Toward scale‐free like behavior under increasing cognitive load

To understand how cognition and response selection processes might emerge from dynamic brain systems, we analyzed reaction times during the performance of both a working memory task and a choice reaction time task at different levels of “cognitive load.” Our findings suggest a continuous transition—tuned by load—from random behavior toward scale‐free like behavior as an expanding connectivity process in a network poised near a critical point. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

From exaptation to radical niche construction in biological and technological complex systems

Biological adaptation assumes the evolution of structures toward better functions. Yet, the roots of adaptive trajectories usually entail subverted—perverted—structures, derived from a different function: what Gould and Vrba called “exaptation.” Generally, this derivation is regarded as contingent or serendipitous, but it also may have regularities, if not rules, in both biological evolution and technological innovation. On the basis of biological examples and examples from the history of technology, the authors demonstrate the centrality of exaptation for a modern understanding of niche, selection, and environment. In some cases, biological understanding illuminates technical exaptation. Thus, the driver of exaptation is not simply chance matching of function and form; it depends on particular, permissive contexts. © 2013 Wiley Periodicals, Inc. Complexity 18: 7–14, 2013     

Nonlinear complementarity as unconstrained and constrained minimization

The nonlinear complementarity problem is cast as an unconstrained minimization problem that is obtained from an augmented Lagrangian formulation. The dimensionality of the unconstrained problem is the same as that of the original problem, and the penalty parameter need only be greater than one. Another feature of the unconstrained problem is that it has global minima of zero at precisely all the solution points of the complementarity problem without any monotonicity assumption. If the mapping of the complementarity problem is differentiable, then so is the objective of the unconstrained problem, and its gradient vanishes at all solution points of the complementarity problem. Under assumptions of nondegeneracy and linear independence of gradients of active constraints at a complementarity problem solution, the corresponding global unconstrained minimum point is locally unique. A Wolfe dual to a standard constrained optimization problem associated with the nonlinear complementarity problem is also formulated under a monotonicity and differentiability assumption. Most of the standard duality results are established even though the underlying constrained optimization problem may be nonconvex. Preliminary numerical tests on two small nonmonotone problems from the published literature converged to degenerate or nondegenerate solutions from all attempted starting points in 7 to 28 steps of a BFGS quasi-Newton method for unconstrained optimization.Dedicated to Phil Wolfe on his 65th birthday, in appreciation of his major contributions to mathematical programming.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grant CCR-9101801.     

Bohr–Sommerfeld–Heisenberg theory in geometric quantization

In the framework of geometric quantization we extend the Bohr–Sommerfeld rules to a full quantum theory which resembles the Heisenberg matrix theory. This extension is possible because Bohr–Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this basis. This permits the definition of appropriate shifting operators. As examples, we discuss the 1–dimensional harmonic oscillator and the coadjoint orbits of the rotation group.     

Almost‐anywhere theories: Reductionism and universality of emergence

Here, we aim to show that reductionism and emergence play a complementary role in understanding natural processes and in the dynamics of science explanation. In particular, we will show that the renormalization group—one of the most refined tools of Theoretical Physics—allows to understand the importance of emergent processes' role in Nature identifying them as universal organization processes, that is, they are scale independent. We can use the syntaxes of Quantum Field Theory and the processes of Spontaneous Symmetry Breaking as a trans‐disciplinary theoretical scenario for many other forms of complexity, especially the biological and cognitive ones. © 2010 Wiley Periodicals, Inc. Complexity, 2010     

Subject design and factors affecting achievement in mathematics for biomedical science

Reports such as Bio2010 emphasize the importance of integrating mathematical modelling skills into undergraduate biology and life science programmes, to ensure students have the skills and knowledge needed for biological research in the twenty-first century. One way to do this is by developing a dedicated mathematics subject to teach modelling and mathematical concepts in biological contexts. We describe such a subject at a research-intensive Australian university, and discuss the considerations informing its design. We also present an investigation into the effect of mathematical and biological background, prior mathematical achievement, and gender, on student achievement in the subject. The investigation shows that several factors known to predict performance in standard calculus subjects apply also to specialized discipline-specific mathematics subjects, and give some insight into the relative importance of mathematical versus biological background for a biology-focused mathematics subject.     

A branch-and-bound algorithm for instrumental variable quantile regression

This paper studies a statistical problem called instrumental variable quantile regression (IVQR). We model IVQR as a convex quadratic program with complementarity constraints and—although this type of program is generally NP-hard—we develop a branch-and-bound algorithm to solve it globally. We also derive bounds on key variables in the problem, which are valid asymptotically for increasing sample size. We compare our method with two well known global solvers, one of which requires the computed bounds. On random instances, our algorithm performs well in terms of both speed and robustness.     

