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The number of students who leave U.S. schools mathematically underprepared has prompted widespread concern. Low achieving students, many of whom have been turned off mathematics, are often placed in low tracks and given remedial, skills-oriented work. This study examines a different approach wherein heterogeneous groups of students were given responsibility and agency and asked to engage in a range of mathematical practices collaboratively. The teaching intervention, which was introduced in the first paper, took place as part of a summer class on algebra, and it gave students the opportunity to participate with mathematics in changed ways. This paper will report evidence that the vast majority responded with increased engagement, achievement, and enjoyment. The students chose collaboration and agency as critical to their improved relationships with mathematics.

New and radically reformative thinking about the enactive and embodied basis of cognition holds out the promise of moving forward age-old debates about whether we learn and how we learn. The radical enactive, embodied view of cognition (REC) poses a direct, and unmitigated, challenge to the trademark assumptions of traditional cognitivist theories of mind— those that characterize cognition as always and everywhere grounded in the manipulation of contentful representations of some kind. REC has had some success in understanding how sports skills and expertise are acquired. But, REC approaches appear to encounter a natural obstacle when it comes to understanding skill acquisition in knowledge-rich, conceptually based domains like the hard sciences and mathematics. This paper offers a proof of concept that REC’s reach can be usefully extended into the domain of science, technology, engineering, and mathematics (STEM) learning, especially when it comes to understanding the deep roots of such learning. In making this case, this paper has five main parts. The section “Ancient Intellectualism and the REC Challenge” briefly introduces REC and situates it with respect to rival views about the cognitive basis of learning. The “Learning REConceived: from Sports to STEM?” section outlines the substantive contribution REC makes to understanding skill acquisition in the domain of sports and identifies reasons for doubting that it will be possible to apply the same approach to knowledge-rich STEM domains. The “Mathematics as Embodied Practice” section gives the general layout for how to understand mathematics as an embodied practice. The section “The Importance of Attentional Anchors” introduces the concept “attentional anchor” and establishes why attentional anchors are important to educational design in STEM domains like mathematics. Finally, drawing on some exciting new empirical studies, the section “Seeing Attentional Anchors” demonstrates how REC can contribute to understanding the roots of STEM learning and inform its learning design, focusing on the case of mathematics.

New and radically reformative thinking about the enactive and embodied basis of cognition holds out the promise of moving forward age-old debates about whether we learn and how we learn. The radical enactive, embodied view of cognition (REC) poses a direct, and unmitigated, challenge to the trademark assumptions of traditional cognitivist theories of mind— those that characterize cognition as always and everywhere grounded in the manipulation of contentful representations of some kind. REC has had some success in understanding how sports skills and expertise are acquired. But, REC approaches appear to encounter a natural obstacle when it comes to understanding skill acquisition in knowledge-rich, conceptually based domains like the hard sciences and mathematics. This paper offers a proof of concept that REC’s reach can be usefully extended into the domain of science, technology, engineering, and mathematics (STEM) learning, especially when it comes to understanding the deep roots of such learning. In making this case, this paper has five main parts. The section “Ancient Intellectualism and the REC Challenge” briefly introduces REC and situates it with respect to rival views about the cognitive basis of learning. The “Learning REConceived: from Sports to STEM?” section outlines the substantive contribution REC makes to understanding skill acquisition in the domain of sports and identifies reasons for doubting that it will be possible to apply the same approach to knowledge-rich STEM domains. The “Mathematics as Embodied Practice” section gives the general layout for how to understand mathematics as an embodied practice. The section “The Importance of Attentional Anchors” introduces the concept “attentional anchor” and establishes why attentional anchors are important to educational design in STEM domains like mathematics. Finally, drawing on some exciting new empirical studies, the section “Seeing Attentional Anchors” demonstrates how REC can contribute to understanding the roots of STEM learning and inform its learning design, focusing on the case of mathematics.

