Related papers: The Combinatorics of Occam's Razor
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
We investigate properties of families $F$ of subsets of a finite set in a situation where subsets are incomparable by the binary inclusion relation and a) for any $A\notin F$, there is such set $A'\in F$ that either $A\subset A'$ or…
For a variety with a finitely generated total coordinate ring, we describe basic geometric properties in terms of certain combinatorial structures living in its divisor class group. For example, we describe the singularities, we calculate…
The goal of Ordinal Regression is to find a rule that ranks items from a given set. Several learning algorithms to solve this prediction problem build an ensemble of binary classifiers. Ranking by Projecting uses interdependent binary…
This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the…
Most questionnaires offer ordered responses whose order is poorly studied via belief functions. In this paper, we study the consequences of a frame of discernment consisting of ordered elements on belief functions. This leads us to redefine…
We use model theoretic techniques to construct explicit first-order axiomatizations for the classes of posets that can be represented as systems of sets, where the order relation is given by inclusion, and existing meets and joins of…
We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a…
We consider functions of natural numbers which allow a combinatorial interpretation as density functions (speed) of classes of relational structures, s uch as Fibonacci numbers, Bell numbers, Catalan numbers and the like. Many of these…
In many situations, the decision maker observes items in sequence and needs to determine whether or not to retain a particular item immediately after it is observed. Any decision rule creates a set of items that are selected. We consider…
The predictability of a sequence is defined as the asymptotic performance of the best performing predictor in a given class. The value of the predictability of a sequence will in general depend on the choice of this predictor class. The…
The union of a collection of $n$ sets is generally expressed in terms of a characteristic (indicator) function that contains $2^{n}-1$ terms. In this article, a much simpler expression is found that requires the evaluation of $n$ terms…
Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed…
To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at…
We analyze combinatorial optimization problems over a pair of random point sets of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set,…
We explore the connection between an agent's decision problem and her ranking of information structures. We find that a finite amount of ordinal data on the agent's ranking of experiments is enough to identify her (finite) set of…
Nonlinear systems are capable of displaying complex behavior even if this is the result of a small number of interacting time scales. A widely studied case is when complex dynamics emerges out of a nonlinear system being forced by a simple…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…