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 Apr3 comment Recursive/Fibonacci Induction This is carried out at mathematica.stackexchange.com/questions/18481/… for Fibonacci numbers defined over any field (whose characteristic is not $5$ or $2$). Apr1 comment Fejer Kernel problem This is really a comment because it's not an answer: although it points out that the integral is invariant with respect to $f'$, it still leaves the O.P. with the problem of finding that constant ("show (1) gives you $1$"). Mar30 comment Detect values in array that are statistically inconsistent Option 1 is statistically terrible, for many reasons, and option 2 requires strong assumptions to be valid. Consult the stats site for better solutions and discussion. Mar30 comment Detect values in array that are statistically inconsistent This is not a mathematical question nor does it have a definite answer. For instance, some processes have regular variation when the observations are expressed as logarithms (as is often the case in chemistry: pH is a log). In your first example the common logs all vary between 3.2 and 1.7, averaging around 2. However, one of them is zero: it's far below all the rest. That would make $1$ the outlier, not $1542$. A good answer to this question will therefore consider the nature of the process, its statistical characteristics, and the objectives of the algorithm. Mar30 comment Difficult and unusual probability problem, how to solve? Hint: consider the case $k=1$. What's the answer? (The answer in the general case is obtained in a similar manner and relies on this one.) Mar30 comment Find $\lim\limits_{(x,y) \to(0,0)} \frac{xy^2}{ x^2 + y^4}$ +1 Nice reference. For some illustrations, please see mathematica.stackexchange.com/questions/21544/…. Mar28 awarded Pundit Mar27 comment Hugely ugly messy determinant - any trick to find it? If you view the determinant as a function of $a$, evidently it's a polynomial of degree $3$. By inspection--and understanding the basic properties of determinants--you can identify three roots immediately, so you know it up to some multiple that depends only on $b$, $c$, and $d$. Repeating this observation for the other variables pins down the determinant up to a constant. By choosing nice values for $a$, $b$, $c$, and $d$, you can compute that constant--and have thereby reduced the problem to finding the determinant of a single matrix with numerical entries. Mar19 comment How to find the trigonometric identities? Did you really mean to post this on a site about Mathematica software? By the way, there is an infinity of trigonometric identities. They can all be determined from a very small number of them in standard ways, such as a statement of the first order linear ODE satisfied by $\cos(x) + i \sin(x)$ (together with their initial conditions) and the definitions of the other trig functions in terms of them, or the power series definition of $\exp(i x)$, or the definition of $\exp(z)$ as the inverse of $\int_1^z \frac{dz}{z}$, or the infinite product representation of $\csc(z)$, etc. Mar1 comment A Cover of an Orientable Manifold is Orientable You used one more thing that you're not explicitly making note of: the existence of a nowhere vanishing form on $M$. Feb6 comment Probability proof and combinatorics I have rolled this question back to its original form because the corrections added in the edit make it incomprehensible: without the typographical errors, there's no question here at all and the comments make no sense. Feb6 revised Probability proof and combinatorics rolled back to a previous revision Feb6 comment Probability proof and combinatorics @Dilip Sorry; I wasn't trying to imply you weren't aware: your comment showed you understood the issues. My comment was intended to help the OP avoid a possible misunderstanding of your comment, which could cause the reader to focus on the second equality in each line without noticing the blatant typographical errors that occurred at the first equality. Feb6 comment Probability proof and combinatorics @Dilip Actually, the $k!$ in the numerator on the first line and the $(n-k+1)!$ on the second line are typos. Upon removing the extraneous "!", all is well. Feb5 comment sum of two random variables +1 There should be no objections from purists: use of $\delta$ can be made rigorous in the (Schwartz) sense of distributions. Integration by parts of the convolution of the PDFs produces an integral which is the product of the derivative of the Gaussian PDF and the Heaviside function, both of which are integrable (even Riemann integrable). Jan13 comment Game with losing and winning a dollar The probability of winning $n$ dollars is $0$ unless $k=0$, because the most that can be won is $n-k$. Perhaps you mean to ask for the probability of reaching a total of $n$ dollars? Jan13 reviewed Close What is gained by computing additional digits of $\pi$? Jan13 comment A system of equations of Vietnamese Mathematical Olympiad 2013 Cross-posted at mathematica.stackexchange.com/questions/17621/…. Jan11 comment A system of equations of Vietnamese Mathematical Olympiad 2013 Hint: add the equations and show that $\sqrt{\sin^2 x + \dfrac{1}{\sin^2 x}} + \sqrt{\cos^2 x + \dfrac{1}{\cos^2 x}}$ is minimized when $\sin^2 x = \cos^2 x = 1/2$. The rest is easy. Jan10 comment Can a symmetric matrix always be represented as the sum of a positive-definite and negative-definite matrix? Intuition: Sylvester's Law of Inertia implies there always exists a basis in which a Real symmetric nondegenerate matrix is diagonal with $p$ ones and $q$ $-1$'s on the diagonal; $(p,q)$ is its signature and is a property of the matrix as an endomorphism of a vector space (that is, it is the same for all such bases). The matrix with the $-1$'s replaced by zeros is obviously positive semidefinite and the matrix with the $1$'s replaced by zeros is obviously negative semidefinite; their sum is the original matrix.