Friday, March 16, 2018

Working with tautological spaces

Reaching for more handling with mathematical expressions found out there was another level to what the math could do, which involved using complex identities I decided to call tautological spaces. Have had this approach since December 1999.

For example the simplest tautological space: x+y+vz = x+y+vz

Where usually show the modular algebra form, but will demonstrate like did long ago, where will show explicit. And use a demonstrative case where here is reference from 2014 for the modular algebra form.

With a bit of algebraic manipulation, have:

x+y = -vz + (x+y+vz), and squaring:

x2 + 2xy + y2 = v2z2 - 2vz(x+y+v) + (x+y+vz)2

Where can see all the growing complexity. Now that is STILL an identity of course, so x, y, z and v are completely free variables. But let's put some conditions on them.

Let: x2 + y2 = z2

Then can subtract to get: 2xy= (v2-1)z2 - 2vz(x+y+vz) + (x+y+vz)2

Where now can get to:

(v2-1)z2 = 2xy + 2vz(x+y+vz) - (x+y+vz)2

With some simple algebraic manipulation. And have the result that (v2 - 1)z2 - 2xy has x+y+vz as a factor, when x2 + y2 = z2, and can also easily realize that v is still free.

From the full explicit form can also just look at expressions that result from setting v, like v = 1, gives:

0 = 2xy + 2z(x+y+z) - (x+y+z)2

And of course then: -2xy = (x+y+z)(z-x-y)

Notice also are not of course limited to integers, as is a trigonometric result if z = 1.

That is: -2cos(x)*sine(x) = (cos(x)+ sine(x) + 1)(1 - cos(x) - sine(x))

And get a better feel for how is a true modular algebra when get to modular form, as does not care about Diophantine or not. Learned then that with an actual modular algebra is not just with integers. Should make sure to emphasize which is why am so doing here.

The modular algebra form of course is: (v2-1)z2 - 2xy = 0 mod (x+y+vz)

So what the modular algebra form gives is compactness, and is MUCH easier to manipulate.

Where can also notice a sense of how you get algebraic manipulations by how you set v, which can be a HUGE benefit. With explicit is just given, but with modular form you can do equivalent, as MAY find in pieces that way. Just then stitch together. But can also get an explicit result too.

The freedom to change v in ways that help analysis, was the point of the discovery for me, wanted something I could control. And that result of also giving algebraic manipulation of the conditional expression was really cool to finally realize. Was a big surprise for me when came across.

Here demonstrated the full complex identity where notice raised to a form where could meaningfully subtract the conditional expression which leaves a residue, which can be analyzed. With the full modular algebra form I call that the conditional residue.

That actually is an important limitation on this approach. Like is meaningless to subtract that conditional from x+y+vz as you do not eliminate any part of the conditional expression. So had to square to get to something useful here. And depends on the conditions being set on the variables.

In my experience, tautological spaces are effective when you maintain roughly the same number of terms or less than the conditional expression, and are ineffective if you end up with more.

The tautological space had to be an asymmetric form I learned in order for it to work. And have generalized methodology based on my analysis of what seemed to be necessary.

And you can do fewer variables by simply setting for example y or z, where can consider by setting z = 1, with full case of three variable quadratics, which recently posted on, as yeah for years that's what I did. Which probably helped me figure things out with a bit less complexity.

So highlighting a bit I learned just on my own. And do wonder how much more others could figure out, as am certain only scratched the surface.

James Harris

Tuesday, March 13, 2018

Attention reality and math usefulness

Maybe should address possibly vague notions about value of math discovery and how changed things for me. For one thing do believe there are people who think a committee of some kind of expert human beings MUST accept and promote you, or you have nothing.

That is of course not the case and actually has not been in the past. Where history is full of instances where ideas won out over time, and can say the web speeds that process up.

Math actually spreads on usefulness have witnessed with my own ideas.

However, humans can simply decide to not talk about certain things in certain ways. The math does not care. Still works and humans are attracted to things that work well for them. And some will even notice who made that possible.

So the discoverer gets attention. And in my case have not often handled it well, while decrying it not being of a certain type. I was very picky about what I wanted. And had not gotten it, and threw some tantrums in the past. Am over that now. Where also better appreciate what I have.

