40,732 research outputs found
The Collatz conjecture and De Bruijn graphs
We study variants of the well-known Collatz graph, by considering the action
of the 3n+1 function on congruence classes. For moduli equal to powers of 2,
these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the
Collatz graph, these graphs are very structured, and have several interesting
properties. We then look at a natural generalization of these finite graphs to
the 2-adic integers, and show that the isomorphism between these infinite
graphs is exactly the conjugacy map previously studied by Bernstein and
Lagarias. Finally, we show that for generalizations of the 3n+1 function, we
get similar relations with 2-adic and p-adic De Bruijn graphs.Comment: 9 pages, 8 figure
Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
In 1862 Wolstenholme proved that for any prime the numerator of the
fraction written in reduced form is divisible by , and the numerator of
the fraction
written in reduced form is divisible by . The first of the above
congruences, the so called {\it Wolstenholme's theorem}, is a fundamental
congruence in combinatorial number theory. In this article, consisting of 11
sections, we provide a historical survey of Wolstenholme's type congruences and
related problems. Namely, we present and compare several generalizations and
extensions of Wolstenholme's theorem obtained in the last hundred and fifty
years. In particular, we present more than 70 variations and generalizations of
this theorem including congruences for Wolstenholme primes. These congruences
are discussed here by 33 remarks.
The Bibliography of this article contains 106 references consisting of 13
textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of
Integer Sequences. In this article, some results of these references are cited
as generalizations of certain Wolstenholme's type congruences, but without the
expositions of related congruences. The total number of citations given here is
189.Comment: 31 pages. We provide a historical survey of Wolstenholme's type
congruences (1862-2012) including more than 70 related results and 106
references. This is in fact version 2 of the paper extended with congruences
(12) and (13
A Divergence Critic for Inductive Proof
Inductive theorem provers often diverge. This paper describes a simple
critic, a computer program which monitors the construction of inductive proofs
attempting to identify diverging proof attempts. Divergence is recognized by
means of a ``difference matching'' procedure. The critic then proposes lemmas
and generalizations which ``ripple'' these differences away so that the proof
can go through without divergence. The critic enables the theorem prover Spike
to prove many theorems completely automatically from the definitions alone.Comment: See http://www.jair.org/ for any accompanying file
Discrepancy bounds for low-dimensional point sets
The class of -nets and -sequences, introduced in their most
general form by Niederreiter, are important examples of point sets and
sequences that are commonly used in quasi-Monte Carlo algorithms for
integration and approximation. Low-dimensional versions of -nets and
-sequences, such as Hammersley point sets and van der Corput sequences,
form important sub-classes, as they are interesting mathematical objects from a
theoretical point of view, and simultaneously serve as examples that make it
easier to understand the structural properties of -nets and
-sequences in arbitrary dimension. For these reasons, a considerable
number of papers have been written on the properties of low-dimensional nets
and sequences
The congruence of Wolstenholme and generalized binomial coefficients related to Lucas sequences
Using generalized binomial coefficients with respect to fundamental Lucas
sequences we establish congruences that generalize the classical congruence of
Wolstenholme and other related stronger congruences.Comment: 23 page
Order regularity for Birkhoff interpolation with lacunary polynomials
In this short paper we present sufficient conditions for the order
regularity problem in Birkhoff interpolation with lacunary polynomials.
These conditions are a generalization of the Atkinson-Sharma theorem.Peer ReviewedPostprint (published version
A generalization of the problem of Mariusz Meszka
Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L
containing n positive integers not exceeding n there exists a near 1-factor in
K_p whose list of edge-lengths is L. In this paper we propose a generalization
of this problem to the case in which p is an odd integer not necessarily prime.
In particular, we give a necessary condition for the existence of such a near
1-factor for any odd integer p. We show that this condition is also sufficient
for any list L whose underlying set S has size 1, 2, or n. Then we prove that
the conjecture is true if S={1,2,t} for any positive integer t not coprime with
the order p of the complete graph. Also, we give partial results when t and p
are coprime. Finally, we present a complete solution for t<12.Comment: 15 page
Supersymmetry and Combinatorics
We show how a recently proposed supersymmetric quantum mechanics model leads
to non-trivial results/conjectures on the combinatorics of binary necklaces and
linear-feedback shift-registers. Pauli's exclusion principle plays a crucial
role: by projecting out certain states/necklaces, it allows to represent the
supersymmetry algebra in the resulting subspace. Some of our results can be
rephrased in terms of generalizations of the well-known Witten index.Comment: 14 pages, 3 figures, text expanded, references adde
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