Reticulation networks are now frequently used to model the history of life for various groups of species whose evolutionary past is likely to include reticulation events such as horizontal gene transfer or hybridization. However, the reconstructed networks are rarely guaranteed to be temporal. If a reticulation network is temporal, then it satisfies the two biologically motivated timing constraints of instantaneously occurring reticulation events and successively occurring speciation events. On the other hand, if a reticulation network is not temporal, it is always possible to make it temporal by adding a number of additional unsampled or extinct taxa. In the first half of the paper, we show that deciding whether a given number of additional taxa is sufficient to transform a non-temporal reticulation network into a temporal one is an NP-complete problem. As one is often given a set of gene trees instead of a network in the context of hybridization, this motivates the second half of the paper which provides an algorithm, called TemporalHybrid, for reconstructing a temporal hybridization network that simultaneously explains the ancestral history of two trees or indicates that no such network exists. We further derive two methods to decide whether or not a temporal hybridization network exists for two given trees and illustrate one of the methods on a grass data set.

Let $${\mathbb {F}}$$ a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action $${(T_{g})_{{g} \in \mathbb {F}^{\omega}}}$$ of the infinite abelian group $${\mathbb {F}^{\omega}}$$ on a probability space $${X = (X, \mathcal {B}, \mu)}$$ is generated by phase polynomials $${\phi : X \to S^{1}}$$ of degree less than C(k) on X, where C(k) depends only on k. In the case where $${k \leq {\rm char}(\mathbb {F})}$$ we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for $${\mathbb {Z}}$$ by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case $${k \leq {\rm char}(\mathbb {F})}$$ , with a partial result in low characteristic.

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber Δ euc . In addition to the metric length it contains information on the direction of the segment. In this paper we study restrictions on the Δ euc -valued side lengths of polygons in Euclidean buildings. The main result is that for thick Euclidean buildings X the set $${\mathcal{P}n(X)}$$ of possible Δ euc -valued side lengths of oriented n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of Δ euc -valued weights of semistable weighted configurations on the Tits boundary ∂ Tits X.The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM2].

In this paper, we study the numbers D n,k which are defined as the number of permutations σ of the symmetric group S n such that σ has no cycles of length j for j ≤ k. In the case k = 1, D n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D n,k generalized derangement numbers. Garsia and Remmel [4] defined some natural q-analogues of D n,1, denoted by D n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues D n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In [4], Garsia and Remmel also suggested an approach to define q-analogues of the numbers D n,k . In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2.

Functional connectivity (FC) estimated from functional magnetic resonance imaging (fMRI) signals is important in understanding neural representation and information processing in cortical networks. However, due to a lack of "ground truth" FC pattern, the reliability and robustness of FC estimates are usually examined in downstream FC analysis tasks, such as performing participant's identification (also known as "fingerprinting"). In this paper, we propose to learn FC via a smooth graph learning framework. In particular, we treat each time frame of the fMRI time series as a graph signal on an underlying functional brain graph, and estimate the smooth graph functional connectivity (SGFC) by learning the weighted graph adjacency matrix based on graph signal smoothness assumption. We demonstrate that our approach gives rise to a natural and sparse graph representation of FC from which reliable graph measures can be extracted. Reliability of SGFC is evaluated in the context of fingerprinting and compared to correlation FC (CFC). SGFC achieves higher fingerprinting accuracy across several different experiment settings; the improvement is even more significant when a shorter fMRI scanning length is used for FC estimation. In addition to being reliable, we also validate the cognitive relevance of SGFC by using it to predict fluid intelligence. Finally, in evaluating topological measures of the sparse graph, SGFC reveals a more small-world and modular structure compared to CFC. Together, our results suggest that the smooth graph learning framework produces a naturally sparse, reliable, and cognitive-relevant representation of functional connectivity.

This paper discusses discusses a certain mathematical constant that was recently proven to be 2-normal -- every m-long string of digits in its binary expansion appear, in the limit, with frequently 1/2^m. This paper establishes the result that this constant is NOT 6-normal -- its expansion base 6 is decidedly not random.

