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Questions tagged [alternative-proof]

If you already have a proof for some result, but want to ask for a different proof (using different methods).

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Alternative proof that $U(n^2-1)$ is not cyclic for $n>2$.

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 4.84 ibid. and I want to solve it using only the tools available in the textbook so far. (A free copy of the book is ...
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1answer
45 views
+50

Expectation and variance of the number of elements of a random non-empty set selected from a finite power set

Let $S$ denote a finite set of cardinality $|S| = N$. Select randomly a non-empty subset of $S$. Let $X$ indicate the number of items belonging to this subset. (a) Describe the probability mass ...
3
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1answer
41 views

Can we multiply both sides of a limit equation?

Compute $\lim_{x \to -1} f(x)$ for a function $f: \mathbb R \to \mathbb R$ such that $$4 = \lim_{x \to -1} \frac{f(x)+2}{x+1} - \frac{x}{x^2-1} \tag{1}$$ $$ = \lim_{x \to -1} \frac{f(x)+2}{x+1} - \...
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5answers
42 views

Proving whether the set of third degree polynomial is not a vector space?

In my book, it says the above set fails the first axiom. It says if I take two sets $p_1(x)=x^3-x^2$ and $q(x)= 1-x^2$. They are not closed under addition. I can understand why that's true by using a ...
3
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2answers
54 views

Prove that if $ab$ is a perfect square and $GCD(a,b)=1$, then $a$ and $b$ are perfect squares

How can I easily prove that if $ab$ is a perfect square and $GCD(a,b)=1$ then $a$ and $b$ are perfect squares. I actually managed to prove that this way: if an integer $n$ is a perfect square, then ...
5
votes
2answers
288 views

Does there exist a real number a given distance from each rational number?

Let $r_n$ be an enumeration of the rational numbers and let $a_n$ be a sequence of positive real numbers that converges to zero. Does there exist $x\in \mathbb{R}$ such that $|x-r_n|>a_n$ for all $...
1
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1answer
26 views

Show that the Image of $0$ under a Linear Mapping is also $0$

In my book of Linear Algebra, I have the following exercises: Let $T: V \to W$ be a linear map from one vector space to another. Show that $T(0) = 0$. I'm somehow having a block. For me, it is ...
6
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1answer
74 views

Suppose $a, b\in G$ such that $\lvert a\rvert$ is odd and $aba^{-1}=b^{-1}$. Show that $b^2=e$.

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 4.30. Suppose $a$ and $b$ belong to a group, $a$ has odd order, and $aba^{-1}=b^{-1}$. Show that $b^2=e$. My Attempt: ...
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0answers
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+50

The smallest subfield of a complete ordered field is dense in itself

The smallest subfield of a complete ordered field is dense in itself. This theorem is part of my attempt to prove that A complete ordered field is unique up to isomorphism. My questions: Could you ...
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2answers
380 views

Integral $\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$

Last year I wondered about this integral:$$\int_0^\frac{\pi}{2} x^2\sqrt{\tan x}\,\mathrm dx$$ That is because it looks very similar to this integral and this one. Surprisingly the result is quite ...
3
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2answers
51 views

Regarding $x < y \Rightarrow x^n < y^n$ proof rigor.

I came across the implication $$x < y \Rightarrow x^n < y^n$$ $$x,y>0, n\in Z^+$$ in a textbook and came up with the following proof. Proof Since $x<y$ the following chain of inequalities ...
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1answer
25 views

Find the number of elements in set $A=\{1,2,\ldots, N\}$ such that the element has multiples in each row of a $N \times N$ square matrix.

For a given $N$, the $N \times N$ grid looks like : $1$ $\qquad$ $N+1$ $\qquad$ $2N+1$ $\qquad$ $3N+1$ $\qquad$...$\qquad$ $N(N-1)+1$ $2$ $\qquad$ $N+2$ $\qquad$ $2N+2$ $\qquad$ $3N+2$ $\qquad$... $...
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1answer
218 views

Have I correctly proved that $\lim_{||(x,y)||\to\infty}\frac1{y-x}\int_x^y\exp(-1/|t|)dt$ equals $1$?

I should prove that, as long as $y\ne x$, $$f(x,y)=\frac1{y-x}\int_x^y\exp(-1/|t|)dt\longrightarrow1 \ \text{as $||(x,y)||\to\infty$}$$ and I would like to do it without $\varepsilon-\delta$ reasoning....
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0answers
49 views

Are these upper and lower bounds for $\frac{x!}{\left\lfloor{x}\right\rfloor!}$ useful? If so, are they already known?

