Mathematicians Galore Project- Augustin Cauchy Biography:
Augustin-Louis Cauchy was a French mathematician known as an “early pioneer of analysis”. He lived from 1789 to 1857 in France. Because of the French Revolution during Cauchy’s childhood, his family briefly left Paris to seek a calmer atmosphere in rural France. His family members shared his Roman Catholic values. He pursued many occupations including author, professor, and engineer. Cauchy was a writer and wrote around eight hundred research articles and five complete textbooks. He was a founder of mathematical analysis and a leader in establishing the theory of substitution groups. He has over 35 mathematic discoveries/theories named after him.
Cauchy’s most renowned contribution to mathematics is the Cauchy Distribution. Standard Cauchy Distribution is the distribution of a random variable that is the ratio of two independent variables and has the probability density function. The probability density function is a function of a continuous (random) variable whose integral across an interval gives the probability that the value of the variable lies within the same interval.
In 1811 Cauchy proved that the angles of a convex polyhedron are determined by its faces. This is known as “Cauchy’s Theorem”. It states: “If P and Q are combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent.” “Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same.” (Wikipedia)Holomorphic functions are complex-valued functions of one or more complex variable that varies at every point in its domain.
Cauchy’s name is one of the 72 engraved on the Eiffel tower. 72 scientists, mathematicians, and engineers have their names on the tower in recognition of their contributions.
Bibliography-
· O'Connor, J. J., and E. F. Robertson. "Augustin Louis Cauchy." The MacTutor History of Mathematics Archive. N.p., Jan. 1997. Web. 05 Nov. 2014. <http://www-history.mcs.st-and.ac.uk/Biographies/Cauchy.html>.
· Wikipedia contributors. "Cauchy distribution." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 3 Oct. 2014. Web. 30 Oct. 2014.
· Wikipedia contributors. "Cauchy's theorem (geometry)." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 12 Sep. 2014. Web. 5 Nov. 2014.
Cauchy’s most renowned contribution to mathematics is the Cauchy Distribution. Standard Cauchy Distribution is the distribution of a random variable that is the ratio of two independent variables and has the probability density function. The probability density function is a function of a continuous (random) variable whose integral across an interval gives the probability that the value of the variable lies within the same interval.
In 1811 Cauchy proved that the angles of a convex polyhedron are determined by its faces. This is known as “Cauchy’s Theorem”. It states: “If P and Q are combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent.” “Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same.” (Wikipedia)Holomorphic functions are complex-valued functions of one or more complex variable that varies at every point in its domain.
Cauchy’s name is one of the 72 engraved on the Eiffel tower. 72 scientists, mathematicians, and engineers have their names on the tower in recognition of their contributions.
Bibliography-
· O'Connor, J. J., and E. F. Robertson. "Augustin Louis Cauchy." The MacTutor History of Mathematics Archive. N.p., Jan. 1997. Web. 05 Nov. 2014. <http://www-history.mcs.st-and.ac.uk/Biographies/Cauchy.html>.
· Wikipedia contributors. "Cauchy distribution." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 3 Oct. 2014. Web. 30 Oct. 2014.
· Wikipedia contributors. "Cauchy's theorem (geometry)." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 12 Sep. 2014. Web. 5 Nov. 2014.
Ring Theory:
Ring theory is a base to abstract algebra. A ring is defined as a set with defined properties and where multiplication and addition of the intergers are defined.
To be considered a ring, the set must satisfy the following four conditions:
1) r forms an abelian group under addition (additive identity)
a. a+b=b+a a-a=0 –a+a=0
2) r is closed under multiplication
3) r is associative under multiplication
a. a(bxc)=(axb)c
4) Distributive property
a. a(b+c)=axb+axc
Things not required by rings: (anything not mentioned above)
Does not need to be commutative under division (there are rings that are commutative under division but it is not required)
Examples of numbers that are rings include irrational numbers, real numbers, complex number, set of all 2x2 matrices whose coefficients are integers
Matrices and polynomials
PROOFS:
1) Theorem 1: Let 𝑅 be a commutative ring with identity, and let 𝑈 be the set of units in 𝑅. Then 𝑈 is a group under multiplication of 𝑅.
