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This paper studies the reach control problem for a coupled affine multi-agent system, which aims to find an affine feedback control for the trajectories of the agents to reach and exit a particular facet of a given simplex in the state space in finite time. The interactions between agents characterized by diffusive coupling prevent the effective construction of controller using the well developed techniques to study similar problems for affine single-agent systems. In fact, the affine feedback control designed for a single affine system may not work for the multi-agent case anymore as some agent can be driven to exit the simplex through a restricted facet under the influence from its coupled peers. A sufficient condition is developed to guarantee that all the agents move continuously in a cone containing the simplex and exit through the exit facets in finite time under an affine feedback control. A numerical example is given to verify the effectiveness of our derived result. [ABSTRACT FROM AUTHOR]

We propose an approach to determine the distribution of particular linear combinations of hybrid censored order statistics which is based on the calculation of volumes of polytopes. For this purpose, we establish efficient and compact volume formulas in terms of B-splines. Further, we illustrate our approach for ten different progressive hybrid censoring schemes under an exponential assumption. [ABSTRACT FROM AUTHOR]

The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison's seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain. [ABSTRACT FROM AUTHOR]

Previous research has found multiplex (MPX) children have an advantage in cognition compared to simplex (SPX) children. However, MPX parent's previous experience with older diagnosed siblings has not been considered. We used a large database sample to investigate the MPX advantage and contribution of birth order. Children from the Autism Genetic Resource Exchange (AGRE) were stratified into first- (MPX1, n = 152) and second-affected MPX (MPX2, n = 143), SPX (n = 111), and only-child SPX (SPXOC, n = 23) groups. Both MPX groups had higher cognitive scores compared to SPX groups, with no differences between MPX1 and MPX2 groups. No differences were found for autism symptoms or adaptive behaviour. These results suggest parent experience due to birth order is an unlikely contributor to the MPX cognitive advantage. [ABSTRACT FROM AUTHOR]

In this note, we construct generalized Bernstein‐Kantorovich–type operators on a triangle. The concern of this note is to present a Voronovskaja‐type and Grüss Voronovskaja‐type asymptotic theorems, and some estimates of the rate of approximation with the help of K‐functional, first and second order modulus of continuity. We also obtain Korovkin‐ and Voronovskaja‐type statistical approximation theorems via weighted mean matrix method. Lastly, we show that the numerical results which explain the validity of the theoretical results and the effectiveness of the constructed operators. [ABSTRACT FROM AUTHOR]

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It is a well known fact that the widely used Nelder–Mead (NM) optimization method soon becomes inefficient as the dimension of the problem grows. It has been observed that part of the reason for the inefficiency is that the search direction becomes practically perpendicular to the direction of the local downhill gradient. In this paper, we identify which operations are responsible for the increase of the angle between the search direction and the direction of the local downhill gradient. We show that it is possible to decrease this angle by randomly perturbing the centroid of the n best vertices around which the original NM algorithm moves the worst vertex. Using a perturbed centroid, the algorithm outperforms the standard NM method as well as the NM method with adaptive parameters for problems with higher dimensions, and works well for problems with dimensions of well over 100. [ABSTRACT FROM AUTHOR]

The volume of a pentahedron is calculated in terms of dihedral angles, sides, face angles of component tetrahedrons (3-simplex) using an exact computation technique (known as signed simplex decomposition). Here the signed simplex decomposition means an established technique to subdivide a polyhedra (a simplicial complex or more strictly CW complex) to component tetrahedral (3-simplex) which is used in computer graphics where meshes with pyramids, prisms or hexahedra must be subdivided into tetrahedral to use efficient algorithms for volume-rendering, iso-contouring and particle advect. Moreover we attempt to give a tensorial notation of the volume of the pentahedron. [ABSTRACT FROM AUTHOR]

