# All Questions

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### Evaluating the indefinite integral $\int\frac{dx}{qx+c}$

Evaluate the indefinite integral (remember to use $\ln |u|$ where appropriate) $$\int\frac{dx}{qx+c}\qquad (q\neq 0)$$ I have no idea how to approach this. But here's what a have so far using ...
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### Algebraic variety in $\mathbb{C^{2}}$

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in ...
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### Prove $\left\lfloor n\right\rfloor = n + O(1)$

Can someone show me why $\left\lfloor n\right\rfloor = n + O(1)$. Using the same logic, can anyone derive a similar proposition for $\left\lceil n\right\rceil$?
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### How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: ...
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### Determining functions types. one - one , onto and bijective

Could anyone give an idea to start up with this question: Let $f$ be a function $f: \mathbb{R}^3 \to \mathbb{R}$ such that $f(x,y,z) = xyz$ . How to verify whether it's one to one or onto function?
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### Prove an $n\times n$ matrix is negative definite

I wonder is there any way to prove the $n\times n$ matrix with elements below is negative definite: $$\sigma_{ij} = \frac{a_ia_j}{\sum_k s_ka_k} \space; i \neq j \text{ (off diagonal terms)}$$ ...
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### Is the empty set a member of itself [duplicate]

I'm taking Discrete Mathematics and am a little confused on the "empty set" or "null set" defined by ∅. I understand that ∅ is not a member of {∅} but is ∅ a member of {∅, {∅}} given that there is a ...
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### Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$  This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as ...
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### visualizing a function from the plane into the reals

If $$(x_1,y_1), (x_2,y_2),(x_3,y_3)$$ are points in the plane and if $a,b$ are fixed real numbers, how can I visualize $$f=(ax_1+b-y_1)^2+(ax_2+b-y_2)^2+(ax_3+b-y_3)^2$$ as a function from the plane ...
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### Prove or Disprove? $\log(n^n)\text{ is } \Theta(\log n)$

I need help confirming that my way of proof is alright. This is my first class in algorithms so I just wanna know if I'm on the right track. :)
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### Limit of $\cfrac{\sin(\theta)}{\theta}$ in degrees

What does $\lim \limits_{\theta\to0}\cfrac{\sin(\theta)}{\theta}$ equal when $\theta$ is expressed in degrees? I know that theta in degrees is $\frac{\pi}{180}$ theta radians, but I don't get the ...
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### Are irreducible polynomials and irreducibles (in an integral domain) different?

There's this theorem that you can factor polynomials (in $\mathbb{Z}$[x]) into polynomials of lower degrees r and s in $\mathbb{Q}$[x] iff you can factor that polynomial into polynomials of the same ...
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### The use of Schauder fixed point in ladyzehskaya

The book Linear and Quasilinear equations of parabolic type gives the uniform parabolic pde theory in the literature. Ladyzhenskaya use Leray-Schauder rather than Schauder fixed point theorem. why? ...
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### Why does $y = x\sin(\frac{180}{x})$ approach $\pi$?

A few days ago I was playing on my scientific calculator and I ran over an interesting little equation: $180\sin(1)$ is extremely close to $\pi$. At first I thought it was a coincidence, but then I ...