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1

Using the idea of Maestro Will: Necessarily $\det(A-I)$ divides $\det(B-I)\not= 0$ or $\det(B-I)=0$. We calculate the Smith normal forms $A-I=SDT,B-I=U\Delta V$. Thus $(U^{-1}XS)D(TYV^{-1})=\Delta$. It suffices to solve the equation $XDY=\Delta$. where $d_i$ divides $d_{i+1}$ and $\delta_i$ divides $\delta_{i+1}$. An instance without any solutions: ...


1

We have: $$ u_n = \sqrt{n}\frac{(2n)!}{4^n n!^2} = \sqrt{n}\frac{(2n-1)!!}{(2n)!!} = \sqrt{n}\prod_{k=1}^{n}\left(1-\frac{1}{2k}\right),$$ so: $$ u_n^2 = \frac{n}{4}\prod_{k=2}^{n}\left(1-\frac{1}{k}+\frac{1}{4k^2}\right)=\frac{n}{4}\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)\prod_{k=2}^{n}\left(1+\frac{1}{4k(k-1)}\right) $$ or just : $$ u_n^2 = ...


0

A number of tableau provers for modal logics at least can be found under http://www.cs.man.ac.uk/~schmidt/tools/ You might be interested in LotREC, Logics Workbench, Mettel (resp. its successor Mettel2) or the Tableau WorkBench. These are generic provers and can be used for many logics, but usually come with calculi for propositional logic (and others) ...


0

It seems like a bug with Mac Grapher. You can try using the Mac version of Graphing Calculator 3D instead:


0

One way to do this is to write an equation for $y$ in terms of a parameter, say $a$: $$y = a^3 - 3a + 5, \quad a = 1$$ is what you would enter as an equation (so it has a check box to the left rather than a green equals sign). This will plot a horizontal line $y = 3$. Then enter in the same equation but using the variable name $x$: $$y = x^3 - 3x + 5$$ ...


2

You can use this online tool. The demand curves are a-x (blue). For different values of a, you can draw the corresponding demand curves. If you want additionally different supply curves, you can draw up to thee different supply curves. My inputs: a-x (blue) x-2 (red) x-1 (green) a from 8 to 10; incrementing by 2. The graph can be very well improved by ...


1

I'd use Share$\LaTeX$ with pgfplots. It takes some time to get used to, but the folks down at $\TeX$ Stack Exchange can help you out.


1

Maple has an environment variable Order which controls this. So, after, Order := 3: you can use the right-click context-menu action for series, and see the effect. See its help-page here.


1

If you are on *NIX platforms you can use asciitex. There is also aamath which does a great job in traslating the equation to ascii text files. If you don't want to install anything locally take a look at http://ascii.gallery/matheq which I believe is a web wrapper over aamath.


0

I'm assuming you are interested in the region $R<1$; the region $R>1$ as shown in the graph. Dominique is right, you do need a second constraint or boundary condition. More about that below. Both the wicked behavior near 0 and the combination $f''+\frac{2}{R}f'$ argue for making the substitution $x = \frac{1}{R}$. This transforms the equation to ...


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Try Sage ! It does all this and much more. Nathann


0

Most sites offer no assistance in entering formulas, these ones do: Excellent online mathjax editor with clean interface http://www.texpaste.com/ Other options, interface a bit more complicated though but that's a personal choice http://www.codecogs.com/latex/eqneditor.php arachnoid.com/latex/


2

You might want to check out geogebra, with which you can graph inequalities on a number line and oh so much more than that. That is, it enables you to do many graphing tasks, simple and complicated. You can download the software (free of charge), and I believe you can work with geogebra online.


5

By examining A103111 you can see that $k=96545,$ found by the late Donovan Johnson in 2008. Otherwise srsieve and pfgw are a pretty good combination.


2

Try using Aspect Ratio, Plot[{-2*Sqrt[1 - x^2], 2*Sqrt[1 - x^2], 2*x - 2*Sqrt[2]}, {x, -2, 2}, PlotStyle -> {Blue, Blue, Red}, AspectRatio -> Automatic]



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