On Ramanujan's Q-function

This study provides a detailed analysis of a function which Knuth discovered to play a central role in the analysis of hashing with linear probing. The function, named after Knuth Q(n), is related to several of Ramanujan's investigations. It surfaces in the analysis of a variety of algorithms ans discrete probability problems including hashing, the birthday paradox, random mapping statistics, the "rho" method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q(n) function. in this way, tight bounds are derived. They answer a question of Knuth (the art of Computer Programming, vol. 1, 1968), itself a rephrasing of earlier questions of Ramanujan in 1911-1913

The Number of Symbol Comparisons in QuickSort and QuickSelect

International audienceWe revisit the classical QuickSort and QuickSelect algo-rithms, under a complexity model that fully takes into account the ele-mentary comparisons between symbols composing the records to be pro-cessed. Our probabilistic models belong to a broad category of informa-tion sources that encompasses memoryless (i.e., independent-symbols) and Markov sources, as well as many unbounded-correlation sources. We establish that, under our conditions, the average-case complexity of QuickSort is O(n log 2 n) [rather than O(n log n), classically], whereas that of QuickSelect remains O(n). Explicit expressions for the implied constants are provided by our combinatorial–analytic methods

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Formal and Informal Financing Decisions of Small Businesses

This study investigates small businesses’ financing decisions. Drawing upon asymmetric information theory, institutional theory and relevant literature on cognitive financial constraints, human capital and social capital, we propose a theoretical framework in which financing determinants come from three dimensions: entrepreneurs’ individual factors, organisational (firm-level) factors and contextual (institutional) factors. We employ this model to distinguish four types of firms: (1) firms that use no external finance, (2) firms that use informal finance only, (3) firms that use formal finance only and (4) firms that use both formal and informal finance. An empirical test on Vietnamese small businesses shows that factors from all three dimensions are important in understanding small businesses’ financing decisions