Nevzat Eren Akkaya
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Finding roots of polynomial $f(x)$ in $\mathbb{Z}_p = \mathbb{Z}/p\mathbb{Z}$, where $p$ is prime.
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3 votes

$x^p \equiv x \bmod p \Rightarrow x^{2p^2}=(x^p)^{2p} \equiv x^{2p} \equiv x^2 \bmod{p}$ $x^p \equiv x \bmod p \Rightarrow x^{p^2} \equiv x \bmod p$ $x^{2p^2}- \overline{2}x^{p^2} - x^p + \overline{2} ...

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Is $\begin{bmatrix} a &b \\0 &1 \end{bmatrix}$ a cyclic group?
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3 votes

$\begin{pmatrix} a &b \\0 &1 \end{pmatrix}\begin{pmatrix} c &d \\0 &1 \end{pmatrix}=\begin{pmatrix} ac &ad+b \\0 &1 \end{pmatrix}$ $\Rightarrow$ $\begin{pmatrix} a &...

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Can the values of the expressions $\frac{1}{\sqrt{2a+1}},\frac{1}{\sqrt{2a-1}},3(a^2-19)\sqrt{2a-1},18\sqrt{2a+1}$ be in G.P.?
3 votes

$\frac{\dfrac{1}{\sqrt{2a-1}}}{\dfrac{1}{\sqrt{2a+1}}}=\frac{18\sqrt{2a+1}}{3(a^2-19)\sqrt{2a-1}}\Rightarrow \frac{6}{a^2-19}=1$ $\frac{6}{a^2-19}=1 \Rightarrow a^2=25$ which means $a=5$ or $a=-5$ but ...

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Changing of Basis Questions
3 votes

Let $B=\{1,x,x^2\}$ then $\forall p \in P_2(R)$ $\exists c_0,c_1,c_2$: $p=c_0+c_1x+c_2x^2$ $T(p)=c_0T(1)+c_1T(x)+c_2T(x^2)$ $\Rightarrow$ $[T(p)]_B=A\begin{pmatrix} c_0 \\ c_1 \\ c_2 \end{pmatrix}$ $...

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How can we apply this simple eigenvector expression 'repeatedly'?
2 votes

Assume that $ABu=(\alpha+2)Bu$ $[\pi(H),\pi(X)] = 2\pi(X) \Rightarrow AB-BA=2B$ $AB-BA=2B \Rightarrow AB^2-BAB=2B^2$ $AB^2-BAB=2B^2 \Rightarrow AB^2u-BABu=-(\alpha+2)B^2u+AB^2u=2B^2u$ $-(\alpha+2)B^2u+...

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Using angle between vectors to find identity of vector new rotated sixty degrees
2 votes

Your method actually could work but there is an easier approach involving the Rotation matrix which is defined as $R_\theta =\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta ...

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Proof $\sum_{k=1}^{2n} (-1)^{1+k}\frac{1}{k} = \sum_{k=1}^{n}\frac{1}{n+k}$ by induction.
2 votes

Note that $$\sum_{k=1}^{n+1}\frac{1}{(n+1)+k}= \sum_{k=1}^{n}\frac{1}{n+k}-\frac1{n+1}+\frac{1}{2n+1}+\frac1{2n+2}$$ $$\begin{align} -\frac1{n+1}+\frac{1}{2n+1}+\frac1{2n+2}&=\frac{1}{2n+1}-\frac{...

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Using modular arithmetic to find the remainder of $12^{157}$ when divided by $10$
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2 votes

You could use Fermat`s Little Theorem and Chinese Remainder Theorem together. Note that $12^{157} \equiv 0 \pmod{2} \Rightarrow 12^{157}=2k$ $12^4 \equiv 1 \pmod{5} \Rightarrow 12^{157} \equiv 2 \pmod{...

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Does $\sum_{n=1}^{\infty}\left(\frac{\sin x}{x}\right)^n$ converges?
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2 votes

Note that $-1<\frac{\sin x}{x}<1$ let $u=\frac{\sin x}{x}$ then, $$\begin{align}\sum_{n=1}^{\infty}\left(\frac{\sin x}{x}\right)^n&=\sum_{n=1}^{\infty}(u)^n\\ &=\lim \limits_{n \to \...

