Dr. Sarah's WebCT Test 1

	1.	Muhammed was a mathematician in the 1700s. He lived in the Katsina area, which is now Nigeria.
	2.	Muhammed said: Do not give up, for that is ignorance and not according to the rules of this art. Those who know the arts of war and killing cannot imagine the agony and pain of a practitioner of this honorable science...You cannot hope to achieve success without infinite perseverance.
	3.	The following is a magic square: 4 9 2 7 5 7 8 1 6
	4.	13 can be the center of a 3x3 magic square.
	5.	13 can be the center of a 5x5 magic square.
	6.	The dihedral group is the symmetry group of the square. It can act on magic squares to form new magic squares. Dihedral groups arise frequently in art and nature.
	7.	A rotation by 90 degrees clockwise followed by a reflection about the line y=x is the same as a reflection about the y axis.

	1.	There was a rich tradition of mental calculations among illiterate people in certain parts of Africa. Yet, the slave trade was extremely destructive to the development of mathematics. Knowledge was sometimes kept secret and only transmitted from father to son. Thus, when an expert was taken prisoner and sold as a slave, his specific professional knowledge might disappear completely from his village or region.
	2.	Even though he was illiterate, with his good visualization and amazing calculation ability, Thomas Fuller was a blaring contradiction to those who tried to prove that African Americans were mentally inferior to whites.
	3.	At the age of 80 Fuller was asked to find the amount of seconds in a man's life that was 70yrs, 17 days and 12 hours old. Astoundingly he answered with precise accuracy in minutes while even including leap years (a total of 17) and gave the correct answer of 2,210,500,800 sec.
	4.	Starting from scratch with the above problem, Thomas Fuller would have been quicker than the first calculating machines which wouldn't even had the capability of showing the complete answer.
	5.	Starting from scratch with the above problem, (ie no advance info on the problem given and measuring the time from the first moment the problem was given until the moment the answer is received), Thomas Fuller would probably have been quicker than the ENIAC, the first general-purpose electronic computer.
	6.	Thomas Fuller would have beaten (in terms of speed) today's computers and calculators.
	7.	Since computers today run millions of clock cycles a second these days, the race for faster speeds is basically over.

	1.	Banneker taught himself astronomy and advanced mathematics. He successfully predicted the solar eclipse that occurred on April 14, 1789, contradicting the forecasts of prominent mathematicians and astronomers of the day.
	2.	In 1792 Secretary of State Thomas Jefferson, white supremacist, and slave owner pronounced Blacks mathematically inferior. In response to Jefferson, Benjamin Banneker sent a copy of his almanac along with a twelve page twelve page letter to Thomas Jefferson refuting his statements and requesting aid in improving conditions for African Americans.
	3.	Single false position is the taking of a guess or "false" assumption for a specific algebra problem and substituting it in the equation. Then the given answer is divided by the random number answer, which gives the actual answer.
	4.	Single false position works to obtain a correct answer for the equation x + x/4 = 15.
	5.	Single false position works to obtain a correct answer for the equation 6x-23=100.
	6.	Double false position uses two guesses instead of 1. Then the true answer is obtained by using the formula (error1*trial2 - error2*trial1) ------------------------------ error1-error2
	7.	Double false position works to obtain a correct answer for the equation 6x-23=100.
	8.	Double false position is really the secant method of numerical approximation in disguise. If we have an equation such as 6x-23=100, then we write f(x)=6x-23, and the error is then f(x)-100. To see that this is the secant method, we apply it to the function g(x)=x-100, with f(x1) and f(x2) as the "x" values in writing the slope of the secant line of g.
	9.	Historically, false position can be viewed as simply a medieval approach to problem solving and algebraic substitution. The early mathematicians, including Banneker, used this approach to finding the x in an algebraic sentence instead of the method we use today.

