# Tagged Questions

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

255 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 $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. ...
27 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 ...
33 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 ...
18 views

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 $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 ... 1answer 197 views ### Proof that \frac{2}{3} < \log(2) < \frac{7}{10} Positive integrals$$\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 $$(http://math.stackexchange.com/a/1618454/134791) prove that ... 5answers 2k views ### Is there an idempotent element in a finite semigroup? Let (G,\cdot) be a finite semigroup. Is there any a\in G such that:$$a^2=a$$It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ... 0answers 18 views ### Extension of co-coercivity in strongly convex functions I am studying strongly convex functions and they mention if f(x) is strongly convex with Lipschitz gradients L, which means \parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y ... 0answers 15 views ### Degrees of vertices in a circuit must be even Let G be a graph with a circuit. Let C denote the subgraph of G consisting of vertices and edges of the circuit. Then for every vertex in C, \deg (v) considered in C is even. I would ... 1answer 29 views ### Function is differentiable in all the points of its domain I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function \in C^k and a function is C^k if is composition of ... 4answers 583 views ### Prove that the symmetric group S_n, n \geq 3, has trivial center. I am trying to prove this: Let \sigma be a non-identity element of S_{n}. If n \geq 3 show that \exists \gamma \in S_{n} such that \sigma\gamma \neq \gamma\sigma. Hint: Let ... 2answers 27 views ### Help Proving the Average is greater than B^(1/n) Let a_1, a_2, \ldots, a_n be positive real numbers. Define the following two numbers: A = (a_1 + a_2 + \cdots + a_n) /n  (The average of the numbers) B = (a_1 + a_2 + \cdots + ... 3answers 35 views ### How to prove these two sets are identical? This is more a question of the methadology one should use to solve these type of questions: Say there is a set V \subseteq X \subseteq Y and U \subseteq Y such that$$X \setminus V = U \cap X $$... 2answers 49 views ### Solving the recurrence A_n = \sum_{k=1}^{n} 2^{k+1} A_{n-k} Let me ask a very simple question: Let (A_n) be a sequence of integers defined by A_0 = 1 and$$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$There is an explicit formula for ... 2answers 19 views ### Decreasing sequence and prove by contradiction I have "solved" the following question using prove by contradiction. But it seems a bit off to me: Let {x_k} be a sequence satisfying x_{k+1}\le(1-\beta)x_k for 0\lt\beta\lt 1 , and x_0\le C. ... 0answers 25 views ### Use Cauchy-Schwarz inequality to prove that \langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C is continuous. Let (a,b) \in \mathscr H \times \mathscr H be fixed. So we have to prove that for a given \epsilon \gt 0, we can find \delta_1 \gt 0 and \delta_2 \gt 0 such that \lvert \langle x,y\rangle - ... 5answers 159 views ### Looking for a non-combinatorial proof that a! \cdot b! \mid (a+b)! (I use a and b to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of$$\frac{(a+b)!}{a! b!}$$as a multinomial coefficients, is there a proof that for ... 1answer 49 views ### \mathbb CP^1 \approx S^2 proof check I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ... 4answers 45 views ### Show that 6^n/n! \le 6^5/5! \times 6/n I want to show that$$\frac{6^n}{n!} \le \frac{6^5}{5!} \cdot \frac 6n$$without using induction, which I've done but is rather clunky. Is there a more straight forward way of doing this? 1answer 67 views ### Proving that product of transpose matrix and the matrix is inversible I need to prove that A^T$$A$is an invertible matrix. $$A= \begin{bmatrix} \vec{a_1} & \vec{a_2} & \ldots & \vec{a_n} \\ \end{bmatrix}$$ Can I prove this using ... 1answer 32 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 ... 1answer 40 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 ... 4answers 78 views ### If$ f \rightarrow c$then prove$\frac{1}{a} \int_{[0,a]} f \rightarrow c$Let$f$be an extended real-valued$\mathcal{M}_{L}$-measurable function on$[0,\infty)$such that$f$is$\mu_L$-integrable on every finite subinterval of$[0,\infty)$, and $$\lim_{x\rightarrow ... 2answers 41 views ### Simple Inequality of Complex Numbers, \left| \frac{a-b}{1-\overline{a}b} \right| <1 Exercise from Ahlfor's Complex: Given a,b \in \mathbb C, with |a| <1, |b|<1, prove:$$\left|\frac{a-b}{1-\overline{a}b}\right| <1.$$My argument: Lemma: If \alpha, \beta \in ... 1answer 32 views ### Show that a piecewise function of two solutions to an ODE is a solution to the ODE Let x=x(t). If the first order ODE x'=f(t,x) (*) is satisfied by u and v, each over I = (a,b), show that$$w(t) := u1_{(a,t_0)} + v1_{[t_0,b)}$$satisfies (*), where u(t_0) = v(t_0) ... 1answer 14 views ### Show code C is a 1 error correcting Let C=\{(0,0,0,0,0),(1,1,1,0,0),(0,0,1,1,1),(1,1,0,1,1)\}\in\mathbb{F}_2. Show C is 1 error correcting. Definition: a code C\subseteq\mathbb{F}^n is t error correcting , if for any two ... 2answers 24 views ### Proof using pumping lemma that \{0^m1^n \mid m \neq n \} is not regular First of all, there's already a question very similar to this, but in my case I just wanted to show my attempt with the hope that if there are any erros or it's wrong completely you can help me in the ... 