This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

learn more… | top users | synonyms

1
vote
0answers
15 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...
3
votes
2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
0
votes
2answers
42 views

an unclear step in a textbook solution of quadratic inequality

We have a quadratic inequality $$Ax^2+Bx+C>0$$ After solving it for cases where $B^2-4AC > 0$, my textbook turns to cases where $B^2-4AC < 0$: Using the perfect square method, let's ...
1
vote
0answers
21 views

Simple clarification- big $O$ and small $O$ notations in Erdos-Kac theorem proof

From The Probabilistic Method by Alon and Spencer. Let $\nu(n)$ be the number of primes $p$ dividing $n$ and set \begin{equation} X_p= \begin{cases} 1, & \text{if}\ p|x, \\ ...
1
vote
1answer
41 views

Longest path technique of proving a graph theory problem

Question: Let G be a simple graph, where the minimum degree of a vertex is k. Show that G contains a path of length at least k and a cycle of length at least k + 1. Proof: Consider the longest ...
0
votes
2answers
39 views

The exterior measure of a closed cube is equal to his volume.

The proof goes like that: Let $Q\subset \mathbb R^d$ a closed cube of $\mathbb R^d$. Since $Q$ covers itself, we must have $m*(Q)\leq Q$. Therefore, it suffice to prove the reverse equality. Let ...
2
votes
1answer
44 views

The topology of $\mathbb{Z}_p$

I don't know much about topology, but anyway... Assuming $\displaystyle\prod{A_n} =\prod_{n\geq 1}{A_n}$, why is $\mathbb{Z}_p$ closed in a product of compact spaces? Googling I found Tychonoff's ...
9
votes
1answer
390 views

Algorithm to answer existential questions - Reduction

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
6
votes
2answers
123 views

Proving that the product of two numbers (in $\mathbb{R}$ or $\mathbb{C}$) is a continuous function.

This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as: $$(x,y)\mapsto xy$$ (Let indicate that map with ...
1
vote
0answers
21 views

Maximum bound of number of prime factors

This comes from the proof of theorem 4.2.1 from The Probabilistic Method by Alon and Spencer. Letting $\nu(n)$ be the number of primes $p$ dividing $n$, here's an excerpt verbatim: Let $x$ be ...
1
vote
0answers
66 views

Prove the convergence of the integral

I have an article to prove the following lemma, but I don't quite understand. Can you explain more? Lemma. We have $N(t)=N_{\alpha_{n}}(t)=\#\lbrace n:\alpha_{n}\leq t\rbrace$. Let $\{\alpha_{n}\}$ ...
4
votes
0answers
49 views

How to prove these claims about ideal sheaves?

The following claims come from the proof of Proposition 3.10 (Page 66) of D.Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry. Since I couldn't find these results in Hartshorne's Algebraic ...
0
votes
0answers
23 views

Explanation of a Proof for the Chevalley-Warning Theorem

I can see how the power sum is applicable in the final paragraph of the proof, but it's still too vague in my head. Can anyone provide more details as to why the power sum of each monomial vanishes? ...
2
votes
2answers
54 views

Spivak Chapter 5, Lemma 3 - Explanation

I'm having some trouble understanding the 3rd lemma in Chapter 5 (page 101 for the 3rd edition) of Spivak's Calculus, that states: If $\ y_{0} \neq 0 $ and $\ |y-y_{0}| < min(\frac{|y_{0}|}{2}, ...
2
votes
2answers
74 views

$\pi(x)$ Proof Clarification

In a proof from a number theory book that $${\pi(x) \over x}\le {2k \over x} + {\phi(k) \over k}$$ Where $x=kl+r$ with $0 \le r\lt k $ It is stated that $$\pi(x) \le k+(l-1)\phi(k) + r \le 2k+{x\over ...
1
vote
1answer
45 views

Question about functions that are not uniformly continuous.

