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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2answers
42 views
Show that a linear map on a finite dimensional complex vector space always have an eigenvalue.
What is an alternative proof that a linear map $T$ on a finite dimensional complex vector space $V$ with dimension $n$ always has an eigenvalue?
Here is the original proof idea:
We take a no zero ...
0
votes
0answers
20 views
Convexity of Sum-of-max-linear-terms
Consider the sum-of-max-terms function $f: \mathbb{R}^m \mapsto \mathbb{R}$:
$$\begin{align}
f(z) = \sum_{k=1}^K \underset{j \in \mathcal{I_k}}{\max} \{z_j \}
\end{align}$$
where $\mathcal{I}_{k} \...
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votes
5answers
31 views
what's the best way to prove the equivalences of such formulas?
I want to prove the following:
$$2^n+2^{n-1}+...+2^1 + 1 = 2^{n+1}-1$$
The only Method that I know of is proof-by-induction but is this the best way to prove the equivalences of such formulas?
2
votes
7answers
70 views
Showing $a^2 + b^2 > 2ab$ without using the fact that $(a-b)^2 = a^2 + b^2 -2ab$?
I am wondering if we can Show that $a^2 + b^2 > 2ab$ without using the fact that $(a-b)^2 = a^2 + b^2 -2ab$?
(I'm particularly interested in $0<a<b<1$ but I don't think restricting $a$ ...
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votes
3answers
76 views
Proving a function $\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)$ is a linear polynomial
Here's the question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function
such that $$\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)\text{ for all
}x\in\mathbb{R},y>0$$ Show that ...
1
vote
1answer
43 views
Proving the divergence of $\sum_2^\infty{\frac{1}{n \log n}}$ using comparison test
It is quite straightforward using the Cauchy Condensation test. But is there any way to solve this problem using some well known comparison test?
I cannot think of any way of my own. Any help/hint ...
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3answers
63 views
Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
This question was asked in ISI BStat / BMath 2018 entrance exam:
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that
for all $x\in\mathbb{R}$ and $t\ge 0$, $$f(x)=f(e^{t}x)$$ ...
0
votes
1answer
33 views
Proving a sequence is bounded under given conditions
This question was asked in ISI BStat / BMath entrance exam:
Let $f :\mathbb{R}\to\mathbb{R}$ be a differentiable function such
that its derivative $f'$ is a continuous function. Moreover, assume
...
3
votes
2answers
87 views
Prove recursive formula for number of spanning trees in complete graph.
When $t(n)$ is number of spanning trees in complete graph $K_n$ prove recursive formula for $t(n)$: $$t(n) = {1\over (n-1)}\sum_{k=1}^{n-1} k(n-k){n-1 \choose k-1}t(k)t(n-k)$$
Could someone prove it ...
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votes
2answers
168 views
Quartic as a product of quadratics
The statement is
Every degree $4$ polynomial with real coefficients is expressible as the product of two degree $2$ polynomials with real coefficients.
This and much more general versions are of ...
1
vote
1answer
53 views
Proving if $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both converge then the series $a_1+b_1+a_2+b_2+\ldots$ converges.
If $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both
converge with sums $\alpha$ and $\beta$, then show that the series
$a_1+b_1+a_2+b_2+a_3+b_3+\ldots$ converges with sum $\alpha + \...
2
votes
1answer
24 views
How can I show these series combinations are the same algebraically?
We have to show:
$$\frac{1}{2}\sum_{r=1}^{n} {\frac{1}{r}} \ + \ \sum_{r=1}^{n} {\frac{1}{2r+1}} \ = \ \sum_{s=1}^{2n}{\frac{1}{s+1}}$$
I understand how this works intuitively and the most formal ...
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votes
1answer
68 views
Sketches of proofs written in English that show that first-order logic is complete?