This paper presents an overview of results for the geostatistical analysis of collocated multivariate data sets, whose variables form a composition, where the components represent the relative importance of the parts forming a whole. Such data sets occur most often in mining, hydrogeochemistry and soil science, but the results gathered here are relevant for any regionalised compositional data set. The paper covers the basic definitions, the analysis of the spatial codependence between components, mapping methods of cokriging and cosimulation honoring compositional constraints, the role of pre- and post-transformations such as log-ratios or multivariate normal score transforms, and block-support upscaling. The main result is that multivariate geostatistical techniques can and should be performed on log-ratio scores, in which case the system data-variograms-cokriging/cosimulation is intrinsically consistent, delivering the same results regardless of which log-ratio transformation was used to represent them. Proofs of all statements are included in an appendix. [ABSTRACT FROM AUTHOR]

Methods in parametric cluster analysis commonly assume data can be modelled by means of a finite mixture of distributions. However, associating each mixture component to one cluster is frequently misleading because different mixture components can overlap, and then, associated clusters can overlap too suggesting a unique cluster. A number of approaches have already been proposed to construct the clusters by merging components using the posterior probabilities. This article presents a generic approach for building a hierarchy of mixture components that integrates and generalizes some techniques proposed earlier in the literature. Using this proposal, two new techniques based on the log-ratio of posterior probabilities are introduced. Moreover, to decide the final number of clusters, two new methods are presented. Simulated and real datasets are used to illustrate this methodology. [ABSTRACT FROM AUTHOR]

Prospective, longitudinal designs utilizing "high‐risk" infant siblings of children diagnosed with Autism Spectrum Disorder (ASD‐sibs) have provided unique and valuable insights regarding the early ASD phenotype. However, it remains unclear whether these cases are representative of all children with ASD. The objective in the present study was to investigate whether the early development of toddlers with ASD from multiplex (MPX) families, who have an affected older sibling, is similar or different to toddlers with ASD from simplex (SPX) families, where there is no affected sibling. A further aim was to examine patterns of association between autism symptom severity and cognitive functioning within each group to inform possible mechanisms for group similarities/differences. Behavioral and cognitive assessment data from a sample of toddlers with ASD was utilized, comprising 45 MPX, 127 first‐born SPX, and 72 later‐born SPX toddlers. Participants in the MPX group had significantly higher developmental quotients on the Mullen Scales of Early Learning compared to those in the SPX groups, who did not differ from each other. However, all three groups were similar on their autism severity scores (measured using the Autism Diagnostic Observation Schedule and the Autism Diagnostic Interview), and the pattern of relationships between cognitive ability and autism symptom severity. The results suggest that caution be exercised in generalizing findings from ASD‐sib samples to other samples of children with ASD. The higher cognitive abilities in the MPX group, in addition to biological differences, may also be an outcome of family environmental factors, which deserves further investigation. Autism Research 2019, 12: 682–693. © 2019 International Society for Autism Research, Wiley Periodicals, Inc. Lay Summary: We sought to establish whether toddlers with autism from families where there is more than one affected child, called multiplex families, are different to children from simplex families, where there is only one affected child, and no other members within the immediate family with an autism diagnosis. We found that while toddlers from multiplex families were similar to those from simplex families in their autism symptoms, they were more developmentally advanced than children in the latter group. [ABSTRACT FROM AUTHOR]

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the ℓ1-norm is not strictly convex. We expect that the above novel geometric interpretation will aid the analysis of rules defined by votewise distances, and the discovery of new rules with desirable properties. [ABSTRACT FROM AUTHOR]

A self-organizing map (SOM) is a non-linear projection of a D-dimensional data set, where the distance among observations is approximately preserved on to a lower dimensional space. The SOM arranges multivariate data based on their similarity to each other by allowing pattern recognition leading to easier interpretation of higher dimensional data. The SOM algorithm allows for selection of different map topologies, distances and parameters, which determine how the data will be organized on the map. In the particular case of compositional data (such as elemental, mineralogical, or maceral abundance), the sample space is governed by Aitchison geometry and extra steps are required prior to their SOM analysis. Following the principle of working on log-ratio coordinates, the simplicial operations and the Aitchison distance, which are appropriate elements for the SOM, are presented. With this structure developed, a SOM using Aitchison geometry is applied to properly interpret elemental data from combustion products (bottom ash, fly ash, and economizer fly ash) in a Wyoming coal-fired power plant. Results from this effort provide knowledge about the differences between the ash composition in the coal combustion process. [ABSTRACT FROM AUTHOR]