I am not asking for any particular question today rather I need some help to understand the concept of a function that is not uniformly continuous. Ok, so If I understand this correct, to prove a ...
3
votes
1answer
18 views

Understanding a matrix bound/inequality

I came across the following statements; For a positive matrix $A$, that is bounded $0 \leq A \leq I,$ where $I$ is the identity matrix and a statement like $Y \leq X$ means that $X-Y$ is positive ...
2
votes
0answers
69 views

Derivation of Backward Kolmogorov Equation

I'm following Kallianpur-Gopinath's textbook "Stochastic analysis and diffusion processes" to study Kolmogorov equations and I got stuck in a step of the derivation of the backward equation. In ...
0
votes
1answer
15 views

Understanding part of a proof, to show commutatitivy of geometric mean for matrices

Ando defined a matrix geometric mean, for two positive $n \times n$ matrices, as follows: $$G(A,B) = B^{1/2}(B^{-1/2}AB^{-1/2})^{1/2}B^{1/2}$$ where all square roots are positive square roots. This ...
2
votes
2answers
87 views

Polynomial in several variables over $GF(2)$

Can anyone please explain how this Lemma has been proved? Lemma: Let $f$ be a nonzero polynomial in variables $x_1,\ldots,x_n$ over $GF(2)$, and let $d$ be the maximum degree of $f$ with respect to ...
7
votes
1answer
143 views

Mean value theorem for random variables (inside an expectation value)

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$E[f(X+Y)]=E[f(X)+E[f^\prime(X+\theta Y)]Y]$$ for real valued random variables $X$ and $Y$ ...
3
votes
2answers
88 views

Theorem 7.2 in General Topology by S. Willard

Theorem 7.2 If $X$ and $Y$ are topological spaces and $f:X \to Y$ , then the following are all equivalent :- I) $f$ is continuous. II) for each E $\subset X$ , $f(\bar E) \subset ...
0
votes
1answer
45 views

Looking for explanation of Banach-Tarski Proof, preferably by visual methods “Video, Pictures, Diagrams…”

could someone please explain the four steps of Banach-Tarski? 1- Find a paradoxical decomposition of the free group in two generators. 2- Find a group of rotations in 3-d space isomorphic to the free ...
-1
votes
2answers
18 views

On Closure of Product subset of $\Bbb R×\Bbb R$

Suppose that $\Bbb R×\Bbb R$ has the standard topology. If $S=\left\{(t,\sin{\frac{1}{t}})\mid t\in R\text{ and }t\gt 0\right\}$. Show that $(0,0)$ $\in \overline{S} $
0
votes
0answers
18 views

Queries on proof of Plancharel's theorem

I have been provided with a proof for plancharel's theorem: $$\int^\infty_{-\infty} |f(t)|^2 \ {\rm d}t = \int^\infty_{-\infty} |\hat{f}(\omega)|^2 \ {\rm d}\omega.$$ proof. Let $$f(t) = ...
1
vote
1answer
66 views

Question 7F from general topology by Stephen willard?

Can someone help me with 7F from Willard? In part two : $\mathbf{7}$F. Functions to and from the plane. The facts presented here for the plane will be proved in more generality for ...
3
votes
2answers
35 views

What is the best way to explain setting a restriction on $\delta$ in $\epsilon$-$\delta$ proofs?

I'm trying to prepare a somewhat informal lesson striving to provide an intuitive understanding of why for some limit proofs, we have to set an upper bound on $\delta$. For example, here's part of ...
1
vote
1answer
47 views

PDE - Don't understand teacher's solution

I'm reading a solution to a problem in PDE class that the teacher gave, and I don't fully understand his solution. The problem is $\frac{dx}{1+\sqrt{z-x-y}}=dy=\frac{dz}{2}$ and what we want is to ...
2
votes
0answers
34 views

$R$ commutative ring with unity. Prove if $R/M$ is a field $\implies M$ is a max ideal.

$R$ commutative ring with unity. Prove if $R/M$ is a field $\implies M$ is a max ideal. Let us propose the opposite: that M isn't the max ideal. So $$I \triangleleft R, M\nsubseteq I$$ (How this is ...
0
votes
1answer
60 views

Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...
1
vote
1answer
32 views

Mittag-Leffler Proof. Rudin notation question and some basic real analysis/topology

Proof is below What is meant by $\sum_{\alpha \in A_n}$? I thought $P_\alpha$ is a sum already. What is this open set he is talking about? Because $A_n$ isn't open. Oh he means ...
2
votes
2answers
39 views

What is meant by “regularized rank-1 approximation”?

Suppose we have the following formula: $$\forall i: \{d_i, x_{[i]}^T\}={\mathrm{argmin}_{d,z}} 1/2 ||E_i-dz^T||_F^2+ \lambda|z||_1 $$ I really don't get how the following sentence is concluded and ...
0
votes
2answers
21 views

What is $P$ and $X$ is supposed to be in this analysis question?