While there is increasing evidence for unique chemical reactions at interfaces, there are fewer data on photochemistry at liquid-vapor junctions. This paper reports a comparison of the photolysis of molybdenum hexacarbonyl, Mo(CO)(6), in 1-decene either as liquid droplets or in bulk-liquid solutions. Mo(CO)(6) photolysis is faster by at least three orders of magnitude in the aerosols than in bulk-liquids. Two possible sources of this enhancement are considered: (1) increased light intensity due to either Morphology-Dependent Resonances (MDRs) in the spherical aerosol particles and/or to increased pathlengths for light inside the droplet due to refraction, which are termed physical effects in this paper; and (2) interface effects such as an incomplete solvent-cage at the gas-liquid boundary and/or enhanced interfacial concentrations of Mo(CO)(6), which are termed chemical effects. Quantitative calculations of the first possibility were carried out in which the light intensity distribution in the droplets averaged over 215-360 nm was obtained for 1-decene droplets. Calculations show that the average increase in light intensity over the entire droplet is 106%, with an average increase of 51% at the interface. These increases are much smaller than the observed increase in the apparent photolysis rate of droplets compared to the bulk. Thus, chemical effects, i.e., a decreased solvent-cage effect at the interface and/or enhancement in the surface concentration of Mo(CO)(6), are most likely responsible for the dramatic increase in the photolysis rate. Similar calculations were also carried out for broadband (290-600 nm) solar irradiation of water droplets, relevant to atmospheric conditions. These calculations show that, in agreement with previous calculations by Mayer and Madronich [B. Mayer and S. Madronich, Atmos. Chem. Phys., 2004, 4, 2241] MDRs produce only a moderate average intensity enhancement relative to the corresponding bulk-liquid slabs when averaged over a range of wavelengths characteristic of solar radiation at the Earth's surface. However, as in the case of Mo(CO)(6) in 1-decene, chemical effects may play a role in enhanced photochemistry at the aerosol-air interface for airborne particles.

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Current educational policies in the United States attempt to boost student achievement and promote equality by intensifying the curriculum and exposing students to more advanced coursework. This paper investigates the relationship between one such effort - California's push to enroll all 8th grade students in Algebra - and the distribution of student achievement. We suggest that this effort is an instance of a "collective effects" problem, where the population-level effects of a policy are different from its effects at the individual level. In such contexts, we argue that it is important to consider broader population effects as well as the difference between "treated" and "untreated" individuals. To do so, we present differences in inverse propensity score weighted distributions investigating how this curricular policy changed the distribution of student achievement. We find that California's attempt to intensify the curriculum did not raise test scores at the bottom of the distribution, but did lower scores at the top of the distribution. These results highlight the efficacy of inverse propensity score weighting approaches for examining distributional differences, and provide a cautionary tale for curricular intensification efforts and other policies with collective effects.

Current educational policies in the United States attempt to boost student achievement and promote equality by intensifying the curriculum and exposing students to more advanced coursework. This paper investigates the relationship between one such effort - California's push to enroll all 8th grade students in Algebra - and the distribution of student achievement. We suggest that this effort is an instance of a "collective effects" problem, where the population-level effects of a policy are different from its effects at the individual level. In such contexts, we argue that it is important to consider broader population effects as well as the difference between "treated" and "untreated" individuals. To do so, we present differences in inverse propensity score weighted distributions investigating how this curricular policy changed the distribution of student achievement. We find that California's attempt to intensify the curriculum did not raise test scores at the bottom of the distribution, but did lower scores at the top of the distribution. These results highlight the efficacy of inverse propensity score weighting approaches for examining distributional differences, and provide a cautionary tale for curricular intensification efforts and other policies with collective effects.

Recently it has become possible to measure fluorescence phase-shift and modulation data over a wide range of modulation frequencies. In this paper we describe the analysis of these data by the method of nonlinear least squares to determine the values of the lifetimes and fractional intensities for a mixture of exponentially decaying fluorophores. Analyzing simulated data allowed us to determine those experimental factors that are most critical for successfully resolving the emissions from mixtures of fluorophores. The most critical factors are the accuracy of the experimental data, the relative difference of the individual decay times, and the inclusion of data measured at multiple emission wavelengths. After measuring at eight widely spaced modulation frequencies, additional measurements yielded only a modest increase in resolution. In particular, the uncertainty in the parameters decreased approximately as the reciprocal of the square root of the number of modulation frequencies. Our simulations showed that with presently available precision and data for one emission bandpass, two decay times could be accurately determined if their ratio were greater than or equal to 1.4. Three exponential decays could also be resolved, but only if the range of the lifetimes were fivefold or greater. To reliably determine closely-spaced decay times, the data were measured at multiple emission wavelengths so that the fractional intensities of the components could be varied. Also, independent knowledge of any of the parameters substantially increased the accuracy with which the remaining parameters could be determined. In the subsequent paper we present experimental results that broadly confirm the predicted resolving potential of variable-frequency phase-modulation fluorometry.

