All Questions

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Is a Lie algebra a complex or a real vector space?

I am trying to learn Lie theory and for this purpose I worked out the Lie algebras of some matrix groups. The examples I worked happened to be complex matrix groups and it lead me to wonder whether, ...
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Evaluate $\lim_{x\to\infty} ((x^5+x^4)^{1/6}-(x^5-x^4)^{1/6})$

I've been struggling with the following: $$\lim_{x\to\infty} ((x^6+x^5)^{1/6}-(x^6-x^5)^{1/6})$$ Tried factoring out $x^{5/6}$ and then using L'hopital- which got me nowhere, tried multiplying by the ...
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Total Probability theorem and Bayes theorem

Two reinforced concrete buildings A and B are located in a seismic region. It is estimated that an impending earthquake in the region might be strong (S), moderate (M), or weak (W) with probabilities. ...
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How to prove that angular velocity is not a derivative of angular displacement?

The angular velocity $\omega$ of a two-dimensional solid body is given by $$\omega = \hat{z} \cdot \frac{\vec{r} \times \vec{v}}{r^2},$$ where $\vec{r}$ and $\vec{v}$ are the position and the velocity ...
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Help on choosing a research topic

I have qualified for a research institute but the problem is I have been asked to give a writeup on a research topic on which I want to work in future.I have never been involved in an active ...
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Find the numbers by XoR

I have 6 numbers M1, M2 and M3 and E1, E2 and E3 such that M1 xor M2 = E1 xor E2 M2 xor M3 = E2 xor E3 ...
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System of exponential equations

If $x,y,z \in \mathbb{R}$ and $$\begin{cases} 2^x+3^y=5^z \\ 2^y+3^z=5^x \\ 2^z+3^x=5^y \end{cases}$$ does it imply that $x=y=z=1$?
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Prove that the circumcenter of $\triangle PIQ$ is on the hypotenuse $AC$.

In a right $\triangle ABC$ right angled at $B,BD$ is the perpendicular on $AC.P,Q,I$ are the incenters of $\triangle s ABD,CBD$ and $ABC$ respectively.Prove that the circumcenter of $\triangle PIQ$ is ...
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How to find $\int_0^{1/4}\frac{1}{x\sqrt{1-4x}}\ln\left({\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}}}\right)dx$

Let $H_n$ be the harmonic series. I want to find the value of $A=\displaystyle\sum_{n=0}^\infty \binom{2n}{n}\left(\frac{1}{4}\right)^n\frac{H_n}{n}$. From this paper : ...
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Given $f (1)= 1$, $f '(1)=1$, $f ''(1)=1$, find $f (1/2)$

Given $f (1)= 1$, $f '(1)=1$, $f ''(1)=1$, find $f (1/2)$. That's it this weird question expects us to solve. Should I assume some function here?
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How to efficiently solve an inequality where the variable occurs in the denominator?

For example, this inequality: $$\frac{2x(x-4)}{x-1} \le 7$$ I can solve this by finding the 'critical values' (or in other words, two values which $x$ can/or cannot equal), then putting them on a ...
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Find all random variables whose distribution satisfies an equation

The problem I have to solve is formulated as follows: Find all random variables such that if $Y$ has the distribution $N(0,1)$ and $X, Y$ are independent then $X+Y$ has the same distribution as ...
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Half life, exponential decaying equation question

If a radioactive substance has a half-life of $10$ days, in how many days will $1/8$ of the initial amount be present? Assume the decaying process is continuous (exponential). Will the answer just be ...
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Can a real function $f$ on $[0,1]\cap\mathbb{Q}$ be differentiable?

can a real function $f$ on $[0,1]\cap\mathbb{Q}$ be differentiable? if so, and if the derivative of $f$ is zero, then is $f$ is a constant function?
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$V$ is irreducible exactly then when $I(V )$ is a prime ideal

If $V$ is an algebraic set of $K^n$, show that $V$ is irreducible exactly then when $I(V )$ is a prime ideal of $K[X_1, X_2, \dots, X_n]$ . Let $V$ be irreducible. We suppose that $I(V)$ is not a ...
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How to solve this task about circles and lines intercepting each other?

We have drawn some lines and circles on a paper. Every two has an interception, but none three goes through the same point. How many lines and circles have we drawn if we have 75 interceptions?
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Homotopy category of chain complexes as a localzation

For an Abelian category $\mathcal{A}$, define the homotopy category of chain complexes $\mathcal{K}(\mathcal{A})=\mathcal{C}(\mathcal{A})/\mathcal{I},$ where $\mathcal{C}(\mathcal{A})$ denotes the ...
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$|A[d]|=d^r \implies A\cong\left(\mathbb{Z}/n\mathbb{Z}\right)^r$

Let $A$ be a finite abelian group of order $n^r.$ Suppose that for every $d|n$ we have $| A[d]|=d^r,$ where $A[d]$ denotes the subgroup consisting of all elements of order $d.$ I want prove that : ...
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Proving that if $8\mid (n^2+2n)$ then $2\mid n$

Let $n\in \mathbb N$ prove that if $8\mid (n^2+2n)$ then $2\mid n$. From the given, there exsits $k\in \mathbb N$ such that $8k= (n^2+2n)$, take $k=1$, and we get $2\cdot 4 = n(n+2)$. Now my ...
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Are these two elliptic integral evaluations identical?

