{"title":"Analysis of carries in signed digit expansions.","authors":"Clemens Heuberger, Sara Kropf, Helmut Prodinger","doi":"10.1007/s00605-016-0917-x","DOIUrl":null,"url":null,"abstract":"<p><p>The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the (<i>q</i>, <i>d</i>)-system and the symmetric signed digit expansion. The results include expectation, variance, covariance between the positive and negative carries and a central limit theorem. Dependencies between the digits require determining suitable transition probabilities to obtain equidistribution on all expansions of given length. A general procedure is described to obtain such transition probabilities for arbitrary regular languages. The number of iterations in von Neumann's parallel addition method for the symmetric signed digit expansion is also analyzed, again including expectation, variance and convergence to a double exponential limiting distribution. This analysis is carried out in a general framework for sequences of generating functions.</p>","PeriodicalId":54737,"journal":{"name":"Monatshefte fur Mathematik","volume":"182 2","pages":"299-334"},"PeriodicalIF":0.8000,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7175708/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte fur Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00605-016-0917-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2016/6/10 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the (q, d)-system and the symmetric signed digit expansion. The results include expectation, variance, covariance between the positive and negative carries and a central limit theorem. Dependencies between the digits require determining suitable transition probabilities to obtain equidistribution on all expansions of given length. A general procedure is described to obtain such transition probabilities for arbitrary regular languages. The number of iterations in von Neumann's parallel addition method for the symmetric signed digit expansion is also analyzed, again including expectation, variance and convergence to a double exponential limiting distribution. This analysis is carried out in a general framework for sequences of generating functions.
期刊介绍：
The journal was founded in 1890 by G. v. Escherich and E. Weyr as "Monatshefte für Mathematik und Physik" and appeared with this title until 1944. Continued from 1948 on as "Monatshefte für Mathematik", its managing editors were L. Gegenbauer, F. Mertens, W. Wirtinger, H. Hahn, Ph. Furtwängler, J. Radon, K. Mayrhofer, N. Hofreiter, H. Reiter, K. Sigmund, J. Cigler.
The journal is devoted to research in mathematics in its broadest sense. Over the years, it has attracted a remarkable cast of authors, ranging from G. Peano, and A. Tauber to P. Erdös and B. L. van der Waerden. The volumes of the Monatshefte contain historical achievements in analysis (L. Bieberbach, H. Hahn, E. Helly, R. Nevanlinna, J. Radon, F. Riesz, W. Wirtinger), topology (K. Menger, K. Kuratowski, L. Vietoris, K. Reidemeister), and number theory (F. Mertens, Ph. Furtwängler, E. Hlawka, E. Landau). It also published landmark contributions by physicists such as M. Planck and W. Heisenberg and by philosophers such as R. Carnap and F. Waismann. In particular, the journal played a seminal role in analyzing the foundations of mathematics (L. E. J. Brouwer, A. Tarski and K. Gödel).
The journal publishes research papers of general interest in all areas of mathematics. Surveys of significant developments in the fields of pure and applied mathematics and mathematical physics may be occasionally included.
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