The seminar will touch briefly on two of these, namely algebraic combinatorics and Hopf algebras, before delving more deeply into a third direction, the theory of quasigroups. Alexandr Kazda, Vanderbilt University Title: Tuesday, November 19, Location: Finding the lowest possible air fare between two cities under only the condition that the database of air fares, routing rules, flights and airports is finite.

Spring Semester Speaker: A chip-firing game on a graph G begins with a configuration the number of 'chips' on each vertex. It was the development of computability theory also known as recursion theory that provided a precise explication of the intutitive notion of algorithmic computability, thus making the notion of recursive enumerability perfectly rigorous.

The algebra A T will be such that A T possesses certain properties if and only if T halts, thus showing that these properties are undecidable in general. The goal of the series is to address certain undecidable problems in algebra by associating to each Turing machine an algebraic structure.

If we allow both bounded-error quantum and classical algorithms, then there is no speedup since a classical probabilistic algorithm can solve the problem with a constant number of queries with small probability of error. It is evident that Diophantine sets are recursively enumerable.

If we restrict the sort of constraints to be used, the complexity of solving the CSP can vary from polynomial to NP-complete. Rings whose multiplicative endomorphisms are power functions Time: But if the reader feels strong enough to tackle the paper, Martin Davis observes that "This remarkable paper is not only an intellectual landmark, but is written with a clarity and vigor that makes it a pleasure to read" Davis in Undecidable, p.

It does this, starting in sequence from 1, by converting the numbers base 8 into strings of seven letters to test. The set of rational numbers is not complete. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.

Also, every polynomial of odd degree admits at least one real root: Alan Turing constructed this paradox with a machine and proved that this machine could not answer a simple question: Computers cannot directly store arbitrary real numbers with infinitely many digits.

Tuesday, October 8 and 15, Location: When must all Boolean algebras have many subalgebras. In fact, more can be said: Tuesday, November 12, Location: Its "computation" is to test all computing machines including itself for "circles", and form a diagonal number from the computations of the non-circular or "successful" computing machines.

When a vertex 'fires', it sends a chip to each adjacent vertex, so the only rule of the game is that a vertex may 'fire' if and only if it has at least deg v chips.

While an impossibility assertion in science can never be absolutely proved, it could be refuted by the observation of a single counterexample.

However, the sentence within quotation marks, which is itself a definition of the alleged number is less than characters in length. In fact, Nagel and Newman required a page introduction to their exposition of the proof. The degree of operations on groups Time: If a is not 1 or 0, and b is irrational number, is a to the power of b transcendental number.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. All these definitions satisfy the axiomatic definition and are thus equivalent.

Will this computing machine lock in a "circle". We determine the P-rings up to isomorphism. We will consider the question of whether an infinite Boolean algebra has the maximum number of subalgebras.

The algorithm determines whether a function f is either constant 0 on all inputs or 1 on all inputs or balanced returns 1 for half of the input domain and 0 for the other half.

One certainly needs to be able to specify probability density functions to support expected values, and presumably some integration software though potentially limiting the choice of parameterized distributions could lead to a simplified problem of looking up moments of those distributions.

Equivalence for 3D manifolds is thought to be decidable. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar.

This theorem is proved by showing that four of the axioms together imply the opposite of the fifth. Two examples of widely accepted impossibilities in physics are perpetual motion machineswhich violate the law of conservation of energyand exceeding the speed of lightwhich violates the implications of special relativity.

Arnold, V.

I. Local problems of analysis. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Meh. 25 no.

2 MR [Ar70b] Arnold, V. I. Algebraic unsolvability of the problem of Ljapunov stability and the problem of the topological classification of the singular points of an analytic system of differential equations.

(Russian) Funkcional. Description Given a set P of multivariate polynomials in nn variables, a CAD is a special partition of R into com ponents, called cells, over which all the polynomials have constant signs.

Undecidability of the Spectral Gap Here we prove that this is an undecidable problem. We construct families of quantum spin systems on a 2D lattice with translationally-invariant, nearest-neighbour interactions for which the spectral gap problem is an algebraic number computable from n, and is a projector.) Then: (i).

The local. Discusses the relations between Hilbert's tenth problem, arithmetic, and algebraic geometry. This book addresses such areas as Hilbert's tenth problem for various rings. Undecidable problems concerning densities of languages Jakub Kozik Our techniques can be extended to show that this problem is undecidable even for languages given by language it is an algebraic number.

This observation (with several reﬁnements) has been used in [Kem80]. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem.

Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the .

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