Research Articles (Mathematical Sciences)

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3 x 3 lemma for star-exact sequences
(INT PRESS BOSTON, INC, 2012) Gran, Marino; Janelidze, Zurab; Rodelo, Diana
ENGLISH ABSTRACT: A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can deﬁne short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 × 3 lemma is equiv- alent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with “exact forks” and then, the corresponding 3 × 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories sat- isfying such 3 × 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be uniﬁed in the context of star-regular categories recently intro- duced in a joint work of A. Ursini and the ﬁrst two authors.
Application of Smirnov words to waiting time distributions of runs
(Electronic Journal of Combinatorics, 2017) Freiberg, Uta; Heuberger, Clemens; Prodinger, Helmut
Consider infinite random words over a finite alphabet where the letters occur as an i.i.d. sequence according to some arbitrary distribution on the alphabet. The expectation and the variance of the waiting time for the first completed h-run of any letter (i.e., first occurrence of h subsequential equal letters) is computed. The expected waiting time for the completion of h-runs of j arbitrary distinct letters is also given.
The asymptotic number of binary codes and binary matroids
(Society for Industrial and Applied Mathematics (SIAM), 2006-08) Wild, Marcel
The asymptotic number of nonequivalent binary n-codes is determined. This is also the asymptotic number of nonisomorphic binary matroids on n elements.
Benchmarking multi-rate codon models
(Public Library of Science, 2010-07-21) Delport, Wayne; Scheffler, Konrad; Gravenor, Mike B.; Muse, Spencer V.; Pond, Sergei Kosakovsky
The single rate codon model of non-synonymous substitution is ubiquitous in phylogenetic modeling. Indeed, the use of a non-synonymous to synonymous substitution rate ratio parameter has facilitated the interpretation of selection pressure on genomes. Although the single rate model has achieved wide acceptance, we argue that the assumption of a single rate of non-synonymous substitution is biologically unreasonable, given observed differences in substitution rates evident from empirical amino acid models. Some have attempted to incorporate amino acid substitution biases into models of codon evolution and have shown improved model performance versus the single rate model. Here, we show that the single rate model of non-synonymous substitution is easily outperformed by a model with multiple non-synonymous rate classes, yet in which amino acid substitution pairs are assigned randomly to these classes. We argue that, since the single rate model is so easy to improve upon, new codon models should not be validated entirely on the basis of improved model fit over this model. Rather, we should strive to both improve on the single rate model and to approximate the general time-reversible model of codon substitution, with as few parameters as possible, so as to reduce model over-fitting. We hint at how this can be achieved with a Genetic Algorithm approach in which rate classes are assigned on the basis of sequence information content. © 2010 Delport et al.
Biocapacity optimization in regional planning
(Nature Research, 2017) Guo, Jianjun; Yue, Dongxia; Li, Kai; Hui, Cang
Ecological overshoot has been accelerating across the globe. Optimizing biocapacity has become a key to resolve the overshoot of ecological demand in regional sustainable development. However, most literature has focused on reducing ecological footprint but ignores the potential of spatial optimization of biocapacity through regional planning of land use. Here we develop a spatial probability model and present four scenarios for optimizing biocapacity of a river basin in Northwest China. The potential of enhanced biocapacity and its effects on ecological overshoot and water consumption in the region were explored. Two scenarios with no restrictions on croplands and water use reduced the overshoot by 29 to 53%, and another two scenarios which do not allow croplands and water use to increase worsened the overshoot by 11 to 15%. More spatially flexible transition rules of land use led to higher magnitude of change after optimization. However, biocapacity optimization required a large amount of additional water resources, casting considerable pressure on the already water-scarce socio-ecological system. Our results highlight the potential for policy makers to manage/optimize regional land use which addresses ecological overshoot. Investigation on the feasibility of such spatial optimization complies with the forward-looking policies for sustainable development and deserves further attention.
Calculating the output distribution of stack filters that are erosion-dilation cascades, in particular LULU-filters
(Cornell University Library, 2011) Anguelov, R.; Butler, P. W.; Rohwer, C. H.; Wild, Marcel
ENGLISH ABSTRACT: Two procedures to compute the output distribution 0S of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields 0S for some important LULU-operators S. Properties of 0S can be used to characterize smoothing properties.
Canonical trees, compact prefix-free codes, and sums of unit fractions: a probabilistic analysis
(SIAM, 2015) Heuberger, Clemens; Krenn, Daniel; Wagner, Stephan
For fixed t ≥ 2, we consider the class of representations of 1 as a sum of unit fractions whose denominators are powers of t, or equivalently the class of canonical compact t-ary Huffman codes, or equivalently rooted t-ary plane “canonical” trees. We study the probabilistic behavior of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property), and the number of leaves at maximum distance from the root (discrete distribution).
Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape
(Frontiers Media, 2015) Shi, Pei-Jian; Huang, Jian-Guo; Hui, Cang; Grissino-Mayer, Henri D.; Tardif, Jacques C.; Zhai, Li-Hong; Wang, Fu-Sheng; Li, Bai-Lian
Tree-rings are often assumed to approximate a circular shape when estimating forest productivity and carbon dynamics. However, tree rings are rarely, if ever, circular, thereby possibly resulting in under- or over-estimation in forest productivity and carbon sequestration. Given the crucial role played by tree ring data in assessing forest productivity and carbon storage within a context of global change, it is particularly important that mathematical models adequately render cross-sectional area increment derived from tree rings. We modeled the geometric shape of tree rings using the superellipse equation and checked its validation based on the theoretical simulation and six actual cross sections collected from three conifers. We found that the superellipse better describes the geometric shape of tree rings than the circle commonly used. We showed that a spiral growth trend exists on the radial section over time, which might be closely related to spiral grain along the longitudinal axis. The superellipse generally had higher accuracy than the circle in predicting the basal area increment, resulting in an improved estimate for the basal area. The superellipse may allow better assessing forest productivity and carbon storage in terrestrial forest ecosystems.
Co-authorship networks in South African chemistry and mathematics
(Academy of Science for South Africa, 2008) Durbach, I. N.; Naidoo, D.; Mouton, J.
Co-authorship networks are graphs in which the nodes of the graph represent authors and two authors are connected by an edge if they have written one or more papers together. When applied to the authorship of scholarly papers, analysing the structure of a co-authorship network can provide useful insights into the way in which research is carried out in a particular field. We examine two co-authorship networks in our article, constructed from papers written on the subjects of chemistry and mathematics during the period 1990 to 2005, in which at least one of the authors was South African. Local results are compared with other studies conducted in much larger discipline-wide networks. We find that many of the same patterns exist locally, with the main difference being a far more fragmented South African mathematics network. We discuss some tentative implications of these results.
Codon test : modeling amino acid substitution preferences in coding sequences
(PLOS Computational Biology, 2010-08) Delport, Wayne; Scheffler, Konrad; Botha, Gordon; Gravenor, Mike B.; Muse, Spencer V.; Pond, Sergei L. Kosakovsky
Codon models of evolution have facilitated the interpretation of selective forces operating on genomes. These models, however, assume a single rate of non-synonymous substitution irrespective of the nature of amino acids being exchanged. Recent developments have shown that models which allow for amino acid pairs to have independent rates of substitution offer improved fit over single rate models. However, these approaches have been limited by the necessity for large alignments in their estimation. An alternative approach is to assume that substitution rates between amino acid pairs can be subdivided into K rate classes, dependent on the information content of the alignment. However, given the combinatorially large number of such models, an efficient model search strategy is needed. Here we develop a Genetic Algorithm (GA) method for the estimation of such models. A GA is used to assign amino acid substitution pairs to a series of K rate classes, where K is estimated from the alignment. Other parameters of the phylogenetic Markov model, including substitution rates, character frequencies and branch lengths are estimated using standard maximum likelihood optimization procedures. We apply the GA to empirical alignments and show improved model fit over existing models of codon evolution. Our results suggest that current models are poor approximations of protein evolution and thus gene and organism specific multi-rate models that incorporate amino acid substitution biases are preferred. We further anticipate that the clustering of amino acid substitution rates into classes will be biologically informative, such that genes with similar functions exhibit similar clustering, and hence this clustering will be useful for the evolutionary fingerprinting of genes.
Combining uncertainties in a court of law using Bayesian networks
(Obiter Law Journal: Nelson Mandela Metropolitan University (NMMU), Faculty of Law, 2017) Muller, M. A.
People generally have difficulty dealing with the counter-intuitive notion of probability, and therefore they often misunderstand aspects of uncertainty. This is particularly significant in a court of law when for example an estimate of the probability of the evidence gets confused with an estimate of the probability of guilt. The circumstantial evidence is especially prone to being handled incorrectly. Professor Fenton at the Queen Mary University of London said, “You could argue that virtually every case with circumstantial evidence is ripe for being improved by Bayesian arguments”.1 In this paper, the evidence in a famous court case is revisited in the context of Bayesian networks.
Compactly generating all satisfying truth assignments of a horn formula
(Delft University in cooperation with IOS Press, 2012-01) Wild, Marcel
ENGLISH ABSTRACT: While it was known that all models of a Horn formula can be generated in outputpolynomial time, here we present an explicit algorithm as opposed to the rather vague oracle-scheme suggested in the proof of [6, Thm.4]. It is an instance of some principle of exclusion that compactly (thus not one by one) generates all models of certain Boolean formulae given in CNF. The principle of exclusion can be adapted to generate only the models of weight k. We compare and contrast it with constraint programming, 0; 1 integer programming, and binary decision diagrams.
