# All Questions

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### When is minimizing the sum of function equivalent to minimizing sum of independent variables

I have to admit I am not good at math, but this is a problem I am having trouble with. What kind of function $f$ can guarantee $min\sum_{i=1}^Kf(x_i)$ is equivalent to $min\sum_{i=1}^Kx_i$. Thank you. ...
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### How does 5.9-7.66=-1.76?

Sorry for such an easy question but this has been confusing me like crazy. I have borrowed like normal and subtracted like every other problem, and the answer I get is -2.24. How is this wrong?
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### Riemann Zeta Function, Stirling's Numbers, and Infinite Series

A while back I was able to prove the following identity, $$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$ where $s(k,n)$ are the Sterling numbers of ...
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### expected value of game involving uniform variable and its square

I am trying to determine the expected value of the following game: Let $u$ be drawn from a uniform distribution on $[0,1]$. We write down $u$ on one side of a piece of paper and $u^2$ on the other ...
Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, $w =\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ I have the following vector: $V = \begin{bmatrix} ... 1answer 15 views ### Proportionality between two quantities Its known that if one variable is proportional to two others than it is also proportional to their product. $$\forall a,b,c\in ℝ:a\propto b\wedge a\propto c\Rightarrow a\propto b\cdot c$$ I think i`ve ... 0answers 38 views ###$i^i$is real and also infinitely many sets in$\mathbb C - \mathbb Ri^i$is real and also infinitely many sets in$\mathbb C - \mathbb R$where operations in proper superset/field maps to a proper subfield. Is this of mapping between superfields to subfields of any ... 1answer 40 views ### Interesting Trigonometry Identity Prelude While studying trigonometry, I came across this very interesting problem. It wasn't very difficult to solve, however it's result was quite interesting. I have given the solution below. Try to ... 1answer 14 views ### Total Probability Theorem / Partition In the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider$B$to be the sample space and$A_1$,$A_2$to be the partition then the theorem says: $$\... 4answers 35 views ### Helping understanding finding a limit without L'hopital's rule? \lim \limits_{x \to 0}x\sin\left(\frac{1}{x}\right) I need to find and prove this limit. I can easily plug it into wolfram alpha, but I want to make sure I learn something. It's been 3 years ... 0answers 34 views ### Concavity when 2nd derivative is zero I was self-studying for a CLEP calculus exam when the following problem came up. It basically asks you to sketch a graph of a function based on the information given. My question is about the part ... 3answers 72 views ### Are there ways to solve equations with multiple variables? I am not at a high level in math, so I have a simple question a simple Google search cannot answer, and the other Stack Exchange questions does not either. I thought about this question after reading ... 2answers 42 views ### Real Numbers Raised to Imaginary Powers? What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples ... 0answers 26 views ### What some good lecture notes for self learning that are complete (nothing but the most trivial is left as an exercise)? Subject that are bit difficult to self learn such as Galois theory, algebraic geometry, etc. I have two months of free time and I'd like a quick entry to these subjects and wish to understand more ... 2answers 30 views ### Two variable definition of derivative Let f:(0,1)\rightarrow \mathbb R be a real valued map from the unit interval. Let$$A:=\left \{a\in (0,1):\exists f^*(a)=\lim_{(x,y)\to(a,a)} \frac{f(x)-f(y)}{x-y}\right \}$$It is true that if ... 1answer 17 views ### Two different remainders for same expression$$\frac{n! + 1}{n} = (n-1)! + \frac{1}{n}$$The remainder is \frac{1}{n}$$n! + 1 \equiv 1 \cdot 2 \cdot 3 \cdot \dotsc (n-1) \cdot 0 + 1 \equiv 1 \mod n$$The remainder is 1 What is going on? ... 1answer 13 views ### Relationship of L_1 distance between CDFs and PDFs? Let F:(-\infty,\infty)\rightarrow[0,1] and G:(-\infty,\infty)\rightarrow[0,1] two CDFs with PDFs f and g, respectively. Is there a connection/inequality between:$$d_1 = \int_{-\infty}^{\... 0answers 14 views ### What is the relationship between the function$\mathbb{E}(Y \mid X = x)$and linear regression? Consider the function $$r(x) = \mathbb{E}(Y \mid X = x)$$ This has been called the regression function in a textbook I'm using. I'm trying to figure out the relationship between this function ... 4answers 65 views ### Compute$ \lim_{x\to\infty}\frac{1}{x}\int_1^x\cos\frac{1}{t}\,dt \$
I have a limit I can't solve. I'm studying for a test in Calculus II. I'm asked to compute the limit: $$\lim_{x\to\infty}\frac{1}{x}\int_1^x\cos\frac{1}{t}\,dt$$ It's hinted that there is no need ...