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Prove the orthogonality relation of Chebyshev polynomials of the first kind
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2 votes

$n=m=0 \Rightarrow \int T_{n}(x)\,T_{m}(x)\,{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}=\int \frac{dx}{\sqrt{1-x^2}}=arcsin x$ $\int \frac{dx}{\sqrt{1-x^2}}=arcsin x \Rightarrow \int_{-1}^{1} T_{n}(x)\,...

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Can we solve $u(x)=-1+x+\frac{x^2}{2}+2e^x-\int_0^xu(t)dt$ with using noise terms phenomenon?
2 votes

$u(x)=-1+x+\frac{x^2}{2}+2e^x-\int_0^xu(t)dt \Rightarrow \frac{du}{dx}=2x+2e^x+1-u$ $\frac{du}{dx}=2x+2e^x+1-u \Rightarrow e^x\frac{du}{dx}+e^xu=2e^xx+2e^{2x}+e^x$ $e^x\frac{du}{dx}+e^xu=2e^xx+2e^{2x}+...

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Find inverse such that $A = A^{-1}$
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2 votes

$A^{-1}$ exists $\Rightarrow$ $Det(A)=-bc \neq 0$ which means both $b$ and $c$ can not be $0$ $A^{-1} = A \Rightarrow A^2=I$ $A = \begin{pmatrix} a & b \\ c & 0 \end{pmatrix} \Rightarrow A^2=\...

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Show that for each $m\in \mathbb{N}$, there exists $n\in \mathbb{N}$ such that $x^2\equiv 1 \pmod n$ has at least $m$ solutions?
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2 votes

let $p$ be a prime number $x^2\equiv 1 \pmod p$ $\Rightarrow$ $x^2-1\equiv (x-1)(x+1)\equiv 0 \pmod p$ $(x-1)(x+1)\equiv 0 \pmod p$ $\Rightarrow$ $x \equiv 1$ or $x \equiv -1$ Let $n=p_1p_2...p_n$ and ...

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Interpretation of eigenvalues and associated eigenvalues
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2 votes

$Bu_3=u_4$ and $Bu_4=u_3$ $\Rightarrow$ $Bu_3+Bu_4=B(u_3+u_4)=u_4+u_3$ hence 1 is an eigenvalue for $u_3+u_4$ $Bu_3=u_4$ $\Rightarrow$ $-Bu_3=B(-u_3)=-u_4$ $B(-u_3)=-u_4$ and $Bu_4=u_3$ $\Rightarrow$ ...

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Proof of Idempotency for Matrices
2 votes

$(I-Y)^2=I-Y$ $\Rightarrow$ $Y^2-2Y=-Y$ $\Rightarrow$ $Y^2=Y$ hence it is idempotent

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Show G=g with constraints
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2 votes

We know that $\frac{\partial G}{\partial x}=\frac{\partial g}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial g}{\partial v}\frac{\partial v}{\partial x}$ $\frac{\partial G}{\partial y}=\frac{\...

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Why are three vectors linearly dependent when one of them is a combination of the other two?
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2 votes

It is trivial actually.Lets say $v_1$ ,$v_2$ and $v_3$ are vectors . If $v_1=v_2 + v_3$, then $v_2 + v_3 - v_1=0$, so there exists a solution for $c_1v_1+c_2v_2+c_3v_3=0$ which is different than $c_1=...

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limit of a $\ln(\cos(x))/x$
1 votes

by the definition of the derivative, we know that $f^`(a)=\lim_{x \to 0}\frac{f(x+a)-f(a)}{x}$ it can be observed that $f(0)=\ln(\cos(0))=0$ so according to definition $f^`(0)=\lim_{x \to 0}\frac{\ln(\...

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Does this property of the anti-symmetric matrices hold in general?
1 votes

Yes, since $A^T=-A$ for all $v\in V$ $\langle Av,v \rangle=0$ Proof: Note that $\langle u,v\rangle=u^Tv$ $A^T=-A \Rightarrow \langle v,AV\rangle= v^TAv=-v^TA^Tv$ $\langle -Av,v\rangle=-v^TA^Tv \...

