Local Uniformization and Free Boundary Regularity of Minimal Singular Surfaces

We continue the study of minimal singular surfaces obtained by a minimization of a weighted energy functional in the spirit of J. Douglas’s approach to the Plateau problem. Modeling soap films spanning wire frames, a singular surface is the union of three disk-type surfaces meeting along a curve which we call the free boundary. We obtain an a priori regularity result concerning the real analyticity of the free boundary curve. Using the free boundary regularity of the harmonic map, we construct local harmonic isothermal coordinates for the minimal singular surface in a neighborhood of a point on the free boundary. Applications of the local uniformization are discussed in relation to H. Lewy’s real analytic extension of minimal surfaces.     

Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

In (Gluskin, Litvak in Geom. Dedicate 90:45–48, [2002]) it was shown that a polytope with few vertices is far from being symmetric in the Banach–Mazur distance. More precisely, it was shown that Banach–Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very large, in fact it satisfies the same lower bound as the Banach–Mazur distance. In a sense it shows the following phenomenon: if a convex polytope with small number of vertices is as close to a symmetric body as it can be, then most of its vertices are as bad as the worst one. We apply our results to provide a lower estimate on the vertex index of a symmetric convex body, which was recently introduced in (Bezdek, Litvak in Adv. Math. 215:626–641, [2007]). Furthermore, we give the affirmative answer to a conjecture by Bezdek (Period. Math. Hung. 53:59–69, [2006]) on the quantitative illumination problem.     

Vertex Degrees of Steiner Minimal Trees in ℓ p d and Other Smooth Minkowski Spaces

We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees (SMTs) in the d -dimensional Banach spaces _p ^d independent of d . This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d [19]. Our upper bounds follow from characterizations of singularities of SMTs due to Lawlor and Morgan [14], which we extend, and certain _p -inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d -dimensional Banach space (i.e. Minkowski space); the same upper bound for Steiner points having been found by Lawlor and Morgan. We obtain a second upper bound for the degrees of vertices in terms of 1 -summing norms. Received April 22, 1997, and in revised form October 1, 1997.     

Extending Persistence Using Poincaré and Lefschetz Duality

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ³. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ³, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds. An erratum to this article can be found at     

Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution

We prove duality results for adjoint operators and product norms in the framework of Euclidean spaces. We show how these results can be used to derive condition numbers especially when perturbations on data are measured componentwise relatively to the original data. We apply this technique to obtain formulas for componentwise and mixed condition numbers for a linear function of a linear least squares solution. These expressions are closed when perturbations of the solution are measured using a componentwise norm or the infinity norm and we get an upper bound for the Euclidean norm.      

Unintended consequences of collocation: using agent-based modeling to untangle effects of communication delay and in-group favor

In studies about office arrangements that have individuals working from remote locations, researchers usually hypothesize advantages for collocators and disadvantages for remote workers. However, empirical findings have not shown consistent support for the hypothesis. We suspect that there are unintended consequences of collocation, which can offset well-recognized advantages of being collocated. To explain these unintended consequences, we developed a multi-agent model to complement our laboratory-based experiment. In the lab, collocated subjects did not perform better than the remote even though collocators had faster communication channels and in-group favor towards each other. Results from the multi-agent simulation suggested that in-group favoritism among collocators caused them to ignore some important resource exchange opportunities with remote individuals. Meanwhile, communication delay of remote subjects protected them from some falsely biased perception of resource availability. The two unintended consequences could offset the advantage of being collocated and diminish performance differences between collocators and remote workers. Results of this study help researchers and practitioners recognize the hidden costs of being collocated. They also demonstrate the value of coupling lab experiments with multi-agent simulation.

Finite-Temperature Coarse-Graining of One-Dimensional Models: Mathematical Analysis and Computational Approaches

We present a possible approach for the computation of free energies and ensemble averages of one-dimensional coarse-grained models in materials science. The approach is based upon a thermodynamic limit process, and makes use of ergodic theorems and large deviations theory. In addition to providing a possible efficient computational strategy for ensemble averages, the approach allows for assessing the accuracy of approximations commonly used in practice.     

Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System

We describe methods for proving bounds on infinite-time averages in differential dynamical systems. The methods rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions. Nonnegativity is enforced by requiring the polynomials to be sums of squares, a condition which is then formulated as a semidefinite program (SDP) that can be solved computationally. Although such computations are subject to numerical error, we demonstrate two ways to obtain rigorous results: using interval arithmetic to control the error of an approximate SDP solution, and finding exact analytical solutions to relatively small SDPs. Previous formulations are extended to allow for bounds depending analytically on parametric variables. These methods are illustrated using the Lorenz equations, a system with three state variables (x, y, z) and three parameters \((\beta ,\sigma ,r)\). Bounds are reported for infinite-time averages of all eighteen moments \(x^ly^mz^n\) up to quartic degree that are symmetric under \((x,y)\mapsto (-x,-y)\). These bounds apply to all solutions regardless of stability, including chaotic trajectories, periodic orbits, and equilibrium points. The analytical approach yields two novel bounds that are sharp: the mean of \(z^3\) can be no larger than its value of \((r-1)^3\) at the nonzero equilibria, and the mean of \(xy^3\) must be nonnegative. The interval arithmetic approach is applied at the standard chaotic parameters to bound eleven average moments that all appear to be maximized on the shortest periodic orbit. Our best upper bound on each such average exceeds its value on the maximizing orbit by less than 1%. Many bounds reported here are much tighter than would be possible without computer assistance.     

