Mathematics

Read at: This Tournament Goes to Eleven VIII: Spinal Tap Takes Manhattan
Written by: Steven Canning of the University of Illinois @ Urbana-Champagne 
Final editing by: Andrew R. Juhl and the University of Iowa Academic Quiz Team
Description: Math, algebra, and other things Andrew doesn’t know much about.

Tossups 

1. An informal statement of this theorem is that the theory of algebraically closed fields is a complete model. The weak form of this theorem states that if I is a proper ideal of a polynomial ring over an algebraically closed field, then the zero set of I is non-empty. This theorem states that the ideal generated by the vanishing set of an ideal is equal to the radical of that ideal. Thus there is a bijection between the affine varieties and the radical ideals. For ten points, identify this theorem with a German name that translates as “zero set theorem.” 
	Answer: Hilbert’s Nullstellensatz

2. Essential to the proof of the fundamental theorem of calculus, this theorem was first discovered by Lagrange in the 18th Century. Cauchy's generalization of this theorem considers two functions instead of one. The function in question must be differentiable on the open interval in question and continuous on the closed interval. Commonly proved using Rolle’s theorem, for ten points, name this theorem that states between two points a function there must exist a third point where the derivative is equal to the slope of a line through the two points.
	Answer: Mean Value Theorem

3. The namesake sums of this integral type are constructed by a subdividing the range of the function, rather than the domain. If a function is Riemann integrable, then its Riemann integral is equal to this integral. However a much larger variety of functions are integrable using this type of integral. Utilizing its namesake measure, for ten points, name this type of integral named for an early twentieth century French mathematician.
	Answer: Lebesgue Integral

4. The justification of many statistical procedures, this theorem was developed early on by de Moivre and Laplace. However, it was first stated in full generality and proven in 1901 by Aleksandr Lyapunov. Lyapunov's namesake condition describes when this theorem holds. For ten points, identify this important theorem of statistical theory that states that the sum of many independent random variables will be approximately normally distributed.
	Answer: Central Limit Theorem
	
5. The divergence of this is zero, which is a statement of the conservation of energy. A rank 2 tensor, this quantity is also symmetric. Giving the momentum flux density in an arbitrary direction, the zero, zero component is equal to the mass-energy density. For ten points, identify this physical quantity that is the source of gravitation in Einstein's field equations.
	Answer: Stress-Energy Tensor
6. Napoleon supposedly spared the city of Gottingen in 1806 because this man, the greatest mathematician in Europe, resided there. One of his greatest early achievements was a ruler and compass construction of the 17-gon, a result that eventually appeared as section VII of his magnum opus Disquisitiones Arithmeticae. Also a gifted astronomer, he predicted the orbit of the asteroid Ceres in 1801 by using his newly discovered least squares method. His greatest achievement, however, might have been when he provided a proof of the fundamental theorem of algebra in his dissertation. For ten points, name this German mathematician that is generally recognized as the greatest mathematician of modern times.
	Answer: Carl Friedrich Gauss

7. His 1935 dissertation, On the Gaussian error function, derived fundamental results of probability theory, though his work would soon go in a different direction. The namesake of a hypothetical computing machine, he developed the theory of computable numbers. However, in 1952 he was convicted of a homosexuality, a crime in Britain at the time. Deciding that prison was not conducive to an academic career, he opted for the hormone injections instead. The hormone injections changed his personality and made his work difficult. In 1954 he committed suicide by eating an apple dipped in cyanide. For ten points, name this mathematician that is commonly considered the founder of computer science.
	Answer: Alan Turing

8. The Fall Out Boy song, "Tell That Mick He Just Made My List Of Things To Do Today" takes its title from a line in this movie. At the beginning, Max has a dream that he is in his math class, and on the board there is the hardest geometry problem in the world, which he disposes of with minimal effort. Max is a poor student, but amazingly involved in extracurricular activities. The plot focuses on an awkward love triangle between a kindergarten teacher, a wealthy businessman, and 15 year old Max Fischer. For ten points, identify this 1998 Wes Anderson film, starring Bill Murray and Jason Schwartzman.
	Answer: Rushmore

