Mathematics Weekly Newsletter - Friday, June 27, 2014

Top new questions this week: How were 'old-school' mathematics graphics created? I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today: … (math-software) (visualization) (art)   asked by TSGM 47 votes answered by MJD 35 votes What is that curve that appears when I use $\ln$ on Pascal's triangle? I made a little program that generates Pascal triangles as images : I first tried it associating to each pixel a color whose intensity was proportional to the number in the Pascal triangle The … (combinatorics) (algebra-precalculus)   asked by Hippalectryon 35 votes answered by Ted Shifrin 16 votes Complex analysis is more "real" than real analysis In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself … (real-analysis) (complex-analysis) (soft-question) (self-learning) (philosophy)   asked by user144542 23 votes answered by Conifold 9 votes Does the series $\sum_{n=1}^\infty\frac{\sin n}{\ln n+\cos n}$ converge? $$\sum_{n=1}^\infty\frac{\sin n}{\ln n+\cos n}$$ My guess is "yes", but I can't prove it. (real-analysis) (sequences-and-series)   asked by CuriousGuest 22 votes answered by Greg Martin 15 votes How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$? I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that n sum ------------- 1 2.7182818 2 4.6707743 3 … (calculus) (sequences-and-series) (proof-strategy) (exponential-function)   asked by Gottfried Helms 18 votes answered by TheOscillator 7 votes Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$? Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis … (real-analysis)   asked by noname 15 votes answered by leonbloy 3 votes Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$ In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it … (sequences-and-series) (numerical-methods) (approximation) (closed-form)   asked by Kirill 15 votes answered by gnasher729 1 vote Greatest hits from previous weeks: What is the intuitive relationship between SVD and PCA Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional dataset into fewer dimensions while retaining important … (linear-algebra) (statistics)   asked by wickedchicken 95 votes answered by Ｊ. Ｍ. 92 votes Is $\frac{dy}{dx}$ not a ratio? In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{dy}{dx}$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the … (calculus) (nonstandard-analysis)   asked by BB_ML 323 votes answered by Arturo Magidin 588 votes Can you answer these? Tool that draws Venn diagram from subset relation Is there a tool (LaTeX, JavaScript, Mathematica..) that allows one to draw Venn diagram automatically from subsets relations, e.g. $$A\subset A+B$$ $$A\subset C$$ $$C\subset C+D$$ $$B \not\subset … (elementary-set-theory) (math-software)   asked by JGab 6 votes Is there any simple proof for this? Let $G$ be a group with $|G:Z(G)|=n$ then $\phi(x)=x^n$ is a homomorphism from $G$ to $Z(G)$. I guess it has a proof using transfer theory, I wonder whether it has an elemantary proof or not. … (abstract-algebra) (group-theory) (finite-groups)   asked by mesel 8 votes What is the name of a graph made of k copies of a 4-cycle connected end to end in a chain, possibly with leaves? Do graphs of the following sort have a specific name? We've been calling them Cactapillars, as they're cacti that look a little like caterpillars (and the name Caterpillar already refers to a … (graph-theory) (terminology)   asked by user160151 5 votes

Subscribe to more Stack Exchange newsletters Unsubscribe from this newsletter or change your email preferences by visiting your subscriptions page on stackexchange.com. Questions? Comments? Let us know on our feedback site. If you no longer want to receive mail from Stack Exchange, unsubscribe from all stackexchange.com emails. Stack Exchange, Inc. 110 William St, 28th Floor, NY NY 10038 <3