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How to prove or disprove if $a_n \leq M$, for some number $M$, then $L \leq M$?
1 votes

Hint:Assume that $L>M$ and let $L-M=\epsilon$ than use $\epsilon-\delta$ definition of limit for given $\epsilon$

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Related to Hom, Images (im) and Kernels (ker)
1 votes

Let $ v \in U$ $ v \in U \Rightarrow f(v) \in im(f)$ $ im(f) ⊂ ker(g) \Rightarrow f(v) \in ker(g)$ $f(v) \in ker(g) \Rightarrow g(f(v))=0$ $g(f(v))=0 \Rightarrow g ◦ f = 0 $

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Unable to solve this differentiation
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1 votes

$$\frac{d(-R\omega \sin(\theta))}{dt}=-R\frac{d(\omega sin(\theta))}{dt}$$ according to product rule and chain rule, $$\frac{d(\omega \sin(\theta))}{dt}=\frac{d\omega}{dt}\sin(\theta)+\omega (\frac{d\...

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Is is true that $ \exists~ \lim(a_n+b_n)~\text{and}~\exists~\lim a_n \Rightarrow \exists ~\lim b_n. $
1 votes

let $lim_{n \to \infty }b_n=b$ and $lim_{n \to \infty}(a_n+b_n)=u$ $lim_{n \to \infty }b_n=b \Rightarrow lim_{n \to \infty }-b_n=-b$ $ lim_{n \to \infty }-b_n=-b$ and $lim_{n \to \infty}(a_n+b_n)=u$ $\...

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if a curve is asked and you are asked to find a specific value in that curve
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1 votes

$y=ux$ and $y(1 + xy) dx = x dy \Rightarrow uxdx+ux^3dx=uxdx+x^2du$ $uxdx+u^2x^3dx=uxdx+x^2du \Rightarrow u^2x^3dx=x^2du$ $u^2x^3dx=x^2du \Rightarrow x dx=\frac{du}{u^2}$ $x dx=\frac{du}{u^2} \...

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How do I solve a limit when direct substitution gives division by zero?
1 votes

Limit does not exist in this case $\lim_{x\to 1^+}\frac{x^2+x+1}{x-1}=\lim_{x\to 1^+}(x+2+\frac{3}{x-1})=\infty$ $\lim_{x\to 1^-}\frac{x^2+x+1}{x-1}=\lim_{x\to 1^-}(x+2+\frac{3}{x-1})=-\infty$

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Cauchy sequence of positive terms, find it's limit.
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1 votes

$|x_{n+2}-x_{n+1}| \leq 0.2|x_{n+1}-x_n| \Rightarrow |x_{n+k}-x_{n}|\leq c^k |x_2-x_1|=c^k$ $s=\frac{1}{|c|}-1 \Rightarrow |c|=\frac{1}{1+s}$ and $Ns \geq \epsilon$ $|c|=\frac{1}{1+s} \Rightarrow |c|^...

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Geometric Series: Value and Common Ratio
1 votes

$x_1=c , x_2=c+4, x_3=c+6 \Rightarrow c+4=cr$ and $c+6=cr^2$ $\frac{c+4}{c}=\frac{c+6}{c+4} \Rightarrow 2c+16=0$ $c=-8$ hence $r=\frac12$

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Calculate the line integral where $F(x,y,z) = (\sin(x), \cos(y), xz)$, $r(t) = (t^3, -t^2, t)$ and $t\in[0,1]$
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1 votes

$F(r(t))=(sin(t^3),cos(-t^2),t^4)$ , $r`(t)=(3t^2,-2t,1)dt \Rightarrow\int F \cdot dr=\int_0^1(3t^2sin(t^3)-2tcos(-t^2)+t^4)dt$ $\int_0^1(3t^2sin(t^3)-2tcos(-t^2)+t^4)dt=sin(-1)-cos(1)+2$

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Consider the quadratic integer ring $A=\Bbb Z[\sqrt-2]$. Show that $U(A)=\{-1,1\}$.
1 votes

$a^2+2b^2=1 \Rightarrow 2b^2=1-a^2=(1-a)(1+a)$ $2b^2 \geq 0 \Rightarrow -1 \leq a \leq 1$ but we know the fact that $a$ is an integer so $ -1 \leq a \leq 1 $ and $a \in \Bbb Z \Rightarrow $ $a=-1$ or ...

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Differentiating multiple products/quotients
1 votes

$(ln(u(x))^`=\frac{u’(x)}{u(x)} $ $y(x) = \frac{u’(x)}{u(x)r(x)}=\frac{d\ln u}{dx} \frac1{r(x)} \Rightarrow \frac{dy}{dx}=\frac{d^2(\ln u)}{dx^2}\frac1{r(x)}+\frac{d\ln u}{dx}\frac{-r^`(x)}{r(x)^2}$

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