9. An ideal is prime if and only if its quotient ring is one of these. This implies that in this type of ring, zero is a prime ideal. It possesses a multiplicative unit element different from zero and is commutative in multiplication. If one of these has a finite number of elements, then it is a field. For ten points, name this type of commutative ring, one with no zero divisors.
	Answer: Integral Domain (do not accept UFD or Unique Factorization Domain)

10. The possible symmetries within a crystal form one of these. These are monoids in which every element is invertible. The symmetric variety of this structure is the set of permutations of n elements. If every element is a multiple of a single element, then this is said to be cyclic. If the operation is commutative it is said to be Abelian. Consisting of a set paired with a binary operation, for ten points, name this algebraic structure.
	Answer: Group

11. In 1696 Newton was challenged to solve this problem, and he managed to solve it in less than a day.  Jakob Bernoulli solved this problem, and his brother, Johann, tried to take that proof as his own. One of the first problems posed in the calculus of variations, it has the cycloid as its solution. It is named for the Greek for 'shortest time.' For ten points, identify this famous problem of finding the shape of the curve on which a bead under the influence of gravity will travel from one point to another in the least time.
	Answer: Brachistochrone Problem

12. For commutative rings, this theorem states that if J and K are ideals of R such that J plus K equals R, then R modulo the intersection of J and K is isomorphic to the Cartesian product of R modulo J and R modulo K. If p and q are coprime integers, this theorem states that every pair of residue classes modulo p and q corresponds to a residue class modulo p times q. For ten points, identify this theorem of elementary number theory that can be utilized to solve a system of simultaneous congruencies.
	Answer: Chinese Remainder Theorem

13. This theorem can be taken as a special case of the law of cosines, though it can be proven easily without trigonometry. Mathematically equivalent to the parallel postulate, it forms the basis for defining distance in the Euclidean plane. For ten points identify this most famous theorem of Euclidean geometry that relates the lengths of the sides of a right triangle.
	Answer: Pythagorean Theorem

14. In Hamiltonian Mechanics, the phase space of a system forms one of these. General Relativistic space-time is also considered to be one of these objects. Around every point in this type of space there exists a neighborhood that is topologically equivalent to a unit ball. For ten points, identify this type of space that appears to be flat on short distance scales.
	Answer: Manifold

15. A non-traditional mathematician, he focused on solving problems rather than developing theories. Working primarily in number theory and combinatorics, he published his first results in 1931 at the age of eighteen. He possessed many peculiarities including his vocabulary; to him, God was the supreme fascist and children were epsilons. All of his possessions were kept in a single suitcase and he lived on the generosity of his colleagues. He was a very prolific mathematician with many of his papers featuring coauthors. A numbering system was even developed in order to characterize a mathematician's academic proximity to this man, with the number 1 reserved for those coauthors. For ten points, identify this eccentric Hungarian mathematician.
	Answer: Paul Erdős

16. While working in St. Petersburg, Leonhard Euler is supposed to have solved this problem. The key to the solution turned out to be a matter of odd and even. Only the starting and ending vertices of the path are allowed to have odd degree. All other vertices must have even degree; an edge to leave on for every edge that leads in. The seven bridges over the Pregel River in the namesake German town caused the townspeople to consider this problem. For ten points, identify this famous graph theory problem that asks if it is possible to cross every bridge exactly once.
	Answer: Konigsberg Bridge Problem

17. In 1950 it was proven that this must exist for all finite games with any amount of players. When this condition is reached, no player can choose a better strategy given their opponents strategies. In the Prisoner's Dilemma, this situation corresponds to both players betraying their accomplice. For ten points, identify this game theoretic concept named for the subject of the 2001 film, A Beautiful Mind.
	Answer: Nash Equilibrium

18. This generalization of the derivative takes a tensor as an argument and returns a tensor with rank increased by one. The divergence of this operation is equal to the Laplacian operation. Pointing in the direction in which the directional derivative has the largest magnitude, this vector operation produces a vector from a scalar function. For ten points, identify this operation that in Cartesian coordinates has components equal to the partial derivative of the function with respect to that coordinate.
	Answer: Gradient

