David
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"Advice to young mathematicians"
15 votes

$ \clubsuit$ Try to view every thing to one dimension or two dimension where we can see geometrically.. $\quad$ It is quite often helps to realize the things and to generalize it. $\clubsuit$ Aim ...

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Find $\lim\frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\ldots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}\right)$
8 votes

From the Hint given by @cameron Williams, I did the following. $$\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\left(\frac{\sqrt{1}-\sqrt{3}}{-2}+\frac{\sqrt{3}-\sqrt{5}}{-2}+\ldots+\frac{\sqrt{2n-1}-\...

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Frobenius Inequality Rank
5 votes

If you know the Sylvester inequality then it's a two lines proof. Sylvester Inequality: Consider $A_{m\times r}$ and $B_{r\times n}$. Then $$r(AB)\geq r(A)+r(B)-r \tag{1}$$ where $r(\cdot)$ ...

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Permutation inverse form
Accepted answer
4 votes

(i) is correct. And for the second one you just see from the co-domain to domain.. that is, \begin{align} (P_1\circ P_2)^{-1}&=\begin{pmatrix}1&2&3&4&5&6\\ 2&4&5&...

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quadratic equation what am I doing wrong?
3 votes

\begin{align} \sqrt{5x+19}&=\sqrt{x+7}+2\sqrt{x-5}\\ 5x+19&=x+7+4(x-5)+4\sqrt{(x+7)(x-5)}\\ 32&=4\sqrt{(x+7)(x-5)}\\ 8&=\sqrt{(x+7)(x-5)}\\ 64&=x^2+2x-35\\ 0&=x^2+2x-99 \end{...

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An example of a problem which is difficult but is made easier when a diagram is drawn
3 votes

The diagram deals with the upper sum of Riemann integral.. The criteria for Riemann integrablity, $$U(P,f)-L(P,f)<\epsilon$$ is hard to realise. But with the diagram its highly easy.

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How many Arrangement of "AMAZED" letter E Positioned between two A's (Not necessarily Flanked)
3 votes

\begin{array}{c|cccccc|c} Case:1\to &A&\underline{4\;ways}&\underline{3\;ways}&\underline{2\;way}&\underline{1\;way}& A&=24 ways\\~\\ Case:2\to &A&\underline{3\;...

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let $(a_n)$ be a sequence of real numbers such that $|a_{n+1}-a_n|\leq \frac {n^2}{2^n}$ for all $n\in \mathbb N$. Then
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3 votes

\begin{align} |a_{n}-a_m|&\leq |a_n-a_{n+1}|+|a_{n+1}-a_{n+2}|+\ldots+|a_{m-1}-a_m|\\ &\leq \frac{n^2}{2^n}+\frac{(n+1)^2}{2^{n+1}}+\ldots+\frac{m^2}{2^m}\\ &=\frac{n^2}{2^n}\left\{1+\frac{...

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Diagonalization and find matrix that corresponds to the given condition
3 votes

The eigen values are $1,3 $ clearly. So it is diagonalizabe(distinct eigen values). And so, there exits $P$ such that $$A=P\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right)P^{-1}.$$ Now ...

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Prove that the group $\mathbb{Z}^{n}$ is generated by at least $n$ elements
3 votes

Suppose every element $z\in\mathbb Z$ can be written as $$z=\sum_{i=1}^k \alpha_i z_i, k<n$$ Every element of $\mathbb Q^n$ also can be writen as $$ \frac p q=\sum_{i=1}^k\alpha _i \frac {p_i}{...

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Prove that a set is open
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2 votes

Remember One can do in this way also, I am going to use these two aspects. $1.$ If $f$ is contiuous and $A$ is open, then $f^{-1} (A)$ is also open. $2.$ Finite intersection of open sets is again ...

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Set of Two Linear Equations
2 votes

Hint: The equation you have quoted is pair of straight lines. The blue and green colour represents the required pair of straight lines and the intersection points have also been quoted...

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Integral inequality for a continuous and decreasing function on an interval
2 votes

\begin{align} \int_0^1 f(x)dx&=\int_0^\frac 1 2 f(x) dx +\int_\frac 1 2 ^ 1 f(x) dx\\ &\geq \int_0^\frac 1 2 f(x) dx+\int_0^\frac 1 2 f(x) dx \\ & = 2\int_0^\frac 1 2 f(x) dx \end{align}...

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Prove that for all $y,z\in\mathbb{R}^+$ it is true that $(y + z) (1 - 8 y z + z^2 + y^2 (1 + 9 z^2))\ge0$
2 votes

Hint: Use completing the square for $-8yz+z^2+y^2$ which gives $$-8yz+z^2+y^2=y^2+z^2-2yz-6yz=(y-z)^2-6yz$$ Now, \begin{align} 1-8yz+z^2+y^2+9y^2z^2&= 1+(y-z)^2-6yz+9y^2z^2\\ &=(y-z)^2+(...

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A question about surjective functions.
Accepted answer
2 votes

\begin{align} f^{-1}([0,1])&=\{x\in X: f(x)\in [0,1]\} \\ &=\{x\in X: x^2\in [0,1]\}\\ %&=\{x\in A: x=\pm\sqrt y\}\\ &=[-1,\,1] \end{align} So, it is not one-one and so you can not ...