Recently it has become possible to measure fluorescence phase-shift and modulation data over a wide range of modulation frequencies. In this paper we describe the analysis of these data by the method of nonlinear least squares to determine the values of the lifetimes and fractional intensities for a mixture of exponentially decaying fluorophores. Analyzing simulated data allowed us to determine those experimental factors that are most critical for successfully resolving the emissions from mixtures of fluorophores. The most critical factors are the accuracy of the experimental data, the relative difference of the individual decay times, and the inclusion of data measured at multiple emission wavelengths. After measuring at eight widely spaced modulation frequencies, additional measurements yielded only a modest increase in resolution. In particular, the uncertainty in the parameters decreased approximately as the reciprocal of the square root of the number of modulation frequencies. Our simulations showed that with presently available precision and data for one emission bandpass, two decay times could be accurately determined if their ratio were greater than or equal to 1.4. Three exponential decays could also be resolved, but only if the range of the lifetimes were fivefold or greater. To reliably determine closely-spaced decay times, the data were measured at multiple emission wavelengths so that the fractional intensities of the components could be varied. Also, independent knowledge of any of the parameters substantially increased the accuracy with which the remaining parameters could be determined. In the subsequent paper we present experimental results that broadly confirm the predicted resolving potential of variable-frequency phase-modulation fluorometry.

To develop a theory of Robust Intelligence (RI), we continue to advance our theory of interdependence on the efficient and effective control of systems of autonomous hybrid teams composed of robots, machines and humans working interchangeably. As is the case with humans, we believe that RI is less likely to be achieved by individual computational agents; instead, we propose that a better path to RI is with interdependent agents. However, unlike conventional computational models where agents act independently of neighbors, where, for example, a predator mathematically consumes its prey or not as a function of a random interaction process, dynamic interdependence means that agents dynamically respond to the bi-directional signals of actual or potential presence of other agents (e.g., in states poised to fight or flight), a significant increase over conventional modeling complexity.  That this problem is unsolved, mathematically and conceptually, precludes hybrid teams from processing information like human teams operating under challenges and perceived threats. To simplify this problem, we use bistable models for interdependence  with a focus on teams and firms as we increase complexity to the level of systems. As part of the problem, in this paper, and countering simplification, sentient multi-agent systems require an aggregation process like data fusion. But the conventional use of fusion for the control of mobile systems hinges on mathematical convergence into patterns, increasing uncertainty whenever divergent information has the potential to process information into knowledge. The goals of our research are: First, to analyze why valid models of interdependence are difficult to build. Second, to reduce uncertainty in decision-making by moderating convergence processes in data aggregation (e.g., fusion) with differential clustering between alternative (orthogonal) views that check convergence processes and promote information processing (e.g., second opinions from independent physicians; prosecutor-defense attorneys; Republicans-Democrats in Congress; opposed scientists, like Bohr-Einstein). Third, in line with our theoretical expectations, we plan to lay the groundwork for agent-based systems to model the stability from the cooperative contexts associated with teams, and the instability from the competitive contexts associated with multiple teams or firms that constitute systems. Our result will be a new theory of interdependence; a new model of data aggregation; and new agent-based models of interdependence.

The paper presents an outline of a unified answer to five questions concerning logic: (1) Is logic in the mind or in the world? (2) Does logic need a foundation? What is the main obstacle to a foundation for logic? Can it be overcome? (3) How does logic work? What does logical form represent? Are logical constants referential? (4) Is there a criterion of logicality? (5) What is the relation between logic and mathematics? © 2010 The Author(s).

The paper presents an outline of a unified answer to five questions concerning logic: (1) Is logic in the mind or in the world? (2) Does logic need a foundation? What is the main obstacle to a foundation for logic? Can it be overcome? (3) How does logic work? What does logical form represent? Are logical constants referential? (4) Is there a criterion of logicality? (5) What is the relation between logic and mathematics? © 2010 The Author(s).

In earlier papers we studied direct limits $${(G,\,K) = \varinjlim\, (G_n,K_n)}$$ of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which $${G_n = N_n\rtimes K_n}$$ with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections $${\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}$$ for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit $${L^2(G/K) := \varinjlim L^2(G_n/K_n)}$$ is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces $${\mathcal{A}(G_n/K_n)}$$ of regular functions on G n /K n and injections $${\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}$$ for m ≧ n related to restriction by $${\nu_{m,n}(f)|_{G_n/K_n} = f}$$ . Thus the direct limit $${\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}$$ sits as a particular G-submodule of the much larger inverse limit $${\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}$$ . Further, we define a pre Hilbert space structure on $${\mathcal{A}(G/K)}$$ derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space $${\mathcal{A}(G/K)}$$ , and also defines a G-invariant inner product on $${\mathcal{A}(G/K)}$$ for which the left regular representation of G is multiplicity-free.

