Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function M ( x ) = ∑ k = 1 x μ ( k ) , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function, namely, based on reorganizing globally in distinct blocks the terms of its series. With this aim, we focus attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős–Kac theorem for square-free numbers, etc. These results point to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk. We also present an argument in favor of the thesis that the validity of the RH also implies the validity of the generalized RH for the Dirichlet L-functions. Next we study the local properties of the Mertens function, i.e. its variation induced by each Möbius coefficient restricted to the square-free numbers. Motivated by the natural curiosity to see how closely to a purely random walk any sub-sequence is extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity; together with several frequency tests within a block, the list of our tests includes those for the longest run of ones in a block, the binary matrix rank test, the discrete Fourier transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc, for a total of 18 different tests. The successful outputs of all these tests (each of them with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive ‘experimental’ confirmations of the Brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however extremely improbable.

Using the longest run subsequence problem within homology-based scaffolding

Genome assembly is one of the most important problems in computational genomics. Here, we suggest addressing an issue that arises in homology-based scaffolding, that is, when linking and ordering contigs to obtain larger pseudo-chromosomes by means of a second incomplete assembly of a related species. The idea is to use alignments of binned regions in one contig to find the most homologous contig in the other assembly. We show that ordering the contigs of the other assembly can be expressed by a new string problem, the longest run subsequence problem (LRS). We show that LRS is NP-hard and present reduction rules and two algorithmic approaches that, together, are able to solve large instances of LRS to provable optimality. All data used in the experiments as well as our source code are freely available. We demonstrate its usefulness within an existing larger scaffolding approach by solving realistic instances resulting from partial Arabidopsis thaliana assemblies in short computation time.

On the exceptional sets in Erdös–Rényi limit theorem of β-expansion

Let [Formula: see text] be a real number. For any [Formula: see text], the run-length function [Formula: see text] is defined as the length of the longest run of 0’s amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. Let [Formula: see text] be a non-decreasing sequence of integers and [Formula: see text], we define [Formula: see text] In this paper, we show that the set [Formula: see text] has full Hausdorff dimension under the condition that [Formula: see text].

Ambiguity Reduction Through Optimal Set of Region Selection Using GA and BFO for Handwritten Bangla Character Recognition

To recognize different patterns, identification of local regions where the pattern classes differ significantly is an inherent ability of the human cognitive system. This inherent ability of human beings may be imitated in any pattern recognition system by incorporating the ability of locating the regions that contain the maximum discriminating information among the pattern classes. In this chapter, the concept of Genetic Algorithm (GA) and Bacterial Foraging Optimization (BFO) are discussed to identify those regions having maximum discriminating information. The discussion includes the evaluation of the methods on the sample images of handwritten Bangla digit and Basic character, which is a subset of Bangla character set. Different methods of sub-image or local region creation such as random creation or based on the Center of Gravity (CG) of the foreground pixels are also discussed here. Longest run features, extracted from the generated local regions, are used as local feature in the present chapter. Based on these extracted local features, together with global features, the algorithms are applied to search for the optimal set of local regions. The obtained results are higher than that results obtained without optimization on the same data set.

A Holistic Approach for Handwritten Hindi Word Recognition

Holistic word recognition attempts to recognize the entire word image as a single pattern. In general, it performs better than segmentation based word recognition model for known, fixed and small sized lexicon. The present work deals with recognition of handwritten words in Hindi in holistic way. Features like area, aspect ratio, density, pixel ratio, longest run, centroid and projection length are extracted either from entire word image or from the hypothetically generated sub-images of the same. An 89-elements feature vector has been designed to represent each word in the feature space and five different classifiers have been used for measuring recognition performances. Considering the complexities of Hindi characters, the technique shows an impressive result using a Multilayer Perceptron (MLP) based classifier. Moreover, the technique shows scale and rotation invariant nature to a significant extent.

Modern Eras and Alternative Futures: The American Elections of 2016 in the Longest Run

On the morning after, the aggregate story of the American elections of 2016 was one of surprise. Professional pundits – not to mention the pollsters, those masters of disaggregation – had been rather strikingly wrong. So, one of the major products of the elections of 2016 was a major research project for these professional disaggregators. But can those who treat elections as aggregate phenomena – as patterned wins and losses across time – fit these results into an ongoing pattern? Within this pattern, what looks familiar and what looks anomalous? Or, said the other way around, when