	1.	Her parents felt that her interest in mathematics was inappropriate for a female and discouraged her. She began studying at night to escape them, but they went to such measures as taking away her clothes once she was in bed and deprived her of heat and light to make her stay in her bed at night instead of studying.
	2.	Sophie was able to obtain the lecture notes for several courses and study from them. This gave her the opportunity to learn from many of the prominent mathematicians of the day. Sophie was particularly interested in the teachings of J. L. Lagrange. Under the pseudonym of M. LeBlanc ( a former student of Lagrange's), Sophie submitted a paper on analysis to Lagrange at the end of the term. He was quite impressed with the work and wanted to meet the student who had written it. Lagrange was amazed that the author of the work was actually a female, but he recognized her abilities and became her mentor. With a male to introduce her, Sophie could enter the circle of scientists and mathematicians that she never before could. While her gender had been a hindrance to her, her middle class social status helped her.
	3.	A Sophie Germain prime is a prime number p so that 2p+1 is also prime.
	4.	Sophie came up with these primes since they arose in her work on Fermat's Last Thoerem, which says that there are no positive whole number solutions to x^n +y^n = z^n for any integer n.
	5.	Sophie Germain was the first person to work on a more generalized version of Fermat's last theorem. Instead of looking at a specific power n, she tried to prove Fermat's last theorem for prime powers p, where p satisfied certain conditions.
	6.	In her work on Fermat's Last Thoerem, she needed to make sure that she could find 2 numbers a and b, and two primes p and q, so that a^p mod q b^p mod q were off by 1 number.
	7.	Recall that a mod b is the remainder of a/b. The following are true: 12 mod 7 = 5 987897 mod 80 = 55
	8.	Sophie Germain primes are still of interest today in coding theory. For certain codes, 2 very large primes > 150 digits each, are used since n=pq has only p and q as nontrivial factors, and it is very hard to factor.

	1.	Sofia's exposure to mathematics began at a very young age. She claims to have studied her father's old calculus notes that were papered on her nursery wall. Her uncle also took the time and interest to talk to her about mathematics.
	2.	Sofia was determined to continue her education at the university level. However, the closest universities open to women were in Switzerland, and young, unmarried women were not permitted to travel alone. To resolve the problem Sofia entered into a marriage of convenience to Vladimir Kovalevsky.
	3.	Sofia Kovalevskaya was the first woman to receive a Ph.D. in mathematics. Yet even with such a prestigious degree and the help of her mentor Weierstrass, she was not able to find employment afterward.
	4.	She worked on the mathematics of a spinning top which was unequally weighted, and had the center of mass away from the spinning point.
	5.	Her mathematics had applications in astronomy.
	6.	She used difference equations in her work.
	7.	A DE is an equation written in terms of derivatives of functions. y=e^(3x) is a solution to the de y' = y
	8.	A PDE is an equation that is written in terms of partials of functions. f(x,y)=sin(xy) is a solution to the PDE x^2 *f +f_(yy)=0
	9.	DEs and PDEs are very useful in real life since they model many real life phenomena

	1.	Woodard became only the second African American to receive a Ph.D. in mathematics.
	2.	"On two dimensional analysis situs with special reference to the Jordan Curve Theorem", by Woodard, was the first research paper (beyond a PhD thesis) published in an accredited mathematics journal by an African American.
	3.	The Jordan Curve Theorem States that every simple closed curve divides the plane up so that there is one inside component and one outside component.
	4.	The curve x^2-3*y^2=1 is an example of a simple closed curve.
	5.	The curve x^2+3*y^2=1 is an example of a simple closed curve.
	6.	Woodard changed Moore's axioms to obtain axioms that contain no assumptions on the character of the boundaries of regions, and then he proved the Jordan curve theorem with his set of axioms.
	7.	While the statement of the Jordan curve theorem is easy to understand, the proof is non-trivial. With certain curves, when you are close to the boundary, it can be hard to know if you are inside or outside the curve.