19. The Riemann mapping theorem states that any region of this type has a bijective, analytic mapping to the unit disk. Cauchy’s Integral Theorem holds on regions of this type. The standard Euclidean plane has this property, but loses it if you remove a single point. For ten points, identify this type of domain in which every simple, closed curve can be shrunk continuously to a point.
	Answer: Simply Connected (do not accept Connected)

20. This probability distribution corresponds to random variables that count the number of discrete occurrences of an event during a given time interval. The limiting case of a binomial distribution, it is independent of the sample size. Its sole parameter is the expected number of successes, a value that also happens to be the mean and the variance of the distribution. For ten points, identify this probability distribution that is used to model many real world situations, including the number of typos per page and radioactive decay counts.
	Answer: Poisson distribution

21. The W(x) [read: W of X] in Abel’s Identity involves this function of two homogenous solutions to second-order linear ODEs.  In a variation of parameters problem, one will appear in the denominator of the solution integrals.  If it equals zero over some range, then the functions being described are linearly dependent somewhere in that range.  The determinant of a square matrix, its first row contains a set of n functions, while the last row contains the n minus 1th derivatives of those n functions.  FTP, name this tool useful in solving differential equations, named after a Polish mathematician.
	Answer: Wronskian

Bonuses - Mathematics

1. For ten points each, identify these things related to the intersections of polynomials.
[10] This term describes a polynomial with all terms having equal degree.
	Answer: homogeneous
[10] Though affine space is fine for some purposes, this space is better used to analyze the intersection properties of polynomials. The points in this space correspond to the lines through the origin in affine space.
	Answer: projective space
[10] This theorem states that the number of intersections of two polynomials in projective space is simply the product of their degrees.
	Answer: Bezout’s Theorem

2. Name these special functions for ten points each.
[10] For positive integer values this function reduces to the factorial function, so it is commonly said to be an extension of the factorial function to the complex numbers.
	Answer: Gamma function
[10] These functions can either be defined recursively or as solutions to their namesake differential equation.
	Answer: Bessel functions
[10] Certain linear combinations of the Bessel functions can generate these functions.
	Answer: Hankel functions

3. How well do you remember your trigonometry? Identify these trig related things for ten points each.
[10] This trigonometric function is equal to the reciprocal of the sine function.
	Answer: Cosecant (prompt on ‘csc’; do not accept secant)
[10] This theorem states that the ratio of the length of the side to the namesake function of its opposite angle is the same for every side of a triangle.
	Answer: Law of Sines
[10] The cosine of i times x is equal to this related function of x.
	Answer: Hyperbolic Cosine (prompt on cosh)

4. For ten points each, answer these related questions about a certain branch of mathematics.
[10] This namesake structure of a branch of mathematics is a collection of subsets of a given set that are to be considered open subsets.
	Answer: Topology
[10] The complement of an open set is this type of set.
	Answer: Closed
[10] The Heine-Borel Theorem guarantees that any subset of Euclidean space that is closed and bounded has this property.
	Answer: Compact or Compactness

5. Quantum mechanics has always involved a heavy amount of math. Identify these mathematical things from quantum mechanics for ten points each.
[10] Wave vectors in quantum mechanics belong to this type of vector space.
	Answer: Hilbert Space
[10] Because the Hamiltonian is a linear operator in quantum mechanics, this principle allows us to add two wave functions together yielding another acceptable state.
	Answer: Principle of Superposition
[10] Heisenberg's Uncertainty Principle arises because the quantum mechanical operators don't necessarily follow this rule of multiplication.
	Answer: Commutative Law (accept anything along the lines of 'they don't commute')

6. Computational Theory is a newer branch of mathematics, and it is the precursor of computer science. For ten points each, name these items from computational theory.
[10] Mathematically this is a 5-tuple that consists of a set of states, a set of symbols understood, a transition function, an initial state, and a set of accepting states. It is a model for a machine that can accept or reject strings by reading them sequentially.
	Answer: deterministic finite automaton
[10] This type of language is accepted by some deterministic finite automaton.
	Answer: Regular Language
[10] There are three operations on languages used in regular expressions. Along with union and concatenation, there is this operation that consists of the strings that can be formed by taking any number of strings in L and concatenating them.
	Answer: Kleene Star or Closure

