## Free Textbooks in Mathematics

Fri, 02/24/2017 - 08:00 — yonatan zilpaThe following is a list of links to useful textbooks in mathematics, available for free on the Internet. All books are legally safe to download, The books are in printable format - Postscript (PS) or Portable Document Format (PDF). You are free to download, read and print them. This is a partial list of "free textbooks" in math, the list is to be updated regularly.- Read more about Free Textbooks in Mathematics
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## The circle of Apollonius

Fri, 02/26/2016 - 17:05 — yonatan zilpaBy definition, a circle is the locus of all points equidistant from a fixed center. However, the circle of Apollonius is defined differently. In the following post I will define and construct an Apollonian circle.- Read more about The circle of Apollonius
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## Steinhaus theorem

Thu, 11/24/2016 - 13:15 — yonatan zilpaLet $\lambda$ be Borel measure.- If $A$ is mesuarable and $\lambda(A)>0$, then $A-A=\left\{ x-y\; :\; x,y\in A \right\}$ contains a segment $I$ such that $0\in I$
- If $A, B$ are measurable sets and $\lambda(A), \lambda(B)>0$, then $A+B$ contains a segment I.

- Read more about Steinhaus theorem
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## Four bottles - Equidistant mouth problems

Sun, 04/17/2016 - 05:29 — yonatan zilpaHow to place four identical bottles in the plane such that the distance between each pair of mouths will be the same?- Read more about Four bottles - Equidistant mouth problems
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## Asymptotic Differentiable Function

Thu, 11/24/2016 - 14:45 — yonatan zilpaLet $f$ be any function asymptotic at zero. Prove/Disprove:

If $f$ is differentiable everywhere in its domain, then $\lim_{x\to \infty}f\;'(x)$ must be equal to zero?- Read more about Asymptotic Differentiable Function
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## Conceptual Language and Learning

Mon, 12/26/2016 - 15:17 — yonatan zilpaNormally when we learn human spoken language we translate words to our own mother tongue language. But when we learn our mother tongue language we have no language to translate new words. How do we understand mother tongue without translating words? Is there some kind of conceptual language where we can translate mother tongue new words? And if so, can we use this conceptual language to become better learner?- Read more about Conceptual Language and Learning
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## Emotions Relationship & Learning

Wed, 12/07/2016 - 14:48 — yonatan zilpaIt's been said that "necessity is the mother of inventions", but necessity is the mother of learning as well. Learning is vital to survival it is a necessity, however necessity determines what we learn. This interaction between necessity and learning is part of "emotional relationship" between the learner his/her topic.- Read more about Emotions Relationship & Learning
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## Minimum way to move all circles

Thu, 11/24/2016 - 13:04 — yonatan zilpaConsider a game with three vertical sticks and 5 circles, where each circle has a hole in the middle. The circles arranged in one stick as shown in the picture: The task is to move all the circles to a nearby stick, circles can be move from one stick to another, but a circle cannot be put over a smaller one. What are the minimum number of steps needed to complete this task?- Read more about Minimum way to move all circles
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- Read more about How many squares can be found here?
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Let $\nu$ be a signed measure over measurable space $(X,\mathcal{M})$. Denote
$\tilde{N}=\left\{ u\in \mathcal{M} \; :\; \mbox{$u$ is $\nu$-negative}
\right\}$ and $\tilde{P}=\left\{ u\in \mathcal{M} \; :\; \mbox{$u$ is
$\nu$-positive} \right\}$

- $\exists N\in \tilde{N}\Big(\exists P \in \tilde{P}\Big( N\cap P=\emptyset \; \mbox{ and }\; N\cup P = X\Big)\Big)$
- $\forall N,N^{'}\in \tilde{N}\Big(\forall P,P^{'}\in \tilde{P} \Big(N\sqcup P =N^{'}\sqcup P^{'}=X \; \Longrightarrow \nu( P\triangle \tilde{P})=\nu( N\triangle \tilde{N})=0\Big)\Big) $

- Read more about Hahn Decomposition Theorem
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