Rather than ask if Intellectual Property is important. Is brand name recognition important to your business? Is marketing important to your business? Are brand names important to you as a customer? When you work, is it important for you to get paid? Is your company name important for your business? Should a company be paid more for higher quality products? Should a company be paid more for innovative products? Should a competing company be allowed to reverse engineer your product and market the exact same thing at the cost of manufacturing until you go out of business?
If you have not tried it, try to inline your simple functions. The cost may be your function call overhead. I am assuming you are using C or C++. Also, it may help to look at my post on the same level as this one.
Information for information works as well as money. Thanks for the iterative CORDIC algorithm information and clarifying reformulating the problem to not use trig was an option. I was using an Intel 386 processor, and I may have purchased a coprocessor, but I don't recall for sure. If I did, my measurement may be of the coprocessor time rather than the algorithm. I am also not sure if I compiled with coprocessor support or not. Thanks for pointing out my possible error. The method in the tan() implementation was to use Chebyshev polynomials (See http://en.wikipedia.org/wiki/Chebyshev_polynomials#Trigonometric_definition) to make the Taylor's series converge faster (See http://en.wikipedia.org/wiki/Approximation_theory). It required some rescaling as well. First, find the Taylor's series of the tan(x) function to the number of terms needed and then choose the Chebyshev polynomials T_i(x) with values smaller than the approximation accuracy you need. Use the largest T_i(x) first since T_(n+1)(x) T_n(x).. Solve for the highest power of "x" and use it to eliminate the power of "x" in the Taylors series. You may have to add up the errors of each Chebyshev polynomial you use. For instance, T_4(x) = 8x^4 -8x^2 + 1 so x^4 = (1/8) * (T_4(x) + 8 x^2 - 1). Use x^4 = 8 x^2 - 1/8 to eliminate x^4. This reduces the number of multiplies and additions needed for the evaluation of the Taylors series to a certain approximation. I did also see a case where someone used fractional powers of "x" to possibly reduce the complexity still further. I am recalling this from memory so a I may have got a few details wrong. One of the 3 books that mentioned this method was a Math book by Korn and Korn. There is a Dover edition.
Back in the late 80's/ early 90's, I considered using a lookup table for some trig calculations. The first thing I did was check the computation time for a "multiply" and for a tan() calculation in a loop dividing by the number of iterations. I was supprised to find the cost of a tan() computation was only the cost of 8 multiplies. I am curious as to how you are beating this using vector math. Are you are using normalized 3 vectors and the dot and cross products to compute sine and cosine? And maybe sin(x) = sqrt( 1 - cos(x) * cos(x)). While I was using closed source code, a friend had access to the open source implementation of tan() and printed it out for me. They might have been different. I might still have it. It was an interesting technique. I eventually found a book that described it. I believe I still have the book. How much would you pay for the information? I like people who don't like math, they pay well!
A year and a half ago I watched a public television show which documented the creation of compatible organs. They stripped an incompatible mouse heart of all but the scaffold which was translucent and then seeded it with stem sells from the mouse they wanted to transplant it into. The cells grew into heart cells and the heart started beating in the lab environment. When transplanted into the mouse it worked fine. They suggested that the same thing could be done with pigs hearts to make them compatible with humans as the scaffold was not what the immune system of the body attacked. Not sure if this has already been done. They also used an inkjet head on a 3D printer to print a mouse heart of the scaffold material, seeded it with stem cells, and it started beating in the lab environment. As I recall, they had also done the similar things with lungs, kidneys, and other body parts. This was the show. "Replacing Body Parts" Aired January 26, 2011 on PBS http://www.pbs.org/wgbh/nova/body/replacing-body-parts.html Transcript on the website.
The actual problem statement for most of us is how to solve math problems while simultaneously making money proportional to the usefulness or proportional to the time spent on this problem and other unsuccessful problem attempts. Otherwise, we end up going out of business.
"Public domain" means any company can produce it and market it anywhere in the world. So, it would not be revenue neutral for the government unless the "Free Trade" treaties are abolished. "Public Domain" should maybe be split into "Public can produce it in the country that funded the invention" and true Public Domain. Taking away the 20 year patent is the same as stealing from small companies and individuals and giving the benefits to the large corporations. A 5 year limit on patents would make it impossible for small companies and individuals to get funding from banks to produce the product they invented. It would also reduce the incentive for a large corporation to purchase the patent at anything like the development cost as they can wait it out. All small companies would have to forget about research and focus innovation on manufacturing, marketing, and distribution.
OpenRave has an open source IKS solver http://openrave.programmingvision.com/en/main/openravepy/ikfast.html http://openrave.programmingvision.com/en/main/_modules/openravepy/ikfast.html which is supposed to create optimized C or C++ code for the IKS. Of course you can get the IKS solutions for most common manipulators from text books.
The 300 farmers should start a company which hires a biotech firm to put a patentable marker modification into their seed. Then, if it gets into their neighbors fields or into the fields Monsanto uses to grow its seeds, they use their company to sue Monsanto and their neighbors for patent infringement. They could also have Monsanto recall all of its seed.
Also, the comparison blogs are less likely to have evaluated the particular part number in the store resulting in reduced sales. When I shopped for a camera, I looked at the physical stores, which only had limited comparison information on the display cards. I also looked online at http://www.photographyblog.com/ and other comparison sites. The blog was a big help in narrowing down the options to the camera I wanted. In the end, Target and Office Max did not carry the camera I wanted. I could have bought it cheaper online but went to Best Buy to get it sooner. The moral of the story is that the online blogs were more of a showroom for the product than the physical stores.