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})$ for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.

The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to p (p a prime) of a finite group G and those of the subgroup N, the normalizer of Sylow p-subgroup. In this paper we observe that MC implies the existence of analogous bijections involving various pairs of algebras, including certain crossed products, and that MC is equivalent to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is equivalent to parallel statements about holomorphic orbifolds V G , V N . There is a uniform formulation of MC covering these different situations which involves quantum dimensions of objects in pairs of ribbon fusion categories.

Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TV-regularizers, making them difficult to compute. The previously introduced Split Bregman method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the Split Bregman concept to image segmentation problems, we build fast solvers which can out-perform more conventional schemes, such as duality based methods and graph-cuts. The convex segmentation schemes also substantially outperform conventional level set methods, such as the Chan-Vese level set-based segmentation algorithm. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions. The primary purpose of this paper is to examine the effectiveness of “Split Bregman” techniques for solving these problems, and to compare this scheme with more conventional methods.

An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how to fill a curvilinear triangle with circles.In this paper we consider the basic building blocks of these rules, irreducible Apollonian configurations. Our main result is to show how to find a small field that can realize such a configuration and also give a method to relate the bends of the new circles to the bends of the circles forming the curvilinear triangle.

This paper shows that a well designed transport system has an embedded exchange value by serving as a market for potential exchange between consumers. Under suitable conditions, one can improve the welfare of consumers in the system simply by allowing some exchange of goods between consumers during transportation without incurring additional transportation cost. We propose an explicit valuation formula to measure this exchange value for a given compatible transport system. This value is always nonnegative and bounded from above. Criteria based on transport structures, preferences and prices are provided to determine the existence of a positive exchange value. Finally, we study a new optimal transport problem with an objective taking into account of both transportation cost and exchange value.

We consider a general family of regularized Navier–Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n≥2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier–Stokes equations, the Navier–Stokes-α model, the Leray-α model, the modified Leray-α model, the simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. We give a unified analysis of the entire three-parameter family of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α→0 limit in α models. Next, we show existence of a global attractor for the general model, and then give estimates for the dimension of the global attractor and the number of degrees of freedom in terms of a generalized Grashof number. We then establish some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. We finish by deriving some new length-scale estimates in terms of the Reynolds number, which allows for recasting the Grashof number-based results into analogous statements involving the Reynolds number. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, and determining operators for the well-known specific members of this family of regularized Navier–Stokes and MHD models, the framework we develop also makes possible a number of new results for all models in the general family, including some new results for several of the well-studied models. Analyzing the more abstract generalized model allows for a simpler analysis that helps bring out the core common structure of the various regularized Navier–Stokes and magnetohydrodynamics models, and also helps clarify the common features of many of the existing and new results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Grönwall-type inequalities.

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski’s second theorem on successive minima with the volume replaced by lattice point enumerator is true up to an exponential factor, whose base is approximately 1.64.

The Hirsch conjecture was posed in 1957 in a question from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n−d. The number n of facets is the minimum number of closed half-spaces needed to form the polytope and the conjecture asserts that one can go from any vertex to any other vertex using at most n−d edges.Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound n−d is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.

The main result of this paper is the following theorem: Let M be a premouse with a top extender, F. Suppose that (a) M is linearly coarsely iterable via hitting F and its images, and (b) if M * is a linear iterate of M as in (a), then M * is coarsely iterable with respect to iteration trees which do not use the top extender of M * and its images. Then M is coarsely iterable.

We apply Nadel’s method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler–Einstein metric. In particular, all del Pezzo surfaces of degree 4, 5, or 6 and certain special del Pezzo surfaces of lower degree are shown to have a Kähler–Einstein metric. These existence statements are not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian–Yau.

In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form $$\mathbf{u}_t -\Delta\left[ \mathbf{\nabla}\Phi(\mathbf{u})\right] = 0,$$ where $${\Phi(z)}$$ is a strictly convex function. I show that when $${\Phi}$$ is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.