Today am potentially known like to say in over 150 countries where relied on something else for baseline. That something else being downloads of an open source project wrote for Java developers and put on SourceForge, an open source repository, back in 2004. Was a reaction to a lot of emotions. Wanted something concrete in a different way. Which helped me with confidence in my math too.

This blog with its old name though had visits according to Google Analytics from 125 countries at its peak that I noticed. Where is just what registered by however Google does those stats. And that was over a decade ago.

That messes with your mind. Do not behave the same trying to process. So I like to say POTENTIALLY known in over 100 countries, where rely then on two sources with data from others, so is objective. And just for my work, with no departments or faculty or other researchers or any academics actually, with any piece of it. Is all mine, alone. Which also messes with your mind.

There is a comfort in having other humans with you with certain things. So much harder when have to go it alone in many ways. While the shared value is still that of humanity with more useful information.

So yeah, alone in the role of discoverer, thankfully with a world of others in the benefits of that knowledge.

The math pulls attention in and of itself. And with good reason. Say some person wants to do something needed with some mathematical tool to solve an important problem, will pick best available if know of, and can, or not?

If you think not, I disagree, as from what I've seen--will pick best available.

Your discovery is not about a committee.

It is not up to other humans to decide the value of your work with their opinions, but for your fellow humans to show they value with use. We do not talk about calculus or algebra as about an opinion.

Obvious to most actually I think, but to the money obsessed? May believe money matters which is so much about celebrity in cultures where celebrity rules. Yet how many track those celebrities over decades? Their moments in the intensity of attention are often barely a few years, or often less.

In mathematics we know people over centuries and even millennia. There is no comparison from modern celebrity. And often am a huge fan, but also do watch modern celebrities come, and modern celebrities go. Audiences endlessly wanting the new. And why not? Our expectations keep rising as learn what is possible.

Was actually such a HUGE advantage for me escaping that noise. And learned what is actually necessary, and turns out it is not. Which has helped me as learned my role. My global position results from my discovery but did not come with an instruction manual. Have worked at figuring out.

No person who understands math, or pursues for the right reasons gets lost on those details for long. Math is not a zone for instant celebrity, and soon thereafter flame-outs. It is an arena where you can help people to know, for as long as there are humans.

Which is comforting really. And I think is important for me to emphasize. Human systems are what change with the new. Like when algebra arrived, mathematics changed rapidly for the better as our species learned more with greater efficiency. The humans back then? Even if their opinion was recorded do we care?

The great thing: great mathematical tools with which can know more. Discovery rules.

James Harris

Sunday, March 11, 2018

Some science speculation

Have tried to refrain from talking physics things much, especially while working at being certain of ideas. But now am more like, why not? But still will emphasize as speculation.

So yeah, was impossible not to think of quarks when considering the entanglement have talked quite a bit, including some recently may as well give a link there.

So we know algebraically there is an entanglement which I found first with cubics, which would maybe mean related equations help determine quark behavior.

But ended up focusing on quadratics as EASIER but realized might mean that there is a quadratic case in physics--undiscovered.

Best guess as to where? Electrons.

Could explain much about electron behavior, if we think is a single particle but is actually two VERY tightly coupled particles, where is just vastly closer than with bosons. Talking as if you have physics knowledge as easier for me, but also still dealing with basics most folks should have.

Then again, problem there is now maybe can develop better quantum chromodynamics where have been problems. Is more difficult than quantum mechanics, but of course is the physics of the nucleus.

That means could get everything from nuclear fusion, to more advanced nuclear weapons.

Like how casually I drop that one, where have pondered for over a decade. And for me, quarks making more sense!--Would be so HUGE is hard to really express. But does the math actually apply there? Takes the real deal physicists to be sure. Just have an undergraduate degree and from two decades ago.

Where I emphasize is speculation, but if true, and some other countries take me seriously, and work it, the United States could wake up to a world where other nations have better nuclear weapons. Where sky is the limit there.

Would change the global order of things, immediately. And as an American? Am not exactly excited at that possibility, of a humbled US having to change how it does all its international politics.

But science does not care. And technology does not care. Where hesitate to note, but my emotion in wanting my situation resolved is also not relevant.

