Axiom Submission: Fill & Download for Free

tl;dr: Make sure your formula is original and accurate. Then publish it on ArXiv.The first step when discovering a math formula is to make sure that what you discovered is unique and not proven before. To find whether it has been discovered, use academic databases such as JSTOR, Gale, EBSCOhost, and Internet Archive (I have found better databases — take a look under edit 3). If it is original, you can proceed.The second step is to publish it and release it for others to review and improve. I disagree with the other Quorans answering this question, saying that you should publish it to Quora or any other publishing journal/answer service. Most of these services are not reliable and may lead to having your formula stolen. Instead, you should publish your idea to arXiv.org e-Print archive. ArXiv is a highly-automated electronic archive and distribution server for research articles. It covers several fields of study such as physics, mathematics, computer science, nonlinear sciences, quantitative biology, quantitative finance, statistics, electrical engineering and systems science, and economics. The reason why you should use ArXiv is that it is maintained by Cornell University Library along with the arXiv Scientific Advisory Board and the arXiv Member Advisory Board. ArXiv is used because it allows for you to publish your research quicker than publishing with peer reviewed journals. ArXiv allows you to share your work openly to the world whereas academic journals may share your work to a smaller audience.To publish your formula on ArXiv, first, register for an account. Get started on arXiv. Type up your formula and proofs as either a TeX File or as a PDF. Make sure the metadata in your files is visible when transferred to ArXiv. Then follow the guidelines on the following link to make sure your formula is published without any missing items: Submission Guidelines.Once you have published your article, a group of moderators within the mathematics field will review your work using The arXiv moderation system. If it is passed, then your math formula will be released for everyone to know under your rights. Your submission will be timestamped, so if anyone else breaks the copyright laws, you will be able to legally take action against them.Publishing on ArXiv gives you several benefits. Your article may be published later on in academic journals and your paper may be very influential. Many mathematicians have posted their discoveries on ArXiv and have won mathematical awards such as the Fields Medal. You, too, may win awards for discovering and proving new formulas.I hope your paper is published and will be very influential! I wish you the best of luck in your mathematical endeavors!Edit 1: If you want to take a look at other mathematical papers, here is the link. Mathematics Archive.Edit 2: Responded to some user comments and updated answer.Edit 3: Found some better academic databases. These databases are specifically for mathematics while the ones posted above are more general and cover a broader set of fields. You can use these to find whether your findings are original or to explore some new concepts.1. Metamath Proof Explorer:The Metamath Proof Explorer has over 20,000 completely worked out proofs, starting from the very foundation that mathematics is built on and eventually arriving at familiar mathematical facts and beyond. Essentially everything that is possible to know in mathematics can be derived from a handful of axioms known as Zermelo-Fraenkel set theory, which is the culmination of many years of effort to isolate the essential nature of mathematics and is one of the most profound achievements of mankind. The Metamath Proof Explorer starts with these axioms to build up its proofs.Proof Explorer - Home Page - Metamath2. Mizar Project:The Mizar project started around 1973 as an attempt to reconstruct mathematical vernacular in a computer-oriented environment. Its database contains more than 9,400 definitions of mathematical concepts and more than 49,000 theorems.Mizar Home Page3. vdash:vdash is a wiki of formalized math which aims to lower the barriers to formalization. It's also one approach towards a math commons — a site with all mathematics in one place, in a common language, and in a way that anyone can edit. The main ingredients are an interactive theorem prover, a library of computer-checked mathematics, and a wiki web interface. Formalized mathematics can provide greater certainty, more detailed explanations, and instant verification. Furthermore, formal proofs can be manipulated by computers in a modular and interactive manner, and could lead to many exciting future directions in education and mathematical practice.vdash: a formal math wiki4. PlanetMath:PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members.PlanetMath.orgGo check out these online resources and learn something new!

The smooth functions [math]f:\R\to\R[/math] do make up a vector space over [math]\R[/math], with pointwise addition and multiplication by scalars.As such – a “naked” vector space, with no additional structure – it does indeed have a basis, but I wouldn’t try to “find” one. This is an infinite-dimensional vector space, but even worse: its dimension is [math]2^{\aleph_0}[/math]. To prove this, observe (for example) that the smooth functions [math]e^{ax}[/math] for arbitrary [math]a[/math] are linearly independent.So your basis is uncountable, and it probably requires the axiom of choice to even prove that it exists. An explicit construction is, in all likelihood, impossible. You simply need far too many functions. This is reminiscent of a Hamel basis for [math]\R[/math] over [math]\Q[/math].You many be interested in a different notion of “basis”, one that is more suitable for function spaces. That would be a set of functions which allows any smooth function to be represented, in an appropriate sense, as an infinite linear combination.And now you are facing a different problem: for this to make sense you need some notion of distance, or convergence – a metric, a norm, an inner product, something. However, none of the usual norms on function space are suitable here, since you didn’t place any limitation on the behavior of the functions at infinity. They are unbounded, they don’t have finite [math]L^2[/math] norm or any other norm. I doubt there’s a useful way to tame that space into submission with the usual Fourier apparatus, and at any rate, without specifying which metric you’re interested in, there’s no way to answer the question. A basis, in this sense, exists for richer structures than a mere vector space.As Sridhar pointed out, we could simply consider pointwise convergence as our sense of convergence of infinite series. But again, I wouldn’t expect you can find any explicit uncountable basis in that sense either.

It would most likely look very much like MMA does today. There is an argument made that in the absence of rules, people will most likely revert to "cheap shots" like groin strikes and eye gouges. The problem with this argument is that these attacks are generally useless unless one has the upper hand and has a dominant position. A strike to the grain is easy enough to block or fend off the same as one would do with a punch or kick to the stomach, an eye gouge is easy enough to block the same one would block a punch to the head, the same applies to a slap to the ear to damage the equilibrium.What MMA training teaches, is similar to what BJJ and submission grappling teaches, and that is, position for submission. In the case of MMA or even streetfighting, the axiom would be position before victory.All of the cheap shots can be effective, dirty fighting is a thing, but without the sound basics, these weapons are not fight stoppers.That being said, these techniques can be very effective in a self-defense, surprise attack situation, but this is a very different situation from fighting. Fighting and self-defense are mutually exclusive things. In the end, though, you're better off knowing solid fighting technique.