Source page 626. Can someone explain what is $\| P\| $ mean? Is that partition or what? Also why bother with $\epsilon/2$ if the giant expression in the middle proves the lemma. Finally, ...
0
votes
0answers
26 views

Existence of nice exhaustion - Rudin.

This is taken from Rudin's Complex Analysis/Real Analysis Can someone tell me why $K_n \subset \Omega$? I agree it is compact, but why does it follow that it is a subset of $\Omega$? WLOG, I ...
1
vote
1answer
78 views

On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
0
votes
0answers
42 views

Minor issue about the proof of the Cauchy Convergence Criterion on Understanding Analysis (Abbott)

I don't understand one small part of the proof of the following statement: If a sequence is a Cauchy sequence, then it converges. PROOF: Let $(x_n)$ be a a Cauchy sequence. Then it is bounded ...
0
votes
1answer
24 views

Discrete Math - Relation among the relations

I need help understanding an assignment involving sets. We're given a few sets, $A=\{1,2\}$, $B=\{1,2,3\}$, $C=\{1,2,3,4\}$, $P=\{(1,1),(2,3),(3,2)\}$, $Q=\{(1,1),(2,2),(3,2)\}$, ...
0
votes
0answers
24 views

blocking set of projective planes

Have a quick question, considering the Desarguesian projective plane of order q, what is an upper bound of the minimal blocking set? Wikipedia says the size of the blocking set is bounded below by ...
1
vote
1answer
44 views

Explanation of a proof about continuity in Spivak's Calculus

I can't see the "it follows that" part of the following proof in Spivak's Calculus book: Given that $f$ is continous at $b$, there is a $\delta>0$ such that; if $|x-b|<\delta$ then ...
1
vote
2answers
65 views

Regarding the proof of $\int_0^xf(u)(x-u)du$

This question has already been asked: For continuous function $f$, prove: $\int_{0}^{x} \; \left[\int_{0}^{t}f(u) \;du \right] \;dt=\int_{0}^{x} f(u)(x-u)du$ but I really don´t understand the part ...
0
votes
2answers
64 views

$\det(aI-T) = 0$ implies $a$ is an eigenvalue of $T$

In Hoffman and Kunze's linear algebra, right after the definition of an eigenvalue for an endomorphism $T: V \to V$ (i.e. $a\in F$ is an eigenvalue if there exists $\alpha\in V$, $\alpha \neq 0$, such ...
0
votes
1answer
31 views

Not understanding a line in a proof concerning Monomorphism and injectivity

In the proof that "in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective" given in Wikipedia ...
3
votes
1answer
76 views

No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$

There is a proof in my textbook for the following claim, which doesn't make a whole lot of sense to me. My annotations are in bold. Could someone perhaps elaborate on what's going on? Claim. If ...
0
votes
0answers
35 views

Theorem 2 of Section 5.4 in Fulton's Algebraic Curves

The Theorem and its proof are: To understand the proof we also need: Finally, Corollary 1 for Bezout's Theorem says that $\sum_{P \in F \cap G} m_P(F) m_P(G) \le deg(F) deg(G)$ for any curves ...
1
vote
1answer
61 views

Prove that the Zariski space $\text{Zar} \space (K,A)$ is compact.

I posted part of the proof from Matsumura's Commutative Ring Theory. I got stuck in the last sentence where it says "Hence the intersection of all the elements of $\mathcal{A}$ is the same thing as ...
5
votes
0answers
71 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
0
votes
2answers
28 views

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$

Let $F$ be a class of sets. Prove that $B - \mathop{\bigcup}_{A\in F} A = \mathop{\bigcap}_{A\in F} (B-A)$ I've started like this: $X= B - \mathop{\bigcup}_{A\in F} A$ $Y= \mathop{\bigcap}_{A\in ...
1
vote
4answers
91 views

Prove that if $a<b$, then $-b<-a$

Prove that if $a<b$, then $-b<-a$ I'm a bit lost in this one, this is what I did: First case: $0<a<b$ $|a|<|b|$, so $|-a|<|-b|$ Since both are negative and $|-b|$ is greater ...
2
votes
1answer
184 views

How to remember all the proofs in mathematics

I have a problem where I forget the proof of a theorem after some time without reworking it out. However, my teacher said that he was able to prove a theorem even without reworking it out for a long ...
0
votes
1answer
45 views

Ten men selected to include a captain vs ten men selected to include at least one officer

I can't see how these are different scenarios, but the answers are different. The general scenario is that there is a company of volunteers that consists of a captain, a lieutenant, an ensign, and ...