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

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0answers
6 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 ...
0
votes
1answer
19 views

Function on a well-ordered set

Let $(W,<)$ be a well ordered set. Let $f : W\rightarrow W$ be a function such that $u < v$ implies $f(u) < f(v)$. Show that $\forall w \in W, w \leq f(w)$. I was thinking to consider $T=\{x ...
0
votes
0answers
11 views

Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
0
votes
1answer
21 views

Provide an example to show that $S$ may not necessarily be a unique factorisation domain when $R$ is a unique factorisation domain.

Let $R$ and $S$ be integral domains, and suppose that $\phi:R \rightarrow S$ is a surjective ring homomorphism. Provide an example to show that $S$ may not necessarily be a unique factorisation domain ...
3
votes
2answers
42 views

Proof that an infinite subset of $\mathbb{N}$ is countable

I want to prove that if $A$ is an infinite subset of the natural numbers, then it is countable. I thought of an informal proof: put the elements of $A$ in increasing order. Then associate the ...
2
votes
1answer
25 views

Determine if n is a prime.

Let $n$ be a positive natural number. You know the following facts about $n$ . Firstly, $n<10^{6}$ . Moreover, not a single integer $k$ between $1$ and $10^{4}$ divides $n$ . Does it ...
0
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1answer
24 views

formal proof that p-values are uniformly distributed

I'm trying to prove that $p$-values under the null hypothesis are uniformly distributed in $[0, 1]$ for an absolutely continuous test statistic $X$. Proof: By continuity of $F_X$, it is sufficient to ...
1
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1answer
23 views

A problem on finding the nearest points to the origin on the intersection of two surfaces

Suppose we are to find the points nearest to the origin on the curve of intersection of the two surfaces $g^{-1}_{1}\{ 0 \}$ and $g_{2}^{-1}\{ 0 \}$, where $g_{1}: (x, y, z) \mapsto x^{2} - xy + y^{2} ...
5
votes
2answers
39 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
0
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0answers
21 views

Argument for finite solution of power Diophantione Equation.

Assume the equation $4x^3=y^2+3$ has infinite positive integer solution. If $x,y$ has general solution then it is clear that for any $x$(rational, integer), there is a $y$. It can be said there is a ...
2
votes
0answers
51 views

Is $\forall n\exists m:\, m^2=n,\text{ where }m,n ∈ \mathbb N$ true or false?

$\forall n\exists m:\, m^2=n,\text{ where }m,n ∈ \mathbb N$. Prove whether this expression is true or false. My attempt: False, take $n=3,$ then there is no such integer $m$, such that $m^2=3$. Thus, ...
3
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1answer
30 views

Prove that square of even integer is even.

Is my proof correct? Suppose $n=2m$ is an even integer. Since $n=2m$ , then $n^{2}=(2m)^{2}$ $n^{2}$ = $(2m)^{2}$ = $4m^{2}$ =$2(2m^{2)}$ Since $(2m^{2})$ is an integer and $2(2m^{2})$ ...
1
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1answer
20 views

Deciphering proof of SLLN

I was looking at a proof of the string law of large numbers, and am having trouble finding where the proof uses the assumption that the random variables are identically distributed. I'll reproduce the ...
7
votes
2answers
113 views

$\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational.

This is my attempt at this question. Is this correct? $\forall x\in\mathbb{R},,$ if $x^{2}$ is rational, then $x$ is rational. This statement is false. Using counterexample, let $x=\sqrt{2}$. Since ...
2
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0answers
30 views

A question regarding Parseval's identity.

In most books/websites, Proposition 2 (see below) is either stated for a Hilbert space or proved via Riesz-Fischer. Does the follow approach (which seems to work in an inner product space) fall down ...
0
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1answer
38 views

$f:U \rightarrow \mathbb{R}$, $U$ is an open conected subset of $\mathbb{R}^n$ and $f \in C^1$ need to show that $f$ is $M$ Lipschitz on any compact

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
2
votes
0answers
31 views

Is $f\colon Y'\to Y$ continuous?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and $T\colon X\to X$ continuous, describing the following dynamics: For $\eta\in X$ let $\eta(y)$ describe the y-th position in the bi-infite sequence ...
0
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0answers
24 views

Precompact and locally finite implies finite intersection

An exercise in Lee's Introduction to Smooth Manifolds asks the following: Let $M$ be a topological manifold, and let $\mathcal U$ be an open cover. Suppose the sets in $\mathcal U$ are precompact ...
0
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0answers
14 views

Let $R$ be a ring with 1 and N be a submodule of R-module M. If $M$ is free of finite rank, is $M/N$ necessarily free of finite rank?

