Royal Society Open Science Mathematics
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Royal Society Open Science RSS feed -- recent Mathematics articles2054-5703Royal Society Open Science<![CDATA[Classification of self-assembling protein nanoparticle architectures for applications in vaccine design]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/4/161092?rss=1
We introduce here a mathematical procedure for the structural classification of a specific class of self-assembling protein nanoparticles (SAPNs) that are used as a platform for repetitive antigen display systems. These SAPNs have distinctive geometries as a consequence of the fact that their peptide building blocks are formed from two linked coiled coils that are designed to assemble into trimeric and pentameric clusters. This allows a mathematical description of particle architectures in terms of bipartite (3,5)-regular graphs. Exploiting the relation with fullerene graphs, we provide a complete atlas of SAPN morphologies. The classification enables a detailed understanding of the spectrum of possible particle geometries that can arise in the self-assembly process. Moreover, it provides a toolkit for a systematic exploitation of SAPNs in bioengineering in the context of vaccine design, predicting the density of B-cell epitopes on the SAPN surface, which is critical for a strong humoral immune response.
]]>2017-04-26T00:53:58-07:00info:doi/10.1098/rsos.161092hwp:master-id:royopensci;rsos.1610922017-04-26Mathematics44161092161092<![CDATA[Optimal strategies for throwing accurately]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/4/170136?rss=1
The accuracy of throwing in games and sports is governed by how errors in planning and initial conditions are propagated by the dynamics of the projectile. In the simplest setting, the projectile path is typically described by a deterministic parabolic trajectory which has the potential to amplify noisy launch conditions. By analysing how parabolic trajectories propagate errors, we show how to devise optimal strategies for a throwing task demanding accuracy. Our calculations explain observed speed–accuracy trade-offs, preferred throwing style of overarm versus underarm, and strategies for games such as dart throwing, despite having left out most biological complexities. As our criteria for optimal performance depend on the target location, shape and the level of uncertainty in planning, they also naturally suggest an iterative scheme to learn throwing strategies by trial and error.
]]>2017-04-26T00:05:43-07:00info:doi/10.1098/rsos.170136hwp:master-id:royopensci;rsos.1701362017-04-26Mathematics44170136170136<![CDATA[On strongly connected networks with excitable-refractory dynamics and delayed coupling]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/4/160912?rss=1
We consider a directed graph model for the human brain’s neural architecture that is based on small scale, directed, strongly connected sub-graphs (SCGs) of neurons, that are connected together by a sparser mesoscopic network. We assume transmission delays within neuron-to-neuron stimulation, and that individual neurons have an excitable-refractory dynamic, with single firing ‘spikes’ occurring on a much faster time scale than that of the transmission delays. We demonstrate numerically that the SCGs typically have attractors that are equivalent to continual winding maps over relatively low-dimensional tori, thus representing a limit on the range of distinct behaviour. For a discrete formulation, we conduct a large-scale survey of SCGs of varying size, but with the same local structure. We demonstrate that there may be benefits (increased processing capacity and efficiency) in brains having evolved to have a larger number of small irreducible sub-graphs, rather than few, large irreducible sub-graphs. The network of SCGs could be thought of as an architecture that has evolved to create decisions in the light of partial or early incoming information. Hence the applicability of the proposed paradigm to underpinning human cognition.
]]>2017-04-05T00:05:32-07:00info:doi/10.1098/rsos.160912hwp:master-id:royopensci;rsos.1609122017-04-05Mathematics44160912160912<![CDATA[Suicides on the Austrian railway network: hotspot analysis and effect of proximity to psychiatric institutions]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/3/160711?rss=1
Railway suicide is a significant public health problem. In addition to the loss of lives, these suicides occur in public space, causing traumatization among train drivers and passengers, and significant public transport delays. Prevention efforts depend upon accurate knowledge of clustering phenomena across the railway network, and spatial risk factors. Factors such as proximity to psychiatric institutions have been discussed to impact on railway suicides, but analytic evaluations are scarce and limited. We identify 15 hotspots on the Austrian railway system while taking case location uncertainties into account. These hotspots represent 0.9% of the total track length (5916 km/3676 miles) that account for up to 17% of all railway suicides (N=1130). We model suicide locations on the network using a smoothed inhomogeneous Poisson process and validate it using randomization tests. We find that the density of psychiatric beds is a significant predictor of railway suicide. Further predictors are population density, multitrack structure and—less consistently—spatial socio-economic factors including total suicide rates. We evaluate the model for the identified hotspots and show that the actual influence of these variables differs across individual hotspots. This analysis provides important information for suicide prevention research and practice. We recommend structural separation of railway tracks from nearby psychiatric institutions to prevent railway suicide.
