mm-1559 === Subject: fourier transforms... help!!! IĒm really puzzled about this: I need to find the Fourier Transform of this function: A.cos ((w+5)t) if (-T/2)I need to find the Fourier Transform of this |>function: |> A.cos ((w+5)t) if (-T/2)X(t)= |> A.cos (wt) if t equals any other value |>I donĒt know how to solve it, this is the case |>of an asyncronous pulse modulated in FM. In Maple 9: > X:= A*cos(w*t) + (Heaviside(t+T/2)-Heaviside(t-T/2))*A*(cos((w+5)*t)-cos(w*t)): > inttrans[fourier](X,t,k) assuming w>0,T>0; / | | A |Pi Dirac(k + w) + Pi Dirac(k - w) T (k + w + 5) + Pi sin(-------------) Dirac(k + w + 5) I 2 T (k + w) sin(---------) T (k - w - 5) 2 + Pi sin(-------------) Dirac(k - w - 5) I - -------------- 2 k + w T (k + w + 5) T (k - w - 5) T (k - w) sin(-------------) sin(-------------) sin(---------)| 2 2 2 | + ------------------ + ------------------ - --------------| k + w + 5 k - w - 5 k - w / > simplify(%, Dirac); # there should be no deltas at +/-(w+5) / T (-k + w) | sin(----------) | 2 Pi A Dirac(k + w) + Pi A Dirac(-k + w) + A |- --------------- -k + w T (-k + w + 5) T (k + w) T (k + w + 5) sin(--------------) sin(---------) sin(-------------)| 2 2 2 | + ------------------- - -------------- + ------------------| -k + w + 5 k + w k + w + 5 / Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: fourier transforms... help!!! Present it as X(t)= A.cos (wt) [-inf,+inf] + (A.cos ((w+5)t)t-A.cos (wt) ) (for (-T/2) shift the relativ sin x/ x function) === Subject: Re: fourier transforms... help!!! Easiest way would be to use the definition from the integral, and separate the infinite region into t<-T/2, |t|T/2. Perform each integral, and simplify the sum of the results. You can exploit symmetry for the two infinite sections. The usual trick of introducing exp(-a^2*|t|) as a factor, with limit as a->0 will make the outer integral convergent. > IĒm really puzzled about this: > I need to find the Fourier Transform of this > function: > A.cos ((w+5)t) if (-T/2) X(t)= > A.cos (wt) if t equals any other value > I donĒt know how to solve it, this is the case > of an asyncronous pulse modulated in FM. > Alejandro === Subject: Re: fourier transforms... help!!! > IĒm really puzzled about this: > I need to find the Fourier Transform of this > function: > A.cos ((w+5)t) if (-T/2) X(t)= > A.cos (wt) if t equals any other value > > I donĒt know how to solve it, this is the case > of an asyncronous pulse modulated in FM. > Alejandro Well as far as I remember Fourier transform is linear, so you will have the fourier transform of three segments separately added together. The first part is simple, you just have to integrate: A.cos((w+5)t)*exp(-iWt) from T/2 to T/2. The second part is the fourier transform of a step cosine, delayed by T/2. You may find it in any table. The last one is exactly the same but with time inverted (t=-t). Have a look at the properties of Fourier transform for finding F{f(-t)} when F{f(t)} is known. Add up the three transforms and you have what you were looking for. === Subject: Re: Plotting with singularity points, maple vs mathemtica behavior, which is more correct? |>in maple 9.01 |>f:= (x,y)-> y*(1- 1/(x^2+y^2) ); |>plot3d(f,x=-2..2,y=-2..2); |>I get a floating point overflow error and an empty plot. As others mentioned, you want either plot3d(f,-2..2,-2..2) or plot3d(f(x,y),x=-2..2,y=-2..2). As for how to deal with the singularity: what Maple does is calculate the values at points on a grid. In this case the grid includes one point very close to (but, due to roundoff error, not exactly at) the singularity, producing a very large value. If you want to see a nicer plot, use the view option: e.g. > plot3d(f,-2..2,-2..2,view=-10..10, grid=[100,100], style=patchcontour); Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Plotting with singularity points, maple vs mathemtica behavior, which is more correct? > |>in maple 9.01 > |>f:= (x,y)-> y*(1- 1/(x^2+y^2) ); > |>plot3d(f,x=-2..2,y=-2..2); > |>I get a floating point overflow error and an empty plot. > As others mentioned, you want either plot3d(f,-2..2,-2..2) or > plot3d(f(x,y),x=-2..2,y=-2..2). > As for how to deal with the singularity: what Maple does is calculate > the values at points on a grid. In this case the grid includes one > point very close to (but, due to roundoff error, not exactly at) the > singularity, producing a very large value. If you want to see > a nicer plot, use the view option: e.g. > plot3d(f,-2..2,-2..2,view=-10..10, grid=[100,100], style=patchcontour); > Robert Israel israel@math.ubc.