A comparison of elasticities of viral levels to specific immune response mechanisms in human immunodeficiency virus infection
(BioMed Central, 2014-10) Showa, Sarudzai P.; Nyabadza, Farai; Hove-Musekwa, Senelani D.; Magombedze, Gesham
Background The presence of an asymptomatic phase in an HIV infection indicates that the immune system can partially control the infection. Determining the immune mechanisms that contribute significantly to the partial control of the infection enhance the HIV infection intervention strategies and is important in vaccine development. Towards this goal, a discrete time HIV model, which incorporates the life cycle aspects of the virus, the antibody (humoral) response and the cell-mediated immune response is formulated to determine immune system components that are most efficient in controlling viral levels. Ecological relationships are used to model the interplay between the immune system components and the HIV pathogen. Model simulations and transient elasticity analysis of the viral levels to immune response parameters are used to compare the different immune mechanisms. Results It is shown that cell-mediated immune response is more effective in controlling the viral levels than the antibody response. Killing of infected cells is shown to be crucial in controlling the viral levels. Our results show a negative correlation between the antibody response and the viral levels in the early stages of the infection, but we predicted this immune mechanism to be positively correlated with the viral levels in the late stage of the infection. A result that suggests lack of relevance of antibody response with infection progression. On the contrary, we predicted the cell-mediated immune response to be always negatively correlated with viral levels. Conclusion Neutralizing antibodies can only control the viral levels in the early days of the HIV infection whereas cell-mediated immune response is beneficial during all the stages of the infection. This study predicts that vaccine design efforts should also focus on stimulating killer T cells that target infected cells.
Comparison of spatial downscaling methods of general circulation model results to study climate variability during the Last Glacial Maximum
(Creative Commons Attribution 4.0 License, 2018) Latombe, Guillaume; Burke, Ariane; Vrac, Mathieu; Levavasseur, Guillaume; Dumas, Christophe; Kageyama, Masa; Ramstein, Gilles
The extent to which climate conditions influenced the spatial distribution of hominin populations in the past is highly debated. General circulation models (GCMs) and archaeological data have been used to address this issue. Most GCMs are not currently capable of simulating past surface climate conditions with sufficiently detailed spatial resolution to distinguish areas of potential hominin habitat, however. In this paper, we propose a statistical downscaling method (SDM) for increasing the resolution of climate model outputs in a computationally efficient way. Our method uses a generalised additive model (GAM), calibrated over present-day climatology data, to statistically downscale temperature and precipitation time series from the outputs of a GCM simulating the climate of the Last Glacial Maximum (19 000–23 000 BP) over western Europe. Once the SDM is calibrated, we first interpolate the coarse-scale GCM outputs to the final resolution and then use the GAM to compute surface air temperature and precipitation levels using these interpolated GCM outputs and fine-resolution geographical variables such as topography and distance from an ocean. The GAM acts as a transfer function, capturing non-linear relationships between variables at different spatial scales and correcting for the GCM biases.We tested three different techniques for the first interpolation of GCM output: bilinear, bicubic and kriging. The resulting SDMs were evaluated by comparing downscaled temperature and precipitation at local sites with paleoclimate reconstructions based on paleoclimate archives (archaeozoological and palynological data) and the impact of the interpolation technique on patterns of variability was explored. The SDM based on kriging interpolation, providing the best accuracy, was then validated on present-day data outside of the calibration period. Our results show that the downscaled temperature and precipitation values are in good agreement with paleoclimate reconstructions at local sites, and that our method for producing fine-grained paleoclimate simulations is therefore suitable for conducting paleo-anthropological research. It is nonetheless important to calibrate the GAM on a range of data encompassing the data to be downscaled. Otherwise, the SDM is likely to overcorrect the coarse-grain data. In addition, the bilinear and bicubic interpolation techniques were shown to distort either the temporal variability or the values of the response variables, while the kriging method offered the best compromise. Since climate variability is an aspect of the environment to which human populations may have responded in the past, the choice of interpolation technique is therefore an important consideration.
A complete study of the ramification for any separable cubic global function field
(Springer, 2019) Marques, Sophie; Ward, Jacob
We explicitly describe the ramified places in any separable cubic extension of a cubic global function field in terms of a unique given parameter. This is all done using the uniqueness of the purely cubic closure, which is a useful new tool for the study of cubic function fields. We give a notion of local standard forms, that is useful for many purposes, including classifying and computing of integral bases. We then determine explicitly the genus of any separable cubic extension of any global function field given the minimal polynomial of the extension. The formulae we obtain is particularly useful for further study owing to the well-understood and straightforward close relation between the parameter we define and ramification within the extension.
