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In our paper ''Algorithms for Finding and Proving Balanced Q2 Identities'' we pointed out that the proof algorithm in that paper had not been able to prove 1,417 of the 97,306 tentative identities which the paper's search algorithm had found. To deal with this problem, we give here a more powerful proof algorithm based on a familiar technique in the theory of modular forms. The first six sections of this paper establish the connection with modular forms. As it happens, a given balanced T2 identity in x is mapped directly to a modular form identity of weight 1 on a particular congruence subgroup. The map itself consists of simply multiplying each term of the T2 identity by the same power of x, written xĪĪ. The proof algorithm, which is then applied to this identity, consists basically of the following steps: Move the terms of the identity onto the left side of the equation and expand this side into a power series up to a predetermined degree L. If the first L ++ 1 terms of this expansion are zero, then the identity is true. When this algorithm was applied to the 1,417 recalcitrant identities, it showed they were all true, as hoped. [ABSTRACT FROM AUTHOR]

The behavior of the multiplicative acceleration of Selmer's algorithm is widely unknown and no general result on convergence has been detected yet. Solely for its 2-dimensional, periodic expansions, there exist some results on convergence and approximation due to Fritz Schweiger. In this paper we show that periodic expansions of any dimension do in fact converge and that the coordinates of the limit points are rational functions of the largest eigenvalue of the periodicity matrix. [ABSTRACT FROM AUTHOR]

We study a card game called Mousetrap, together with its generalization He Loves Me, He Loves Me Not. We first present some results for the latter game, based, on one hand, on theoretical considerations and, on the other one, on Monte Carlo trials. Furthermore, we introduce a combinatorial algorithm, which allows us to obtain the best result at least for French card decks (52 cards with 4 suits). We then apply the algorithm to the study of Mousetrap and Modular Mousetrap, improving recent results. Finally, by means of our algorithm, we study the reformed permutations in Mousetrap, Modular Mousetrap and He Loves Me, He Loves Me Not, attaining new results which give some answers to several questions posed by Cayley and by Guy and Nowakowski in their papers. [ABSTRACT FROM AUTHOR]

Motivated by the study of Mahonian statistics, in 2000, Babson and Steingríímsson [[Séém. Lothar. Comb]] introduced the notion of a ''generalized permutation pattern'' (GP) which generalizes the concept of ''classical'' permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingríímsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Méélou et al. [[J. Comb. Theory A]] introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets. In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects. [ABSTRACT FROM AUTHOR]

For and a == ( a1, . . . , ad) T ∈∈ ℤℤ d, let ττ r( a) == ( a2, . . . , ad, --⌊⌊ r T a⌋⌋) T. Further, let . In this paper we prove that if some roots of the polynomial Xd ++ rdX d--1 ++ ⋯⋯ ++ r2 X ++ r1 are t -th roots of unity and the others lie in the open unit disc, then | a k++ t -- ak| < c1 with a constant c1 which does not depend on k. Under some conditions this yields an algorithm to decide whether the sequence is, for all a, ultimately periodic, or becomes divergent for some a. We study the boundary of the closure of 3 and show that large parts of it belong to 3, while others lie outside 3. [ABSTRACT FROM AUTHOR]

This paper investigates representations of real numbers with an arbitrary negative base -- ββ < --1, which we call the (-- ββ)-expansions. They arise from the orbits of the (-- ββ)-transformation which is a natural modification of the ββ-transformation. We show some fundamental properties of (-- ββ)-expansions, each of which corresponds to a well-known fact of ordinary ββ-expansions. In particular, we characterize the admissible sequences of (-- ββ)-expansions, give a necessary and sufficient condition for the (-- ββ)- shift to be sofic, and explicitly determine the invariant measure of the (-- ββ)-transformations. [ABSTRACT FROM AUTHOR]