History is full of such stupid stories. I do not want to wake up to learning should have sounded the alarm, when a nuclear explosion in my country reveals things have changed. Where may not even know who set it off. The potential technology may be there.

We think nuclear arms need uranium, like to get to plutonium, but what if they do not?

Could be a simple innovation maybe, if have the math you need, to get the physics, and entire world different as a result. Ad then the United States no longer the dominant world power.

Since am speculating may as well give scarier scenario where wouldn't have to worry about delivery, if weapons didn't give off radiation. Then could just use a global shipping company.

Could get REALLY wild speculating as hey, I like to write and could be a fiction scenario some day, but here feel like made my point, so will stop myself there, as could make it scarier.

Where think is understandable would want to be VERY sure about my math before putting such speculation out. Think have said some before and pulled back. But do have these innovations with math which could simplify quite a bit. History shows such things cannot be simply ignored.

Why did I spend so much space on negatives though? Just mentioned maybe nuclear fusion, which would just be cool, I think, and went on and on about other. Guess is too easy to focus on negatives for me? But maybe is natural here.

Such things have pondered for years. The dragging from mathematicians I now believe is just typical against such stunning innovation as have found where talk example with rideshare on my other blog.

That the math I found might be used by our reality though? Of course as someone with a physics degree find that intriguing, but also note, is speculation. May as well put down though, like with the electron, if true?

Then THAT would be yet another thing for me in the history books. And one of the greatest as well.

Yeah definitely need that in writing, just in case am right.

James Harris

Saturday, March 03, 2018

When proof rules emotion

One benefit appreciate now more than could in the past was the questioning of certainty. Before did have situations where thought that proof was about a certainty of having checked everything. And could go over mathematical equations over and over again, thinking were correct. But thankfully had that collapse more than once.

People can talk mathematical proof, yet how do you know something is a mathematical proof? Answering that question for me became much about identities.

But still there is that fear of being a person who can look at something believing is true, and mind can play tricks on you. Which I think is how the mathematical discipline has turned to other eyes and checking by others. But human fallibility remains possible. Especially with complex arguments where I DO think computers will handle all proof checking, eventually.

For me though practical reality after looked into computerized proof checking, years ago, even contacting an expert in the area and getting nowhere, pushed me first with functional ideas, and then to number authority.

So defined mathematical proof functionally. But also now like to look at things like:

(462 + 482 + 722)(1722 + 258+ 430+ 6022 + 17622) = 

            615+ 30752 + 141452 + 159902  + 1884972   =  774*210

Which is my favorite one lately. It was found using the BQD Iterator. Where was easy to figure out, but the cool thing is the numerical perfection. Once calculated you know must be perfect. Checking is easy enough with modern systems.

Compare though with:

(x2 + 2y2)(u2 + 3v2)(x'2 + 4y'2)(u'2 + 5v'2) = p2 + 359q2

And finding integer solutions for all the variables is easy.

x = 1, y = 2, u = 2, v = 2, x' = 3, y' = 2, u' = 4, v' = 2, p = 358, q = 2

First iteration is easiest.

Both examples also should say demonstrate modular in that am using BQD Iterator with basic form u2 + Dv2 = F.

So that module is used twice for first, where expanded with a technique for having as many squares as wanted, and four times for second. So went less fancy with second example and left in variables versus putting in solutions like with prior one.

But feels so different to me regardless. There is logic and there is the feeling around looking at things. Now a step backwards to symbols, and is not so easy for me at least to feel certainty. Though mathematical proof IS there. And is interesting I think the emotion can feel despite the proof. Which in my experience?

Is great! The proof does not care about my emotion. When I do the math, it works.

Contrast with emotion the other way, where for instance trusting humans, you find that trust disappointed. Would rather be skeptical and find my skepticism overturned than go that route, as have done before.

I prefer to talk to the math.

The math will never make a mistake. The math will never be wrong.

The math is always right.

To me, when proof rules emotion you know as you go upside down. Trusting not your faith in your ability to look over the mathematics. Which is weird, huh?

So I go through the checks. Like use my definition of mathematical proof to check, every aspect from beginning to end. So I know have a proof, intellectually. Logically it is perfect. Yet part of me still is hesitant, until the relief, when run some numbers.

That burst of positive have found works vastly better to continue, than the other ever did.

Numbers rule.