2answers 2k views ### Proving graph connectedness given the minimum degree of all vertices I know that this is a repeat of a previous question asked with a similar title, but I didn't want to revive an old thread. The solution presented in that thread seems to be the common one, but I was ... 1answer 60 views ### For all x, f(x)=\int_{0}^{x}f(t)dt. Prove that f(x)=0 for all x. Suppose that the function f:\mathbb{R}\rightarrow\mathbb{R} is continuous and that$$f(x)=\int_{0}^{x}f(t)dt\qquad\text{for all$x$}$$Prove that f(x)=0 for all x. Attempt Since ... 0answers 48 views ### Monomorphisms of monoids are stable under coproducts Let M,N,K be three monoids (or even groups, if you like) and let N \to K be an injective homomorphism. Then, the induced morphism M \sqcup N \to M \sqcup K is also injective. This is easy to ... 2answers 61 views ### Alternative proof: G group of order p^2, p prime \Rightarrow H is normal in G. Let G be a group of order p^2 where p is prime. If H is a subgroup of order p, show that H is normal in G. I would like to prove this with the tools that the book has provided up to ... 0answers 32 views ### Solving a Diophantine Equation with 2 variables This is my answer for the following question: Find all natural numbers (a,b) for which a^b-b^a=1. When a or b equals 1, (a,b)=(2,1) is trivial. If a,b>1, I generalized the problem ... 1answer 13 views ### On a Simple Theorem from Hilbert's *The Foundations of Geometry* I want my proof writing skills to get better. I am trying to do this through proving theorems from Hilbert's axioms for Euclidean Geometry. I found Hilbert's The Foundations of Geometry here, a ... 2answers 37 views ### Is there a name for this double summation identity? What is the shortest way to illustrate that it holds? Say I have the following the expression:$$\sum\limits_{j=0}^{i-1} \sum\limits_{u=0}^{j} g(u)$$By enumeration, it is easy to see that: in the case when j=0 we have$$\sum\limits_{u=0}^{j} g(u) ... 0answers 35 views ### Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ... 2answers 47 views ### Can one show that$\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||}$is always on the range of$\cos \theta $? A basic property of the dot product of two vectors is that $$\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||} = \cos \theta$$ Where$\theta$is the angle between the two vectors. Since there is ... 1answer 55 views ### Prove$T$is invertible If$T\in L(X,X)$where$X$is a Banach space and$L(X,X)$is denoted as the space of bounded linear maps, and$\|I-T\|<1$where$I$is the identity operator, then$T$is invertible? Here ... 0answers 18 views ### Easy computations using the functional equations for Riemann and Gamma functions Let$\zeta(z)$the Riemann Zeta function and$\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ... 0answers 12 views ### Direct proof of the existence of optimal memoryless deterministic policies in MDP It is well known that (finite-state, finite-action, discrete time) MDPs admit an optimal policy that is memoryless and deterministic (sometimes called pure). The proof of this fact for ... 2answers 152 views ### Proving finite additivity for this semi-algebra (infinite coin flips) Background copied and pasted from another one of my questions: Background: Consider flipping a coin$n$times. Define the sample space as $$\Omega = \{(r_1,r_2,r_3,\dots); r_i = 0 \text{ or }1\}$$ ... 1answer 27 views ### Generalisation of Binomial Theorem, Leibniz Formula and similar theorems Since the beginning of the year, our maths teacher showed us the Binomial Theorem in$\mathbb{R}$\, then in$\mathbb{C}$\, in$M_n(\mathbb{K)}$with two matrices which commute, and now the Leibniz ... 0answers 33 views ### Proof$f(x,y)=x_1+e^{x_{2}}$is strictly convex I am trying to show that$f(x,y)=x_1+e^{x_2}$is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ... 3answers 78 views ### Proving that$\sqrt{a_1^2} + \sqrt{a_2^2} +…+ \sqrt{a_n^2} > \sqrt{a_1^2 + a_2^2 +…+a_n^2}$using Pythagoras I think I have a proof using Pythagoras for$\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$. I'm interested in whether there's a way to use that proof with Pythagoras to prove the general ... 1answer 145 views ### How to Prove the Chain Rule for Limits Using a$\varepsilon-\delta$Argument? I came across the chain rule for limits the other day and it interested me quite a bit and surprisingly I couldn't find the proof on the internet anywhere. From what I understand the chain rule for ... 3answers 131 views ### Isomorphism of Non-Symmetric Matrices$A, B$are non-symmetric matrices of dimension$m \times n$where$m=n$or$m \neq n$. Example: An example of$6 \times 3$non-symmetric matrix is$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & ... 2answers 65 views ### Dummit and Foote exercise verification? I was working on the following problem: Let$\sigma$be the m-cycle$(1 2...m)$. Show that$\sigma^{i}$is also an m-cycle iff$\gcd(i,m)=1$A solution to this problem is given here. But the ... 1answer 38 views ### Verify proof that if$M,N$are$R$-modules and$M$is Noetherian,$N$is finitely generated, then$M\otimes_R N$is Noetherian I have to prove that If$M,N$are$R$-modules and$M$is Noetherian,$N$is finitely generated, then$M\otimes_R N$is Noetherian We let$S$be a non-finitely generated submodule of$M\otimes_R ...
I know two proof about the approximation of Euler-Mascheroni constant $\gamma$, but very technical. So I would like to know if someone has a strategic proof to show that $0,5<\gamma< 0,6.$ ...