Imagine you have a transmitter and receiver that send signals using exactly one frequency and no others. That is, the signal is a perfect sine wave of a particular frequency. How much information can you send on this frequency?
The answer is none; you can't change the signal at all, so you can't send information. Once you start changing the signal, (i.e. change the amplitude) you are actually adding in more frequencies - this is Fourier 101.
To send information, you have to use a band of frequencies. The wider the band, the more information per channel, but the fewer channels. So there is a limited amount of information that can be sent.
What about reflections, or multipath signals? You have to admit, its a pretty hard problem with an expensive solution, especially compared to a $5 brain-dead receiver you can buy today.
Sure, he's a former MIT professor...of CS! That makes him about as well qualified as I (a college math professor) to talk about EM. Actually, though, I do research on numerical methods for wave propagation, and am currently PI on a $220,000 NSF grant studying waves in the Earth's magnetic field, so I do know something about waves. Your post, of course, contains no actual content, so its hard to respond to the substance of it since there isn't any.
I have a Ph.D. in applied mathematics and am an expert in numerical methods for wave propagation, so I do know something about waves. Yes, one can imagine a different technology such as directional antennae or spead-spectrum, but how much more complex do your receivers have to be?
Clearly there is no such thing as limitless bandwidth; Shannon's theory tells us there is maximum amount of information that can be transmitted over any one channel, and simple physics tells us that there are a limited number of channels, no matter how you slice it.
The analogy isn't a complete one. What if you can only see the green light that is reflecting off of diffuse clouds overhead - can you still separate the signals?
Sure, you can build in directional antennae, but then your radio has to know what direction the station is in, and be able to keep the antenna pointed in the right direction. Can your walkman keep its antenna pointed in the right direction while you are vigorously jogging? Not for $20 it couldn't.
This article is complete bunk. Yes, its true that radio frequencies are like colors. So imagine this scenario: you are receiving signals from someone who is using 'green'. They are flashing a huge green light, and you can pick up the pulses they are sending by being bathed in the green light. Now someone else comes along and also starts flashing a huge green light. You can't read the signal any more, because there are now two huge green lights bathing you with their signals. How can you tell which pulse is coming from which light? You can't! That's interference.
The Symbolic Math Toolbox combines the symbolic mathematics and variable precision arithmetic capabilities of the Maple 8symbolic engine with the powerful numeric and visualization capabilities of matlab.
i.e. the matlab symbolic toolbox is actually just , you guessed it, maple!
If your matlab code is mostly level 2 or level 3 linear algebra computations involving medium-sized problems, then you are probably write -- matlab is as fast as c/c++ code.
For solving lots of really small problems, c/c++ can be way faster due to pipelining/loop-unrolling that can't happen in interpreted matlab code.
For solving really large problems, matlab's memory management is too inefficient; you have to carefully control memory usage for large problems, so you need to write c/c++/fortran code for that.
I'm a research mathematician, and I use lots of tools:
matlab for design prototypes of numerical algorithms and for visualizing data.
mathematica for doing messy algebra/calculus/differential equations.
my own c/c++ code, with a lapack backend, for doing large-scale computations (matlab and mathematica are too slow for big computations). So, the answer is e) all of the above!
I was by far the smartest person in my high school (now I have a Ph.D. in mathematics). I wasn't a nerd though, and none of the nerds even approached my intellectual capacity. For the most part they were just stupid AND ugly.
I'm a big sci-fi fan, and somehow the series of novels by Patrick O'Brian appealed to the same part of me that likes sci-fi. The novels are set in the early 19th century, and the main characters are officers in the british navy. His writing is very good, and the immersion you get when reading his stuff is total. Not sci-fi, but definitely not standard fiction either (and not historical fiction really, though it does play on historical events)..
Your formula does indeed compute Pi, but it does not, as you imply, "produce each next digit [from the previous one]". If that were true, then each new term in the series would add a new correct digit to the approximation, and would leave the previous digits unchanged, which it clearly does not. For example, to get 14 digits correct with that formula, you need approximately 10 million terms in the series.
Don't get me wrong, I absolutely love Mathematica, but if your numerical problem is so huge that it benefits greatly from running on a cluster, then you shouldn't be doing it in Mathematica in the first place. Nothing can match custom written Fortran, C, or C++ code for performance and efficient use of memory (Mathematica is a huge memory hog). Non-numerical problems, on the other hand, might benefit greatly from this.