]]>2017-03-08T00:05:29-08:00info:doi/10.1098/rsos.160711hwp:master-id:royopensci;rsos.1607112017-03-08Mathematics43160711160711<![CDATA[The elastic theory of shells using geometric algebra]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/3/170065?rss=1
We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.
]]>2017-03-08T00:05:29-08:00info:doi/10.1098/rsos.170065hwp:master-id:royopensci;rsos.1700652017-03-08Mathematics43170065170065<![CDATA[Modelling human perception processes in pedestrian dynamics: a hybrid approach]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/3/160561?rss=1
In this paper, we present a hybrid mathematical model describing crowd dynamics. More specifically, our approach is based on the well-established Helbing-like discrete model, where each pedestrian is individually represented as a dimensionless point and set to move in order to reach a target destination, with deviations deriving from both physical and social forces. In particular, physical forces account for interpersonal collisions, whereas social components include the individual desire to remain sufficiently far from other walkers (the so-called territorial effect). In this respect, the repulsive behaviour of pedestrians is here set to be different from traditional Helbing-like methods, as it is assumed to be largely determined by how they perceive the presence and the position of neighbouring individuals, i.e. either objectively as pointwise/localized entities or subjectively as spatially distributed masses. The resulting modelling environment is then applied to specific scenarios, that first reproduce a real-world experiment, specifically designed to derive our model hypothesis. Sets of numerical realizations are also run to analyse in more details the pedestrian paths resulting from different types of perception of small groups of static individuals. Finally, analytical investigations formalize and validate from a mathematical point of view selected simulation outcomes.
]]>2017-03-01T01:27:27-08:00info:doi/10.1098/rsos.160561hwp:master-id:royopensci;rsos.1605612017-03-01Mathematics43160561160561<![CDATA[Parapatric speciation in three islands: dynamics of geographical configuration of allele sharing]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/2/160819?rss=1
We studied the time to speciation by geographical isolation for a species living on three islands connected by rare migration. We assumed that incompatibility was controlled by a number of quantitative loci and that individuals differing in loci by more than a threshold did not mix genetically with each other. For each locus, we defined the geographical configuration (GC), which specifies islands with common alleles, and traced the stochastic transitions between different GCs. From these results, we calculated the changes in genetic distances. As a single migration event provides an opportunity for transitions in multiple loci, the GCs of different loci are correlated, which can be evaluated by constructing the stochastic differential equations of the number of loci with different GCs. Our model showed that the low number of incompatibility loci facilitates parapatric speciation and that migrants arriving as a group shorten the waiting time to speciation compared with the same number of migrants arriving individually. We also discuss how speciation rate changes with geographical structure.
]]>2017-02-22T00:05:23-08:00info:doi/10.1098/rsos.160819hwp:master-id:royopensci;rsos.1608192017-02-22Mathematics42160819160819<![CDATA[Prediction limits of mobile phone activity modelling]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/2/160900?rss=1
Thanks to their widespread usage, mobile devices have become one of the main sensors of human behaviour and digital traces left behind can be used as a proxy to study urban environments. Exploring the nature of the spatio-temporal patterns of mobile phone activity could thus be a crucial step towards understanding the full spectrum of human activities. Using 10 months of mobile phone records from Greater London resolved in both space and time, we investigate the regularity of human telecommunication activity on urban scales. We evaluate several options for decomposing activity timelines into typical and residual patterns, accounting for the strong periodic and seasonal components. We carry out our analysis on various spatial scales, showing that regularity increases as we look at aggregated activity in larger spatial units with more activity in them. We examine the statistical properties of the residuals and show that it can be explained by noise and specific outliers. Also, we look at sources of deviations from the general trends, which we find to be explainable based on knowledge of the city structure and places of attractions. We show examples how some of the outliers can be related to external factors such as specific social events.
]]>2017-02-15T01:33:41-08:00info:doi/10.1098/rsos.160900hwp:master-id:royopensci;rsos.1609002017-02-15Mathematics42160900160900<![CDATA[Calibration with confidence: a principled method for panel assessment]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/2/160760?rss=1
Frequently, a set of objects has to be evaluated by a panel of assessors, but not every object is assessed by every assessor. A problem facing such panels is how to take into account different standards among panel members and varying levels of confidence in their scores. Here, a mathematically based algorithm is developed to calibrate the scores of such assessors, addressing both of these issues. The algorithm is based on the connectivity of the graph of assessors and objects evaluated, incorporating declared confidences as weights on its edges. If the graph is sufficiently well connected, relative standards can be inferred by comparing how assessors rate objects they assess in common, weighted by the levels of confidence of each assessment. By removing these biases, ‘true’ values are inferred for all the objects. Reliability estimates for the resulting values are obtained. The algorithm is tested in two case studies: one by computer simulation and another based on realistic evaluation data. The process is compared to the simple averaging procedure in widespread use, and to Fisher's additive incomplete block analysis. It is anticipated that the algorithm will prove useful in a wide variety of situations such as evaluation of the quality of research submitted to national assessment exercises; appraisal of grant proposals submitted to funding panels; ranking of job applicants; and judgement of performances on degree courses wherein candidates can choose from lists of options.
