mathbb{R}^n$ be a box, $f: Q \to \mathbb{R}$ be bounded, integrable on $Q$. Suppose $g: Q \to \mathbb{R}$ is another bounded function such that $f(x) = g(x)$ for any $x \in Q ...
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37 views

Consecutive natural numbers [duplicate]

Please I want to know what is the most appropriate expression that if it is asked to find the counterexample of "The product of any three consecutive natural numbers is divisible by 9" My expression ...
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24 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
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1answer
25 views

Lang's proof of Cauchy's Theorem

In proving Cauchy's theorem in his 'Algebra', Lang first prove[s] by induction that if $G$ has exponent $n$ then the order of $G$ divides some power of $n$. Let $b \in G, b \ne 1$, and let $H$ be ...
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1answer
18 views

Some questions about proof of Theorem 2.43 in Baby Rudin

I will include the proof here and highlight the parts that are giving me trouble. Theorem $\hspace{5 pt}$ Let $P$ be a nonempty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable. Proof ...
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1answer
35 views

Where did this “+1” term come from for this inductive proof?

Where did this "+1" term come from for this inductive proof? It is in boxed in black. For context, We are trying to prove this sequence: has the following solution: $$x_{ n }=\frac { 3^{ n+1 ...
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1answer
72 views

Proof of The Associative Law.

The associative law of multiplication for three positive integers $a,b$ and $c$ can be proved$^1$ from the Commutative Law and the property of "Number of things" easily. We can prove$^2$ the ...
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1answer
40 views

Gelfand transform on disk algebra

I tried to prove that if $A$ is the disk algebra then the Gelfand transform is the identity map. The statement can be found here in Theorem 4.4 but it is given without proof. Please can someone check ...
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1answer
53 views

The Limit: $\lim_{x \to \infty}\frac{e^{f(x+a)}}{e^{f(x)}}$

I'm doing some challenge review problems and I was wondering whether this proof looked correct: Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $\lim_{x \to ...
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4answers
78 views

$f[A]\cap f[B]\supsetneq f[A\cap B]$ - Where does the string of equivalences fail ? [Chartrand 3E 9.12(b), 9.29]

I only realised that equality may fail in $f[A]\cap f[B]\supseteq f[A\cap B]$ (i.e., that we can have $A,B,f$ for which $f[A]\cap f[B]\neq f[A\cap B]$) after checking the answer. I don't see any ...