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

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0
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2answers
16 views

Simplification of a function

I have to simplify the following function : $g(x)$ = ${\sin x\over \sin 1} \cos (x-1)-{\sin x\over \sin 1} \cos 1 + {\sin (x-1)\over \sin 1} \cos 1-{\sin (x-1) \over \sin 1} \cos x$ My attempt: => ...
0
votes
0answers
10 views

Proving a PDE has a particular weak form (check my proof please!)

Let $u_t - \Delta u = f$ hold in $L^2(0,T;H^{-1})$ for a solution $u \in L^2(0,T;H^1_0)$ with $u_t \in L^2(0,T;H^{-1})$. This means $$\int_0^T \left(\langle u_t(t), v(t)\rangle + \int_\Omega \nabla ...
0
votes
3answers
41 views

Is my arithmetical proof using induction correct?

The exercise 2.b of my textbook ask me to prove that: $$\text{(P): }\;\forall n\in \mathbb{N}, 13\;|\;(3^{n+2}+4^{2\cdot n+1})$$ I would like to know if my proof is correct and if not what I need to ...
0
votes
1answer
13 views

Analysis: Proof checking and help on 2nd part (Integrals)

So I have the question, $f(x)= x$ if $0$ $\leq$ $x$ $\leq$ $1$ and $f(x)= x+2$ if $1<x$ $\leq$ $2$ (the same f(x) I just couldn't figure out how to do the big bracket) Part 1 is asking me to ...
-1
votes
2answers
31 views

Verify the Identity [duplicate]

$\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}$ So far I have gotten $\frac{n!}{(k-1)!\big(n-(k-1)\big)!} + \frac{n!}{ k! (n-k)!}$ But I quickly lose myself once I have to start making the ...
0
votes
4answers
33 views

Verifying my proof that if $|S(x)| \leq 1$, then $\lim_{x \to 0} x\cdot S(1/x) = 0$

Question: Suppose that $S : R \to R$ is a function so that for all $x$, $−1 \leq S(x) \leq 1$. Prove from the limit definition that $$\lim_{x\to0} x \cdot S(1/x) = 0.$$ This is my ...
2
votes
1answer
17 views

Epsilon-N method - Proof verification

Prove (using the epsilon-N method) that the sequence of numbers $\dfrac{5n^3-2}{n^3}$ converges. Calculate the limit first. First we calculate the limit: $\lim_{n \to \infty} \dfrac{5n^3-2}{n^3} ...
0
votes
0answers
10 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
3
votes
2answers
48 views

$\varepsilon-\delta$ proof of $\lim_{x \to -\infty} \frac{1}{1+x}=0$

I do not have a clue about where to start. If I'm right, I need to find a relation between $\varepsilon$ and $\delta$ such that $0<|x + \infty|<\delta$ implies $|\frac{1}{1+x}|<\varepsilon$. ...
2
votes
1answer
33 views

Limit of Inner products.

I had the following appear on an exam, and I can't see why I'm wrong and have no clear explanation from the professor. My answer is below. Let$ X$ be a Hilbert space, and let $\mathcal{E}$ be an ...
-1
votes
1answer
40 views

Prove a=v*dv/dx

Using calculus, and assuming a particle moving along the x-axis is concerned, prove that $a=v*dv/dx$ ~~~~~~~~~~~~ this is what I did, but im not sure it's rigorous enough: $a=dv/dt$ $t=x/v$ ...
2
votes
0answers
36 views

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$ Proof: $f_G:G_1\rightarrow G_2$ and $f_H:H_1\rightarrow H_2$. Question 1: Is the following statement valid? Does ...
0
votes
1answer
34 views

¬p ⊬ ⎕(p → q): Where's the mistake in my proof?

My professor noted on one of his slides that ¬p ⊬ ⎕(p → q). Intuitively, this seems correct; however, I can only prove that it is false. I suspect I've made a mistake in my proof. Where have I gone ...
2
votes
3answers
69 views

Is my proof by contradiction about the empty set correct?

I am trying to learn about proofs and one of the exercice in my book (Maths ABC) is about proof by contradiction. I think I understand the concept but I would like to have a feedback on the following ...
0
votes
0answers
8 views

Proving properties of nth roots

First let me define some things. Let $x \gt 0$ and $n \ge 1$. Now $x^{\frac{1}{n}}:=\sup\ [ y\in \mathbb R : y \ge 0 \text{ and } y^n \le x]$ (a) If $x \gt 1$ then $x^{\frac{1}{k}}$ is a decreasing ...
-2
votes
0answers
44 views

Will this limit apply both to primes and to composites? [on hold]

In searching for the next prime after any natural number, there is a limit for the distance from any given natural number n to the next prime. Whether it is Bertrand's postulate or Riemann's ...
0
votes
0answers
6 views

Proof: Monotonicity of heuristic

I'm trying to proof the consistency of a heuristic for a problem, namely the chebyshev distance: The heuristic is given by: $$H(N,G) = \max(|N_x-G_x|, |N_y - G_y|)$$ And represents the estimated ...
1
vote
0answers
22 views

Prove that ordinal multiplication is left distributive

Suppose $\alpha, \beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
0
votes
1answer
36 views

Prove that the following function tends to infinity as z approaches i (complex analysis)?