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For every real number $a$ and for every $\epsilon > 0 $ there are infinitely many rationals between $a$ and $a$ + $\epsilon $
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2 votes

there is another way of proving this... Suppose there are only finitely many rationals in between $a$ and $a+\epsilon$, say $x_1,x_2\ldots,x_n$ such that $$a<x_1<x_2<\ldots<x_n<a+\...

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$How to determine if this series is convergent?
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2 votes

all the terms are positive, So, $a_n=\frac 1 {\sqrt n}=\frac 1 p$ where $p^2=n$. So, it contains a sub-series $\sum \frac 1 n$ which is diverges... so the original series must be diverge.

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What does it mean for a function to be holomorphic?
1 votes

Holomorphic is just talking differentiability in the complex plane. For example, in Real analysis, we look at the left and right limits only. whereas here we are looking in all possible directions. ...

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Requesting abstract algebra book recommendations
1 votes

Topics in Algebra by I.N. Herstein is good book to read. My professors also suggesting the same.

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How to show that following relation is true for matrix multiplication?
Accepted answer
1 votes

Let us first calculate $A^H b$ where $H$ is the hermitian operator. That is, $$ (A^H)_{ij} = A_{ji}^*$$ where $*$ represents the complex conjugate. \begin{align} A^H b b^H A & = (A^H b) (b^H A) \...

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If 𝑓 is measurable and $𝑎\leq 𝑓(𝑥)\leq 𝑏$ for $𝑥\in𝐸$, and if $\mu(E)<\infty$, then $a\mu(E)\leq \int_E fd\mu \leq b \mu(E)$.
1 votes

Using the defn: Let $0\leq s\leq f$ where $s$ is any simple function. $$\int f d\mu= sup \int s d\mu = sup \sum c_i\mu(E_i)\leq b sup \sum \mu (E_i)=b\mu(E)$$ For the lower bound, consider the ...

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Why is $x^{-1} = \frac{1}{x}$?
1 votes

Its not true always... infact. Its entirely depends on which set you are working. One more notable thing is $x^{-1}$ is just a notation to represent a inverse of an element $x$(mostly for ...

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How would you find the roots of the following equation?
Accepted answer
1 votes

$$x^{13}+1=0$$ $$x^{13}=-1$$ $$x=(-1)^{\frac 1{13}}$$ $$ x= (\cos \pi+i\sin \pi )^{\frac 1{13}}$$ Then by De Moivre's formula, add $2k\pi$ with the arguments, and then you can use as $$ x= (\cos (...

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Space span of $f=\cos^2(x)$ and $g=\sin^2(x)$?
1 votes

Clearly $(1)$ and $(3)$ are in the linear span of $f$ and $g$, since $\cos 2x=\cos^2 x-\sin^2 x$ $\cos^2x+\sin^2x=1$ For the other options, Checking for $\sin x$, if $a\cos^2 x+b\sin^2 x=\sin x$ ...

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homogenous linear equation
1 votes

Yes..its homogenious.. $$ y' =\frac{y^2 + x \sqrt{4x^2 + y^2}}{xy} $$ For this, their is well known method, running by substituting, $$y=vx\implies y'=v+xv'$$ So, it will became as, \begin{align} ...

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Show that $\int_0^{\pi/2}\frac{|ab|dx}{a^{2}\cos^{2}x + b^{2}\sin^{2}x} = \frac{\pi}{2}$
1 votes

\begin{align}\int_0^{\pi/2}\frac{|ab|dx}{a^{2}\cos^{2}x + b^{2}\sin^{2}x}&=|ab|\int_0^{\pi/2}\frac{\sec^2x}{a^2+b^2 \tan^2 x}dx\\ & =|ab|\int_0^{\pi/2}\frac 1 {a^2+b^2\tan^2x}d(\tan x) \\ &...

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If for an entire function $f(1)=i$ then find the value of $f(i)$
1 votes

$f$ is a linear polynomial as quoted by you. So, $$f(z)=a_0+a_1z$$ and $|f(z)|\leq K|z|\implies |f(0)|\leq K|0|\implies f(0)=0$ Hence, $a_0=0$ and $f(1)=i\implies a_1=i$ therefore, $f(i)=a_0+a_1....

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Partial fractions expansion problem $\frac{x^3-1}{4x^3-x}$
Accepted answer
1 votes

Divide the numerator by denominator first.. $$x^3-1=\frac 1 4 (4x^3-x)-\frac 1 4 (x-4)\tag{division algorithm}$$ $$\frac{x^3-1}{4x^3-x}=\frac 1 4 \frac{4x^3-x}{4x^3-x}-\frac 1 4\frac{x-4}{4x^3-x} $$ ...

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$f:X\rightarrow Y$ be a continuous bijection.$X$ and $Y$ are Banach spaces and $f$ is linear
1 votes

Observe that $f:X\to Y$ is bounded linear map and bijective. Claim: $f^{-1}:Y\to X$ also continuous... let $G$ open in $X$ then $$f^{-1}(G)=\{y\in Y: f^{-1}(x)=y,x\in G\}$$ $$=\{y\in Y: f(x)=y,x\in ...

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Derivative of $x^{x^{\cdot^{\cdot}}}$?
1 votes

its given that $y=f(x)=x^{x^{x^{x^\cdots}}}$ I can write this function as, $$y=x^{y}$$ Taking $log_e$ on both sides, we have, $$\log y=y\log x$$ Now, differentiate w.r.to $x$ on both sides, $$\...

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