I am asked to use mathematica to plot the fixed-points of the following system $$ \frac{dN}{dt} = -\gamma N \left( 1 – \left( \beta M + N\right) \right) $$ $$\frac{dM}{dt} = M \left( 1 – \left( \alpha N + M\right) \right) $$ for the case where $\alpha = 2,\ \beta =2,\ \gamma=1$. I assume it […]

I asked Mathematica to evaluate $$\sum_{i=1}^{\infty} \cos(i a)\cos(i b)$$ where $a$ and $b$ are unspecified real numbers, and it told me the answer was simply $-\frac{1}{2}$. This is clearly wrong, as I expect the series to fail to converge for almost any choice of $a,b$ — but is there some formal sense in which Mathematica’s […]

I have been using Mathematica for almost a year, and the most indispensable feature in Mathematica to me has been the ability to write really nice documents and print them into PDF or TEX files. In fact, because of the escape button feature ( i.e. $\boxed{\text{esc}}$ b $\boxed{\text{esc}}$ instantaneously prints $\beta$ and $\boxed{\text{esc}}$ un $\boxed{\text{esc}}$ […]

The following is a JEE (A national level entrance test) question: Find the largest value of the non-negative integer ( a ) for which: $$ \displaystyle \lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4 $$ On solving this, we get $ […]

I’m playing a video game right now and in it is a puzzle (see here). There are solutions to solving it (see here) on the Internet, but I’d like to know if this path is the shortest path (least amount of moves) possible to solve the puzzle. The rules of the game are clear from […]

Oblongs of size $ \frac{1}{1} \times \frac{1}{2}$, $ \frac{1}{2} \times \frac{1}{3}$, $ \frac{1}{3} \times \frac{1}{4}$, $ \frac{1}{4} \times \frac{1}{5}$, … have a total area of 1. $\sum\limits_{a=1}^\infty \frac{1}{a (a+1)} =1$ I believe it’s still an unsolved question whether the infinite rectangles can fit in the unit square. If it’s been solved, I’d love to see […]

I would like to obtain a numerical evaluation of the series $$S=\sum_{n=1}^{\infty}(-1)^{\frac{n(n+1)}{2}}\frac1{n!}\int_0^1x(x+1)\cdots(x+n-1)\: dx$$ to five significant digits. I’ve used Mathematica, but some results puzzle me: $$\text{NSum}\left[(-1)^{\frac{n*(n+1)}{2}}*\frac{\int_0^1 \text{Pochhammer}[x,n] \, dx}{n!},\{n,1,23\}\right]=-0.530724…$$ $$\text{NSum}\left[(-1)^{\frac{n*(n+1)}{2}}*\frac{\int_0^1 \text{Pochhammer}[x,n] \, dx}{n!},\{n,1,24\}\right]=-0.298186…$$ Any help is welcome.

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535…$$ I wonder if it is possible to express the exact solution in terms of known mathematical constants and elementary or special functions (I am especially interested in those implemented in Mathematica)?

I need matrix H. G is working, but H is just derivative?? How to obtain? g = Integrate[x^(p + q – 2*(m + n + 1)), {x, -1, 1}]; h = Integrate[D[x^(p + q – 2*(m + n + 1)), {x, 2}], {x, -1, 1}]; c[r_, n_] := ((-1)^n (2 r – 2 n – […]

Just doing a quick plot of the cuberoot of x, but both Mathematica 9 and R 2.15.32 are not plotting it in the negative space. However they both plot x cubed just fine: Plot[{x^(1/3), x^3}, {x, -2, 2}, PlotRange -> {-2, 2}, AspectRatio -> Automatic] http://www.wolframalpha.com/input/?i=x%5E%281%2F3%29%2Cx%5E3 plot(function(x){x^(1/3)} , xlim=c(-2,2), ylim=c(-2,2)) Is this a bug in […]

Intereting Posts

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Perfect Map $p:\ X\to Y$, $Y$ compact implies $X$ compact
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If a theory has a countable $\omega$-saturated model does it need to have only countable many countable models?
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Showing $\gamma < \sqrt{1/3}$ without a computer
Help solving a limit
Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.
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Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$
Quadratic reciprocity via generalized Fibonacci numbers?
Alternative ways to write $ k \binom{n}{k} $
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An integral and series to prove that $\log(5)>\frac{8}{5}$