]]>2017-02-08T01:51:00-08:00info:doi/10.1098/rsos.160760hwp:master-id:royopensci;rsos.1607602017-02-08Mathematics42160760160760<![CDATA[Symmetry, topology and the maximum number of mutually pairwise-touching infinite cylinders: configuration classification]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/1/160729?rss=1
We provide a complete classification of possible configurations of mutually pairwise-touching infinite cylinders in Euclidian three-dimensional space. It turns out that there is a maximum number of such cylinders possible in three dimensions independently of the shape of the cylinder cross-sections. We give the explanation of the uniqueness of the non-trivial configuration of seven equal mutually touching round infinite cylinders found earlier. Some results obtained for the chirality matrix, which is equivalent to the Seidel adjacency matrix, may be found useful for the theory of graphs.
]]>2017-01-18T00:05:48-08:00info:doi/10.1098/rsos.160729hwp:master-id:royopensci;rsos.1607292017-01-18Mathematics41160729160729<![CDATA[A non-equilibrium formulation of food security resilience]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/1/160874?rss=1
Resilience, the ability to recover from adverse events, is of fundamental importance to food security. This is especially true in poor countries, where basic needs are frequently threatened by economic, environmental and health shocks. An empirically sound formalization of the concept of food security resilience, however, is lacking. Here, we introduce a general non-equilibrium framework for quantifying resilience based on the statistical notion of persistence. Our approach can be applied to any food security variable for which high-frequency time-series data are available. We illustrate our method with per capita kilocalorie availability for 161 countries between 1961 and 2011. We find that resilient countries are not necessarily those that are characterized by high levels or less volatile fluctuations of kilocalorie intake. Accordingly, food security policies and programmes will need to be tailored not only to welfare levels at any one time, but also to long-run welfare dynamics.
]]>2017-01-18T00:05:48-08:00info:doi/10.1098/rsos.160874hwp:master-id:royopensci;rsos.1608742017-01-18Mathematics41160874160874<![CDATA[Not Normal: the uncertainties of scientific measurements]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/1/160600?rss=1
Judging the significance and reproducibility of quantitative research requires a good understanding of relevant uncertainties, but it is often unclear how well these have been evaluated and what they imply. Reported scientific uncertainties were studied by analysing 41 000 measurements of 3200 quantities from medicine, nuclear and particle physics, and interlaboratory comparisons ranging from chemistry to toxicology. Outliers are common, with 5 disagreements up to five orders of magnitude more frequent than naively expected. Uncertainty-normalized differences between multiple measurements of the same quantity are consistent with heavy-tailed Student’s t-distributions that are often almost Cauchy, far from a Gaussian Normal bell curve. Medical research uncertainties are generally as well evaluated as those in physics, but physics uncertainty improves more rapidly, making feasible simple significance criteria such as the 5 discovery convention in particle physics. Contributions to measurement uncertainty from mistakes and unknown problems are not completely unpredictable. Such errors appear to have power-law distributions consistent with how designed complex systems fail, and how unknown systematic errors are constrained by researchers. This better understanding may help improve analysis and meta-analysis of data, and help scientists and the public have more realistic expectations of what scientific results imply.
]]>2017-01-11T00:45:45-08:00info:doi/10.1098/rsos.160600hwp:master-id:royopensci;rsos.1606002017-01-11Mathematics41160600160600<![CDATA[Measure for degree heterogeneity in complex networks and its application to recurrence network analysis]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/1/160757?rss=1
We propose a novel measure of degree heterogeneity, for unweighted and undirected complex networks, which requires only the degree distribution of the network for its computation. We show that the proposed measure can be applied to all types of network topology with ease and increases with the diversity of node degrees in the network. The measure is applied to compute the heterogeneity of synthetic (both random and scale free (SF)) and real-world networks with its value normalized in the interval [0,1]. To define the measure, we introduce a limiting network whose heterogeneity can be expressed analytically with the value tending to 1 as the size of the network N tends to infinity. We numerically study the variation of heterogeneity for random graphs (as a function of p and N) and for SF networks with and N as variables. Finally, as a specific application, we show that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from the time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors.
]]>2017-01-11T00:45:45-08:00info:doi/10.1098/rsos.160757hwp:master-id:royopensci;rsos.1607572017-01-11Mathematics41160757160757