Prove: $\lim_{z\to i} [2 /({1 + z^2})] = \infty$ My attempt: We want $M > 0$ such that if $0 < |z - i| < \delta$ then $|2/(1 + z^2)]>M$ Now $|2/(1 + z^2)|>M$ ...
2
votes
0answers
27 views

Every graph G contains a path with d(G) edges (proof critique).

Curious to see whether my proof below is acceptable or not. Any feedback will be well received. Many thanks. Every graph G contains a path with $\delta(G)$ edges. $\mathbf{Proof.}$ Let $G$ be a ...
0
votes
2answers
141 views

A New, Possible Proof of the Infinitude of the Primes?

$$1=1$$ $$2=2$$ $$3=3$$ $4=2\cdot2$ At $4$, the first prime number, $2$, is there as a factor. So I say that at the square of $2$, $2$ comes into play as a prime factor. At this point, $2$ is the ...
1
vote
1answer
32 views

Show that $\frac{\mathrm{d^{2}}B }{\mathrm{d} A^{2}}> 0 $ if $U''<0$.

Given, $A = W_0 - L_0 + I - qI$, $B = W_0 - qI$, and $EU = p U(A) + (1-p) U(B) = k$, where $k$ is a constant. $\frac{\mathrm{d} B}{\mathrm{d} A}\bigg|_{}^{EU=k} = \frac{\frac{\partial EU }{\partial ...
1
vote
4answers
13 views

Differentiability question ends up in contradiction.

Let $f(x)=x^3cos\frac{1}{x}$ when $x\neq0$ and $f(0)=0$. Is $f(x)$ differentiable at $x=0$? My first attempt Definition: A function is differentiable at $a$ if $f'(a)$ exists. $$f'(x)=\lim_{h ...
0
votes
1answer
18 views

transitivity of subformula relation - proof

Problem: prove that the relation "is a subformula of" is transitive for propositional formulas. let $\phi \in Sub(\psi)$ and $\psi \in Sub(\chi)$ prove that $\phi \in Sub(\chi)$. my proof: if $\phi ...
0
votes
1answer
18 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
2
votes
2answers
234 views

Is it true that $3$ is the only prime of the form $n^2-1$? [on hold]

One less than a perfect square is prime if and only if the prime is 3. Is this really, really true and do we have proof?
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0answers
60 views

Where Are the Primes In Relation To the Perfect Squares? How Are the Perfect Squares Arranged Along the Natural Number Line? [on hold]

The question is concerning the location of any given prime which satisfies Legendre's conjecture, or simply, any given prime. Do they not all? All primes > 3 are in the pair of arithmetic progressions ...
1
vote
0answers
21 views

Check my proof - Linear Algebra

Still not completely confident with my capabilities in writing formal proofs so I thought I would ask for a check of this proof. Theorem Let $V$ and $W$ be vector spaces, and let $T$ and $U$ be ...
2
votes
3answers
84 views

What is wrong with this calculation of $\binom{\frac{1}{2}}{k}$?

The reason I ask this question is that I want to show that: \begin{equation*} \binom{\frac{1}{2}}{k} = -2\frac{1}{k}\binom{2k-2}{k-1}\left(-\frac{1}{4} \right)^k \end{equation*} \begin{align*} ...
0
votes
1answer
16 views

To Prove That Field of Fractions of Given Rings is Same. Proof Verification.

I am trying to solve Q. 8a in Section 9.1 from Abstract Algebra by Dummit & Foote. The problem is: Let $F$ be a field and $R=F[x,x^2y,x^3y^2,...,x^ny^{n-1},...]$ be a subring of $F[x,y]$. ...
1
vote
1answer
20 views

Show that $\lim_{n\rightarrow \infty } Var(Y_{n}) = 0$.

Given $Var(Y_{n}) = (\theta_{2}-\theta_{1})^2\frac{n}{(n+2)(n+1)^2}$. My work: $$lim_{n\rightarrow \infty } Var(Y_{n}) = \lim_{n\rightarrow \infty } (\theta_{2}-\theta_{1})^2\frac{n}{(n+2)(n+1)^2} = ...
0
votes
1answer
39 views

Equicontinuity if the sequence of derivatives is uniformly bounded.

I would really appreciate if someone could look over this proof for me. Let $ \left\{ g_m \right\} $ be a sequence of functions defined on an interval $ [a,b] \subset \mathbb{R}^n$. Let $ \left\{ ...
-2
votes
0answers
29 views

Solution to Quartic, Pentic, Hexic and Sietic Polynomials? [on hold]

Is there a mistake in this article: Solution to Quartic, Pentic, Hexic and Sietic Polynomials and isn't it in contradiction with Galois theory?
0
votes
2answers
48 views

Find the largest natural number m such that n$^3$-n is divisible by m for all n$\in$ $\mathbb{N}$.

