The talk is divided into two parts, the first part will show how to use the bootstrap method to get a formula to calculate the arguments of and a spectacular formula for the ‘th zero of the Zeta function using LambertW function.
The second part will show new formulas for primes like
At the same time, the prime is well approximated with the formula
In fact, the prime is given exactly by
Using the bootstrap method, one can do the same for many primes. This leads to a conjecture about the representation of all the primes using and a simple function of . And speaking of primes, I will show a set of formulas that can generate an infinity of primes using a recurrence equation function. If is the rounded value of and , then will generate an infinity of primes, beginning with
Here, the exponent can be made as close as we want to .
Looking under the hood of Artificial Intelligence: About cookies, blood, language, and some mathematics
What does the little girl is asking to the little boy that he took from a jar on a shelf in the kitchen while his mother is washing dishes unaware that the sink is overflowing? And, while I am at it, what is this little twisted tube moving weirdly with a wire inside that suddenly disappear out of view provoking great despair for those who were looking?
The first sentence of this abstract describes a common test used to detect dementia and the answer is in the title of this presentation. The second one is the partial description of a cardiac catheterization surgery on newborns using contrast agent. These two sentences themselves are also quite obvious examples of why it is still difficult for computers to understand natural languages (although I am sure you struggled a bit too). They are also examples of two research projects using Artificial Intelligence (AI).
Where are the mathematics? They are, of course, under the hood of AI, and its application to solve these problems here at ÉTS.
I don’t want to sell the punchline so I will throw at you two images and two formulas here.
(1)
Formula (1) (taken from [1]) is an asymmetric coverage measure (inspired by [2]) used to distinguish the discourses of patients during the “Cookie Theft Picture Description Task” [3]. is a function that measures the similarity between a referent, (healthy population) and a subject, (with cognitive decline). The parameters are used to associate a weight to each simplified linguistic pattern, , that we identified as relevant for the task.
Figure 1: Patients’ discourses evolving through time [4]
Figure 1 illustrates a Principal Component Analysis (components 1 and 3) of patients’ discourses evolving through time (10 years). The label near each point indicates the participant ID-interview number. Interviews 1, 2 and 3 were held in 2005, 2012 and 2015, respectively (see [5, 6] for the data). The hue difference indicates normal or cognitively declined aging processes. Circle, square and rhomboid markers indicate healthy control (HC), mild cognitive impairment (MCI) and severe CI, respectively, at the time of the interview.
(2)
Formula (2) points to the well-known difficulty of analyzing symbolically natural languages by associating binary trees to sentences (= parsing trees). On this account (presented in [7] for natural languages), our first sentence can theoretically produce an extravagant number of syntactic trees; while humans discard most of them without even thinking, computers find the task phenomenally troublesome.
Figure 2: Left: X-ray frame without (1) and with (2) contrast agent (from [8]). Right: Tracking of cardiac artery during movement (from [9]).
Finally, figure 2 illustrates the challenges of tracking coronary arteries to help surgeons during cardiac catheterization. There are two challenges here. First, as in the case of sentences analysis, you must be able to recognize the real vessel within the noise surrounding it (two figures on the left, from [8]). Second, the patient is breathing and his heart is beating (hopefully!), so that twisted tube is moving (two figures on the right, from [9]).
My intention is to use these applications to introduce you to natural language processing and machine learning. We will finish our journey pointing to a sample of research themes related to AI at ÉTS, and why education in mathematics and ethics is so important in this new world obsessed with AI.
Computer Exploration in Algebraic Combinatorics via SageMath
This talk is divided into two parts. The first will be an introduction to the SageMath project from a personal perspective. From the SageMath website:
SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Access their combined power through a common, Python-based language or directly via interfaces or wrappers.
Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.
SageMath has become an essential tool in my field of research, algebraic combinatorics.
The scope of algebraic combinatorics has grown so much as to encompass any area of
mathematics “where the interaction of combinatorial and algebraic methods is
particularly strong and significant” [Wikipedia]. This significant interaction between combinatorics and algebra is what makes many of the problems in this field amenable to computer exploration.
The first part of this talk will focus on the history and some features of the SageMath project. The second part will highlight a few examples of how computer exploration is used as a research tool in algebraic combinatorics.
There are now several comprehensive programs that, given a floating point number such as , can return concise non-float constants such as that closely approximate the float. Surprisingly often such a result is the exact limit that is approached as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation.
Some such programs can be used freely online. For example:
Inverse Symbolic Calculator by Simon Plouffe, Jon and Peter Borwein, et al,
Wolfram|Alpha,
On-line Encyclopedia of Integer Sequences by Neil Sloane and Simon Plouffe.
Other such programs are functions built into a computer algebra system. For example:
the Maple identify function adapted by Kevin Hare from Alan Meichsner’s M.S. thesis,
the identify and findpoly functions in MPMath, hence also SymPy and Sage.
Other such programs are freely downloadable. For example:
Plouffe’s inverter Maple program,
the Java MESearch program developed by Jon Zurutuza Salsamendi,
the C ries program developed by Robert Munafo,
the Mathematica AskConstants program developed by me.
The presentation will demonstrate some of these programs and describe their varied underlying algorithms. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the types of constants that they can return.
, the primes and the LambertW functionand a spectacular formula for the
‘th zero of the Zeta function using LambertW function.
is well approximated with the formula
is the rounded value of
and
, then
will generate an infinity of primes, beginning with
can be made as close as we want to
.
is a function that measures the similarity between a referent,
(healthy population) and a subject,
(with cognitive decline). The parameters
are used to associate a weight to each simplified linguistic pattern,
, that we identified as relevant for the task.
, can return concise non-float constants such as
that closely approximate the float. Surprisingly often such a result is the exact limit that is approached as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation.