Mathematics Weekly Newsletter - Friday, January 30, 2015

Top new questions this week: Is there something special about 2015? Is there some property which is satisfied only by the number 2015 (among natural numbers, say) or is there a relatively simple question for which the answer is, surprisingly, 2015? This is inspired ... (algebra-precalculus) (elementary-number-theory) (recreational-mathematics)   asked by Joonas Ilmavirta 37 votes answered by Jack D'Aurizio 64 votes An easy example of a non-constructive proof without an obvious "fix"? I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ... (logic) (examples-counterexamples) (constructive-mathematics)   asked by Valentin Golev 34 votes answered by Neil Strickland 57 votes Connection between the Laplace transform and generating functions As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ... (differential-equations) (recurrence-relations) (generating-functions) (laplace-transform)   asked by oldrinb 19 votes answered by maxerize 3 votes How to make a "function"? I dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ... (algebra-precalculus) (functions)   asked by mathnoob757 14 votes answered by Peter Woolfitt 13 votes Is the Law of Large Numbers empirically proven? Does this reflect the real world and what is the empirical evidence behind this? Layman here so please avoid abstract math in your response. The Law of Large Numbers states that the average of the ... (probability) (statistics) (applications) (law-of-large-numbers)   asked by user1891836 14 votes answered by Erick Wong 20 votes Numbers that are divisible by the number of primes smaller than them Let $\pi(n)$ denote the number of primes less than or equal to $n$ (a.k.a the prime-counting function). For certain values of $n$, the value of $\frac{n}{\pi(n)}$ is integer. Here are the first few ... (number-theory) (prime-numbers)   asked by barak manos 14 votes answered by Nacho Darago 11 votes How can we think and/or write rigorously about integration by substitution? Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ... (calculus) (integration) (proof-writing) (definition)   asked by goblin 13 votes answered by Jack M 9 votes Greatest hits from previous weeks: Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ... (calculus) (nonstandard-analysis)   asked by BB_ML 385 votes answered by Arturo Magidin 660 votes Eigenvectors of real symmetric matrices are orthogonal Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ... (linear-algebra) (reference-request) (eigenvalues-eigenvectors)   asked by Phonon 12 votes answered by Arturo Magidin 31 votes Can you answer these? An integral with respect to the Haar measure on a unitary group Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ... (probability) (statistics) (algebraic-geometry) (differential-geometry) (random-matrices)   asked by Peter 4 votes Bass' paper on 1963 I am currently reading paper by Hayman Bass: On the ubiquity of Gorenstein rings. I found difficulty to understand the proof of Proposition (7.2) Under the the following setting: $A$: commutative ... (commutative-algebra) (projective-module) (gorenstein)   asked by rstarmidi 4 votes Max possible number of sets that have 1 and only 1 member in common I have a set of 25 things that I want to group into sets of 6, with the following conditions: Every set shares one, and only one, member in common with every other set No object can appear twice in ... (combinatorics)   asked by Ryan Ballantyne 4 votes New blog post: Welcome the new trio of moderators of 2014 by Community Blog on Jan 26 The MSE elections are over. Sure, this is a bit late, but that’s no excuse not to be excited. Welcome the new moderators! We should also take a moment to thank the retiring moderators Alex ...

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