Find the largest natural number m such that n$^3$-n is divisible by m for all n$\in$ $\mathbb{N}$. Prove your assertion. So my basis that I have is: Notice that (1)$^3$-(1)=0, and m(0)=0, so m ...
0
votes
1answer
13 views

Please help me finish this proof - the midpoints of the 4 sides of any quadrilateral are the vertices of a parallelogram

a) Let $A$ and $B$ be 2 points in the plane. Show that if $M$ is the midpoint of the line segment $\overline {AB}$, then $\vec{OM} = \frac{1}{2} (\vec{OA}+\vec{OB})$ where $O$ is the origin. I think ...
0
votes
3answers
43 views

How is $A\sin\theta +B\cos\theta = C\sin(\theta + \phi)$ derived?

I have come across this trig identity and I want to understand how it was derived. I have never seen it before, nor have I seen it in any of the online resources including the many trig identity cheat ...
1
vote
2answers
61 views

Proof irrationality $n\sqrt{11}$

Prove that $\sqrt{11}$ is irrational, subsequently prove that $n\sqrt{11}$ is also irrational for every $n \in \mathbb{N}$. You are allowed to use that if $p$ is prime, and $p | a^2$, then $p|a$. ...
1
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0answers
10 views

Proof that the span of a list is equal to the span of any reordering of the list

Claim: If $(w_1,w_2,...w_m)$ is an arbitrary reordering of $(v_1,v_2,...v_m)$, then $span(w_1,w_2,...w_m) = span(v_1,v_2,...v_m)$. Proof By definition, ...
5
votes
1answer
53 views

A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$

I am not sure whether this is right. Can anyone verify, whether this proof is valid? Thanks! Define a sequence $\{a_n\}_{n\ge0}$ as follows: ...
0
votes
1answer
81 views

Does Bertrand's Postulate Give Us the Tightest Proven Upper Bound For Prime Gaps? [on hold]

What has actually been proven? There will always be a prime between n and 2n? There will always be a prime between n and 2n-2? Is Bertrand's postulate the tightest proven upper bound for prime gaps? ...
1
vote
1answer
13 views

Unbiased estimator - Poisson Distribution

Suppose that $X_1, . . . , X_n$ is a random sample of size $n$ from a Poisson distributed population with mean $\lambda$. Assume that $n = 2k$ for some integer $k$. Consider the estimator $$\ ...
1
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0answers
29 views

Power of a point problem

There was a Finnish matriculation examination there was the following question: Consider a circle and a point $P$ outside the circle. From the point $P$ draw two lines such that each of the line ...
1
vote
0answers
32 views

Proof concerning specific class of Proth numbers

Is this proof acceptable ? Theorem Let $N = k\cdot 2^n+1$ with $n>1$ , $k<2^n$ , $k$ odd and $3 \nmid k $ , thus $N$ is prime iff $3^{\frac{N-1}{2}} \equiv -1 \pmod N$ Proof Necessity ...
0
votes
1answer
28 views

Show that | and $\downarrow$ are the only binary connectives \$ such that {$} is functionally complete.

I've been reading and coping with van Dalen's Logic and Structure for a a few days. However, I've getting problems to solve his Exercise 6 from Ch 1 Sec 1.3 (p.28). In this exercise, van Daken asks ...
0
votes
1answer
49 views

Is this the correct conclusions about a quotient topology on $\mathbb R$?

Can someone tell me if my conclusions about this space is correct? I have an equivalence relation for $x,y \in \mathbb R$ given by $$x\sim y \Longleftrightarrow (x = y) \lor (x,y \in (-a,a])$$ for ...
1
vote
1answer
29 views

Statement about solutions for diff. equations.

Let $f_1,f_2:\mathbb{R}^2\to\mathbb{R}$ be $C^{\infty}$ functions such that $f_1(x,y)\leq f_2(x,y)\; \forall(x,y)\in\mathbb{R}^2$. Suppose that $\psi_1:I_1\to \mathbb{R}$ and $\psi_2:I_2\to ...
3
votes
1answer
27 views

Show that $\log \left| z \right|$ is harmonic and find its the conjugate harmonic function.

Is the form correct for the conjugate harmonic? Attempt: First, we are given \begin{align*} \log \left| z \right| &= u(x,y) + iv(x,y) = \log \sqrt{x^2 + y^2} + i \cdot 0 \\ u(x,y) &= \log ...
1
vote
2answers
23 views

Prove the existence of a greatest lower bound of $X$ if $X \subset \mathbb{R}$ is a non-empty set that is bounded below

Attempt: Let $C \subset \mathbb{R}$ be the set of all lower bounds of $X$. Since $C$ is not empty and bounded above, every $x \in X$ is an upper bound of every element $c \in C$. Thus, there exists ...
1
vote
1answer
29 views

Alternative to the Frattini argument

If $G$ is a finite group with $H \trianglelefteq G$, and $P$ is a Sylow $p$-subgroup of $H$, then we can show that $G = N_G(P)H$. While I'm now aware of the (admittedly much simpler/nicer) Frattini ...
0
votes
1answer
30 views

How to Prove: If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.

I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go ...