7. 30-20-10 Identify the mathematician.
[30] Napoleon read his Celestial Mechanics, and asked him how he could have neglected to mention God. He simply replied that he had no need for that hypothesis.
[20] His Théorie Analytique des Probabilités published in 1812 was the first text to put probability on a firm mathematical footing.
[10] Both the electric scalar potential and the gravitational potential obey his namesake equation in empty space.
	Answer: Pierre-Simon Laplace

8. Math sometimes makes its way into fiction. For ten points each, identify these plays that involve math in some way or another.
[10] In this Tom Stoppard play, the intelligent, young Thomasina discusses things ranging from fractals to determinism in Newtonian mechanics.
	Answer: Arcadia
[10] While looking through the deceased Robert's notebooks, Hal discovers a mathematical proof of historic proportions in this Pulitzer Prize winning play by David Auburn.
	Answer: Proof
[10] This Michael Frayn work deals with a 1941 meeting between Werner Heisenberg and Niels Bohr in the titular location.
	Answer: Copenhagen

9. Mathematics can even be found on a show like the Simpsons. For ten points each, answer these questions about the relation between mathematics and the Simpsons.
[10] In the episode "Bart The Genius," Bart is transferred to a school for the gifted after cheating on a math exam. All the students, except Bart, laugh at this funny derivative of y equals r cubed over 3.
	Answer: r dr r
[10] After Homer has a crayon removed from his brain, he becomes very intelligent. While working on a proposal for a flat tax, he accidentally proves this. Ned says "can't let this little doozy get out," and burns the proof.
	Answer: nonexistence of God (accept equivalent)
[10] When Mrs. Krabappel asks "Now whose calculator can tell me what 7 times 8 is?", Milhouse gives this answer.
	Answer: low battery

10. For ten points each, identify these elementary methods used to find the roots of polynomials.
[10] When looking for roots of its namesake form, we can use this theorem in which we look at the quotients of the factors of the constant term with the factors of the leading coefficient.
	Answer: Rational Root Theorem
[10] If we count the number of times the signs of the coefficients change in f of x, this rule will tell us the possible numbers of positive real roots that f of x can have.
	Answer: Descartes’ Rule of Signs
[10] This method from elementary calculus can be employed to estimate the value of a polynomial’s root. Starting with a guess, we subtract the value of the function divided by the value of the derivative, and iterate this until the value converges.
	Answer: Newton’s Method (or Newton-Raphson method)

11. Plane geometry can become quite complicated. For ten points each, identify these advanced things from plane geometry.
[10] This theorem states the condition under which three lines drawn from the vertices of a triangle to the opposite sides are concurrent.
	Answer: Ceva’s Theorem
[10] For any triangle you can construct this circle that passes through the midpoint of every side, the foot of each altitude, and the midpoints of the segment between each vertex and the orthocenter.
	Answer: Nine-Point Circle or Euler's Circle or Feuerbach's Circle
[10] This formula expresses the area of any cyclic quadrilateral as the square root of the products of the semiperimeter minus the side lengths.
	Answer: Brahmagupta's Formula

12. Maybe the stereotype of the mathematician is correct. For ten points each identify these mathematicians given a famous anecdote.
[10] After winning the 1994 Nobel Prize for Economics he remarked "I hope that getting the Nobel will improve my credit rating, because I really want a credit card."
	Answer: John Nash Jr.
[10] In his later life this mathematician became convinced that people were trying to poison him. In an effort to avoid being poisoned, he ended up starving himself to death, dying in 1978.
	Answer: Kurt Gödel
[10] This English mathematician once sent a postcard to a friend with his New Year's Resolutions. They were to prove the Riemann hypothesis, prove the non-existence of God, make a critical play in a cricket match, become the first man to climb Everest, and kill Mussolini.
	Answer: G. H. Hardy
13. For ten points each, identify these terms from combinatorics.
[10] This principle states that given n boxes and m objects with m greater than n, at least one box must contain more than one object.
	Answer: Pigeonhole Principle or Dirichlet's Box Principle
[10] This is a collection of disjoint subsets of a set S whose union is S.
	Answer: Set Partition
[10] The nth one of these numbers gives the number of possible partitions of a set of n elements.
	Answer: Bell Number