Computational and theoretical analysis of human diseases associated with infectious pathogens
(Hindawi, 2015) Noutchie, Suares Clovis Oukouomi; Kwuimy, Cedrick Aurelien Kitio; Tewa, Jean Jules; Nyabadza, Farai; Bildik, Necdet
Mathematical models and computer simulations are useful experimental tools for building and testing theories, assessing quantitative conjectures, answering specific questions, determining sensitivities to changes in parameter values, and estimating key parameters from data. Understanding the transmission characteristics of infectious diseases in communities, regions, and countries can lead to better approaches to decreasing the transmission of these diseases.
Computing the output distribution of a stack filter from the DNF of its positive Boolean function
(Cornell University Library, 2010) Wild, Marcel
ENGLISH ABSTRACT: The majority of nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function. The so called selection probabilities can be computed along the way.
Computing various types of lattices freely generated by posets
(Edizioni CERFIM Centro di Ricerche in Fisica e Matematica, 2000) Wild, Marcel
see item for abstract
A conceptual map of invasion biology : integrating hypotheses into a consensus network
(Wiley, 2020-03-25) Enders, Martin; Havemann, Frank; Ruland, Florian; Bernard-Verdier, Maud; Catford, Jane A.; Gomez-Aparicio, Lorena; Haider, Sylvia; Heger, Tina; Kueffer, Christoph; Kuh, Ingolf; Meyerson, Laura A.; Musseau, Camille; Novoa, Ana; Ricciardi, Anthony; Sagouis, Alban; Schittko, Conrad; Strayer, David L.; Vilà, Montserrat; Essl, Franz; Hulme, Philip E.; Van Kleunen, Mark; Kumschick, Sabrina; Lockwood, Julie L.; Mabey, Abigail L.; McGeoch, Melodie A.; Estibaliz, Palma; Pysek, Petr; Saul, Wolf-Christian; Yannelli, Florencia A.; Jeschke, Jonathan M.
Background and aims: Since its emergence in the mid-20th century, invasion biology has matured into a productive research field addressing questions of fundamental and applied importance. Not only has the number of empirical studies increased through time, but also has the number of competing, overlapping and, in some cases, contradictory hypotheses about biological invasions. To make these contradictions and redundancies explicit, and to gain insight into the field’s current theoretical structure, we developed and applied a Delphi approach to create a consensus network of 39 existing invasion hypotheses. Results: The resulting network was analysed with a link-clustering algorithm that revealed five concept clusters (resource availability, biotic interaction, propagule, trait and Darwin’s clusters) representing complementary areas in the theory of invasion biology. The network also displays hypotheses that link two or more clusters, called connecting hypotheses, which are important in determining network structure. The network indicates hypotheses that are logically linked either positively (77 connections of support) or negatively (that is, they contradict each other; 6 connections). Significance: The network visually synthesizes how invasion biology’s predominant hypotheses are conceptually related to each other, and thus, reveals an emergent structure – a conceptual map – that can serve as a navigation tool for scholars, practitioners and students, both inside and outside of the field of invasion biology, and guide the development of a more coherent foundation of theory. Additionally, the outlined approach can be more widely applied to create a conceptual map for the larger fields of ecology and biogeography.
Correcting the bias of empirical frequency parameter estimators in codon models
(Public Library of Science -- PLOS, 2010-07) Kosakovsky Pond, Sergei; Delport, Wayne; Muse, Spencer V.; Scheffler, Konrad
Markov models of codon substitution are powerful inferential tools for studying biological processes such as natural selection and preferences in amino acid substitution. The equilibrium character distributions of these models are almost always estimated using nucleotide frequencies observed in a sequence alignment, primarily as a matter of historical convention. In this note, we demonstrate that a popular class of such estimators are biased, and that this bias has an adverse effect on goodness of fit and estimates of substitution rates. We propose a ‘‘corrected’’ empirical estimator that begins with observed nucleotide counts, but accounts for the nucleotide composition of stop codons. We show via simulation that the corrected estimates outperform the de facto standard F3|4 estimates not just by providing better estimates of the frequencies themselves, but also by leading to improved estimation of other parameters in the evolutionary models. On a curated collection of 856 sequence alignments, our estimators show a significant improvement in goodness of fit compared to the F3|4 approach. Maximum likelihood estimation of the frequency parameters appears to be warranted in many cases, albeit at a greater computational cost. Our results demonstrate that there is little justification, either statistical or computational, for continued use of the F3|4-style estimators.

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