I came across the following in a textbook:
$\mathcal{L}$ is a language. $\alpha \in \mathcal{L}$ is a formula. $\alpha \frac{c}{x}$ means: replace every instance of $c$ in $\alpha$ by $x$.
I am ...
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2answers
53 views
A question regarding the Bolzano Weierstrass Theorem
This is probably a very stupid/absurd question....
The theorem states that every bounded real sequence has a convergent subsequence. We further know that every sequence of real numbers has a monotone ...
1
vote
0answers
39 views
How do I prove this inequality of a convex function on an interval
Suppose h is a continuous function on the interval $[0:1]$ such that for any $t_1, t_2 \in
[0,1]$
$$ h\left(\frac{t_1 + t_2}{2}\right) \leq \frac{h(t_1) + h(t_2)}{2}$$
Show that for all natural ...
0
votes
1answer
57 views
Proof by induction involving symmetric groups
I am aware that i have posted this question before, but the comments i received did not really help silly old me. So i would appreciate it if someone can walk me through some parts of the problem. I ...
0
votes
3answers
28 views
About cluster points in a proof
I am reading in a proof, but there's only one part that I do not understand
Theorem: Let $\Omega$ be an open subset of $\mathbb{C}$, let $z_1,z_2,\dots$ be a sequence of different points from $\...
7
votes
1answer
235 views
Is induction neccessary for proving that every injective mapping of a finite set into itself is a mapping onto itself?
Upon reviewing the basic theorem that the number of elements in a fixed finite set is unique, I tried to determine what part of this proposition is in need of proof. It seems axiomatic. Nonetheless, ...
6
votes
0answers
79 views
If $f$ is continuous on $[a,b]$ and $g$ equals $f$ except at $x=c\in(a,b)$ then $\int_{a}^{b} f = \int_{a}^{b} g$.
If $f$ is continuous on $[a,b]$ and $g$ equals $f$ except at $x=c \in (a,b)$ where $g$ is defined arbitrarily. Then prove that $g$ is integrable on $[a,b]$ and $\int_{a}^{b} f =
\int_{a}^{b} g$.
Here'...
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votes
1answer
116 views
Evaluating $I(x)=\int_{0}^{\frac{\pi}{2}}{\frac{du}{x^2\cos^2u+\sin^2u}}$
The following is from a national junior contest of an african country.
Find the integral $$I(x)=\int_{0}^{\frac{\pi}{2}}{\frac{du}{x^2\cos^2u+\sin^2u}}$$
$$\underline{\textbf{My attempt:}}$$
...
2
votes
2answers
76 views
Calculate $\sum_{n=1}^{\infty}\left(\frac{1}{3}\frac{2}{5}\cdots\frac{n}{2n+1}\frac{1}{n+1}\right)$
Denote $a_n=\frac{1}{3}\cdot\frac{2}{5}\cdots\frac{n}{2n+1}\cdot\frac{1}{n+1}$.
Please prove
$$\sum_{n=1}^{\infty}a_n=\frac{\pi^2}{8}-1$$
This is the answer given by my friend. He also used ...
1
vote
1answer
63 views
Proof verification: Squeeze theorem for integrals
This was a problem in my analysis textbook-
Suppose that $f(x)\le g(x)\le h(x)$ for all $x\in[a,b]$ and that
$\int_{a}^{b}f$ and $\int_{a}^{b}h$ exists and are equal. Prove that
$\int_{a}^{b}g$ ...
1
vote
1answer
29 views
Proof Verification, family of sets
Given $\left\{ A_i \right\}_{i\in\mathbb{N}}$ a family of non empty sets, $A$ a non empty set such that $\forall i\, \in \mathbb{N}, A_i\cap A\neq\emptyset$. The family $\left\{B_j \right\}_{j\in\...
3
votes
1answer
58 views
Find $P(n+1)$ for a polynomial P
Let $P(x)$ a n-degree polynomial such that for all $k\in\{0,1,...n\}$, we have $P(k)=\frac{k}{k+1}$. Find $P(n+1)$.