James Harris

Tuesday, February 27, 2018

Puzzling over modular effectiveness

Modular is a concept with wide applicability in a variety of domains. And was lucky that weirdly enough after Gauss, development of modular with mathematics apparently lagged. Which I know because was able to introduce first true modular algebra, which then not surprisingly, revolutionized much.

However modular algebra was not my first step into modular with mathematics, as was kind of a relief to notice was key to an earlier puzzling to me result, as SO simple. Where considered packing of spheres in a modular way. Then it makes more sense that I could easily prove that distorting that one piece from hexagonal close packing would lead to a lower density. Since full space is filled with same modules, where any distortion leads to lower density, had an easy proof of optimum packing.

Wrote up a two page paper. Having that rejected by an editor for the Proceedings of the AMS as too simple is just kind of telling to me now.

Would go on to develop a full modular algebra where like to give as easy for context:

x+y+vz = 0(mod x+y+vz)

Which is the simplest form I found with some research, and I call, a tautological space. Have created terminology over more advanced forms.

And have noted that remarkably, can use to study algebraic expressions where can give things which are algebraic manipulations, done by the modular algebra. As the modular algebra is doing all the work, is perfect and can reveal things humans never found before.

Who knew? How could anyone? Watching modular algebra do algebra? Is surreal.

My best example is what I call the BQD Iterator lots. A very simple relation which I've used quite a bit, which the modular algebra gave me, with my method for generally reducing binary quadratic Diophantine equations. My method remarkably is better than that from Gauss.

BUT can now also note that Gauss was am sure doing something I don't need to do any more--complex algebraic manipulations.

Letting the math do the work is more effective and complete. Is faster, and is of course easier.

The math easily checks infinite possibilities in a modular way. The math is complete intelligence.

Those are the kinds of things that tend to show you where human beings will go, in time.

Why do we not notice certain things? Probably is about how we are built.

Of course, how do we know what we do not know?

With the math performing better in a way that humans have tried with our intelligence, can sense are watching an intelligence, but is an infinite one which is perfect, makes no mistakes, and gets to the best answer.

Which then lets us look at what we humans did not find on our own, and wonder.

Could go on and on about it. And have deferred on what I call the social problem as to acceptance until 2028, which is not that big of a deal for me. Figured out my modular approach to packing sphere back in 1996. And had finally the evidence that the method I found around what I decided to call tautological spaces, was doing algebraic manipulations back in 2008.

And that was September 2008, with: Quadratic Diophantine Result

Figure that is decent matter-of-fact naming for that post. At THAT time did not say to myself, have a true modular algebra which can do algebraic manipulations. Didn't even realize had actually handled three variables as only focused on two. That is wild, eh? Recently talked the three variable.

So much did not know then, where would learn in time. At the time, was more like, wow, what's this thing? That's SOMETHING when you first realize that the mod can just go away. Where thank God that was years ago, and have settled down much since then.

Takes years to properly process discovering such things have learned with myself, as slowly you adjust, to what changed for you so quickly.  So much a sense of things, of what I thought I knew got shattered, so quickly.

There was often elation as well, but learned to be suspicious of it. Like to say that math and emotion do not mix well. But that effort is more to maintain an even keel. As can end up with massive high's and intense low's which to me? Is weird.

Took YEARS as slowly rebuilt a firm sense of reality. And noticed was better than before, but so much work to get there. The world seems more crisp to me now, often. And appreciate so much more as well.

And web has helped me greatly and lucky was here evolving as I found these things. As web allowed me a more gradual sense of certain things, while being aware, yeah, people around the globe...what have you all been doing? Web analytics only tell me so much. Guess you would be the people reading this post though.

Where is strange yes, can have simple but powerful ideas, but one reason became fascinated with modular elsewhere. Like would stare at cargo ships loaded down with shipping containers when lived in San Francisco. And would ponder.

Noted was lucky for me that further development of modular seemed to have waited until now. Still there has been some resistance, as now over 18 years since I introduced what I call tautological spaces. With over 21 since my first paper with modular ideas and packing of spheres.

Regardless the ideas are flowing, which web metrics lets me know. So I guess something has changed for humanity allowing the possibility and simply live in the right time.

My view is the web has made the difference as notice, how are you reading these words?

James Harris