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

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15 views

alternative proof that a function is homogeneous of degree one

Given a (profit) function of the form $$ \pi(p) = \sup \{p.y:y \in Y\} $$, where $p \in R_+^k$ is a positive (price) vector and $Y \in R^k$ is a (production-possibility) set. I need to proof that $\...
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1answer
33 views

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$. Now let $z$ be the point $l \cap m$. Let $n$ be the ...
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1answer
54 views

$10$-digit perfect squares that contain each of the digits $1, 2, 3, 4, 5$ twice

Are there any 10-digit perfect squares that contain each of the digits $1, 2, 3, 4, 5$ twice? Perfect squares belong to the set $\{0, 1, 4, 7\}$ modulo $9$ and any such number will be equal to $3$ ...
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7answers
112 views

Proof that $e^{-x} \ge 1-x$

My aim is to prove that $e^{-x} \geq 1-x$ for any $x \geq 0$. What I found so far is Bernoulli's inequality, which states that $$1+x\leq\left(1+\frac{x}{n}\right)^n\xrightarrow [n\to\infty]{} e^x$$ ...
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3answers
102 views

Alternate Proof for the Cancellation Laws

I will first state the theorem: Given a group $(G,*)$ the following laws apply for $a, b, c \in S $ If $a*b=a*c$ then $b=c$ If $b*a=c*a$ then $b=c$ Attempt at alternate Proof: Consider the ...
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2answers
76 views

Possibly not an acceptable proof for uncountablity of countable product of countable sets

Here is a text from the book Topology by Munkres: Theorem 7.7. $ \ \ \ $ Let $X$ denote the two element set $\{0,1\}.$ Then the set $X^\omega$ is uncountable. Proof. $\ \ \ \ $ We show that, ...
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2answers
64 views

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$. What are the possible values of $$\frac{x^2+y^2-1}{xy}$$? I have discovered ...
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0answers
56 views

Simplifying Fraction for Nested Radicals

A while back, a problem asked me to simplify $$\dfrac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\dots+\sqrt{10+\sqrt{98}}+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\dots+\sqrt{10-\sqrt{98}...
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93 views

Proof in Algebraic Topology without appeal to intuition

My question arose from the proof of proposition 1.26 in Hatchers Algebraic Topology. There he constructs a space $Z$ from a path-connected space $X$ as follows: attach a set of 2-cells $e_\alpha$ ...
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4answers
58 views

If $f,g: X\to Y$ be two functions continuous on $X$ then show that $\{x:f(x)=g(x)\}$ is closed in $X$

Problem. Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces and $f,g:X\to Y$ such that $f$ and $g$ both are continuous on $X$. Show that the set $E:=\{x:f(x)=g(x)\}$ is closed ...
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1answer
47 views

Show $\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$

I am trying to give a non-algebraic proof for this equality: $$\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$$ So far, I could only use the identity $\dbinom{x}{y}=\dbinom{x}{x-y}$. ...
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1answer
142 views

Determinant of $n\times n$ matrix with parameter

Problem: Let $\delta \in \mathbb{R}^+$ and $n\in \mathbb{N}$. The matrix $A_n = (a_{i,j}) \in \mathbb{R}^{n\times n}$ is defined as $$ a_{i,j} = \prod_{k=0}^{i-2}\left((j-1)\delta +n-k\right) ...
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0answers
88 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
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1answer
76 views

How to prove this complex inequality elegantly?

Question Let $z_{1,2}\in U(0,1)\subset \Bbb C$, prove that $$\frac{|z_1|-|z_2|}{1-|z_1||z_2|}\le\left|\frac{z_1+z_2}{1+\overline{z_1}z_2}\right|\le\frac{|z_1|+|z_2|}{1+|z_1||z_2|}$$ Actually I haven'...
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1answer
17 views

Improving the proof for: $A$ is equicontinuous, show that $\overline A$ is equicontinuous

Let $A \subset C^0([a,b],\mathbb{R})$ Show that if $A$ is equicontinuous, then $\overline A$ is equicontinuous. Preliminary Proof: Let $(f_n) \subset \overline A$, let $\epsilon >0$ be given, ...
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0answers
47 views