Let $R$ be a ring with 1 and N be a submodule of R-module M. If $M$ is free of finite rank, is $M/N$ necessarily free of finite rank? My idea: No. Consider $R = M = Z_6$ and $N = 2Z_6 = \{2,4\}$. ...
0
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2answers
36 views

Prove an eigenvector for two matrices is also the eigenvector for the product of those matrices. [duplicate]

So let's assume that A and B are both nxn matrices, and that u is an eigenvector for both A corresponding to lambda one and B corresponding to lambda 2. I need to prove that u is also the eigenvector ...
2
votes
2answers
37 views

Proof for $0a = 0$

Is this a valid proof for $0a =0$? I am using only Hilbert's axioms of the real numbers (for simplicity). $(a+0)(a+0) = a^2 + 0a + 0a + 0^2 = (a)(a) = a^2$ Assume that $0a$ does not equal zero. ...
1
vote
2answers
41 views

Summation Proof Dealing With 3s Multiples [duplicate]

So the problem is as follows: Prove that if the sum of digits of a decimal $n$ is three's multiple, then n is three's multiple by direct proof. For example, $11234567$ is 3's multiple because ...
0
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0answers
11 views

Show that $v(E)=\text{sup}\sum_{j=1}^{n}|\mu(A_j)|$ is a measure.

Background A family $\textbf{X}$ of subsets of $X$ is a $\sigma$ algebra in case: $\phi, \mathbb{R} \in \textbf{X}$ $X \setminus A \in \textbf{X}$ if $A \in \textbf{X}$ If $(A_n) \in \textbf{X}$, ...
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0answers
47 views

Summation Direct Proof Help [on hold]

Prove that if the sum of digits of a decimal n is three's multiple, then n is three's multiple by direct proof. For example, 11234567 is 3's multiple because 1+1+2+3+4+5+6+7=24, and in fact, 11234567 ...
0
votes
0answers
26 views

Determine all $n \in \mathbb{N}$ such that $GCD(n,48)=6$, $14|n$ and $|Div^+(n)|=12$.

Determine all $n \in \mathbb{N}$ such that $\gcd(n,48)=6$, $14|n$ and $|Div^+(n)|=12$. What I did: $14|n$ then $2|n$ and $7|n$ so $n=2\cdot7\cdot q$, $q \in \mathbb{Z}$. Then $6|n$ implies $2|n$ and ...
0
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0answers
27 views

Every primitive of an odd function is even (proof)

I'd like to prove that every primitive of an odd function is even. This is my reasoning; FACT: 1: if f(x) is even, then f'(x) is odd [easy to prove]; 2: if f(x) is odd, then f'(x) is even ...
2
votes
1answer
46 views

Proving $(p\to q)\land(p\to r) \equiv p\to(q\land r)$ using logic laws — short cut or incorrect?

Working through this problem: Using logic laws, show that the following are logically equivalent: $$(p\to q)\land(p\to r)\qquad\text{and}\qquad p\to(q\land r).$$ The way I did the problem is ...
2
votes
0answers
28 views

Determine whether this series converges (proof verification)

Determine whether the following series converges: $$\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt{n}-(-1)^{[\sqrt{n}]}}$$ where $$[x]=\max\{k\in\mathbb{Z}: k\leq x\}$$ My attempt: First I write ...
0
votes
1answer
14 views

Inversions and Multiplicativity of the Sign of a Permutation

The question is mainly about showing, for two permutations $\sigma, \pi \in S_{n}$, that $\mathrm{sgn}(\sigma \pi) = \mathrm{sgn}(\sigma) \mathrm{sgn}(\pi)$ using inversions of permutations (i.e. a ...
0
votes
3answers
39 views

Find $GCD(n^2+1,n+1)$

$GCD(n^2+1,n+1)$, $n\in \mathbb{N}$ What I did: $n^2+1=(n-1)(n+1) + 0$ So I thought $(n^2+1:n+1)=n+1$ But that doesn't seem to be the case: $n=2$ $n^2+1=5$ $n+1=3$ $GCD(5,3)=1$ Why is the ...
0
votes
0answers
24 views

I'm having troubles to find this parametrization.

I'm reading the Reid's Undergraduate Algebraic Geometry book of algebraic geometry for undergraduates and I have two questions about a proof of an example on the page 19: Red question: Reid said ...
3
votes
0answers
48 views

Proving a strictly decreasing sequence which tends to zero is positive

Suppose $(a_n)$ is a strictly decreasing sequence such that $a_n\underset{n\to\infty}{\rightarrow}0$. I'm asked to prove that $(a_n)$ is positive. My approach: suppose there is a negative element ...
1
vote
0answers
23 views

Show that the fix points of a function couldn't be in the interior

I want to solve the following problem: Show that the fix points of a function $f:\mathbb B^n\rightarrow \mathbb B^n$ could possibly not be in the interior. By this, Show that the Brouwer fixed-point ...
1
vote
1answer
23 views

I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
2
votes
1answer
18 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
1
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2answers
52 views

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$?