A 2 dimensional sphere is one where the sphere is "locally" the same as a flat 2-dimensional plane. That's what we call a sphere. A 3-dimensional sphere is really a four-dimensional object, because it consists of those points in four dimensional euclidean space that are equidistant from the center of the 3 dimensional sphere. So, a 3-dimensional sphere looks "locally" like flat 3-dimensional space. Its hard to visualize.
For floating point, it means you have greater precision in hardware (allowing things like real physics and shapes to be modelled without noticable issues caused by subtle number creep). Since most systems use IEE-784 (64bit double precision floating point), it means a speedup to that software since you're not working with it as 2 32-bit operations.
Actually, most CPUs today (including G4 and P4) do double-precision in hardware. The G4 does 64-bit FP multiply-add with a throughput of one operation per cycle (I'm pretty sure the P4 does too). Even the loads and stores are operating on 64-bit chunks. Going to a 64-bit processor won't change any of that. The only thing different for FP operations will be (1) you can hold a heck of a lot more numbers in memory! and (2) it might be possible for extended precision (128-bit) to be done easily in hardware.
Ahem, "some" high-performance computing includes of course SMP code (using pthreads) and cluster computing, via PVM. I have designed algorithms and written industrial-strength commercial code that does both. It was fun! The general principles of CS are pretty straightforward, and follow naturally from mathematics.
As far as compiler design goes, I certainly haven't designed my own, but I do know how existing compilers work (you can't do high-perf computing without knowing that).
The point of my post was that everything from "insalling linux" to designing and implementing parallel high-performance code and understanding compiler design is not very hard to pick up on your own if you already know a lot of mathematics.
Funny, I'm someone with a lot of mathematical training (Ph.D. in Applied Math) but only a few courses in computer science. Somehow, I've managed to pick up a humungous amount of CS along the way, things like algorithm design and analysis, designing and coding industrial-strength C/C++ libraries and applications (yes I get paid for this), high-performance computing, OpenGL coding to roll my own volume visualization apps, doing all of my own unix system administration, setting up all of my own hardware...I've always thought that the best way to become really good at coding and software engineering is to first get a degree in mathematics. If you can do that, the rest is easy.
(Okay, I am a bit biased; I'm a college math professor, and in addition I do a lot of research and consulting related to numerical computation).
matlab actually will run under 10.2. What won't run is the graphical desktop interface, but the old-style terminal window interface will run just fine (just do "matlab -nojvm"). Take a look at my posting on macmerc.com for more info.
Sorry, I meant Fortran 77. Fortran 90 of course has pointers, but if I'm not mistaken Fortran 90 has facilities for avoiding pointer aliasing (perhaps that is what your code is showing? I've never written Fortran 90 code...)
...of the simple loop structures. In C, you can have a for( ; ; ) statement that does basically all sorts of weird crap in here ( ; ; ), and you can also do things like pointer aliasing (impossible in Fortran since there are no pointers at all).
Fortran loops, on the other hand, are very very simple - all you can do is increment the loop variable, and repeat. That allows for very heavy loop optimization by the compiler - comparable to what the best assembly programmers might be able to do. Furthermore, a Fortran compiler can more easily generate optimized loop code using vector instruction sets like Altivec or SSE2, whereas C compilers have a much harder time (again, because of the wide variety of loop structures possible in C, and things like pointer aliasing, etc.)
I'm *not* putting down colleges by ANY stretch of the imagination. I'm just saying that colleges tend to focus more on "pratical mathematics" (e.g. "here is the math you need to be an engineering tech"...) whereas a University math department will focus on "theoretical mathematics" (I feel silly typing that.. but you get the point). It really just comes down to what you're interested in learning, and what you want to do with that knowledge.
You seem confused about the difference between a university, a community college, and a 4-year undergraduate college. When you say "college," it appears you mean "community college." Most of what you say doesn't actually apply to 4-year undergraduate colleges.
There is no such thing as understanding mathematics without doing mathematics. You will never understand mathematics without knowing how to do mathematics, that is without knowing the tricks and techniques and methods for solving problems. Likewise, you cannot be functional without comprehension of the concepts, otherwise you hit a brick wall the second you try to do something different than what was assigned for your homework. I say this based on my own experience as a professional research mathematician, scientific consultant, and professor of mathematics at a small liberal arts college.