My attempt:
I consider a new polynomial $Q(x)=(x+1)P(x)-x$. $Q$ is a n+1-degree ...
2
votes
3answers
34 views
Show if all continuous functions $f$ from $X \to \{0,1\}$ with the discrete topology are constant, then $X$ is connected.
This problem was similar to this, but I couldn't make it work with the direction I want to show:
I tried showing the contrapositive (the inverse of the linked problem), which is showing "if $X$ is ...
7
votes
2answers
67 views
Where does poisson distribution functional form come from?
I know that the punctual probability function of a random variable $X$ with a Poisson distribution is:
$$P(X=k)= e^{-\lambda }\frac{\lambda ^{k}}{k!}.$$
Also, I've learned that the formula can be ...
3
votes
2answers
68 views
Show that there exists a set $U$ which is both open and closed and $x \in U \subseteq V$.
Let $X$ be a compact topological space. Suppose that for any $x, y \in X$ with
$x \neq y$, there exist open sets $U_x$ and $U_y$ containing $x$ and $y$, respectively, such that $$ U_x \cup U_y = X\...
6
votes
1answer
107 views
What is the correct way to solve the equation: $x^4-x^3+x^2-x+1=0$
Given the equation: $x^4-x^3+x^2-x+1=0$ we need to find both its real and complex roots. What is the easiest and correct method for solving the equation?
Here is my approach, but it gives wrong ...
0
votes
0answers
29 views
Differentiability and continuity while partials have different conditions
The relevant things I read and will discuss are in this snapshot (from Folland Advanced calculus, and Wolfram Alpha, and this answer by zhw for an old question).
Also, let me add two other links ...
3
votes
3answers
379 views
Analytic proof of area probability
$X,Y$ are i.i.d. $unif(-1,1)$ random variables.
Prove that
$$P(X^2+Y^2\leq 1)=\frac{π}{4}$$
Geometrically, I understand how that happens. $(X,Y)$ is a random point in square having centre at origin ...
1
vote
2answers
102 views
Fundamental theorem of algebra avoiding complex analysis
Suppose you have to give a proof of fundamental theorem of algebra to someone who did not know complex analysis. You can’t therefore use Liouville’s Theorem. How would you proceed avoiding complex ...
2
votes
0answers
36 views
What is a method to prove this trigonometric recursion without mathematical induction?
tl:dr at the end:
So I started of scribbling this simple function in my notebook
$ f(\theta) = (1+\sin\theta + \cos\theta)$
$ = \sin^2\theta + \cos^2\theta + \sin\theta + \cos\theta$
$ = (\sin\...
2
votes
0answers
39 views
What are alternative ways to show this infinite sum is the absolute error between $\pi$ and 22/7
tl:dr at the end.
In my first year undergraduate course having definite integrals, I studied about the $\beta$ and $\Gamma$ functions.
There was a problem from this particular exam which I never had a ...
1
vote
0answers
52 views
Is there better method than this one to solve quadratic diophantine equations?
Say we have the quadratic Diophantine equation :
$4x^2+9y^2=100$
I'm not really sure what the usual method to find integer solutions to these equations are , but one way I thought of was the ...
2
votes
3answers
53 views
Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.
I encountered this problem in Sims' "Computation with Finitely Presented Groups".
Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.
The book uses coset enumeration or ...
2
votes
1answer
42 views
How to prove this simple property for two sets?
We are given two vectors, $a = (a_1,\dots,a_n)$ and $b = (b_1,\dots,b_n)$ such that $0 \le a_i=b_i \le \varDelta$ for $i=1,\dots,n$. We want to modify each of these vectors in an iterative procedure ...
2
votes
1answer
105 views
Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$
I have to prove the inequality
$$
\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}
$$
when $x^2+y^2=1$, using Cauchy-Schwarz Inequality.