The affine Dynkin diagram of type A n (1) has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type A n crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type A n crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.

This paper considers two nonlocal regularizations for image recovery, which exploit the spatial interactions in images. We get superior results using preprocessed data as input for the weighted functionals. Applications discussed include image deconvolution and tomographic reconstruction. The numerical results show our method outperforms some previous ones.

This paper considers two nonlocal regularizations for image recovery, which exploit the spatial interactions in images. We get superior results using preprocessed data as input for the weighted functionals. Applications discussed include image deconvolution and tomographic reconstruction. The numerical results show our method outperforms some previous ones.

The Dirichlet problem $$ \left\{ \begin{array}{l}\Delta _\infty u - |Du|^2 = 0 \quad {\rm on} \, \Omega \subset {{\mathbb R}^n} \\ u|\partial \Omega = g \\\end{array} \right. $$ might have many solutions, where $${\Delta_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}}$$ . In this paper, we prove that the maximal solution is the unique absolute minimizer for $${H(p,z)={\frac{1}{2}}|p|^2-z}$$ from calculus of variations in L ∞ and the minimal solution is the continuum value function from the “tug-of-war” game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting question is how to interpret those intermediate solutions. Most of our approaches are based on an idea of Barles–Busca (Commun Partial Differ Equ 26(11–12):2323–2337, 2001).

The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in $${F_{p^2}}$$ and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.

We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205–227, 2002; 202:577–601, 2005). This paper is therefore intended as a guide for those interested in using the GFM for Poisson-type problems (and by consequence diffusion-like problems and Stefan-type problems) by providing the pros and cons of the different choices for defining the ghost values and locating the interface. We found that in order to obtain second-order-accurate gradients, both a quadratic (or higher order) extrapolation for defining the ghost values and a quadratic (or higher order) interpolation for finding the interface location are required. In the case where the ghost values are defined by a linear extrapolation, the gradients of the solution converge slowly (at most first order in average) and the convergence rate oscillates, even when the interface location is defined by a quadratic interpolation. The same conclusions hold true for the combination of a quadratic extrapolation for the ghost cells and a linear interpolation. The solution is second-order accurate in all cases. Defining the ghost values with quadratic extrapolations leads to a non-symmetric linear system with a worse conditioning than that of the linear extrapolation case, for which the linear system is symmetric and better conditioned. We conclude that for problems where only the solution matters, the method described by Gibou, F., Fedkiw, R., Cheng, L.-T. and Kang, M. in (J. Comput. Phys. 176:205–227, 2002) is advantageous since the linear system that needs to be inverted is symmetric. In problems where the solution gradient is needed, such as in Stefan-type problems, higher order extrapolation schemes as described by Gibou, F. and Fedkiw, R. in (J. Comput. Phys. 202:577–601, 2005) are desirable.

The parity of p(n), the ordinary partition function, has been studied for at least a century, yet it still remains something of a mystery. Although much work has been done, the known lower bounds for the number of even and odd values of p(n) for n ≤ N still appear to have a great deal of room for improvement. In this paper, we use classical methods to give a new lower bound for the number of odd values of p(n).

The original motivation for the study of hook length polynomials was to find a combinatorial proof for a hook length formula for binary trees given by Postnikov, as well as a proof for a hook length polynomial formula conjectured by Lascoux. In this paper, we define the hook length polynomial for plane forests of a given degree sequence type and show that it can be factored into a product of linear forms. Some other enumerative results on forests are also given.

In this paper, we introduce the use of interacting particle systems in the computation of probabilities of simultaneous defaults in large credit portfolios. The method can be applied to compute small historical as well as risk-neutral probabilities. It only requires that the model be based on a background Markov chain for which a simulation algorithm is available. We use the strategy developed by Del Moral and Garnier in (Ann. Appl. Probab. 15:2496–2534, 2005) for the estimation of random walk rare events probabilities. For the purpose of illustration, we consider a discrete-time version of a first passage model for default. We use a structural model with stochastic volatility, and we demonstrate the efficiency of our method in situations where importance sampling is not possible or numerically unstable.

This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against self-intersection is independent from a distance metric that determines the overall character of the interpolation sequence. This decoupled approach provides a powerful control mechanism for determining how the interpolation should appear, while still assuring against intersection and guaranteeing termination of the algorithm. Our algorithm also allows additional control by accommodating a set of algebraic constraints that can be weakly enforced throughout the interpolation sequence.