Surface Area and Volume of a Torus Using Polar Coordinates

Can the volume and surface area of a torus be derived using double integrals and a coordinate transformation to polar coordinates where $x = rcos(\theta)$ and $y = rsin(\theta)$? Equation for the ...
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1answer
47 views

Reference request for “Elementary” Proofs of Picard's Great Theorem

This is Picard's Great Theorem; $\textbf{Great Picard Theorem.}$ Suppose an analytic function $f$ has an essential singularity at $z=a$. Then in each neighbourhood of $a$, $f$ assumes each complex ...
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1answer
32 views

proof - $\gcd(a, m) = \gcd(b, m) = 1 \implies \gcd(ab, m^2) =1$

I have a proof and want to know if its correct. Prove that $\gcd(a, m) = \gcd(b, m) = 1 \implies \gcd(ab, m^2) =1$ Proof: $ax_0 + my_0 = 1$ and $bx_1 + my_1 = 1$ $ax_0 = 1 - my_0$ and $bx_1 = 1- ...
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1answer
38 views

Proving that $z^4-6z^2+4z-3 = y^2$ has only one integer solution

I'm trying to prove the following result. Conjecture. If $z$ is an integer, and $z^4-6z^2+4z-3$ is a square, then $z=3$. A quick check modulo $9$ shows that $z=9w+3$ for some integer $w$. So for ...
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2answers
85 views

square matrix $\mathbf{A}$ with $\mathbf{A}^\intercal = -\mathbf{A}$, proof $\mathbf{A}$ is not invertible.

I tried proving that given a square $\mathbf{A}$ over $\mathbb{R}$ so that $\mathbf{A}^\intercal = -\mathbf{A}$, $\mathbf{A}$ is not invertible. I know that because the matrix is real and its ...
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0answers
8 views

How to prove mathematically these two different definitions of background-position property as equivalent?

I've been reading about how percentage values work for background-position position property. The official definition is that ...
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0answers
96 views

Looking for the most elementary proof that $48X^4+12X^2+1=Y^2$ has no non-trivial integer solution.

As relayed in this question of mine (which is more general in scope), I believe I have found a relatively easy, and completely elementary, way to show that the equation $$48X^4 + 12X^2+1 = Y^2$$ has ...
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1answer
216 views

Does the functional equation $p(x^2)=p(x)p(x+1)$ have a combinatorial interpretation?

A recent question asked about polynomial solutions to the functional equation $p(x^2)=p(x)p(x+1)$. Subsequently, Robert Israel posted an answer showing that solutions are necessarily of the form $p(x)=...
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0answers
39 views

Proving (without using complex numbers) that a real polynomial has a quadratic factor

The Fundamental Theorem of Algebra tells us that any polynomial with real coefficients can be written as a product of linear factors over $\mathbb{C}$. If we don't want to use $\mathbb{C}$, the best ...
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1answer
328 views

Is there a more rigorous way to show these two sums are exactly equal?

I would like to have a more rigorous proof of the hypothesis: The Crandall eta derivative series is equal to the following more elementary one. The following two series give the same sum. $$-\sum _{...
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44 views

Linear Algebra Friedberg Th1.9

I have been trying to prove Th1.9 rigorously since Friedberg didn't do so. Here is my attempt at a rigorous proof. Th 1.9 Restated: If a vector space V, S has n elements and span(S)=V then some ...
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3answers
52 views

Elegant proof that maximum of sums is, at most, sum of maximums

I'm looking for an elegant way to show that, among non-negative numbers, $$ \max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\} $$ I can show that $\max \{a+...
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0answers
36 views

Is there a statement inside of mathematics that is proven only with mathematical induction and with none other method?

Well, the point is that although the method of mathematical induction can be useful and is useful for proving certain statements, I somehow always like things to be proved in some other way than with ...
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1answer
50 views

Solving $x^e =c$ in $\mathbb{F}_{p}$

Find all solutions to the equation $x^3=7$ in $\mathbb{F}_{13},\mathbb{F}_{19}$ and $\mathbb{F}_{35}$. In An Introduction to Mathematical Cryptography (Hoffstein et al), we have that proposition 3.2....
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1answer
98 views

How many elementary ways are there to prove that $\displaystyle\left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $?