The structural and dynamical properties in sodium silicate liquid were investigated by molecular dynamics method. To clarify the distribution of sodium atoms in model, characteristics of simplex have been investigated. The simulation results reveal that Na2O⋅4SiO2 (NS4) liquid has a lot of simplexes with four sodium atoms inside but about half of simplexes do not have sodium. The spatial distribution of sodium is nonuniform, sodium tends to be in the nonbridging oxygen-simplexes and in larger-radius simplex. Moreover, the sodium density for nonbridging oxygen region is significantly higher than the one for Si-region. Namely, link-cluster function F l k (r, t) has been used to clarify dynamical heterogeneity in NS4 liquid. The F l k (r, t) for sets of random, immobile and mobile network atoms is quite different, which indicates that the dynamics of network atoms is heterogeneous. The Si–O network has the structure with two separated domains (immobile and mobile domains). These types of domain are significantly different in local microstructure, mobility of atoms and chemical composition. [ABSTRACT FROM AUTHOR]

The family of lattice simplices in Rn formed by the convex hull of the standardbasis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative algebraic geometry and mirror symmetry. From this perspective, it is useful to have formulae for their discrete volumes via Ehrhart h*-polynomials. Here we show, via an association with numeral systems, that such simplices yield h*-polynomials with properties that are also desirable from a combinatorial perspective. First, we identify n-simplices in this family that associate via their normalized volume to the nth place value of a positional numeral system. We then observe that their h*-polynomials admit combinatorial formula via descent-like statistics on the numeral strings encoding the nonnegative integers within the system. With these methods, we recover ubiquitous h*-polynomials including the Eulerian polynomials and the binomial coefficients arising from the factoradic and binary numeral systems, respectively. We generalize the binary case to base-r numeral systems for all r ≤ 2, and prove that the associated h*-polynomials are real-rooted and unimodal for r ≤ 2 and n ≤ 1. [ABSTRACT FROM AUTHOR]

'Pseudo‐profound bullshit' (PPBS) is a class of meaningless statements designed to appear profound. Profundity ratings for PPBS have been found to be negatively related to analytical thinking and positively related to epistemically suspect beliefs (e.g. belief in the paranormal). Conceptually similar traits within the Openness/Intellect (O/I) domain form a simplex, whereby traits are arranged along a single dimension from intelligence to apophenia (i.e. observing patterns or causal connections were none exist). Across two studies (total N = 297), we attempted to replicate the O/I simplex and determine how it relates to perceiving PPBS as profound. Participants completed questionnaires measuring traits from the O/I simplex and provided profundity ratings for PPBS. Profundity ratings of PPBS tended to correlate negatively with intelligence and positively with apophenia. The association with intelligence generally reflected a greater ability to discriminate the profound from the pseudo‐profound, whereas the association with apophenia reflected poorer discrimination in Study 1, with less conclusive results in Study 2. In both studies, the O/I simplex was closely replicated. The results suggest a link between the O/I domain and perceiving PPBS as profound and tentatively support the theory that intelligence may protect against apophenia. © 2018 European Association of Personality Psychology [ABSTRACT FROM AUTHOR]

In this paper, we provide counterexamples to Ganikhodjaev's conjecture on mean ergodicity of quadratic stochastic operators acting on the finite dimensional simplex. [ABSTRACT FROM AUTHOR]

Blind source separation is addressed, using a novel data-driven approach, based on a well-established probabilistic model. The proposed method is specifically designed for separation of multichannel audio mixtures. The algorithm relies on spectral decomposition of the correlation matrix between different time frames. The probabilistic model implies that the column space of the correlation matrix is spanned by the probabilities of the various speakers across time. The number of speakers is recovered by the eigenvalue decay, and the eigenvectors form a simplex of the speakers’ probabilities. Time frames dominated by each of the speakers are identified exploiting convex geometry tools on the recovered simplex. The mixing acoustic channels are estimated utilizing the identified sets of frames, and a linear umixing is performed to extract the individual speakers. The derived simplexes are visually demonstrated for mixtures of two, three, and four speakers. We also conduct a comprehensive experimental study, showing high separation capabilities in various reverberation conditions. [ABSTRACT FROM AUTHOR]

