Free cumulants,Schröder trees,and operads

The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.     

On weighting two criteria with a parameter in combinatorial optimization problems

Two criteria in a combinatorial problem are often combined in a weighted sum objective using a weighting parameter between 0 and 1. For special problem types, e.g., when one of the criteria is a bottleneck value, efficient algorithms are known that solve for a given value of the weighting parameter.     

Método de elementos de borde jerárquico basado en el árbol de Barnes-Hut aplicado a flujo reptante exterior

In this work, a hierarchical variant of a boundary element method and its use in Stokes flow around three-dimensional rigid bodies in steady regime is presented. The proposal is based on the descending hierarchical low-order and self-adaptive algorithm of Barnes-Hut, and it is used in conjunction with an indirect boundary integral formulation of second class, whose source term is a function of the undisturbed velocity. The solution field is the double layer surface density, which is modified in order to complete the eigenvalue spectrum of the integral operator. In this way, the rigid modes are eliminated and both a non-zero force and a non-null torque on the body could be calculated. The elements are low order flat triangles, and an iterative solution by generalized minimal residual (GMRES) is used. Numerical examples include cases with analytical solutions, bodies with edges and vertices, or with intricate shapes. The main advantage of the presented technique is the possibility of considering a greater number of degrees of freedom regarding traditional collocation methods, due to the decreased demand of main memory and the reduction in the computation times.     

Orthonormal bases generated by Cuntz algebras

We show how some orthonormal bases can be generated by representations of the Cuntz algebra; these include Fourier bases on fractal measures, generalized Walsh bases on the unit interval and piecewise exponential bases on the middle third Cantor set.     

Several Jordan-algebraic aspects of optimization†

We describe some concepts from the theory of Euclidean Jordan algebras and their use in optimization theory. This includes: primal-dual algorithms, optimality conditions, convexity of spectral functions, proof of some inequalities and a Jordan-algebraic version of Horn–Schur theorem     

The q-Calkin-Wilf tree

We define a q-analogue of the Calkin-Wilf tree and the Calkin-Wilf sequence. We show that the nth term f(n;q) of the q-analogue of the Calkin-Wilf sequence is the generating function for the number of hyperbinary expansions of n according to the number of powers that are used exactly twice. We also present formulae for branches within the q-analogue of the Calkin-Wilf tree and predecessors and successors of terms in the q-analogue of the Calkin-Wilf sequence.     

字符串匹配算法在DNA序列比对中的应用

高通量测序技术的飞速发展让生物信息领域迎来了大数据时代。新技术在提供海量生物遗传信息的同时,也给分析这些数据带来了新的挑战。DNA序列比对是信息分析流程中的关键步骤,为后续的变异检测提供序列比对信息。2015“深圳杯”数学建模夏令营B题以DNA序列比对为研究课题,希望参赛学生给出序列快速比对的最佳方案。本文简要点评了各参赛队伍的解答情况,然后介绍了现有DNA序列比对软件中用到的算法和数据结构。     

An improved ensemble learning machine for biological activity prediction of tyrosine kinase inhibitors

Boosting is one of the most important strategies in ensemble learning because of its ability to improve the stability and performance of weak learners. It is nonparametric, multivariate, fast and interpretable but is not robust against outliers. To enhance its prediction accuracy as well as immunize it against outliers, a modified version of a boosting algorithm (AdaBoost R2) was developed and called AdaBoost R3. In the sampling step, extremum samples were added to the boosting set. In the robustness step, a modified Huber loss function was applied to overcome the outlier problem. In the output step, a deterministic threshold was used to guarantee that bad predictions do not participate in the final output. The performance of the modified algorithm was investigated with two anticancer data sets of tyrosine kinase inhibitors, and the mechanism of inhibition was studied using the relative weighted variable importance procedure. Investigating the effect of base learner's strength reveals that boosting is only successful using the classification and regression tree method (a weak to moderate learner) and does not have a significant effect using the radial basis functions partial least square method (a strong base learners). Copyright © 2015 John Wiley & Sons, Ltd.     

Induction of decision trees using genetic programming for modelling ecotoxicity data: adaptive discretization of real-valued endpoints

Recent literature has demonstrated the applicability of genetic programming to induction of decision trees for modelling toxicity endpoints. Compared with other decision tree induction techniques that are based upon recursive partitioning employing greedy searches to choose the best splitting attribute and value at each node that will necessarily miss regions of the search space, the genetic programming based approach can overcome the problem. However, the method still requires the discretization of the often continuous-valued toxicity endpoints prior to the tree induction. A novel extension of this method, YAdapt, is introduced in this work which models the original continuous endpoint by adaptively finding suitable ranges to describe the endpoints during the tree induction process, removing the need for discretization prior to tree induction and allowing the ordinal nature of the endpoint to be taken into account in the models built.