Truncating the infinite series for the derivative of the Digamma function $$ \psi'(x) = \sum_{n=0}^\infty\frac{1}{(x + n)^2} $$ after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ ...
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0answers
14 views

How to show the relationship between the two sequences in Pollard's $\rho$ factoring algorithm?

$\DeclareMathOperator{\mdc}{gcd}$This is about Pollard's $\rho$ factoring algorithm. I'd like to show the relationship between the sequence of numbers modulo $p$ and the sequence of numbers modulo $N$...
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2answers
46 views

For $a=\frac 14$, $f$ is continuous and differentiable.

Let $f(x,y) = \begin{cases} \frac{xy}{(x^2+y^2)^a}, & \text{if $(x,y)\neq (0,0)$ } \\[2ex] 0, & \text{if $(x,y)= (0,0)$} \end{cases}$ Then which one of the ...
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0answers
110 views

How can I prove that clust-p is NP-complete?

CLUST-P: Instance: A a non-empty set, α : A × A → N, p, s ∈ N Question: Does A have a partition, A1, A2, . . . , Ap, such that max α(u,v) <=s, u,v∈Ai ∀1 <= i <= p? It is obvious that A ...
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0answers
42 views

Every complex polynomial is a product of first degree polynomials, alternative proofs?

Fundamental Theorem of Algebra. Every non-constant single-variable polynomial with complex coefficients has at least one complex root. Using the this we can easily show the result with successive ...
3
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1answer
66 views

Combinatorial Proof that $p(n)/(1+\epsilon)^n \to 0$

I was thinking this morning about the identity $ \prod_{n=1}^{\infty} \left( \frac{1}{1-q^n} \right) = \sum_{n=0}^{\infty} p(n) q^n$. The product on the left converges for $|q|<1$, which implies ...
3
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2answers
84 views

Alternative proof for $\zeta\left(2,\frac14\right)=\psi^{(1)}\left(\frac14\right)=\pi^2+8G$

On the German Wikipedia page of the Hurwitz Zeta Function I have come across the following formula $$\zeta\left(2,\frac14\right)~=~\pi^2+8G\tag1$$ where $G$ is Catalan's Constant. Even though I ...
0
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1answer
60 views

Cayley-Hamilton says that evaluating an endomorphism's characteristic polynomial over that endomorphism gives zero. Isn't this true by substitution?

Given an endomorphism $f$ of a vector space $V$, its characteristic polynomial, say $P(x)$, is defined as follows: $P(x) = \det(f -xI)$, where $I$ is the identity endomorphism. It is well known that, ...
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0answers
13 views

If the integral is positive then show that there is a subinterval and $m>0$ such that $f(x)\ge m$ on the subinterval.

Suppose $\int_{a}^{b} f$ exists and is positive. Prove that there exists a subinterval $J\subset [a,b]$ and a constant $m>0$ such that $f(x)\ge m$ for $x\in J$. The above question is from my ...
8
votes
1answer
71 views

On $\sup|\varphi^{-1}(n)|=+\infty$

I am trying to find an elementary proof of the following fact: Given some $N\geq 2$, there are $N$ distinct integers $a_1,\ldots,a_N$ such that $\varphi(a_1)=\ldots=\varphi(a_N)$ with $\varphi$ ...
1
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1answer
32 views

Special cases of Szemeredi's Theorem?

Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial? To clarify, I'm looking for sets of integers where we can show that they contain ...
0
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2answers
75 views

Find a group $G$ with $a\in G$ such that $|a|=6$ but $C_G(a)\neq C_G(a^3)$.

This is part of Exercise 46 of Chapter 3 of Gallian's "Contemporary Abstract Algebra". Notation 1: The centraliser of $g$ in a group $G$ is denoted $C_G(g)$. Notation 2: The dihedral group $...
20
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3answers
357 views

Integral $\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}$

I have stumbled upon the following integral:$$I=\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}=-\frac{\pi}{24}$$ Although I could solve it, I am not quite comfortable with the ...
5
votes
2answers
84 views

Rectangles in a $n \times n$ grid and the sum of the first $n$ cubes: a geometric connection?