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$? I thought that by monotonicity, which states that if $A \subseteq B$ then $P(A) \le P(B)$, then: If $P(A) < P(A \cup ...
3
votes
2answers
59 views

Prove that $x^3 + y^2$ is irreducible in $\mathbb{Q}[x,y]$.

Prove that $x^3 + y^2$ is irreducible in $\mathbb{Q}[x,y]$. My proof: $\mathbb{Q}[x,y] = \mathbb{Q}[x][y]$. Suppose $x^3 + y^2$ is reducible. Then $x^3 + y^2 = (y + g(x))(y + h(x)) = y^2(1 + h(x) + ...
0
votes
0answers
14 views

If $\lim \limits_{x,y\to \infty}f(x,y)=l$ then $\lim \limits_{x,y\to \infty}|f(x,y)|=|l|$

Let $f:\mathbb R^2\to \mathbb R$ if $\lim \limits_{x,y\to \infty}f(x,y)=l$ then $\lim \limits_{x,y\to \infty}|f(x,y)|=|l|$ My attempt: Let $\epsilon>0$, we know that $\exists M>0$ such that ...
0
votes
0answers
54 views

Prove that $\int_{a}^{b} f(x)g'(x) dx = 0$ iff $f$ is constant

Given that $f$ is continuously differentiable and increasing on $[a, b]$, $g$ is differentiable on $[a, b]$, and $g'$ integrable on $[a, b]$. If $g$ is positive and $g(a) = g(b) = 0$, show that ...
1
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1answer
62 views

Determine a basis for $\mathbb{Z} \oplus \mathbb{Z}$ which determines a basis for the submodule $N$ generated by $(6,9)$

Proof Clearly the rank of $\mathbb{Z} \oplus \mathbb{Z}$ is $2$, so we must have that the rank of $(6,9)$ is $\leq 2$.Let $e_1 = (1,0)$ and $e_2 = (0,1)$ be a basis for $\mathbb{Z} \oplus ...
0
votes
1answer
83 views

Covariance inequality for $n$ exchangeable random variables

Let $n \in \mathbb{N}$, $n \geq 2$, assume that $X_1,\ldots, X_n$ are exchangeable, square integrable random variables with $\mathbf{E}\bigl[X^2_1\bigr] < \infty$. Prove that the following ...
0
votes
0answers
19 views

Showing that the 2-torus is parallelizable

Here is the question Let $$ \widehat{\xi}: \mathbb{R}^2 \to \mathbb{R}^2 $$ be a smooth function satisfying $$ \widehat{\xi}(x,y)=\widehat{\xi}(x+m, y+n) $$ for all $x,y\in \mathbb{R}, ...
2
votes
0answers
60 views

Proof that $1 > 0$ using the field and order axioms

This is the problem that you see the first time, and you say: this is the easiest math problem, but you are not quite correct. My reasoning is based on the Peano axioms also. Basically, if 2 numbers ...
2
votes
2answers
50 views

Proof that $-(-x) = x$ using just the field axioms

This is my attempt based on some stuff I have been seeing around: Let $y = -x$, then $-y = -(-x)$. Now, lets sum $y + x = (-x) + x = 0$, then we have $y + x = 0$. If we had the additive inverse of ...
0
votes
2answers
25 views

Problem understanding the proof of a limit

We claim that the limit of the sequence $d_n$ = $2n+4\over 5n+2$ is $2\over 5$ . Proof: Given $\alpha > 0$, let N = $1\over 5$*($16\over 5\alpha$-2) . Then for all n ≥ N, we have n ≥$1\over ...
2
votes
3answers
38 views

A polynomial in $g$ approximates every $f$ iff $g$ is injective

Prove or disprove the following statement: There exists a continuous function $g$ defined on $[a,b]$ with $g(x)\neq x $ for at least one $x\in[a,b]$ such that for every continuous function ...
2
votes
0answers
43 views

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$ My attempts, $\coth^2 (x)-1\equiv(\frac{e^x+e^{-x}}{e^x-e^{-x}})^2-1$ $\equiv \frac{e^{2x}+e^{-2x}+2}{e^{2x}+e^{-2x}-2}-1$ $\equiv ...
1
vote
2answers
33 views

Simplification of powers

I think this is a really simple question, but for some reason my brain can't get round it. I am proving a combinatorial result by probabilistic method and the last step has got me really confused. ...
0
votes
1answer
12 views

Cross Product and Scalar Product proof

Hi i'm trying to prove the following equality where P,Q and R are any 3D vectors: PxQxR = (P.R)Q - (Q.R)P I find it easier by proving that the x coordinate of the left side is equal do the right ...
2
votes
0answers
26 views

Proving a result using Riemann-Stieltjes integration

Let $(\alpha_n)_{n=1}^\infty$ be a sequence of monotonically increasing functions on $[a,b]$ such that the series $\sum_{n=1}^\infty \alpha_n(a)$ and $\sum_{n=1}^\infty \alpha_n(b)$ converge. I must ...