# All Questions

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### Combinations Help

I have an application where I iterate through all k-combinations of a set of size n. For example here I have listed all k-combinations for when n is 4. Also I have separated each list of combinations ...
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### Numb3rs Challenge

I am by no means a mathmatician. I guess you could say that I am a mathematically inclined individual, but never made anything of it until I was in my 30's and became a software engineer. Although ...
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### Simple estimation $e^{a\sqrt{r}} > r$

I want to prove a simple theorem about contour integration via residues and I need the following estimation: $e^{a\sqrt{r}} > r$ for any real a > 0 and r >> 0. Is this true? If so, what is an ...
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### True, false, or meaningless?

Are the following two assertions always true, always false or meaningless? $\exists i \in \emptyset$ $\forall i \in \emptyset$ Because it seems that one encounters expressions of this kind fairly ...
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### Closed-form Expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$? (problem with Mathematica)

I need to calculate a closed-form expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$. This isn't particularly difficult, and I do it by hand pretty much routinely. However I found out ...
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### $(a_{1}+ a_{2} + …+a_{k})^{n}$ where $k >2$, what does it generate?

Binomial expansion generates the Pascal triangle but what does it generate when you have different amount of terms there? You can see here the geometric generation with only 2 terms. I am interested ...
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### calculus essay assistance

I am writing an essay for my calculus class and one of the requirements to meet within the essay is to demonstrate an understanding of integration by explaining a metaphor that uses integration. ...
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### How do I calculate $I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2dt}$?

$$I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2dt}$$ $x_a = x(t_a)$ and $x_b = x(t_b)$ I haven't integrated anything like this since a long time. Lost my powers of integration. How ...
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### Characterize entire functions $f$ such that $|f(z)| \leq |\sin(z)|$

I want to determine all entire functions $f$ such that $|f(z)| \leq |\sin(z)|$. I searched around on MathSEx and I found the following question from which I tried to get inspired but I think it ...
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### Are satellite knots prime?

Which satellite knots are prime? I do know that connected sum of knots is a satellite operation, but I found this statement: "the satellite knots all have structures which are well known and ...
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### how do we prove this calculus statement?

Prove that if we write $z = re^{i\theta}$, then $d$ for derivative, $$dz=e^{i\theta}\,dr + ire^{i\theta}\,d{\theta}$$ (reference/context)
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### $(\mbox{Sh,Sh-map})$ represents the category of sheaves on a stack

I'm trying to understand the following theorem, and I think I don't understand the definitions. Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid ...
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### What is an efficient way to get blur from source and blurred images?

I'm doing little program to get blur from source image and blurred image. But I haven't learned so much things about math in school yet. The equation used for blurring the image A into B: ...
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### The Hessian of the Determinant

It is well known how to take the derivative of the determinant: let $A(s)$ be a family of square matrices smoothly parametrised by the variable $s$ (in other words, $A:\mathbb{R}\to \mathbb{R}^{N^2}$ ...
### Homotopy groups of $S^2$
I'd like to understand higher homotopy groups better and I guess there's no simpler way than understanding them for as simple spaces as possible; therefore $S^2$. My question essentially has two ...