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2 I like the fact that in maple I can rotate the 3d plot with the mouse. This is something I can not do with mma. Once maple can support zooming using the mouse (like matlab can) then its plotting will be even more powerfull. === Subject: Coming soon -- new proof checking software Within a week or so, I will be releasing a free beta version download for my new proof checking software, DC Proof 1.0. In the mean time, here is a sampler from the User Guide: http://members.allstream.net/~dchris/DCProofT.chm It contains a tutorial that illustrates many of the main features of DC Proof. Readers may be interested in both theoretical and a pedagogical aspects of this application. Example 3, is a resolution of Russell's Paradox without the usual prohibition on self-reference. Enjoy. Dan Christensen Toronto, Canada === Subject: Re: Coming soon -- new proof checking software > Within a week or so, I will be releasing a free beta version download for my > new proof checking software, DC Proof 1.0. > In the mean time, here is a sampler from the User Guide: > http://members.allstream.net/~dchris/DCProofT.chm > It contains a tutorial that illustrates many of the main features of DC > Proof. Readers may be interested in both theoretical and a pedagogical > aspects of this application. Example 3, is a resolution of Russell's Paradox > without the usual prohibition on self-reference. I presume a .chm file is a DC Proof file. But if I don't have DC Proof, how can I read it? Disassembling it, I can see a little text, but a lot of binary. --Ron Bruck === Subject: Re: Coming soon -- new proof checking software > Within a week or so, I will be releasing a free beta version download for my > new proof checking software, DC Proof 1.0. > In the mean time, here is a sampler from the User Guide: > http://members.allstream.net/~dchris/DCProofT.chm > It contains a tutorial that illustrates many of the main features of DC > Proof. Readers may be interested in both theoretical and a pedagogical > aspects of this application. Example 3, is a resolution of Russell's Paradox > without the usual prohibition on self-reference. > I presume a .chm file is a DC Proof file. But if I don't have DC > Proof, how can I read it? > Disassembling it, I can see a little text, but a lot of binary. > --Ron Bruck .chm files are a Windows standard readable in any recent version of Windows. It will bring up the standard Microsoft HTML Help browser with tabs for a TOC, index and search. For non-Windows users, I have since put together a purely HTML version at http://members.allstream.net/~dchris/Banner.htm Dan === Subject: Re: Coming soon -- new proof checking software It would be nice if you could compare your proof checking software with other software with similar goals that has been around for a while. You might compare it to Automath (deBruin) http://www.cee.hw.ac.uk/~fairouz/automath2002/ which has been around for 36 years or so... OTTER http://www-unix.mcs.anl.gov/AR/otter/ and also see other links on that page.. RJF > Within a week or so, I will be releasing a free beta version download for my > new proof checking software, DC Proof 1.0. > In the mean time, here is a sampler from the User Guide: > http://members.allstream.net/~dchris/DCProofT.chm > It contains a tutorial that illustrates many of the main features of DC > Proof. Readers may be interested in both theoretical and a pedagogical > aspects of this application. Example 3, is a resolution of Russell's Paradox > without the usual prohibition on self-reference. > Enjoy. > Dan Christensen > Toronto, Canada === Subject: Re: Coming soon -- new proof checking software Richard, theorem prover like Otter. It is an educational tool to teach the fundamentals of formal proof to the non-specialist. The user selects which rule of inference to use for each line. Some time ago, I surveyed the available proof checkers. The output from all of them was largely incomprenhensible to the reader more used to proofs in textbooks. There was also no easy way to document a proof with commentary or to manage the mass of detail in longer proofs. Compared to informal proofs, formal proofs of even fairly simple theorems can be quite lengthy. Although it won't be obvious from this brief tutorial -- each example there is quite short -- I have tried to address each of these problems in DC Proof. As soon as I can get my website online at my new domain name (bureaucratic delays), I will make a fully functional beta version available free of charge. Dan > It would be nice if you could compare your proof checking > software with other software with