My aim is to find the number of rectangles (squares included) in a $n \times n$ grid. It is easy to get that result with combinations. The usual answer to this problem is given by $$\binom{n+1}{2}^2$$...
16
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1answer
173 views

Prove: $\lim_{n\to\infty}{\sum_{m=0}^{n}{\sum_{k=0}^{n-m}{\frac{2^{n-m-k}}{n-m+1}\,\frac{{{2k}\choose{k}}{{2m}\choose{m}}}{{{2n}\choose{n}}}}}}=\pi$

Consider the following limit: $$\lim_{n\to\infty}{\sum_{m=0}^{n}{\sum_{k=0}^{n-m}{\left[\frac{2^{n-m-k}}{n-m+1}\,\frac{{{2k}\choose{k}}{{2m}\choose{m}}}{{{2n}\choose{n}}}\right]}}}=\pi$$ I cooked this ...
6
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1answer
150 views

If $x^3 = y^2$, why is $y/x$ transcendental?

Let the ring $A=\mathbb{k}[x,y]/(x^3-y^2)$, and set $t = \frac{y}{x}$. We can form the subring $\mathbb{k}[t]\subset \operatorname{Frac}(A)$, the smallest ring containing $t$. We have identities like $...
0
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1answer
52 views

Interesting way of determining that every integer is divisible by a prime? [closed]

Is is possible to deduce that every integer is divisible by a prime from the fact that the set of integers not divisible by a prime has natural density zero? Preferably, I would not be looking for, "...
3
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1answer
50 views

Determinant of $n\times n$ matrix - two diagonals and a corner

Suppose we have matrix which has zeroes everywhere with the exception of bottom left corner, main diagonal and right above the main diagonal $$A_n=\begin{pmatrix} b_1 & c_1 & 0 & \ldots &...
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2answers
201 views

Is there a simpler proof of this fact in analysis?

Suppose that $f:(0,1)\to\mathbb{R}$ is differentiable, and that $f(x_1)=f(x_2)=0$ and $f’(x_1)>0$ and $f’(x_2)>0$ for some $0 <x_1<x_2<1$. Then there must exist an $x_0\in(x_1,x_2)$ ...
2
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3answers
40 views

Sufficiency for proof that if $P \in \mathcal{L}(V)$, such that $P^2 = P$ then $V = \text{null}(P) \oplus \text{range}(P).$

I have seen numerous proofs of this result, and understand why they are true. For instance here and here use the same method - writing $v = Pv + (I - P)v$ and then continuing on in a straightforward ...
2
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0answers
36 views

Show that if $K \subseteq M \subseteq L$, then the normal closure $N'$ of $L:M$ is contained in the normal closure $N$ of $L:K$.

I'm trying to show in an easy manner that if $K \subseteq M \subseteq L$, then the normal closure $N'$ of $L:M$ is contained in the normal closure $N$ of $L:K$. My book uses the following definition ...
1
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1answer
44 views

If $f$ be non-negative, bounded and Riemann Integrable on $[a, b]$, then $\sqrt{f}$ is R- integrable.

The easiest way to solve this is to use the theorem: If $g$ be a continuous function in $[M, m] \subset [j, k]$ and $f$ be bounded and Riemann integrable, then $g \circ f$ is Riemann integrable too. ...
11
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1answer
174 views

Show that $\sum\limits_{n=1}^{\infty}\frac1{n^2}=\sum\limits_{n=1}^{\infty}\frac3{n^2\binom{2n}n}$ without actually evaluating both series

$$\sum_{n=1}^{\infty}\frac1{n^2}=\sum_{n=1}^{\infty}\frac3{n^2\binom{2n}n}\tag1$$ Note that $(1)$ is true since the LHS equals $\zeta(2)$ whereas the RHS equals $6\arcsin^2 1$ which both turn out to ...
2
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1answer
60 views

I have a question on subgroups and generators

Let $G$ be a group and $\left\{ H_i | i \in I\right\}$ be a family of subgroups of G. State and prove a condition which makes $\cup_{i \in I} H_{i}$ a subgroup of $G$, that is that $\cup _{i \in I} H_{...
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1answer
38 views

Proving mean of sample minimum of U[0,1] is 1/(n+1) without calculus

Let $U : \mathbb{R} \times \mathbb{R} \nrightarrow \mathrm{dist}[\mathbb{R}]$ denote the parametrized family of uniform distributions where $U(a, b)$ is the uniform distribution with minimum $a$ and ...
2
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1answer
75 views

Prove constructively that $\log_2 3$ is irrational.

