math maniac.
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isn't right to prove that $\sqrt{2}$/4 is irrational number?
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6 votes

Hint $:$ If $\frac {\sqrt 2} {4}$ is a rational number then so is $\sqrt 2 = \frac 1 2 \times \frac {4} {\sqrt 2}.$ Is $\sqrt 2$ a rational number?

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Hello please answer my question
2 votes

I think that you mean $f$ is a differentiable function of $X$ and $Y$ and $X$ and $Y$ are themselves differentiable functions of $x$ and $y$ defined by $$X(x,y) = x, Y(x,y) = x+y,\ (x,y) \in D \...

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Series $q^{2k}$
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2 votes

Observe that $$\sum\limits_{k=1}^{n} q^{2k} = \frac {q^2 (1 - q^{2n} )} {1 - q^2},\ \text {for any}\ n \geq 1.$$ Also observe that $q^{2n} \to 0$ as $n \to \infty$ for $0 < q < 1.$

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Show that there exists $M \gt 0$ such that for all $f \in S,$ $\|f\|_{\infty} \leq M \|f\|_2.$
1 votes

First of all note that the graph of $T$ is precisely $$\mathscr G(T) : = \{(f,f)\ :\ f \in S \} \subseteq \left (S,\|\cdot\|_2 \right) \times \left (S, \|\cdot\|_{\infty} \right ).$$ To prove that $\...

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Calculate the integrals for $f(x) = x^2 + 1$
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1 votes

By integration by parts we have \begin{align*} I & = \int_{0}^{1} (x^2 + 1)^7\ dx \\ & = x(x^2 + 1)^7 \ \bigg |_{0}^{1} - 14 \int_{0}^{1} \left [(x^2 + 1)^7 - (x^2 + 1)^6 \right ]\ dx \end{...

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Disprove that $\lim_{x\to 0^+}\frac{1}{x} = c \in R$ using formal finite limit definition
1 votes

Hint $:$ Take $\epsilon = 1.$ Suppose for that $\epsilon$ ($>0$) you have $\delta > 0$ such that $$\left |\frac 1 x - c \right | < 1,\ \ \text {whenever}\ 0 < x < \delta.\ \ \ \ \ \ \ \...

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How to show either $\left \lfloor {\frac{m-1}{2}} \right \rfloor$ or $\left \lfloor {\frac{m+1}{2}} \right \rfloor$ odd and other is even?
1 votes

Hint $:$ Break the entire problem in two cases. Where $m$ is odd and $m$ is even. If $m$ is odd. Then $m$ is of the form $2k+1$ for some $k \in \Bbb Z.$ Then observe that $\left \lfloor {\frac {m-1} ...

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Finding the order of the product of disjoint cycles in $S_n$.
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1 votes

I have started from the stage where I got stuck in proving the above lemma. It is easy to show what I just mentioned in the edit is that $\text {Ord}\ (ab)\ \big |\ \text {lcm}\ \left (\text {Ord}\ (a)...

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$1.$Find the number of elements in $V$
1 votes

Observe that $V \cong {\Bbb Z_3}^4$ and hence $|V| = 3^4 = 81.$

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Every abelian $p$-group is the direct product of cyclic groups.
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1 votes

We need only to take any element $b \in G \setminus \left \langle a \right \rangle$ which has the minimal order amongst all the elements of $G \setminus \left \langle a \right \rangle.$ Then it will ...

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What will be $\text {Ord}_n\ q$?
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1 votes

If ${[p]_n}^m = [1]_n$ then $\text {Ord}_n\ p = \text {Ord}_n\ q = m$ since $\text {Ord}_n\ q = m\ \bigg |\ \text {Ord}_n\ p.$ Now since $q = p^e$ so $\langle q \rangle \subseteq \langle p \rangle.$ ...

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If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?
0 votes

Let $J : X \longrightarrow X^{**}$ be the natural embedding of $X$ into $X^{**}.$ Suppose that weak-topology and weak$^{*}$-topology on $X^*$ coincides i.e. $\sigma (X^*, J(X)) = \sigma (X^*, X^{**}).$...

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check continuity of a function
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This function has a removable discontinuity at $x=2.$ By defining $f(2) = \frac 1 5$ you can make $f$ continuous at $x=2.$ But the discontinuity at $x=-3$ is not removable and in fact $f$ has an ...

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Giving Formula For Recursion given a1 and a2
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Observe that for any $n \in \Bbb N$ $$\begin{pmatrix} a_{2n} \\ a_{2n-1} \end{pmatrix} = {\begin{pmatrix} 7 & 6 \\ 2 & 3 \end{pmatrix}}^{n-1} \begin{pmatrix} a_2 \\ a_1 \end{pmatrix}$$ and ...