14. For ten points each name these things from set theory.
[10] This term is used to describe two sets that have no elements in common.
	Answer: disjoint
[10] These laws of logic have analogs in set theory. They show how to distribute an intersection over a union or vice versa.
	Answer: De Morgan’s Laws
[10] This binary set operation gives the set of elements that belong to one of the sets but not both.
	Answer: Symmetric Difference

15. Name these things from multivariable calculus for ten points each.
[10] This is defined as the derivative of a function with respect to a certain variable, when all other variables are held fixed.
	Answer: Partial Derivative
[10] When performing a change of variables, the integral gains a factor in the form of the determinant of this matrix of partial derivatives that characterizes the transformation.
	Answer: Jacobian
[10] This theorem of multivariable calculus states that every multiple integral can be evaluated as an iterated one.
	Answer: Fubini's Theorem (accept Tonelli's Theorem)

16. For ten points each, identify these terms from graph theory.
[10] If you take the number of vertices in a graph, add the number of faces, and subtract the number of edges you will get this property of the graph, usually symbolized by the Greek character chi.
	Answer: Euler Characteristic or Euler Number
[10] If a graph is planar it must have this as its Euler Characteristic.
	Answer: two
[10] This theorem states that a graph is planar if and only if it does not contain a subgraph that is homeomorphic to the complete graph of five vertices or the complete bipartite graph K 3, 3.
	Answer: Pontryagin-Kuratowski('s) Theorem 

17. For ten points each, identify these terms from number theory.
[10] This important theory of elementary number theory, states that for any integer a, a to the p is congruent to a modulo p.
	Answer: Fermat's Little Theorem (prompt on 'Fermat's Theorem' or 'Fermat')
[10] Denoted by phi, this number theoretic function gives the number of positive integers less than a given number that are relatively prime to that number.
	Answer: Euler's Totient Function
[10] If this conjecture is true, then for every positive integer m, there exist primes p and q such that the totient of p plus the totient of q is twice m.
	Answer: Goldbach's Conjecture.

18. Identify these terms from logic for ten points each.
[10] An implication and this are always equivalent statements. It is formed by taking the converse of an implication and negating both sides.
	Answer: contrapositive
[10] A type of valid argument in the propositional calculus, this is the logical basis for proof by contrapositive.
	Answer: Modus Tollens
[10] The primary rule of inference in propositional calculus, it states that if p implies q and p is true, then q is true.
	Answer: Modus Ponens

19. For ten points each, answer these related questions.
[10] This major theorem, first proved in 1799 by Gauss, states that a polynomial of degree n has exactly n roots in the complex numbers.
	Answer: Fundamental Theorem of Algebra
[10] Gauss’s proof would not meet today’s standards of rigor. The first rigorous proof of the fundamental theorem of algebra was produced in 1813 by this man. He also lends his name to a type of diagram used to visualize complex numbers.
	Answer: Jean-Robert Argand
[10] A relatively simple, but certainly non-elementary, proof of the theorem utilizes this powerful theorem from complex analysis. It states that if the modulus of f is strictly greater than the modulus of g on a simple closed curve, then f and f plus g have the same number of zeroes inside the curve.
	Answer: Rouché’s Theorem

20. Identify the following things from probability theory for ten points each.
[10] This property of a probability distribution is calculated by subtracting the square of the mean from the second raw moment and then taking the square root.
	Answer: Standard Deviation
[10] The normal distribution is a continuous approximation to this probability distribution that computes that models a finite number of Bernoulli trials.
	Answer: Binomial Distribution
[10] Given a discrete probability distribution, this function gives the probability that the described random variable is equal to a certain value.
	Answer: Probability Mass Function

21. For ten points each, answer these related questions.
[10] This is a sequence of random variables for which the future value is dependent only on the present value.
	Answer: Markov Chain
[10] This general computational method consists of generating random numbers, and then accepting or rejecting them based on certain properties or a probability distribution.
	Answer: Monte Carlo Method
[10] The most common application of the Monte Carlo method is this mathematical process which can be used to find the area under a curve.
	Answer: Integration