The usual proof that $\log_2 3$ is irrational is by contradiction. For instance: Assume the negation: that $\log_2 3 = m/n$ for some integers $m$ and $n$. Then, by the property of logarithms, $2^{m/...
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0answers
86 views

Is there an elementary proof of Fourier's Theorem?

Fourier's Theorem An arbitrary periodic function $F(t)$ with period $T$ can be decomposed into a linear combination of the functions $f_n(t)$ and $g_n(t)$ where, $$f_n(t)=\sin \frac {2πnt}{T}$$ $$g_n(...
5
votes
0answers
49 views

Proving that any given element $hk$ appears $|H \cap K|$ times as a product in the list of $HK$

I have been trying to prove that if $H, K$ are finite subgroups of $G$ then $|HK|=\dfrac{|H| |K|}{|H \cap K|}$. I saw the proofs in Herstein and Gallian textbook and they are essentially the same. ...
0
votes
1answer
89 views

Is there a proof of the real form of Fourier's Theorem?

Complex form of Fourier's Theorem Any periodic function can be decomposed into a linear combination of complex exponentials. Proof Consider a complex exponential with period $T_0$: $$e^{i\frac {2π}{...
3
votes
3answers
85 views

Proof verification: For $a$, $b$, $c$ positive with $abc=1$, show $\sum_{\text{cyc}}\frac{1}{a^3(b+c)}\geq \frac32$

I would like to have my solution to IMO 95 A2 checked. All solutions I've found either used Cauchy-Schwarz, Chebyshevs inequality, the rearrangement inequality or Muirheads inequality. Me myself, I've ...
5
votes
2answers
199 views

Newtonian potential expansion identity

Preliminaries Consider the Newtonian potential $$\frac{1}{|\vec x - \vec y|}$$ with $\vec{x}, \vec{y} \in \mathbb{R}^3$ and $|\vec{x}| = x > y = |\vec{y}|$. Its Taylor expansion is given by $$\...
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2answers
50 views

If $f : M \to N$ is $C^k$, then $\Gamma(f)$ is a closed embedded submanifold of $M \times N$

The following problem is part of an assignment in my Differential Topology course. Exercise. Show the graph of a $C^k$ function $f : M \to N$ between two $C^k$ manifolds $M$ and $N$ defined by $\...
2
votes
2answers
56 views

Suppose $\lim_{n \to \infty}a_n=a$, Prove $\lim_{n\to \infty} \frac{1}{2^n} \sum_{k=0}^n \binom{n}{k}a_k=a$.

Suppose $\lim_{n \to \infty}a_n=a$. Prove that $$\lim_{n\to \infty} \frac{1}{2^n} \sum_{k=0}^n \binom{n}{k}a_k=a\,.$$ Here is a proof, Proof: Consider $t_{n,k}=\frac{1}{2^n}\binom{n}{k}$,$0 \leq t_{...
2
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0answers
12 views

Conditions for a modular multiplication graph to contain $k$-cycles

Let the multiplication graph $n:m$ be the graph with points $0,\dots,m\text{ - } 1$ and a line between points $a$ and $b$ when $n\cdot a \equiv b\operatorname{mod} m$. From the answers to two other ...
1
vote
6answers
136 views

Collecting proofs for $\sum_{n=2}^{\infty} \, \frac{n-1}{2^n} = 1$ [duplicate]

Update: The summation I came across has the form shown in title, and that exact question appears to be new. I could ask for proofs that take on this summation directly (without reducing it to ...
4
votes
3answers
63 views

Simpler ways to show that $n^2$ divides a polynomial?

I want to show that $n^2 \mid P(n)$, where $$P(n) = \frac{n^2(n+1)^2(n+2)(n+3)}{48}$$ for every odd positive integer $n$. The approach I took involved showing that $\cfrac{P(n)}{n^2}$ is always an ...
1
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1answer
51 views

Finding $\frac{d^{n-1}}{dx^{n-1}}(1+\sqrt x)^{2n}|_{x=1}$

How can we find $$\frac{d^{n-1}}{dx^{n-1}}(1+\sqrt x)^{2n}\Big|_{x=1},$$where $n\in\mathbb{N}$? Attempt of complex integration $$\begin{aligned}&\frac{d^{n-1}}{dx^{n-1}}(1+\sqrt x)^{2n}\Big|_{x=1}...