It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. For those interested in the theory, the fact that cases 1, 2 and 3 yield convergence to the Gaussian distribution is a consequence of the Central Limit Theorem under the Liapounov condition. More specifically, and because the samples produced here come from uniformly bounded distributions (we use a random number generator to simulate uniform deviates), all that is needed for convergence to the Gaussian distribution is that the sum of the squares of the weights -- and thus Stdev(S) as n tends to infinity -- must be infinite. More generally, we can work with more complex auto-regressive processes with a covariance matrix as general as possible, then compute S as a weighted sum of the X(k)'s, and find a relationship between the weights and the covariance matrix, to eventually identify conditions on the covariance matrix that guarantee convergence to the Gaussian destribution.

While having myself a strong mathematical background, I have developed an entire data science and machine learning framework (mostly for data science automation) that is almost free of mathematics, and known as deep data science. You will see that you can learn serious statistical concepts (including limit theorems) without knowing mathematics, much less probabilities or random variables. Anyway, for algorithms processing large volume of data in nearly real-time, computational complexity is still very important: read my article about how bad so many modern algorithms are and could benefit from some lifting, with faster processing time allowing to take into account more metrics, more data, and more complicated metrics, to provide better results. It looks like f(n), as n tends to infinity, is infinitely smaller than log n, log(log n), log(log(log n))), and so on, no matter how many (finite number of) nested log's y

Some leading scientists like Sir Roger Penrose even argue that Goedel showed with his Incompleteness Theorem that today's computers can never reach human level intelligence or consciousness, that humans will always be smarter than current computers or any computer algorithm can ever be and that computers will never in the true sense of the word "understand" anything like higher level mathematics, especially not mathematics that deals with trans-finite sets and numbers. Many famous mathematicians (and physicists) created fascinating new theories and discovered deep and far reaching mathematical results. He proved this by using his famous "diagonal" construction (see pic below) that showed that any supposedly complete enumerated list of irrational or real numbers R will always miss some irrational numbers, thereby proving that a complete enumeration of the real numbers by the natural numbers is impossible. Cantor has actually shown that there are even an infinite number of ever bigger infinities by showing that the set of all subsets of any given infinite set is always substantially bigger (cannot be put into a 1-1 relation) than the set itself.

Think of set theory as a hinterland containing strange creatures capable of doing unknown things. Left out in the cold, Gödel's incompleteness made its home in set theory, the formalized study of collections of objects and different levels of infinity. Think of it as a hinterland containing strange creatures capable of doing unknown things--the land beyond the wall, if you're a Game of Thrones fan. When he was just starting to read, at age 4 or 5, Friedman remembers pointing to a dictionary and asking his mother what it was.

A few important unsolved mathematical conjectures are presented in a unified approach, and some new research material is also introduced, especially an attempt at generalizing and unifying concepts related to data set density and limiting distributions. Finally, we provide an algorithm that computes quantities related to densities, for a number of integer families, including prime numbers, and integers that are sum of two squares. The last section discusses potential areas for additional research, such as a probabilistic number theory, generating functions for composite numbers (possibly leading to a generating function for primes) as well as strong abnormalities in the continued fraction expansions for many constants, including for the special mathematical constants (Pi, K, e, and other transcendental numbers) mentioned in this article. Moreover, the family of limiting functions n, log n, log log n, log log log n etc.

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. For those interested in the theory, the fact that cases 1, 2 and 3 yield convergence to the Gaussian distribution is a consequence of the Central Limit Theorem under the Liapounov condition. More specifically, and because the samples produced here come from uniformly bounded distributions (we use a random number generator to simulate uniform deviates), all that is needed for convergence to the Gaussian distribution is that the sum of the squares of the weights -- and thus Stdev(S) as n tends to infinity -- must be infinite. More generally, we can work with more complex auto-regressive processes with a covariance matrix as general as possible, then compute S as a weighted sum of the X(k)'s, and find a relationship between the weights and the covariance matrix, to eventually identify conditions on the covariance matrix that guarantee convergence to the Gaussian destribution.

A few important unsolved mathematical conjectures are presented in a unified approach, and some new research material is also introduced, especially an attempt at generalizing and unifying concepts related to data set density and limiting distributions. Finally, we provide an algorithm that computes quantities related to densities, for a number of integer families, including prime numbers, and integers that are sum of two squares. The last section discusses potential areas for additional research, such as a probabilistic number theory, generating functions for composite numbers (possibly leading to a generating function for primes) as well as strong abnormalities in the continued fraction expansions for many constants, including for the special mathematical constants (Pi, K, e, and other transcendental numbers) mentioned in this article. If the infinite sequence of positive integers s(n) is growing slowly enougth without sharp gaps between s(n 1) and s(n) (this assumption applies both to prime numbers and numbers that are sum of two squares) then one would expect that the first element in each successive absolute forward differences E(S), E 2(S), E 3(S), ... is either 0 or 1.

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. Case 1: a(k) 1, corresponding to the classic version of the Central Limit Theorem, and with guaranteed convergence to the Gaussian distribution. Case 2: a(k) 1 / log 2k, still with guaranteed convergence to the Gaussian distribution Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.) Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.)

A few weeks ago Lydia Dishman wrote a fantastic piece characterizing the top tech companies as defined by the "quality" of their talent using the Paysa CompanyRank algorithm and how these companies change in rank over time. Figure 1 depicts the Paysa CompanyRank time-series of Uber, Facebook, Google and Zynga over time. If a company loses those from top companies or begins hiring from less quality companies, their score (and relative ranking) will decrease. The analog of top publishers linking to other top publishers holds with talent moving from one top company to another.

And yet, Yokoyama and Patey's proof shows that mathematicians are free to use this infinite apparatus to prove statements in finitistic mathematics--including the rules of numbers and arithmetic, which arguably underlie all the math that is required in science--without fear that the resulting theorems rest upon the logically shaky notion of infinity. This was partly because earlier work proved that Ramsey's theorem for triples, or RT23, is not finitistically reducible: When you color trios of objects in an infinite set either red or blue (according to some rule), the infinite, monochrome subset of triples that RT23 says you'll end up with is too complex an infinity to reduce to finitistic reasoning. In a shocking result, Gödel proved that no system of logical axioms (or starting assumptions) can ever prove its own consistency; to prove that a system of logic is consistent, you always need another axiom outside of the system. Gödel's theorem meant that Hilbert's program was doomed: The axioms of finitistic mathematics cannot even prove their own consistency, let alone the consistency of set theory and the mathematics of the infinite.