Neutral Fermion with a Magnetic Moment in External Electromagnetic Fields

The interaction between a massive neutral fermion with a static (spin) magnetic dipole moment and an external electromagnetic field is described by the Dirac–Pauli equation. Exact solutions of this equation are obtained along with the corresponding energy spectrum for an axially symmetric external magnetic field and for some centrally symmetric electric fields. It is shown that the spin–orbital interaction of a neutral fermion with a magnetic moment determines both the characteristic properties of the quantum states and the fermion energy spectrum. It is found that (1) the discrete energy spectrum of a neutral fermion depends on the projection of the fermion spin on a certain quantization axis, (2) the ground energy level of a fermion in these electric fields as well as the energy levels of all bound states with a fixed value of the quantum number characterizing the projection of the fermion spin in the electric field E = er is degenerate and the degeneration order is countably infinite, and (3) the energy spectra of neutral fermions and antifermions with spin magnetic moments are symmetric in centrally symmetric fields. Bound states of a neutral fermion with a magnetic moment in an external electric field do exist even if the Dirac–Pauli equation does not explicitly contain the term with the fermion mass. In addition, in centrally symmetric electric fields, there exist a countably infinite set of pairs of isolated charge-conjugate zero-energy solutions of the Dirac–Pauli equation.     

Error Bounds for R0-Type and Monotone Nonlinear Complementarity Problems

The paper generalizes the Mangasarian–Ren (Ref. 1) error bounds forlinear complementarity problems (LCPs) to nonlinear complementarity problems(NCPs). This is done by extending the concept of R ₀-matrixto several R ₀-type functions, which include a subset ofmonotone functions as a special case. Both local and global error bounds areobtained for R ₀-type NCPs and some monotone NCPs.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

New Constrained Optimization Reformulation of Complementarity Problems

We suggest a reformulation of the complementarity problem CP(F) as a minimization problem with nonnegativity constraints. This reformulation is based on a particular unconstrained minimization reformulation of CP(F) introduced by Geiger and Kanzow as well as Facchinei and Soares. This allows us to use nonnegativity constraints for all the variables or only a subset of the variables on which the function F depends. Appropriate regularity conditions ensure that a stationary point of the new reformulation is a solution of the complementarity problem. In particular, stationary points with negative components can be avoided in contrast to the reformulation as unconstrained minimization problem. This advantage will be demonstrated for a class of complementarity problems which arise when the Karush–Kuhn–Tucker conditions of a convex inequality constrained optimization problem are considered.     

A Comparison of Large Scale Mixed Complementarity Problem Solvers

This paper provides a means for comparing various computercodes for solving large scale mixed complementarity problems. Wediscuss inadequacies in how solvers are currently compared, andpresent a testing environment that addresses these inadequacies. Thistesting environment consists of a library of test problems, along withGAMS and MATLAB interfaces that allow these problems to be easilyaccessed. The environment is intended for use as a tool byother researchers to better understand both their algorithms and theirimplementations, and to direct research toward problem classes thatare currently the most challenging. As an initial benchmark, eightdifferent algorithm implementations for large scale mixedcomplementarity problems are briefly described and tested with defaultparameter settings using the new testing environment.     

Interdisciplinary education – a predator–prey model for developing a skill set in mathematics,biology and technology

The science of biology has been transforming dramatically and so the need for a stronger mathematical background for biology students has increased. Biological students reaching the senior or post-graduate level often come to realize that their mathematical background is insufficient. Similarly, students in a mathematics programme, interested in biological phenomena, find it difficult to master the complex systems encountered in biology. In short, the biologists do not have enough mathematics and the mathematicians are not being taught enough biology. The need for interdisciplinary curricula that includes disciplines such as biology, physical science, and mathematics is widely recognized, but has not been widely implemented. In this paper, it is suggested that students develop a skill set of ecology, mathematics and technology to encourage working across disciplinary boundaries. To illustrate such a skill set, a predator–prey model that contains self-limiting factors for both predator and prey is suggested. The general idea of dynamics, is introduced and students are encouraged to discover the applicability of this approach to more complex biological systems. The level of mathematics and technology required is not advanced; therefore, it is ideal for inclusion in a senior-level or introductory graduate-level course for students interested in mathematical biology.     

