Extender-based Radin forcing

We define extender sequences, generalizing measure sequences of Radin forcing.

Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing.

We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness.

Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.

     

Power function on stationary classes

We show that under certain large cardinal requirements there is a generic extension in which the power function behaves differently on different stationary classes. We achieve this by doing an Easton support iteration of the Radin on extenders forcing.     

Extender-based Magidor-Radin forcing

We define the extender-based Magidor-Radin forcing notion from a Mitchell increasing sequence of extenders. We prove the basic properties of this forcing.     

Catalan-like number sequences and Hausdorff moment sequences

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.     

Radin forcing and its iterations

We provide an exposition of supercompact Radin forcing and present several methods for iterating Radin forcing. This work was partially supported by FWF project number P16790-N04.     

Concatenation of pseudorandom binary sequences

In the applications it may occur that our initial pseudorandom binary sequence turns out to be not long enough, thus we have to take the concatenation or merging of it with other pseudorandom binary sequences. Here our goal is study when we can form the concatenation of several pseudorandom binary sequences belonging to a given family? We introduce and study new measures which can be used for answering this question.     

More fine structural global square sequences

We extend the construction of a global square sequence in extender models from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.     

On the existence of uniformly distributed sequences in compact topological spaces II

We characterize those probability measures on the Bohr compactification of a metrizable, abelian group which admit a u. d. sequence in the original group. We show that the set of u. d. sequences on a nonmetrizable compact space can have the measure zero or one or it can be non-measurable. Finally we show that the existence of a u. d. sequence does not imply the existence of a well distributed sequence.     

On the pseudorandomness of binary and quaternary sequences linked by the gray mapping

Binary and quaternary sequences are the most important sequences in view of many practical applications. Any quaternary sequence can be decomposed into two binary sequences and any two binary sequences can be combined into a quaternary sequence using the Gray mapping. We analyze the relation between the measures of pseudorandomness for the two binary sequences and the measures for the corresponding quaternary sequences, which were both introduced by Mauduit and Sárközy. Our results show that each ‘pseudorandom’ quaternary sequence corresponds to two ‘pseudorandom’ binary sequences which are ‘uncorrelated’.     

Limiting relationships and comparison theorems for sequences

The aim of this paper is to define some mathematical concepts which are useful to measure the speed of convergence of a sequence and to compare two converging sequences. In that way we define the order, the relative order and the α-equivalence of sequences. The asymptotic expansion of a series is studied and an application to Aitken acceleration process is given. A theorem similar to l'Hospital's rule is also proved for sequences.     

Cayley graphs and automatic sequences

We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For 2-automatic sequences, we find a characterization in terms of what we call homogeneity, and among homogeneous sequences, we single out those enjoying what we call self-similarity. It turns out that 2-self-similar sequences (viewed up to a permutation of their alphabet) are in bijection with many interesting objects, for example dessins d’enfants (covers of the Riemann sphere with three points removed). For any p, we show that, in the case of an automatic sequence produced “by a Cayley graph,” the group and indeed the automaton can be recovered canonically from the sequence. Further, we show that a rational fraction may be associated with any automatic sequence. To compute this fraction explicitly, knowledge of a certain graph is required. We prove that for the sequences studied in the first part, the graph is simply the Cayley graph that we start from, and so calculations are possible. We give applications to the study of the frequencies of letters.     

The catenary degrees of elements in numerical monoids generated by arithmetic sequences

We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

A dominant subset of V-shaped sequences for a class of single machine sequencing problems

In this note we define a subset of V-shaped sequences, ‘V-shaped about T’, which generalize ‘V-shaped about d’ sequences. We derive a condition under which this subset contains an optimal sequence for a class of single machine sequencing problems. Cost functions from the literature are used to illustrate our results.     

The short extenders gap three forcing using a morass

We show how to construct Gitik??s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type.     

Rough convergence of double sequences of fuzzy numbers

In this paper, we define the concepts of rough convergence and rough Cauchy sequence of double sequences of fuzzy numbers. Then, we investigate some relations between rough limit set and extreme limit points of such sequences.     

On a family of 2-automatic sequences generating algebraic continued fractions in characteristic 2

We present family of automatic sequences that define algebraic continued fractions in characteristic 2. This family is constructed from ultimately period words and contains the period-doubling sequence.     

Double sequences and Orlicz functions

In this paper we define strong A-convergence with respect to an Orlicz function for double sequences. We show, on bounded double sequences, that statistical convergence and strong A-convergence with respect to any Orlicz function are equivalent. This eliminates a condition of Demirci for bounded single (ordinary) sequences.     

The hopf algebra of linearly recursive sequences

We explain how the space of linearly recursive sequences over a field can be considered as a Hopf algebra. The algebra structure is that of divided-power sequences, so we concentrate on the perhaps lesser-known coalgebra (diagonalization) structure. Such a sequence satisfies a minimal recursive relation, whose solution space is the subcoalgebra generated by the sequence. We discuss possible bases for the solution space from the point of view of diagonalization. In particular, we give an algorithm for diagonalizing a sequence in terms of the basis of the coalgebra it generates formed by its images under the difference-operator shift. The computation involves inverting the Hankel matrix of the sequence. We stress the classical connection (say over the real or complex numbers) with formal power series and the theory of linear homogeneous ordinary differential equations. It is hoped that this exposition will encourage the use of Hopf algebraic ideas in the study of certain combinatorial areas of mathematics.     

Suballowable sequences and geometric permutations

We define suballowable sequences of permutations as a generalization of allowable sequences. We give a characterization of allowable sequences in the class of suballowable sequences, prove a Helly-type result on sets of permutations which form suballowable sequences, and show how suballowable sequences are related to problems of geometric realizability. We discuss configurations of points and geometric permutations in the plane. In particular, we find a characterization of pairwise realizability of planar geometric permutations, give two necessary conditions for realizability of planar geometric permutations, and show that these conditions are not sufficient.