Slashdot Mirror
Imagine you have a transmitter and receiver that send signals using exactly one frequency and no others. That is, the signal is a perfect sine wave of a particular frequency. How much information can you send on this frequency?
The answer is none; you can't change the signal at all, so you can't send information. Once you start changing the signal, (i.e. change the amplitude) you are actually adding in more frequencies - this is Fourier 101.
To send information, you have to use a band of frequencies. The wider the band, the more information per channel, but the fewer channels. So there is a limited amount of information that can be sent.
What about reflections, or multipath signals? You have to admit, its a pretty hard problem with an expensive solution, especially compared to a $5 brain-dead receiver you can buy today.
Sure, he's a former MIT professor...of CS! That makes him about as well qualified as I (a college math professor) to talk about EM. Actually, though, I do research on numerical methods for wave propagation, and am currently PI on a $220,000 NSF grant studying waves in the Earth's magnetic field, so I do know something about waves. Your post, of course, contains no actual content, so its hard to respond to the substance of it since there isn't any.
I have a Ph.D. in applied mathematics and am an expert in numerical methods for wave propagation, so I do know something about waves. Yes, one can imagine a different technology such as directional antennae or spead-spectrum, but how much more complex do your receivers have to be?
Clearly there is no such thing as limitless bandwidth; Shannon's theory tells us there is maximum amount of information that can be transmitted over any one channel, and simple physics tells us that there are a limited number of channels, no matter how you slice it.
The analogy isn't a complete one. What if you can only see the green light that is reflecting off of diffuse clouds overhead - can you still separate the signals?
Sure, you can build in directional antennae, but then your radio has to know what direction the station is in, and be able to keep the antenna pointed in the right direction. Can your walkman keep its antenna pointed in the right direction while you are vigorously jogging? Not for $20 it couldn't.
So being a spreadsheet developer makes you an expert in electromagnetics?
This article is complete bunk. Yes, its true that radio frequencies are like colors. So imagine this scenario: you are receiving signals from someone who is using 'green'. They are flashing a huge green light, and you can pick up the pulses they are sending by being bathed in the green light. Now someone else comes along and also starts flashing a huge green light. You can't read the signal any more, because there are now two huge green lights bathing you with their signals. How can you tell which pulse is coming from which light? You can't! That's interference.
i.e. the matlab symbolic toolbox is actually just , you guessed it, maple!
For solving lots of really small problems, c/c++ can be way faster due to pipelining/loop-unrolling that can't happen in interpreted matlab code.
For solving really large problems, matlab's memory management is too inefficient; you have to carefully control memory usage for large problems, so you need to write c/c++/fortran code for that.
Explain to me how you would do calculus with matlab?
matlab for design prototypes of numerical algorithms and for visualizing data.
mathematica for doing messy algebra/calculus/differential equations.
my own c/c++ code, with a lapack backend, for doing large-scale computations (matlab and mathematica are too slow for big computations).
So, the answer is e) all of the above!
I was by far the smartest person in my high school (now I have a Ph.D. in mathematics). I wasn't a nerd though, and none of the nerds even approached my intellectual capacity. For the most part they were just stupid AND ugly.
I'm a big sci-fi fan, and somehow the series of novels by Patrick O'Brian appealed to the same part of me that likes sci-fi. The novels are set in the early 19th century, and the main characters are officers in the british navy. His writing is very good, and the immersion you get when reading his stuff is total. Not sci-fi, but definitely not standard fiction either (and not historical fiction really, though it does play on historical events)..
Your formula does indeed compute Pi, but it does not, as you imply, "produce each next digit [from the previous one]". If that were true, then each new term in the series would add a new correct digit to the approximation, and would leave the previous digits unchanged, which it clearly does not. For example, to get 14 digits correct with that formula, you need approximately 10 million terms in the series.
Don't get me wrong, I absolutely love Mathematica, but if your numerical problem is so huge that it benefits greatly from running on a cluster, then you shouldn't be doing it in Mathematica in the first place. Nothing can match custom written Fortran, C, or C++ code for performance and efficient use of memory (Mathematica is a huge memory hog). Non-numerical problems, on the other hand, might benefit greatly from this.
