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Besides his academic achievements, he was very talented in organisational matters. Liouville founded the Journal de Mathématiques Pures et Appliquées which retains its high reputation up to today, in order to promote other mathematicians’ work. He was the first to read, and to recognize the importance of, the unpublished work of Évariste Galois which appeared in his journal in 1846. Liouville was also involved in politics for some time, and he became a member of the Constituting Assembly in 1848. However, after his defeat in the legislative elections in 1849, he turned away from politics.
I knew it. I recognized the Galois theory when I was studying that one thing in Complex… darn it but I can’t remember.[learn_press_profile]

Cerebral cortex
Jahangir Moini, Pirouz Piran, in Functional and Clinical Neuroanatomy, 2020
Specialized language areas
Language processing is very complicated and occurs in both cerebral hemispheres. However, it varies between individuals. Related to primarily the left hemisphere are Wernicke’s area and Broca’s area. The first, Wernicke’s area, is close to the auditory cortex, and located in the superior temporal gyrus. It is associated with language comprehension, receiving information from the sensory association areas. It is important in a person’s personality since it integrates sensory information and coordinates the access to auditory and visual memories.
Broca’s area is also known as the motor speech area. It is near the motor cortex and utilized in speech production, located in the inferior frontal gyrus. This area regulates breathing patterns while speaking and vocalizations required for normal speech. It coordinates the activities of the muscles of respiration, the larynx, the pharynx, as well as those of the cheeks, lips, jaws, and tongue. If a person has damage to Broca’s area, sounds can be made, but words cannot be formed. The receptive speech area is another name for the auditory association area. It utilizes feedback to adjust motor commands from the motor speech area. Many different speechrelated problems can occur because of damage to a specific sensory area. Some patients have problems with speaking but understand the usage of correct words, while others can speak consistently yet use many incorrect words. Please note that these two (Broca’s and Wernicke’s areas) are the main language centers. However, in order to be able to read, speak, and write, other areas of the brain need to function in coordination.
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Online Course: https://canvas.harvard.edu/courses/90105/assignments/syllabus
https://canvas.harvard.edu/courses/101719/assignments/syllabus
References:
https://people.math.harvard.edu/~elkies/M55a.10/index.html
https://people.math.harvard.edu/~elkies/M55b.10/
https://people.math.harvard.edu/~ctm/home/text/class/harvard/55a/09/html/index.html
https://people.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/index.html[learn_press_profile]

I just discovered the WordPress app.
So I guess we start with a circle, which I don’t know how to find the area of either.
I guess we start with a square.
Area is a measure of how much space something is taking up. Physicists call this Mass. We will use these interchangeably for now. Area and Mass are now mathematically interchangeable.
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In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the nonpositive integers. For any positive integer n,
Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:
The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
The gamma function has no zeroes, so the reciprocal gamma function 1/Γ(z) is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probabilitydistribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
The gamma function can be seen as a solution to the following interpolation problem: “Find a smooth curve that connects the points (x, y) given by y = (x − 1)! at the positive integer values for x.”
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Required Reading:
Introduction to Topological Manifolds, John M. Lee
https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7.image
https://zenodo.org/record/2063679#.YknrXuiZNrU
https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s132.1.277
http://www.numdam.org/numdambin/feuilleter?id=BSMF_1902__30_ (Page inexistante)
https://gallica.bnf.fr/ark:/12148/bpt6k1074672/f170.image.r=Journal%20math%C3%A9matiques%20pures%20appliqu%C3%A9es.langFR
Translating these papers satisfies the foreign language requirement, solves the statement and will grant you a phd. Good luck!
Proof: For Stevie Wonders Eyes Only[learn_press_profile]