I'm reading a paper on the Schwarz D minimal surface, and I'm wondering whether the authors have made a mistake. They evaluate the integral $$\int_0^z \frac{2t\;\mathrm{d}t}{\sqrt{t^8-14t^2+1}},$$ ...
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Doubt on presumably divergent series with primes

I am wondering if my reasoning is correct. I want to determine if the following series converges or not: $$\sum_{n=1}^\infty\frac{1}{(\ln p_n)^2}$$ where $p_n$ is the ...
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Why is abelian group structure needed in the definition of a ring? [duplicate]

Why do we require $(R, +)$ to be an abelian group for $(R, +, \cdot)$ to be a ring? Why don't we study $(S, +, \cdot)$ where $(S, +)$ is a group and $(S, \cdot)$ is a semigroup and distributivity ...
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Solving the Sequence of this question on Putnam Exam

Problem: Solution: Solution for 2003 A1 Putnam $ka_1 = a_1 + a_1 ... a_1 \le n \le a_1 + (a_1 + 1) + (a_1 + 1) ... (a_1 + 1)$ $= ka_1 + k - 1$ I know these then: What should I do next? Without ...
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Help demonstrate how to arrive at the implication of some given inequalities and equations

Given: $0<x<y<1$ $z=x+y$ $x=u$, $y=z-u$ $0<u<z-u<1$ I need to show that this implies: $0<u<\frac{z}{2}$, if $0<z<1$, and $z-1<u<\frac{z}{2}$, if $1<z<2$ ...
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How to find the number of permutations with offset restriction

First question. Okay I have this problem that I've been trying to figure out for a while. I'm writing a computer program I need to quickly calculate the permutations of a set with 'n' elements with a ...
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Find $\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$

Find $$\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$$ with $n \in \mathbb{N}$ My tried: I think that, needing to find the value of $$I_1=\int_{0}^1 \frac{dx}{(1+x^n)\sqrt[n]{1+x^n}}$$ because: ...
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how to find $a_{50}$ from a recursive term

Given $a_{n+1}=a_n+2n+3,a_1=3$ How can I find $a_{50}$? I can compute $a_2,a_3,...,a_{50}$ But it's a long way. Is there any smart technique to compute? Thanks.
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How can these three equations be the equivalent?

I am trying to recreate these equations, but I can't find the operations I need to make to get to the equivalent solution below. Can you help?
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How can I see that the space curve given by $\gamma(s) = (\frac 4 5 \cos s, 1 - \sin s, - \frac 3 5 \cos s)$ forms a circle in space ?

Let $\gamma: \mathbb R \rightarrow \mathbb R^3$ be a space curve given by $\gamma(s) = (\frac 4 5 \cos s, 1 - \sin s, - \frac 3 5 \cos s)$. How do I see that $\gamma$ has image in $\mathbb R^3$ that ...
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Parallel transport

Suppose I want to parallel transport a vector $v$ living in the tangent space of my surface at a point $p$ along a closed geodesic polygonal path. On each regular component I keep the angle between ...
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Triple XoR - Find relation between the numbers.

I have a = b^c; b = a^c; Is it possible to eliminate c and find a relation between a and b? I have 3 different ...
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Is it possible that a finitely generated ring has an ideal that is not finitely generated

Sorry if this is duplicated. I couldn't find an exact answer of my question. One definition of Noetherian ring is: A ring $R$ is Noetherian if all its ideals are finitely generated. I know there are ...
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Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
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Showing that something is the zero matrix

I'm trying to prove that if the linear system of equations $Ax=b$, where $A$ is a matrix with $m$ rows and $n$ cloumns, only has a solution when $b=0$ then $A$ is the zero-matrix. My idea goes like ...
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$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals (For reference the $\cos$ Fresnel integral is $\int^\infty_0 \cos(x^2)\, dx = \frac{\sqrt{2 \pi}}{4}$) I've tried ...
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Understanding directional derivatives

I am confusing myself when it comes to directional derivatives and gradients. The gradient of a function shows the direction of the greatest change. So when we chose a unit vector as the direction to ...
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Product of symmetric matrices

Let $A \in \mathbb{R}^{n \times n}$ be symmetric. I am trying to understand under which conditions on $B \in \mathbb{R}^{n \times n}$ the product $AB$ is also symmetric. It is clear that if $B$ is ...
I have seen two definitions for the norm in the Clifford algebra $\mathcal{Cl}_{p, q, r}$ According to one of them $||x|| = <x. x^\dagger>_0$, where the dagger stands for the reversal of the ...
How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$ , where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix.
How to prove $A( v \times w)$ = $\det A\cdot(Av \times Aw)$, where $A\in \text {Mat}_{3 \times 3}(\mathbb R)$ is an orthogonal matrix ($\det A = \pm 1$) and $v,w\in \mathbb R^3$. I've tried writing ...