In a comment under this answer, a user boldly asserts that there is ONLY ONE way to prove that $$ \left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $$ where $\zeta$ is Riemann's zeta function. ...
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1answer
81 views

Trace of matrix is sum of eigenvalues (positive semi-definite case)

Let $A \in \mathbb{R}^{n \times n}$. It is well-known that $\text{tr}(A)$ is equal to the sum of the eigenvalues of $A$. Let us know restrict $A$ to being positive semi-definite. Obviously, it is ...
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4answers
59 views

Proof verification : every algebraic set is the union of finitely many irreducible algebraic subsets

I have found various proofs of the result but I have come up with something very different and I wonder whether it is a valid argument: Let $W$ be an algebraic set. Let $I=\mathcal{I}(W)$. We have $I=...
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1answer
15 views

Proof improvement for $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ given $(a+ib)(c+id)(e+if)(g+ih) = A + iB$

If $(a+ib)(c+id)(e+if)(g+ih) = A + iB$, prove that $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ My approach is pretty straightforward: $$(a+ib)(c+id)(e+if)(g+ih)$$ $$((ac-bd)+i(ad+bc))(e+if)(...
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1answer
41 views

If $A$ Is an Upper Triangle Matrix, the Adjoint Is Also Upper Triangular

I already proved it, but it was really laborious. I am wondering if any one has a shorter proof? Write $A = [a_{ik}]$ and let $\overline{A}_{rs} = [c_{ik}]$ denote the minor with row $r$ and column $...
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3answers
404 views

Finding the shortest distance between two Parabolas

Recently, a problem asked me to find the minimum distance between the parabolas $y=x^2$ and $y=-x^2-16x-65$. I proceeded with the problem as thus. Let $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. $\...
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0answers
54 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
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1answer
41 views

Prove $ f(c)\int_{a}^{b}g(x)dx=\int_{a}^{b}g(x)f(x)dx$

Assume that $f:[a,b]\rightarrow\mathbb{R}$ is continuous on $[a,b]$ and $g:[a,b]\rightarrow\mathbb{R}$ is integrable and $g(x)\geq0$ for all $x\in[a,b]$. Then there exists a $c\in(a,b)$ such that $$ f(...
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1answer
23 views

Alternative methods to solve DLP for $GL_{3}(\mathbb{F}_2)$

Is there (or rather what is) a more elegant/efficient way to solve the DLP for $g^x=h$ in $GL_3(\mathbb{F}_2)$ where $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 \end{...
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1answer
27 views

Extension of co-coercivity in strongly convex functions

I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y \...
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1answer
40 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of $C^...
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1answer
239 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that $$3<\pi&...
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2answers
33 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + a_n)^{1 / ...
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2answers
58 views

Solving the recurrence $A_n = \sum_{k=1}^{n} 2^{k+1} A_{n-k}$

Let me ask a very simple question: Let $(A_n)$ be a sequence of integers defined by $A_0 = 1$ and $$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$ There is an explicit formula for $...
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2answers
20 views

Decreasing sequence and prove by contradiction

I have "solved" the following question using prove by contradiction. But it seems a bit off to me: Let {$x_k$} be a sequence satisfying $x_{k+1}\le(1-\beta)x_k$ for $0\lt\beta\lt 1$ , and $x_0\le C$. ...
3
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3answers
40 views

How to prove these two sets are identical?

This is more a question of the methadology one should use to solve these type of questions: Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X $$ ...
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1answer
59 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
2
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4answers
59 views

Show that $6^n/n! \le 6^5/5! \times 6/n$

I want to show that $$\frac{6^n}{n!} \le \frac{6^5}{5!} \cdot \frac 6n$$ without using induction, which I've done but is rather clunky. Is there a more straight forward way of doing this?
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5answers
183 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
0
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1answer
33 views

Alternative proof of $\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it}$

Let $t\in \mathbb{R}$. I want an alternative proof of the following identity $$\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it} \quad(\star)$$ I've came up with this identity observing ...
3
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1answer
47 views

Is there any way to gain some insight into a proof by simply looking at a graphic?

My school is using Pinter's "A Book of Abstract Algebra" for both semesters of Modern Algebra. For a class assignment a couple weeks ago, regarding rings, I was tasked with the following problem (...