# All Questions

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### Having objects trajctories and directions how to find where objects traverse same path?

I have N objects that travel on some trajectories (unique for each object). At each agent curve point we can get object speed (direction). Having some distance ...
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### Find triangel-area with Cavalieri's principle

A triangle is given by $A=(0,0), B=(5,1)$ and $C=(2,4)$. I already know $\lambda^2(\Delta ABC)=9$. Now I want to compute the area by using Cavalieri's principle. I know how to start when I have to ...
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### Not able to solve $({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$

I'm not able to solve $$({\frac{1}{2}})^p + ({\frac{1}{3}})^p + ({\frac{1}{7}})^p - 1 = 0.$$ If you put values of $p$ (like $\frac{1}{2}$ or 2) back in the equation it doesn't satisfy! So please ...
116 views

### Proving something is not differentiable

I am looking for confirmation so that I can be sure I understand what is being asked here. I need to show that the following function $f(x,y)$ is not differentiable at $(0,0)$ but that ...
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### Conics in Complex Projective Spaces

I was reading classification of complex hyperquadrics, I am stuck in $\mathbb CP^2$ what is $X_0^2+X_1^2=0$ in $\mathbb CP^2$, ok in $\mathbb CP^1$ this represents just two points, my attempt if $X_1$ ...
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### At what times, t, does an airplane (vector defined) intersect with a radar beam (2D plane defined)?

A radar beam can be defined as a plane in $3D$ space. If the beam is moving such that the basis is $[1\text{ } 0 \text{ } \cos(wt)]^T$ and $[1 \text{ } \sin(wt) \text{ } 0]^T$ where $t=time$. ...
147 views

### Proving that a set is measurable and has a zero area

I want to prove that each of the following sets is measurable and has zero area. a) A set consisting of a single point. b) A set consisting of a finite number of points in a plane. c) The union of a ...
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### Exercise 24.13 of T. Jech's *Set Theory*

Having struggled my way through most of chapter 24 of Jech's Set Theory, I'm stuck on the very last part of the very last question, 24.13: Let $I=I_{NS}$ be the nonstationary ideal on $\omega_1$, ...
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### Finding another way of doing this integral $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$

Problem : Integrate : $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$ I have the solution : We can substitute $\sqrt{x}= \cos^2t$ and proceeding further, I got the the answer which is ...
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### Orthogonality and linear independence

[Theorem] Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent. Also, orthogonal set and linearly ...
321 views

### Changing Summation Index Question

I'm sorry if this seems like a very novice question, but I am still relatively new to the world of discrete math ( still in 9th grade). I've been reviewing some of the concepts I learned in a ...
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### Is This A Derivative?

I am in a little over my head. This all began with my reading how each level of pascals triangle adds to $2^n$, where n=row# starting with n=0. I then though, "wouldn't it be clever if the rows added ...
170 views

### Derivative of Trace of Matrix wrt parameters

I have the following function which I need to find the derivative of $$L=trace(\Sigma K^{-1})$$ where $K$ is a function of $\theta$ and $\Sigma$ is constant. If I'm correct what I need to do to find ...
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### $\sup_{x\in A}x \sup_{y\in B}y=\sup_{x\in A,y\in B}xy$

Let $A$ and $B$ be two sets of nonnegtive numbers. Prove that $\sup_{x\in A}x \sup_{y\in B}y=\sup_{x\in A,y\in B}xy$. Thanks for your help.
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### number theory: Let $m>n$ for $m,n\in\mathbb{Z}$, prove if $k$ divides $m$ and $k$ divides $n$ then $k$ divides $m\bmod{n}$

Let $m>n$ for $m,n\in\mathbb{Z}$, prove if $k$ divides $m$ and $k$ divides $n$ then $k$ divides $m\bmod{n}$. How should I approach this question? I only got $m=qk$ and $n=pk$ if ...
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### How to compute $\mathbb{P}(\lambda X>4)$ directly?

Given a random variable $X$ which is exponentially distributed i.e. $X\sim E(\lambda)$. Calculate $\mathbb{P}(X-\frac{1}{\lambda}>\frac{3}{\lambda})$. My working: ...
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### for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$

How to show that for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$ I've tried triangle inequality couldn't arrive at any conclusion. Please help me.
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### Minimize the distance in the Euclidean space

The objective is to minimise the distance $d_{0}+d_{1}$. The points $c_{0}$ and $c_{1}$ are given. I need to locate the point $c$ which minimises the distance $d_{0}+d_{1}$. I have worked like this. ...
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### Special case of the hodge decomposition theorem [duplicate]

I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$. any vector can be written as the ...
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### Solve the following limit of the sequence

Another sequence limit I'm stuck. $$\lim_{n\rightarrow\infty}{\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}}$$ Any idea ?
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### Difficulty in Quadratic equation and realtion with irrational roots

One root of the quadratic equation $ax^2 +bx + c=0$ is $\dfrac{2}{\sqrt{3} + \sqrt{5}}$. If $\frac{c}{a}$ is rational, then how do we find the other root. the answer given is that the other root is ...
For the function $\displaystyle h \sin \left(\frac{1}{h}\right)$ when it is evaluated at $h=0$, is it $0$ or is it undefined?
Let $f(x)=x-[x]$ and $g(x)=\tan x$. How could we see that $f(x)-g(x)$ is not a periodic function? This will show that the sum of two periodic functions need not be a periodic function. I hope ...