A 2 dimensional sphere is one where the sphere is "locally" the same as a flat 2-dimensional plane. That's what we call a sphere. A 3-dimensional sphere is really a four-dimensional object, because it consists of those points in four dimensional euclidean space that are equidistant from the center of the 3 dimensional sphere. So, a 3-dimensional sphere looks "locally" like flat 3-dimensional space. Its hard to visualize.
For floating point, it means you have greater
precision in hardware (allowing things like real physics and shapes to be modelled without noticable issues caused by subtle number creep). Since most systems use IEE-784 (64bit double precision floating point), it means a speedup to that software since you're not working with it as 2 32-bit operations.
Actually, most CPUs today (including G4 and P4) do double-precision in hardware. The G4 does 64-bit FP multiply-add with a throughput of one operation per cycle (I'm pretty sure the P4 does too). Even the loads and stores are operating on 64-bit chunks. Going to a 64-bit processor won't change any of that. The only thing different for FP operations will be (1) you can hold a heck of a lot more numbers in memory! and (2) it might be possible for extended precision (128-bit) to be done easily in hardware.
Ahem, "some" high-performance computing includes of course SMP code (using pthreads) and cluster computing, via PVM. I have designed algorithms and written industrial-strength commercial code that does both. It was fun! The general principles of CS are pretty straightforward, and follow naturally from mathematics.
As far as compiler design goes, I certainly haven't designed my own, but I do know how existing compilers work (you can't do high-perf computing without knowing that).
The point of my post was that everything from "insalling linux" to designing and implementing parallel high-performance code and understanding compiler design is not very hard to pick up on your own if you already know a lot of mathematics.
Funny, I'm someone with a lot of mathematical training (Ph.D. in Applied Math) but only a few courses in computer science. Somehow, I've managed to pick up a humungous amount of CS along the way, things like algorithm design and analysis, designing and coding industrial-strength C/C++ libraries and applications (yes I get paid for this), high-performance computing, OpenGL coding to roll my own volume visualization apps, doing all of my own unix system administration, setting up all of my own hardware...I've always thought that the best way to become really good at coding and software engineering is to first get a degree in mathematics. If you can do that, the rest is easy.
(Okay, I am a bit biased; I'm a college math professor, and in addition I do a lot of research and consulting related to numerical computation).
matlab actually will run under 10.2. What won't run is the graphical desktop interface, but the old-style terminal window interface will run just fine (just do "matlab -nojvm"). Take a look at my posting on macmerc.com for more info.
Sorry, I meant Fortran 77. Fortran 90 of course has pointers, but if I'm not mistaken Fortran 90 has facilities for avoiding pointer aliasing (perhaps that is what your code is showing? I've never written Fortran 90 code...)
...of the simple loop structures. In C, you can have a for( ; ; ) statement that does basically all sorts of weird crap in here ( ; ; ), and you can also do things like pointer aliasing (impossible in Fortran since there are no pointers at all).
Fortran loops, on the other hand, are very very simple - all you can do is increment the loop variable, and repeat. That allows for very heavy loop optimization by the compiler - comparable to what the best assembly programmers might be able to do. Furthermore, a Fortran compiler can more easily generate optimized loop code using vector instruction sets like Altivec or SSE2, whereas C compilers have a much harder time (again, because of the wide variety of loop structures possible in C, and things like pointer aliasing, etc.)
I'm *not* putting down colleges by ANY stretch of the imagination. I'm just saying that colleges tend to focus more on "pratical mathematics" (e.g. "here is the math you need to be an engineering tech"...) whereas a University math department will focus on "theoretical mathematics" (I feel silly typing that.. but you get the point). It really just comes down to what you're interested in learning, and what you want to do with that knowledge.
You seem confused about the difference between a university, a community college, and a 4-year undergraduate college. When you say "college," it appears you mean "community college." Most of what you say doesn't actually apply to 4-year undergraduate colleges.
There is no such thing as understanding mathematics without doing mathematics. You will never understand mathematics without knowing how to do mathematics, that is without knowing the tricks and techniques and methods for solving problems. Likewise, you cannot be functional without comprehension of the concepts, otherwise you hit a brick wall the second you try to do something different than what was assigned for your homework. I say this based on my own experience as a professional research mathematician, scientific consultant, and professor of mathematics at a small liberal arts college.