This paper analyzes a conjecture due to Joseph Buhler that the product from k = 1 to n of tan(k) converges to zero.

This paper presents part of the results of a completed study entitled A Longitudinal Study of Minority Ph.D.s from 1980-1990: Progress and Outcomes in Science and Engineering at the University of California during Graduate School and Professional Life. It focuses particularly on the graduate school experience and degree of preparation for the professoriate of African American doctoral students in the sciences and engineering, and presents the results of a survey of 33 African American STEM Ph.D.s from the University of California earned between 1980-1990. Relationships with thesis advisors and principal investigators are evaluated by the study participants in fifteen specific areas from highly-ranked intellectual development to low-ranked training in grant writing. Deficits in training and socialization are discussed along with the tension between being both an African American and a graduate student. Career choices and outcomes are presented. These findings, in conjunction with current analyses of graduate education in STEM, suggest ways in which graduate training for all could be improved.

We are involved in the development of photodynamic therapy (PDT) as a minimally invasive method for treating dysfunctional uterine bleeding, one of the primary clinical indications for hysterectomy. In this paper, we analyse light propagation through the uterus in order to specify the requirements for a light delivery system capable of effectively performing endometrial PDT. Our approach involves developing an analytical model based on diffusion theory to predict optical fluence rate distributions when cylindrical and spherical optical applicators are placed in the uterine cavity. We apply the results of our model calculations to estimate the thermal effects of optical irradiation and the effective photodynamic optical dose. Theoretical fluence rate calculations are compared to fluence rate measurements made in fresh, surgically removed human uteri. Our results show that a trifurcated cylindrical optical applicator inserted into the human uterus can provide a light dose that is sufficient to cause photodynamic destruction of the entire endometrium. When the optical power per unit length of each cylindrical applicator is 100 mW cm-1 (at 630 nm), a fluence rate of 40 mW cm-2 is delivered to the boundary layer between the endometrium and the myometrium (a depth of about 4-6 mm). The optical fluence delivered to the boundary layer after 20 min of exposure is 50 J cm-2, a level that is generally accepted to cause tissue damage throughout the endometrium in most patients.

We are involved in the development of photodynamic therapy (PDT) as a minimally invasive method for treating dysfunctional uterine bleeding, one of the primary clinical indications for hysterectomy. In this paper, we analyse light propagation through the uterus in order to specify the requirements for a light delivery system capable of effectively performing endometrial PDT. Our approach involves developing an analytical model based on diffusion theory to predict optical fluence rate distributions when cylindrical and spherical optical applicators are placed in the uterine cavity. We apply the results of our model calculations to estimate the thermal effects of optical irradiation and the effective photodynamic optical dose. Theoretical fluence rate calculations are compared to fluence rate measurements made in fresh, surgically removed human uteri. Our results show that a trifurcated cylindrical optical applicator inserted into the human uterus can provide a light dose that is sufficient to cause photodynamic destruction of the entire endometrium. When the optical power per unit length of each cylindrical applicator is 100 mW cm-1 (at 630 nm), a fluence rate of 40 mW cm-2 is delivered to the boundary layer between the endometrium and the myometrium (a depth of about 4-6 mm). The optical fluence delivered to the boundary layer after 20 min of exposure is 50 J cm-2, a level that is generally accepted to cause tissue damage throughout the endometrium in most patients.

This paper discusses a calcium-dependent inactivation of alamethicin-induced conductance in asymmetric lipid bilayers. The bilayers used were formed with one leaflet of phosphatidyl ethanolamine (PE) and one of phosphatidyl serine (PS). Calcium, initially confined to the neutral lipid (PE) side, can pass through the open alamethicin channel to the negative lipid (PS) side, where it can bind to the negative lipid and reduce the surface potential. Under appropriate circumstances, the voltage-dependent alamethicin conductance is thereby inactivated. We have formulated a model for this process based on the diffusion of calcium in the aqueous phases and we show that the model describes the kinetic properties of the alamethicin conductance under various circumstances. EGTA on the PS side of the membrane reduces the effects of calcium dramatically as predicted by the model.