Communitive under addition and multiplication.
2) Theorem 2 (cancellation law): Let 𝐷 be an integral domain and let
𝑎, 𝑏, 𝑐 ∈ 𝐷 such that 𝑎 ≠ 0. If 𝑎𝑏 = 𝑎𝑐, then 𝑏 = 𝑐.
Proof: If 𝑎𝑏 = 𝑎𝑐, then 𝑎𝑏 − 𝑎𝑐 = 0. By the distributive law of rings, it follows that 𝑎 𝑏 − 𝑐 = 0. Since 𝐷 is an integral domain, then 𝑎 is not a zero divisor. Thus, 𝑏 − 𝑐 = 0, and so 𝑏 = 𝑐. ∎
1) A ring is a set with multiplication and addition which have similar properties to the intergers. To be considered a ring, the set must satisfy the following four conditions:
1) r forms an abelian group under addition (additive identity)
a. a+b=b+a a-a=0 –a+a=0
2) r is closed under multiplication
3) r is associative under multiplication
a. a(bxc)=(axb)c
4) Distributive property
a. a(b+c)=axb+axc
RING THEORY PROBLEMS:
If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ2, … , ζp−2).[5]
If d is a square-free integer and K = Q(√d) is the corresponding quadratic field, then OK is a ring of quadratic integers and its integral basis is given by (1, (1 + √d)/2) ifd ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).[6]
p and d are rings.
Sources:
http://jesse.jaksin14.angelfire.com/Proofs/Ring_Theory.pdf
http://www.math.niu.edu/~beachy/aaol/rings.html
To be considered a ring, the set must satisfy the following four conditions:
1) r forms an abelian group under addition (additive identity)
a. a+b=b+a a-a=0 –a+a=0
2) r is closed under multiplication
3) r is associative under multiplication
a. a(bxc)=(axb)c
4) Distributive property
a. a(b+c)=axb+axc
Things not required by rings: (anything not mentioned above)
Does not need to be commutative under division (there are rings that are commutative under division but it is not required)
Examples of numbers that are rings include irrational numbers, real numbers, complex number, set of all 2x2 matrices whose coefficients are integers
Matrices and polynomials
PROOFS:
1) Theorem 1: Let 𝑅 be a commutative ring with identity, and let 𝑈 be the set of units in 𝑅. Then 𝑈 is a group under multiplication of 𝑅.
Communitive under addition and multiplication.
2) Theorem 2 (cancellation law): Let 𝐷 be an integral domain and let
𝑎, 𝑏, 𝑐 ∈ 𝐷 such that 𝑎 ≠ 0. If 𝑎𝑏 = 𝑎𝑐, then 𝑏 = 𝑐.
Proof: If 𝑎𝑏 = 𝑎𝑐, then 𝑎𝑏 − 𝑎𝑐 = 0. By the distributive law of rings, it follows that 𝑎 𝑏 − 𝑐 = 0. Since 𝐷 is an integral domain, then 𝑎 is not a zero divisor. Thus, 𝑏 − 𝑐 = 0, and so 𝑏 = 𝑐. ∎
1) A ring is a set with multiplication and addition which have similar properties to the intergers. To be considered a ring, the set must satisfy the following four conditions:
1) r forms an abelian group under addition (additive identity)
a. a+b=b+a a-a=0 –a+a=0
2) r is closed under multiplication
3) r is associative under multiplication
a. a(bxc)=(axb)c
4) Distributive property
a. a(b+c)=axb+axc
RING THEORY PROBLEMS:
If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ2, … , ζp−2).[5]
If d is a square-free integer and K = Q(√d) is the corresponding quadratic field, then OK is a ring of quadratic integers and its integral basis is given by (1, (1 + √d)/2) ifd ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).[6]
p and d are rings.
Sources:
http://jesse.jaksin14.angelfire.com/Proofs/Ring_Theory.pdf
http://www.math.niu.edu/~beachy/aaol/rings.html