The RHS is equal to $\frac{12}...
1
vote
0answers
24 views
If $G$ is a graph having $p$ vertices and min degree $\delta(G)\ge\frac{p-1}{2}$, then the edge connectivity is $\delta(G)$.
Theorem: If $G$ is a graph having $p$ vertices and $\delta(G)\ge\frac{p-1}{2}$, then the edge connectivity of $G$ is $\delta(G)$.
$\delta\left(G\right)$ denotes the minimum degree of $G$.
...
1
vote
1answer
27 views
number of ways to order 5 english and 4 french hits
the question is as follows: suppose a dj has 5 english hits and 4 french hits.
in how many ways can these songs be played if no two english hits should follow each other?
in how many ways can these ...
0
votes
3answers
32 views
Inequality involving absolute values.
I want to ask is whether there is a method to solve following inequality more easily and compactly or it is the only method.
$$|x-2|+|x-8|\le x-2$$
What I know is taking $x<2,8>x>2,x>...
0
votes
0answers
71 views
Baby Rudin 6.11 and 6.8
Theorem 6.8
If $f$ is continuous on $[a, b]$ then $f \ ; \epsilon \; R(\alpha)$$f$ is Riemann integrable on$ [a, b]$.
$f \epsilon R(\alpha)$ means $f$ is Riemann integrable w.r.t. $\alpha$
...
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votes
1answer
33 views
Show that if $A$ is star convex, $A$ is simply connected.
It is answered a couple times as I can see but I want to know if my proof is working. I think I didn't understand the definition of trivial fundamental group very well.
Here it goes:
A is simply ...
1
vote
1answer
55 views
Is there a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle?
Can anyone provide a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle, if there is one? Here, $h_i$ refers to the distance from the side $i$ to the ...
2
votes
1answer
42 views
Parallelogram Law Geometric Proof
So, I've dealt with the parallelogram law in various ways a bunch of times now. And algebraically (or maybe I mean arithmetically) it makes perfect sense to me--I can prove it, understand it, and I ...
15
votes
6answers
672 views
Very indeterminate form: $\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x \longrightarrow (\infty-\infty)^{\infty}$
Here is problem:
$$\lim_{x \to \infty} \left(\sqrt{x^2+2x+3} -\sqrt{x^2+3}\right)^x$$
The solution I presented in the picture below was made by a Mathematics Teacher
I tried to solve this Limit ...
2
votes
1answer
92 views
Taking $\int_{-\infty}^{\infty} \frac{e^{i \omega x}}{1 + ix}dx$ through Residue Theory?
In the text "Basic Complex Analysis" Third Edition by Jerrold E. Marsden and Michael J. Hoffman I'm inquiring if there's an alternate way through Complex-Analysis to evaluate $\text{Example (4.38)}$ ? ...
1
vote
0answers
23 views
Understanding logical implication and alternate proofs
I just don't get exactly how logical implications work.
When using truth tables (albeit impractical for large amounts of variables), it can be quite simple to show truths equate for something such as ...
4
votes
2answers
62 views
Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and $x^2 + 5x - 10^{2013} = 0$ have common roots.
Find the last three digits of $p$ if the equations $x^6 + px^3 + q = 0$ and
$x^2 + 5x - 10^{2013} = 0$ have common roots.
Let $a,b $ be the solutions of second equation, then by Vieta we have $a+...
5
votes
1answer
62 views
sum of the first $n^2$ natural numbers closed form
Before I get downvoted I am still a beginner so please bare with me. I know the summation of the first n are $\frac{n(n+1)}{2}$.
Does that imply the sum of the first $n^2$ is $\frac{n^2(n^2+1)}{2}$?
0
votes
2answers
52 views
Derivations of the trapezoid rule
I know the general method to derive the trapezoid rule is with Taylor series, or, you know, to just look at the trapezoids and figure out the rule. However, I feel that for such a simple rule, there ...