You might try suggesting something like the following: If your motivation in the SOPA/PIPA legislation was to stem the tide of counterfeit goods by blocking the marketing of suspected counterfeit goods on the internet and the ability to order the suspected counterfeit goods via the internet, then I would like to suggest an alternative. In my view, preventing traffic in counterfeit goods is primarily a smuggling prevention problem. Assumptions: 1. It may be legal in some countries to make counterfeits of materials/works which are protected by copyright in other countries. 2. It is still illegal to smuggle the counterfeit works into the country where it is protected by copyright or other intellectual property laws. 3. U.S. Customs and Border Protection (http://www.cbp.gov/) has the task of regulating the flow of goods across the border. 4. The ports or entry which include internet-pipes-entering-the-country, airports, sea ports, and the border roads intersecting the border, are the natural place o interdict products ordered or smuggled via the internet. 5. Some items are for profit some are not for profit. 6. Some items are delivered over the internet others are not delivered over the internet. The copyright holder could give a vendor a unique digitally signed certificate for each copy of a electronic song or book for which a royalty had been paid. The U.S. Customs would provide one of the keys for the signature and the copyright holder would provide the other key. When a sale occurred over the internet and before the download could occur, it would have to pass U.S. Customs. To pass U.S. Customs the vendor would have to provide evidence (The certificate from the copyright holder.) that the royalties for the copy of the electronic song or book had been paid. The copyright holder would have to let U.S. Customs know when the certificates were issued or the U.S. Customs would have to verify the certificate with the copyright holders website as listed in the certificate. U.S. Customs and Border control could test vendors sites by placing an order and verifying the certificates. If the vendor did not supply the certificates the U.S. Customs would then have to notify the website/company that their products were held up in customs and post a banner across the internal-to-the-U.S. website stating that the website is not compliant with U.S. Customs and shipping of the product may be held up in customs pending proof of compliance with U.S. intellectual property laws. Possibly, new orders from within the U.S. could be disabled. Possibly, SSL (Secure Sockets Layer starting with https) sites related to the ordering of products could be blocked. Since purchasers don't want to send unencrypted credit card information over the internet, this would block new orders. Alternatively, once the banner is there, state that future orders through the website may make the buyer/downloader subject to payment of the royalties to the copyright holder. Perhaps, require that the buyer purchase the royalty certificate from the copyright holder independently of the download/purchase from the vendor. Perhaps hold up the credit card transactions at the border pending proof of intellectual property law compliance. http://en.wikipedia.org/wiki/Digital_signature http://en.wikipedia.org/wiki/Digital_Signature_Algorithm Similarly, the production and sale of drugs would require signed certificates of production and testing from the approved factory and labs, which tested the products, to be allowed entry into the U.S. I would also ban the practice of rebranding without verification of the product quality via an independent lab. I believe that newspaper marketing, magazine marketing, radio marketing, TV marketing, and physical stores are not required to verify the accuracy of statements in their advertisements. Why should the internet marketers have to do so? Often times, the internet marketers are intermediaries or resellers of used goods and may not have the ability to verify everything either. I believe it is important to not block sites dedicated to free speech and to focus the legislation solely on the smuggling of items affecting commerce. I also do not believe immunity from prosecution should be given to any commercial entity for blocking the access to the market of another commercial entity. There is a conflict of interest.
As I recall, the MySQL server install is only free for non-commercial use. Read the license agreement. Non-profit may not mean non-commercial so check with your lawyer. I also recommend using postgresql. The online documentation (http://www.postgresql.org/docs/9.1/interactive/index.html) is great so no need to pay for any training. I last used postgresql four years ago for a GIS application.
I learned swing dancing, country western dancing, and a little Latin dancing after undergraduate college. I wish I would have learned it sooner. But then, I was working at 14 (20 hrs/wk government program converting railroad beds to horse and bike trails). If you choose a dancing camp, I would recommend swing dancing over ballroom and that you write-down/diagram the moves after you learn them. Also, find a lady to practice with between lessons. I never went to any camps, but I would suggest considering debate camps, public-speaking camps, self-defense (karate, judo) camps, swimming camps, start-your-own-business camps, and creative-writing-publishing camps as it does not sound like you need much help in studying the tech stuff and these are also useful things to know. Of course, your goal may be to network with people with the same interests so you may want to go to the tech camps.
I also recall trying to make notes so fast that I could not focus on the lecture until after the class. The best lecture classes covered material from many different text books and research papers so they did not duplicate the text book. My best class was in grad school where the professor switched the class from meeting 2 days a week for 1.5 hours a day to 3 days a week for 2 hours a day. The first three hours of the week was spent on lecture. For the second three hours of the week, the professor would pull a card from his deck of cards of student names. If your card came up, you were required to present a problem solution / proof on the chalk board. The rest of the class was required to try to find errors in your proof. Solved problems were immediately retired. It was fair to find the solution by literature search or by solving it yourself as both are valuable skills. Weekly problem sets would consist of about 25 problems and would be in play for 2 - 3 weeks. The professor scored the problems as to difficulty from 1 (least difficult) to 5 (most difficult). The problems were non-trivial. I recall only one person ever solved a 5 point problem. I recall solving mostly 2 and 3 point problems. I may have solved a 4 point problem or two. I progressed from falling asleep reading a linear algebra book at the beginning of the semester to being excited reading the same linear algebra book at the end of the semester. Which is to say, the material sunk in. For me, the material only sinks in when I struggle trying to solve problems. It would have been educational to see examples of how people used the knowledge from the class in a profitable business, perhaps in a handout.
Several professors told me girls tend to lose interest in STEM in the 6th grade. I conjecture that the 6th grade is the level at which their parents cannot help them understand the math as their parents don't remember it. I suspect parents often tell their kids that they never needed math so it is by implication not important. Since tech jobs are fairly high paying jobs, this would also explain why math skill might be correllated with poverty.