We establish a new refinement of the right-hand side of the Hermite-Hadamard inequality for convex functions of several variables defined on simplices. [ABSTRACT FROM AUTHOR]

This 122-page book is intended to be a practical guide to CoDa analysis and its easyto-read format and didactic layout are designed for students and researchers alike from different fields. For more insight, the interested reader can find other books that present the subject in a more up-to-date manner and cover more multivariate techniques with applications and examples from geochemistry. [ABSTRACT FROM AUTHOR]

A Duval triangle is a diagram used for fault type identification in dissolved-gas analysis of oil-filled high-voltage transformers and other electrical apparatus. The proportional concentrations of three fault gases (such as methane, ethylene, and acetylene) are used as coordinates to plot a point in an equilateral triangle and identify the fault zone in which it is located. Each point in the triangle corresponds to a unique combination of gas proportions. Diagnostic pentagons published by Duval and others seek to emulate the triangles while incorporating five fault gases instead of three. Unfortunately the mapping of five gas proportions to a point inside a two-dimensional pentagon is many-to-one; consequently, dissimilar combinations of gas proportions are mapped to the same point in the pentagon, resulting in mis-diagnosis. One solution is to replace the pentagon with a four-dimensional simplex, a direct generalization of the Duval triangle. In a comparison using cases confirmed by inspection, the simplex outperformed three ratio methods, Duval triangle 1, and two pentagons.

• The largest Si-rich region over 4.8 times larger than the largest B-rich region. • 99.78% SiN, 97.58% BN, and 99.96% SiBN simplexes form SiN, BN, and SiBN regions, respectively. • B atoms induce the network structure to have more extensive voids. • The clusters of large simplexes form the STZs. • The growth and coalescence of large simplexes in SB, causing fracture propagation. We conducted molecular dynamics simulations to study an amorphous model of Si 3 B 3 N 7. The model consists of SiN x (x = 3, 4, and 5) and BN y (y = 2, 3, and 4) units, with 93.50% SiN 4 and 99.13% BN 3 , forming a network structure. The splitting of the first peak of the g S i − S i (r) is caused by corner- and edge-sharing bonds, while N atoms belonging to NSi m linkages or NB n and NSi p B q linkages cause the first peak splitting of the g N − N (r). The model includes Si-rich and B-rich regions, and three SiN, BN, and SiBN regions that can be identified through simplex analysis. Overlapping simplexes with R S ≥2.0 Å form a chain that spreads throughout the model. B atoms cause more large simplexes with R S ≥2.6 Å than Si atoms. We subjected the model to a uniaxial tensile test, which resulted in elastic and plastic deformations. Under tensile loading, the bond lengths of atoms belonging to large simplexes are stretched more than the rest. Si-N bonds are more easily broken than B-N bonds, indicating heterogeneous deformation. Large simplexes gather to form shear transformation zones, which evolve into shear band at strains above 0.143. The cluster of large simplexes grows in the shear band, causing crack propagation at high strains. [ABSTRACT FROM AUTHOR]