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Show that $2^{n-1}$ divides $n!$ whenever $n=2^k$ for some $k \in\mathbb{N}$
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Hint $:$ The power of $2$ in the prime factorization of $n!$ is given by $$\sum\limits_{i = 1}^{k} \left \lfloor \frac {n} {2^i} \right \rfloor$$ where $k = \left \lfloor \frac {\ln n} {\ln 2} \right \...

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Logical problem -- AMC 2013 (Australia)
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Bairnsdale is the King and Darlinghurst is the Treasurer. There are only twenty possibilities which are easy to figure out. All of them strike out except the one I have mentioned in my answer. Try ...

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Finding the difference $\log (2) - \sum\limits_{n=1}^{100} \frac {1} {2^n \cdot n}.$
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0 votes

With the help of Ragib Zaman I finally able to do it. Here it is $:$ We first observe that $$\log (2) = - \log \left ( \frac 1 2 \right ) = \sum\limits_{n=1}^{\infty} \frac {1} {2^n \cdot n}.$$ So $$...

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Proving a theorem regarding extensions of embeddings.
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0 votes

Since $L|K$ is a finite field extension then so is $L|M$ where $M$ is any intermediary field between $K$ and $L;$ so that any $x \in L$ is algebraic over $M.$ Let $\mu_{x,M}$ be the minimal polynomial ...

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Find the recurrence relation solution. $a_n$ = $a_{n-1} + 3n - 5$
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Observe that $$a_n - a_0 = \sum\limits_{k=1}^n (a_k - a _{k-1}) = 3 \sum\limits_{k=1}^{n} k - 5 n.$$

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Question on drawing 4 tickets form 7 tickets
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We can view $4$ drawings to be $4$ different boxes and the number on the ticket drawn at $i$-th drawing is the number of balls in the $i$-th box after the balls are thrown to the boxes where $1 \leq i ...

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Finding discriminant of a monic polynomial.
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What I observed is that the actual problem lies in the definition of discriminant of a monic polynomial. Below is a way to prove the desired proposition by redefining the discriminant of a monic ...

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some confusion about open set?
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Observe that the open sets of $[0,1]$ are precisely of the form $U \cap [0,1]$ for some open set $U$ of $\Bbb R.$ Now take any $a \in (-\infty,0)$ and consider the open interval $I=\left ( a,\frac 1 2 ...

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If $\frac f g$ is symmetric then what conclusion can we make about $f$ and $g$?
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I have found the answer. It's quite easy. Take for instance $f=X_1^2X_2 \in K[X_1,X_2]$ and $g=X_1 \in K[X_1,X_2].$ Neither $f$ nor $g$ is symmetric. But $\frac f g = X_1X_2 \in K [X_1,X_2] \subseteq ...

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How many prime factors will $\Phi_n$ have in its prime factorization?
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0 votes

Let $\pi$ be any prime factor of $\Phi_n$ in $\Bbb F_q[X].$ WLOG we may assume that $\pi$ is monic. Let $\zeta$ be a zero of $\pi$ in $\Bbb F_q^{(n)}.$ Then $\pi$ is the minimal polynomial of $\zeta$ ...

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Prove that $\chi_y = \prod\limits_{\sigma \in G} \left (X-\sigma(y) \right ).$
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0 votes

I have an answer according to my idea. Let me give my argument regarding this. Let $L'$ be the splitting field of $F.$ I will prove that $\chi_y\ \big |\ F$ in $L'[X].$ Let $w \in Gy.$ Then $\exists$ $...

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Can we say that $\mu_x$ is a prime polynomial in $\Bbb K[X]$?
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0 votes

I got an example. $\Bbb R^2$ is not an integral domain but it is an algebra over $\Bbb R.$ Take $x=(1,0) \in \Bbb R^2.$ Then $\mu_x = X(X-1) \in \Bbb R[X]$ which is not irreducible in $\Bbb R[X].$

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Why is the map $\varphi$ closed?
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0 votes

There is a more elegant way to think that problem. First we note that $S/J$ is integral over $R/I.$ So the map $\psi : \text {Spec} (S/J) \longrightarrow \text {Spec} (R/I)$ defined by $$\psi (\...

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If two continuous functions are equal almost everywhere on $[a,b]$, then they are equal everywhere on $[a,b]$
-1 votes

If $f = g$ almost everywhere on $[a,b]$ then $|f - g| = 0$ almost everywhere on $[a,b].$ Let $\mu$ denote the Lebesgue measure on $[a,b].$ Then $$\int_{a}^{b} |f-g|\ dx = \int_{[a,b]} |f-g|\ d\mu = 0.$...

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If $f(x)=\lim_{n\to \infty }\left(1+\frac{x}{n}\right)^n$ prove that $f$ is continuous.
-1 votes

Hint $:$ Observe that $f(x) = \lim\limits_{n \rightarrow \infty} \left (1 + \frac x n \right )^n = e^x,$ for all $x \in \Bbb R.$

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