We measured fluorescence phase shift and modulation data for one-, two- and, three-component mixtures of fluorophores at modulation frequencies ranging from 1 to 140 MHz. These data were analyzed using the least-squares procedure described in the preceding paper (Lakowicz, J. R., G. Laczko, M. Cherek, E. Gratton, and M. Limkeman, 1984, Biophys. J., 46:463-477). Using data obtained at a single emission bandpass, the lifetimes and preexponential factors of two-component mixtures could be easily resolved if the lifetimes differed by a factor of 2. With currently available instrumental stability, three-component mixtures could be resolved when the overall range of decay times was 10-fold, (e.g., 1.3, 4.4, and 12 ns). Measurement of phase and modulation data at several emission wavelengths, where the ratio of the preexponential factors varied, enhanced our ability to resolve closely spaced two and three-component decays. Two-component mixtures could then be resolved if the lifetimes differed by 30% (4.4 and 6.2 ns). Also, the multiple-wavelength data allowed the lifetimes and emission spectra of the three-components of a mixture to be resolved. These results demonstrated that resolution of multiexponential decay laws was possible using frequency-domain phase-modulation fluorometry.

The fluorescence lifetime value of tryptophan residues varies by more than a factor of 100 in different proteins and is determined by several factors, which include solvent exposure and interactions with other elements of the protein matrix. Because of the variety of different elements that can alter the lifetime value and the sensitivity to the particular environment of the tryptophan residue, it is likely that non-unique lifetime values result in protein systems. The emission decay of most proteins can be satisfactorily described only using several exponential components. Here it is proposed that continuous lifetime distributions can better represent the observed decay. An approach based on protein dynamics is presented, which provides fluorescence lifetime distribution functions for single tryptophan residue proteins. First, lifetime distributions for proteins interconverting between two conformations, each characterized by a different lifetime value, are derived. The evolution of the lifetime values as a function of the interconversion rate is studied. In this case lifetime distributions can be obtained from a distribution of rates of interconversion between the two conformations. Second, the existence of a continuum of energy substates within a given conformation was considered. The occupation of a particular energy substate at a given temperature is proportional to the Boltzmann factor. The density of energy states of the potential well depends upon the width of the well, which determines the degree of freedom the residue can move in the conformational space. Lifetime distributions can be obtained by association of each energy substate with a different lifetime value and assuming that the average conformation can change as the energy of the substate is increased. Finally, lifetime distributions for proteins interconverting between two conformations, each characterized by a quasi-continuum of energy substates, are presented. The origin of negative components of the lifetime distribution is also discussed. In the companion paper that will follow (Alcala, J. R., E. Gratton, and F. J.Prendergast, 1987, Biophys. J., in press) lifetime distributions obtained here are used to fit experimental data.

Enzyme action is the result of a large numberof discrete steps involving a great variety ofprocesses such as cooperative conformationalchanges, acid-base catalysis, nucleophylic and/orelectrophylic attack from properly positionedgroups, etc.; it is widely recognized that in orderto be useful for catalysis, the various elementaryprocesses must be space- and time-controlledduring enzyme function. In the past decade greatprogresses have been made in understanding thechemistry and the stereochemistry of enzymeaction, with particular emphasis on the role of thespatial effects. Obviously, an analysis of thetemporal aspects of enzyme action is equallyimportant. The ultimate goal is the description ofthe concomitance and/or sequence of individualelementary steps in the catalytic act. This ambitiousbut difficult goal can be approached byfocusing the attention on the time constants of thevarious elementary processes and assessing theirmicroscopic mechanism by comparative studies onrepresentative model systems. This approach was introduced in enzymology with the developmentof fast relaxation methods and will be followed inthis paper, with the understanding that it suffersfrom the same intrinsic limitations as an analysisof a musical piece restricted to a list of the soundfrequencies occurring in it but devoid of anyinformation about their temporal sequence andrelative intensity.Our aims are:l. To review time events detected inenzymes using a proper physical framework, i.e.,the theory of the random processes.2. To identify these events at a molecularlevel by comparison with processes occurring inappropriate model systems.3. To discuss the statistical significance ofthe detected events.We shall start with the simpler model systemsand shall then proceed to analyze situations ofincreasing complexity and eventually considerenzyme-substrate complexes. For each class ofevents some data will be critically reviewed andtheir relevance to enzyme catalysis stressed. All data will then be comparatively discussed accordingto their time scale and some mechanisticconclusions will be derived.The representative enzymes considered in thisreview were chosen among those which can workas separate entities in an aqueous medium becausethey are simpler and better known.