Let \(n\in {\mathbb N}\), and let \(Q_n=[0,1]^n\) be the \(n\)-dimensionalunit cube. For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by\(\sigma S\) we denote the homothetic image of \(S\)with the center of homothety in the center of gravity of S and theratio of homothety \(\sigma\). We apply the followingnumerical characteristics of the simplex.Denote by \(\xi(S)\) the minimal \(\sigma>0\) with the property\(Q_n\subset \sigma S\). By \(\alpha(S)\) we denote the minimal\(\sigma>0\) such that \(Q_n\) is contained in a translateof a simplex \(\sigma S\).By \(d_i(S)\) we mean the \(i\)th axial diameter of \(S\), i.\,e.the maximum length of a segment contained in \(S\) and parallelto the \(i\)th coordinate axis. We apply the computationalformulae for\(\xi(S)\), \(\alpha(S)\), \(d_i(S)\) which have been proved by the firstauthor. In the paper we discuss the case \(S\subset Q_n\).Let\(\xi_n=\min\{ \xi(S): S\subset Q_n\}. \)Earlier the first author formulated the conjecture:{\it if\(\xi(S)=\xi_n\), then \(\alpha(S)=\xi(S)\).} He proved this statementfor \(n=2\) and the case when \(n+1\) is an Hadamard number, i.\,e.there exists an Hadamard matrix of order \(n+1\). The followingconjecture is a strongerproposition: {\it for each \(n\),there exist \(\gamma\geq 1\), not depending on \(S\subset Q_n\), such that\(\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n).\)}By \(\varkappa_n\) we denote the minimal\(\gamma\) with such a property.If \(n+1\) is an Hadamard number, then the precise value of \(\varkappa_n\)is 1. The existence of \(\varkappa_n\) for other \(n\)was unclear. In this paper with the use of computer methods we obtainan equality$$\varkappa_2 = \frac{5+2\sqrt{5}}{3}=3.1573\ldots $$Also we prove a new estimate$$\xi_4\leq \frac{19+5\sqrt{13}}{9}=4.1141\ldots,$$which improves the earlier result \(\xi_4\leq \frac{13}{3}=4.33\ldots\)Our conjecture is that \(\xi_4\) is precisely\(\frac{19+5\sqrt{13}}{9}\). Applying this valuein numerical computations we achive the value$$\varkappa_4 = \frac{4+\sqrt{13}}{5}=1.5211\ldots$$Denote by \(\theta_n\) the minimal normof interpolation projection on the space of linear functions of \(n\)variables as an operator from\(C(Q_n)\)in \(C(Q_n)\). It is known that, for each \(n\),$$\xi_n\leq \frac{n+1}{2}\left(\theta_n-1\right)+1,$$and for \(n=1,2,3,7\) here we have an equality.Using computer methods we obtain the result \(\theta_4=\frac{7}{3}\).Hence, the minimal \(n\) such that the above inequality has a strong formis equal to 4.%, a principal architecture of common purpose CPU and its main components are discussed, CPUs evolution is considered and drawbacks that prevent future CPU development are mentioned. Further, solutions proposed so far are addressed and new CPU architecture is introduced. The proposed architecture is based on wireless cache access that enables reliable interaction between cores in multicore CPUs using terahertz band, 0.1-10THz. The presented architecture addresses the scalability problem of existing processors and may potentially allow to scale them to tens of cores. As in-depth analysis of the applicability of suggested architecture requires accurate prediction of traffic in current and next generations of processors we then consider a set of approaches for traffic estimation in modern CPUs discussing their benefits and drawbacks. The authors identify traffic measurements using existing software tools as the most promising approach for traffic estimation, and use Intel Performance Counter Monitor for this purpose. Three types of CPU loads are considered including two artificial tests and background system load. For each load type the amount of data transmitted through the L2-L3 interface is reported for various input parameters including the number of active cores and their dependences on number of cores and operational frequency.

Abstract This paper reports some new results in relation to simplicial algorithms considering continuities of approximate fixed point sets. The upper semi-continuity of a set-valued mapping of approximate fixed points using vector-valued simplicial methods is proved, and thus one obtains the existence of finite essential connected components in approximate fixed point sets by vector-valued labels; examples are given to show that this is very different from the property for integer-valued labeling simplicial methods. The existence of essential sets is also proved focusing on both perturbations of domains and functions.

The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finite-dimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.

This work presents a systematic approach to determining the significance of the individual factors affecting the analytical performance of in-situ film electrode (FE) for the determination of Zn(II), Cd(II), and Pb(II). Analytical parameters were considered simultaneously, where the lowest limit of quantification, the widest linear concentration range, and the highest sensitivity, accuracy, and precision of the method evidenced a better analytical method. Significance was evaluated by means of a fractional factorial (experimental) design using five factors, i.e., the mass concentrations of Bi(III), Sn(II), and Sb(III), to design the in situ FE, the accumulation potential, and the accumulation time. Next, a simplex optimization procedure was employed to determine the optimum conditions for these factors. Such optimization of the in situ FE showed significant improvement in analytical performance compared to the in situ FEs in the initial experiments and compared to pure in situ FEs (bismuth-film, tin-film, and antimony-film electrodes). Moreover, using the optimized in situ FE electrode, a possible interference effect was checked for different species and the applicability of the electrode was demonstrated for a real tap water sample.