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

learn more… | top users | synonyms

1
vote
0answers
13 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 ...
1
vote
1answer
29 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 ...
0
votes
1answer
53 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
votes
7answers
103 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 $$1+x\leq\left(1+\frac{x}{n}\right)^n\xrightarrow [n\to\infty]{} e^x$$ ...
4
votes
3answers
100 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 ...
1
vote
2answers
75 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, ...
4
votes
2answers
62 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 ...
2
votes
0answers
51 views

Simplifying Fraction for Nested Radicals

A while back, a problem asked me to simplify ...
5
votes
0answers
87 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 $e_\alpha$ ...
1
vote
4answers
55 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 ...
4
votes
1answer
46 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 $\dbinom{x}{y}=\dbinom{x}{x-y}$. ...
8
votes
1answer
140 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) ...
4
votes
0answers
78 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 ...
4
votes
1answer
68 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 ...
0
votes
1answer
16 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, ...
0
votes
0answers
25 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 ...
1
vote
1answer
41 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 ...
2
votes
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- ...
0
votes
1answer
37 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 ...
2
votes
2answers
79 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 ...
1
vote
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 ...
5
votes
0answers
94 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 ...
11
votes
1answer
210 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 ...
2
votes
0answers
36 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 ...
0
votes
1answer
262 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 ...
1
vote
0answers
42 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 ...
0
votes
3answers
43 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 ...
2
votes
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 ...
1
vote
1answer
42 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 ...
1
vote
1answer
82 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. ...
0
votes
1answer
55 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 ...
2
votes
4answers
52 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 ...
1
vote
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: $$(a+ib)(c+id)(e+if)(g+ih)$$ ...
0
votes
1answer
36 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 ...
4
votes
3answers
332 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 $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. ...
1
vote
0answers
50 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 ...
1
vote
1answer
40 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 ...
0
votes
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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
17 views

Degrees of vertices in a circuit must be even

Let $G$ be a graph with a circuit. Let $C$ denote the subgraph of $G$ consisting of vertices and edges of the circuit. Then for every vertex in $C$, $\deg (v)$ considered in $C$ is even. I would ...
0
votes
1answer
33 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 ...
4
votes
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 ...
1
vote
2answers
31 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 + ...
2
votes
2answers
56 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 ...
0
votes
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
votes
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 $$X \setminus V = U \cap X $$ ...
1
vote
1answer
55 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
votes
4answers
58 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?
5
votes
5answers
176 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
votes
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 ...