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two completely different questions......... $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{array} \right) \left( \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array} \right) \left( \begin{array}{rrr} 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 ... 0 If A is a symmetric positive definite matrix then B^TAB is a diagonal matrix iff B is an orthogonal matrix i.e. B^T=B^{-1}. If \exists such a matrix B then B^TAB=diag(\lambda_1,\lambda_2,...,\lambda_n) where \lambda_i are the eigenvalues of A. And the columns of B are the corresponding eigenvectors of A. So to find the B s.t. det(B)<... 1 PARI GP has such a function. From its documentation: matsnf(X,{flag = 0}) If X is a (singular or non-singular) matrix outputs the vector of elementary divisors of X, i.e. the diagonal of the Smith normal form of X, normalized so that d_n | d_{n-1} | ... | d_1. The binary digits of flag mean: 1 (complete output): if set, outputs [U,V,D], ... 1$$0.001 b \le x \le U b where I assume $0 \le x \le U$. I don't see how this particular formulation is related to NP-completeness. If you are worried about the complexity of this don't use a MIP model at all but rather use a heuristic (even this may be overstating things as state-of-the-art MIP solvers actually use lots of heuristics internally).

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It's an "order of operations" issue, definitely, like Zachary Selk mentioned. To add some detail to it: in mathematics, $-1$ is a number to itself. That is, the - is just a way to remind us that this number is negative; however, computers (well, expression parsers) don't see it that way. It will treat exponentiation first, and then negate that result, ...

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The problem I encountered was twofold: 1) $f(x)=-1^x$, won't show oscillatory motion because $=-1^x=-(1^x)-(1)=-1$ 2) $f(x)=(-1)^x$, won't plot because if $\{x|x \in \mathbb R\} \land x=a/2b$ where $b \in \mathbb N$ "$x$ is in the domain of all real numbers and $x=a/$some even number" where $a \neq 2b \lor a \notin \mathbb Q \lor a=2n+1$ where $... 2 What you wrote according to order of operations is$-1^x=-1$. You meant$(-1)^x$probably. See: http://m.wolframalpha.com/input/?i=%28-1%29%5Ex&x=0&y=0 1 You could edit the image to have a horizontal line run through the vase, and then use any photo editor to identify points along the curve of the vase. Then, you could plot some of the pixel coordinates relative to the horizontal line on a new graph, and then, using polynomial regression, identify a piecewise function that best fits the points. Needless to ... 0 I guess you want a function (polynomial) for trhe radius of the vase at different points so that you can work out the surface area to determine the rate of evaporation...? You can do this with geogebra. Look at this http://mrhonner.com/archives/3923 2 Some quick hints: For (A1), you need to know the algorithms. The Prime pages provide a good start, including the underlying mathematics and references. I recommend following those references too. For (A2),$100$digits are easily handled with the right algorithms. Primality proving for general numbers below$400\$ decimal digits takes less than half a ...

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This database exists and is called the web. Googling Continuous non-differentiable function leads to Weierstrass function: In mathematics, the Weierstrass function is an example of a pathological real-valued function on the real line. The function has the property of being continuous everywhere but differentiable nowhere. Smooth but non-...

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