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## AN EXTENSION OF THE SUBSE UENCE PRINCIPLE

Studia Scientiarum Mathematicarum Hungarica
Author:
E. Péter

We prove ha any tigh sequence of random variables con ains a subsequence which satis .es,after any permu a ion of its erms,a large class of limit heorems valid for i.i.d.random variables.This extends a basic heorem of Aldous on subsequence behaviour.

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## Interpolation by Differences In H ∞

Studia Scientiarum Mathematicarum Hungarica
Authors:
Francesc Tugores
and
Laia Tugores

We pose an interpolation problem for the space of bounded analytic functions in the disk. The interpolation is performed by a function and its di˛erence of values in points whose subscripts are related by an increasing application. We impose that the data values satisfy certain conditions related to the pseudohyperbolic distance, and characterize interpolating sequences in terms of uniformly separated subsequences.

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## Analogues of some Tauberian theorems for bounded variation

Acta Mathematica Hungarica
Author:
R. Patterson

## Abstract

In this paper definitions for “bounded variation”, “subsequences”, “Pringsheim limit points”, and “stretchings” of a double sequence are presented. Using these definitions and the notion of regularity for four dimensional matrices, the following two questions will be answered. First, if there exists a four dimensional regular matrix A such that Ay = Σ k,l=1,1 ∞∞ a m,n,k,l y k,l is of bounded variation (BV) for every subsequence y of x, does it necessarily follow that x ∈ BV? Second, if there exists a four dimensional regular matrix A such that Ay ∈ BV for all stretchings y of x, does it necessarily follow that x ∈ BV? Also some natural implications and variations of the two Tauberian questions above will be presented.

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## Weighted Measures of Pseudorandom Binary Lattices

Studia Scientiarum Mathematicarum Hungarica
Authors:
Huaning Liu
and
Yinyin Yang

In cryptography one needs pseudorandom sequences whose short subsequences are also pseudorandom. To handle this problem, Dartyge, Gyarmati and Sárközy introduced weighted measures of pseudorandomness of binary sequences. In this paper we continue the research in this direction. We introduce weighted pseudorandom measure for multidimensional binary lattices and estimate weighted pseudorandom measure for truly random binary lattices. We also give lower bounds for weighted measures of even order and present an example by using the quadratic character of finite fields.

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## Interval Chains and Completeness in Ultrapowers of Ordered Sets

Mathematica Pannonica
Authors:
Zoltán Boros
and
Péter Tóth

., and Parsons , T. D . On monotone subsequences . Amer. Math. Monthly 95 , 1 ( 1988 ), 44 – 45 . [7] Stroyan , K. D ., and Luxemburg , W. A. J . Introduction to the theory of infinitesimals . Academic Press , New York

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## On square-free vertex colorings of graphs

Studia Scientiarum Mathematicarum Hungarica
Authors:
János Barát
and
Péter Varjú

A sequence of symbols a 1 , a 2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x 1 , …, x m , x 1 , …, x m . A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Hałuszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k -tree have a coloring of this kind using O ( c k ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

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## Algebra of Data Reconciliation

Studia Scientiarum Mathematicarum Hungarica
Authors:
Elod P. Csirmaz
and
Laszlo Csirmaz

With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.

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## Restricting statistical convergence

Acta Mathematica Hungarica
Authors:
Santanu Bhunia
,
Pratulananda Das
, and
Sudip Kumar Pal

. 120 2647 – 2654 . [11] Miller , H. I. 1995 A measure theoretical subsequence characterization of statistical convergence Trans. Amer

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## Two valued measure and summability of double sequences in asymmetric context

Acta Mathematica Hungarica
Authors:
Santanu Bhunia
and
Pratulananda Das

. , Patterson , R. F. 2008 Core theorems for double subsequences and rearrangements Acta Math. Hungar. 119 71 – 80 10.1007/s10474-007-7005-6 . [15] Móricz , F. 2003

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## On the Maximal Operators of T Means with Respect to Walsh–Kaczmarz System

Studia Scientiarum Mathematicarum Hungarica
Authors:
Nata Gogolashvili
and