similar goals that has > been around for a while. > You might compare it to > Automath (deBruin) http://www.cee.hw.ac.uk/~fairouz/automath2002/ > which has been around for 36 years or so... > OTTER http://www-unix.mcs.anl.gov/AR/otter/ > and also see other links on that page.. > RJF > Within a week or so, I will be releasing a free beta version download for my > new proof checking software, DC Proof 1.0. > In the mean time, here is a sampler from the User Guide: > http://members.allstream.net/~dchris/DCProofT.chm > It contains a tutorial that illustrates many of the main features of DC > Proof. Readers may be interested in both theoretical and a pedagogical > aspects of this application. Example 3, is a resolution of Russell's Paradox > without the usual prohibition on self-reference. > Enjoy. > Dan Christensen > Toronto, Canada === Subject: Re: Coming soon -- new proof checking software > In the mean time, here is a sampler from the User Guide: > http://members.allstream.net/~dchris/DCProofT.chm My Mac doesn't know what a .chm file is. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Coming soon -- new proof checking software > In the mean time, here is a sampler from the User Guide: > http://members.allstream.net/~dchris/DCProofT.chm > My Mac doesn't know what a .chm file is. Sorry, about that. See my HTML preview. Dan === Subject: New proof checking software -- HTML preview In a few days (bureaucratic delays), I will be releasing a free beta version download of my new proof checking software, DC Proof 1.0. See the following links for a sampler of the User Guide that includes a tutorial illustrating the main features of DC Proof. Readers may be interested from both a theoretical and pedagogical perspective. Example 3 of the tutorial has a resolution of Russell's Paradox that does not rely on the usual prohibition of self-reference. If you are using a recent version of Windows go to http://members.allstream.net/~dchris/DCProofT.chm This will bring up the standard Microsoft HTML Help browser with TOC and index tabs. Otherwise, go to http://members.allstream.net/~dchris/Banner.htm Without a high-speed connection, pages may take a while to download various screen shots (bmp format). Dan Christensen Toronto, Canada === Subject: Re: Comparison of Mathematica on Various Computers > ''Comparison of Mathematica on Various Computers'' > is now on (or href=http://www2.staff.fh-vorarlberg.ac.at/~ku/karl/mma.html>) > Isn't it an overpriced piece of software on any computer? If you pay full list, it is, arguably, overpriced. Although you'll find nothing cheaper with the same capabilities. (Sadly, Maple has crept up to almost match Mathematica in price.) Maxima may eventually catch up, and it's free, but I can't get it to work on any of my operating systems except Windows, and I ain't willing to make that much of a sacrifice (running Windows). At my university, a copy of Mathematica costs about $350 (full copy, for faculty), and student versions are even cheaper (about $200, I think). If you're not willing to pay THAT for software, you're not serious. --Ron Bruck === Subject: Re: MuPAD: it does not plotfunc2d > I have just installed MuPAD Pro 2.5.3 on my Windows2000 machine. > Everything seems to be working fine, except the command plotfunc2d, > which does not plot anything. I always get the following error > message: > It was not possible to create the object. Make sure that the > application is in the registry of the system. there seems to be something broken in your registry. Please execute the batch script binfixinst.bat in your MuPAD installation. Afterwards it should work. -- *---* MuPAD -- The Open Computer Algebra System *---*| |*--|* Ralf Hillebrand (tonner@mupad.de) *---* === Subject: Re: MuPAD: it does not plotfunc2d > I have just installed MuPAD Pro 2.5.3 on my Windows2000 machine. > Everything seems to be working fine, except the command plotfunc2d, > which does not plot anything. I always get the following error > message: > > It was not possible to create the object. Make sure that the > application is in the registry of the system. > there seems to be something broken in your registry. Please > execute the batch script binfixinst.bat in your MuPAD installation. > Afterwards it should work. Ralf, Paulo === Subject: Re: Which one to choose: MuPAD or Maple? > I would like to know the strong and the weak points of these two > program. Could somebody here please help me? > Paul I have both systems, and of the two I prefer Maple. Probably because i) It runs on both Linux and Unix (and the Mac?) ii) there are more resources avaiable. Nim