On Nov 30, 11:42 am, FredJeffries <fredjeffr...@gmail.com> wrote: > On Nov 30, 8:39 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > > > > > You've had this function for 13 years now and you STILL can't > > > calculate the area of a triangle with it. > > > Fred Jeffries who I respect: I'd like to think that's in the context > > of modeling Dirac's delta with triangles or radial basis functions, > > but what's important to describe of EF as plotted is this: removing > > all the space between the integers and plotting the elements in the > > range it would look like f(x) = x from zero to one, half a square and > > a triangle, but the F-Sigma Lebesgue integral of EF evaluates to one > > not one half, now that's the surprise. > > > EF: CDF: of the uniform distribution of the natural integers. > > Sorry, I can't decipher the above two paragraphs. All I see is > 13 years and 3 math degrees and still can't calculate the > area of a triangle
The area of a triangle is base times height over two.
A CDF ranges from zero to one over the range of the elements in the statistical/probabilistic distribution and is increasing. A uniform discrete distribution would have for any m, n that CDF(m+1) - CDF(m) = CDF(n+1) - CDF(n), constant monotone. Where EF is this CDF, putatively, 1/d = 1/d which is true, satisfying these requirements.
The notion of a uniform probability distribution over all the naturals is not necessarily intuitive, and I described how to build one in ZFC besides that EF has the concomitant properties of being the CDF of a uniform distribution of the naturals.
The reference to real functions modeling Dirac's delta a.k.a. the unit impulse function is that this function is a spike to infinity at zero, zero elsewhere as defined in the reals, whose integral evaluates to one. It's standardly modeled as triangles or radial basis functions or any other function really that have area equal to one and diminishes to point width at zero as parameterized by an unbounded free variable. Similarly Heaviside's step is so modeled with a parameterized arctan() and etcetera.
Here, EF's family of functions so modeling it is simply parameterized by d as it is unbounded.
Then, I went deeper to the foundations than that. Simply working up a mutual definition of the real numbers as constructively at once complete ordered field, and, partially ordered ring, with, rather restricted transfer principle, as for example we know from Cauchy/ Weierstrass and Bishop/Cheng, then, it's possible to have the comprehension of the function as a: primitive function, in fact _defining_ the unit line segment. A corresponding geometry of points and spaces to complement Euclid's of points and lines is initially defined, with a fundamental space-filling curve defining shapes via simple properties.
Then of course there are the set-theoretic results extra the number- theoretic results re: cardinality, an axiomless system of natural deduction with natural definitions of sets and ordinals following deductively gives a theory with an empty and universal set in the dually-self-infraconsistent dialetheic and paraconsistent.
So, yeah, in the time between noting the simple construction of EF and today, there's quite a bit of development. Dogged determination, as it were, for me partially satisfied in a great appreciation of the fundamental philosophical tenets.
No, I only have a Bachelor's of Science degree (in mathematics thank you and I know computer science). The guy who wrote a dissertation to convince soi-disant set theorists that half the integers are even has a Ph.D. from M.I.T. He got it for writing a dissertation in set theory that half of the integers are even.
I wonder your familiarity with Nyquist, Shannon, Huffman et alia and how Nyquist's sampling theorems in the discrete would apply, here to the continuous or to sets dense in the real numbers. I don't know of much work in that area.
And Fred Jeffries, I respect you even where you claim not to make sense of this, thank you please for not making no sense of it.
Basically these notions are very fundamental to what is continuous and what is discrete.
So, matter as the atom is particle and wave. What then is our simple point?