1) A,B in R implies that A+B in R

2) A+B=B+A for all A,B in R

3) (A+B)+C=A+(B+C) for all A,B,C in R

4) There exists an element 0 in R such that A+0=A for every A in R

5) Given A in R, there exists B in R such that A+B=0

6) A,B in R implies that A*B in R

7) A(BC)=(AB)C for all A,B,C in R

8) A(B+C)=AB+AC

And

  (B+C)A=BA+CA for all A,B,C in R

AND R must satisfy the ascending chain condition on ideals such that there is no infinite increasing chain of ideals.

     Recall that an Ideal, I, of a ring R is a subring that absorbs elements from R.  To be a an Ideal the subring must satisfy the following  two conditions.

1)   For all A,B in I, A-B is in I.

2)   For all A in I and r in R, rA is in I and Ar is in I.

	1.	Because she was a woman, the university refused to let Emmy Noether take classes They granted her permission to audit classes. She sat in on classes for two years, and then took the exam that would permit her to be a doctoral student in mathematics. She passed the test, and finally was a student in good standing at the University. After five more years of study, she was granted the second degree to a woman in the field of mathematics there.
	2.	After Emmy Noether had her doctorate in mathematics, she was ready to find a job teaching. The University of Erlangen would not hire her, as they had a policy against hiring women professors.
	3.	She made many contributions to the field of mathematics. She spent her time studying abstract algebra, with special attention to rings, groups, and fields. Because of her unique look on topics, she was able to see relationships that traditional algebra experts could not. She published over 40 papers in her lifetime. She was also a teacher that was able to inspire her students to make their own contributions to the field of mathematics. She is known as the mother of algebra.
	4.	Noetherian rings have applications in physics, and a modified definition of them can be viewed as a generalization of vector spaces.
	5.	The set of nxn matrices is a ring with the usual matrix operations + and *
	6.	The integers are a ring even if we switch the operations and make + be integer multiplication and * be integer addition.
	7.	The set of continous functions from [0,1] to the reals is a ring with (f+g)(x)=f(x)+g(x) (f*g)(x)=f(g(x))
	8.	The set of nxn matrices with + defined as matrix multiplication does not satisfy axioms 2) and 5).
	9.	2Z+1 is an ideal of the ring of integers Z
	10.	The integers are not a Noetherian ring because the sequence of ideals 2Z, 4Z, 8Z, 16Z, 32Z, ... fails to satisfy the ascending chain condition.
	11.	F={continuous functions from [0,1] to the reals} is not a neotherian ring because the sequence of ideals I_1, I_2, I_3, ... where I_j = {f in F such that f(1/n) = 0 for all n bigger than or equal to j} fails to satisfy the ascending chain condition.

	1.	Cox had COLORED printed across his transcript. He was the first African American to obtain a PHD in mathematics. He sent his dissertation to Universities in England and Germany which turned Cox down (possibly for reasons of race), but Japan's Imperial University of San Dei accepted the dissertation.
	2.	His Ph.D. at this time was remarkable, as no place or institution was a friend to Negroes. Indeed, there were just 28 Ph.D.'s in Mathematics awarded in all the country in 1925, but 31 black men were lynched that year.
	3.	Difference equations are equations in terms of the derivatives of functions.
	4.	The Fibonacci sequence begins 0, 1, 1, 2, 3, and follows the rules y_0=0, y_1=1 y_k = y_(k-1) + y_(k-2)
	5.	In the Fibonnaci sequence, y_8=23
	6.	The solution to the difference equation formed from the Fibonnaci sequence is F_k = 1/sqrt(5) [((1+sqrt(5))/2)^k - ((1-sqrt(5))/2)^k]
	7.	A difference equation that could be used for a traffic flow problem is N_(t)=N_(t-1) +F - G, where F is the number of cars coming in, and G is the number leaving.
	8.	Difference equations should be used in situations where we have data given in terms of differentiable functions.
	9.	Cox worked on difference equations of the type aN_(t+1) + bN_t = f(t), where a and b are nonzero complex numbers that don't add to 0 and f(t) is a polynomial of any order.
	10.	Difference equations are very useful in real life since they can be used in situations of discrete data that prevents us from taking the derivative. Instead, we can take the divided difference representing the slope of the secant line, which gives an approximation to the derivative.