An inexactQP-based method for nonlinear complementarity problems

Summary. We consider a quadratic programming-based method for nonlinear complementarity problems which allows inexact solutions of the quadratic subproblems. The main features of this method are that all iterates stay in the feasible set and that the method has some strong global and local convergence properties. Numerical results for all complementarity problems from the MCPLIB test problem collection are also reported. Received February 24, 1997 / Revised version received September 5, 1997     

A modified damped Newton method for linear complementarity problems

We present a modified damped Newton method for solving large sparse linear complementarity problems, which adopts a new strategy for determining the stepsize at each Newton iteration. The global convergence of the new method is proved when the system matrix is a nondegenerate matrix. We then apply the matrix splitting technique to this new method, deriving an inexact splitting method for the linear complementarity problems. The global convergence of the resulting inexact splitting method is proved, too. Numerical results show that the new methods are feasible and effective for solving the large sparse linear complementarity problems.     

Beliefs,Practical Knowledge,and Context: A Longitudinal Study of a Beginning Biology Teacher's 5E Unit

The purpose of this three‐year case study was to understand how a beginning biology teacher (Alice) designed and taught a 5E unit on natural selection, how the unit changed when she took a position in a different school district, and why the changes occurred. We examined Alice's developing beliefs about science teaching and learning, practical knowledge, and perceptions of school context in relation to the 5E unit. Data sources consisted of interviews, classroom observations, and lesson materials. We found that Alice placed more emphasis on the explore phase, less emphasis on the engage and explain phases, and removed the elaborate phase over time. Alice's beliefs about science teaching and learning acted as a filter for making sense of practical knowledge and perceptions of context. Although her beliefs were student centered, they aligned with discovery learning in which little intervention from the teacher is required. We discuss how her beliefs, practical knowledge, and perceptions of context explained the changes in her practice. This study sheds insight into the nature of beliefs and how they relate to the 5E lesson phases, as well as the different lenses for viewing the 5E instructional model. Implications for science teacher preparation and induction programs are discussed.     

Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems

We consider a nonlinear complementarity problem defined by a continuous but not necessarily locally Lipschitzian map. In particular, we examine the connection between solutions of the problem and stationary points of the associated Fischer–Burmeister merit function. This is done by deriving a new necessary optimality condition and a chain rule formula for composite functions involving continuous maps. These results are given in terms of approximate Jacobians which provide the foundation for analyzing continuous nonsmooth maps. We also prove a result on the global convergence of a derivative-free descent algorithm for solving the complementarity problem. To this end, a concept of directional monotonicity for continuous maps is introduced.     

More results on the convergence of iterative methods for the symmetric linear complementarity problem

In an earlier paper, the author has given some necessary and sufficient conditions for the convergence of iterative methods for solving the linear complementarity problem. These conditions may be viewed as global in the sense that they apply to the methods regardless of the constant vector in the linear complementarity problem. More precisely, the conditions characterize a certain class of matrices for which the iterative methods will converge, in a certain sense, to a solution of the linear complementarity problem for all constant vectors. In this paper, we improve on our previous results and establish necessary and sufficient conditions for the convergence of iterative methods for solving each individual linear complementarity problem with a fixed constant vector. Unlike the earlier paper, our present analysis applies only to the symmetric linear complementarity problem. Various applications to a strictly convex quadratic program are also given.The author gratefully acknowledges several stimulating conversations with Professor O. Mangasarian on the subject of this paper. He is also grateful to a referee, who has suggested Lemma 2.2 and the present (stronger) version of Theorem 2.1 as well as several other constructive comments.This research was based on work supported by the National Science Foundation under Grant No. ECS-81-14571, sponsored by the United States Army under Contract No. DAAG29-80-C-0041, and was completed while the author was visiting the Mathematics Research Center at the University of Wisconsin, Madison, Wisconsin.     

Wallace's unfinished business: The “Other Man” in evolutionary theory

After a century in the shadows, Alfred Russel Wallace (1823–1913) has recently become the subject of increasing attention. It is suggested here, expanding on observations made by anthropologist Gregory Bateson some years ago, that Wallace's cybernetics‐like view of the operation of natural selection—as a governor‐like principle tending to keep species unvarying—can be expanded to a more complete evolutionary understanding by exploring in modern context Wallace's idea that “more recondite forces” are driving the process. Specifically, when the environment is regarded as a final cause (but not a deterministic force), individual adaptations may be viewed as entropy‐relaying structures (acting in response to, and as a part of, larger scale biogeochemical agenda), whereas negentropy is accumulated by nonrandomly directed organism‐ and population‐level forms of ecological engagement. Thus, range change in particular is viewed as a process that is both driven and nonrandom, and ultimately connected to the derivation of more and more organized individual, population, and community structures. © 2004 Wiley Periodicals, Inc. Complexity 10:25–32, 2004     

Power iteration and inverse power iteration for eigenvalue complementarity problem

In this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.