REFE-Acceptable Mappings: Necessary and Sufficient Condition for the Nonexistence of a Regular Exceptional Family of Elements

By considering the notion of regular exceptional family of elements (REFE), we define the class of REFE-acceptable mappings. By definition, a complementarity problem on a Hilbert space defined by a REFE-acceptable mapping and a closed convex cone has either a solution or a REFE. We present several classes of REFE-acceptable mappings. For this, neither the topological degree nor the Leray-Schauder alternative is necessary. By using the concept of REFE-acceptable mappings, we present necessary and sufficient conditions for the nonexistence of regular exceptional family of elements. These conditions are used for generating several existence theorems and existence and uniqueness theorems for complementarity problems. The authors are grateful to Prof. A.B. Németh for many helpful conversations. The research of S.Z. Németh was supported by Hungarian Research Grants OTKA T043276 and OTKA K60480.     

Calabi–Yau Threefolds and Moduli of Abelian Surfaces I

We describe birational models and decide the rationality/unirationality of moduli spaces A _d (and A _d ^lev ) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.     

RETRACTED ARTICLE: Convergence of Weighted Sums for Arrays of Negatively Dependent Random Variables and Its Applications

We discuss the complete convergence of weighted sums for arrays of rowwise negatively dependent random variables (ND r.v.’s) to linear processes. As an application, we obtain the complete convergence of linear processes based on ND r.v.’s which extends the result of Li et al. (Stat. Probab. Lett. 14:111–114, 1992), including the results of Baum and Katz (Trans. Am. Math. Soc. 120:108–123, 1965), from the i.i.d. case to a negatively dependent (ND) setting. We complement the results of Ahmed et al. (Stat. Probab. Lett. 58:185–194, 2002) and confirm their conjecture on linear processes in the ND case.     

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f ₁,…,f _m } in a Hilbert space H\mathcal {H}, and given a family C\mathcal {C} of closed subspaces of H\mathcal {H}, the subspace clustering problem consists in finding a union of subspaces in C\mathcal {C} that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C\mathcal {C} for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C⁺\mathcal {C}^{+}. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families C\mathcal {C} of unitary representations of Abelian groups is derived.     

FirmNet: the scope of firms and the allocation of task in a knowledge-based economy

The increasing knowledge intensity of jobs, typical of a knowledge economy, highlights the role of firms as integrators of know how and skills. As economic activity becomes mainly intellectual and requires the integration of specific and idiosyncratic skills, firms need to allocate skills to tasks and traditional hierarchical control may result increasingly ineffective. In this work, we explore under what circumstances networks of agents, which bear specific skills, may self-organize in order to complete tasks. We use a computer simulation approach and investigate how local interaction of agents, endowed with skills and individual decision-making rules, may produce aggregate network structure able to perform tasks. To design algorithms that mimic individual decision-making, we borrow from computer science literature and, in particular, from studies addressing protocols that produce cooperation in P2P networks. We found that self-organization depends on imitation of successful peers, competition among agents holding specific skills, and the structural features of, formal or informal, organizational networks embedding both professionals, holding skills, and project managers, holding access to jobs.

The work of Douglas Munn and its legacy

We give an account of some of the highlights of the mathematical research of Douglas Munn.     

Compact Browder maps and equilibria of abstract economies

We obtain a partial resolution of a conjecture raised by Ben-El-Mechaiekh; that is, for a convex subset X of a Hausdorff t.v.s., any compact Browder map T:X ? X (a multimap with nonempty convex values and open fibers) has a fixed point. From this new result, we deduce a collectively fixed point theorem with applications to existences of equilibrium points and maximal elements of an abstract economy. Consequently, some known results are extended.     

Agency and structure: a social simulation of knowledge-intensive industries

Modern knowledge-intensive economies are complex social systems where intertwining factors are responsible for the shaping of emerging industries: the self-organising interaction patterns and strategies of the individual actors (an agency-oriented pattern) and the institutional frameworks of different innovation systems (a structure-oriented pattern). In this paper, we examine the relative primacy of the two patterns in the development of innovation networks, and find that both are important. In order to investigate the relative significance of strategic decision making by innovation network actors and the roles played by national institutional settings, we use an agent-based model of knowledge-intensive innovation networks, SKIN. We experiment with the simulation of different actor strategies and different access conditions to capital in order to study the resulting effects on innovation performance and size of the industry. Our analysis suggests that actors are able to compensate for structural limitations through strategic collaborations. The implications for public policy are outlined.     

Ramanujan and extensions and contractions of continued fractions

If a continued fraction K _n=1^∞ a _n /b _n is known to converge but its limit is not easy to determine, it may be easier to use an extension of K _n=1^∞ a _n /b _n to find the limit. By an extension of K _n=1^∞ a _n /b _n we mean a continued fraction K _n=1^∞ c _n /d _n whose odd or even part is K _n=1^∞ a _n /b _n . One can then possibly find the limit in one of three ways:

Modularity and incremental innovation: the roles of design rules and organizational communication

As little attention has been paid to the relationship between modularity and near decomposability, extant studies have not unveiled the impact of modularity on incremental innovation completely. We argue that the modular structure is a special case of nearly decomposable structure, in which the interdependencies between modules are specified by design rules, and the degree of modularity is defined by the level of near decomposability and the extent to which intermodule dependencies are specified. The results of our simulation experiments show that in the term of near decomposability, the increase of modularity leads to higher innovation advantage in the short term, but effective communication between modules can help systems with moderate and low modularity gain more innovation benefits in the long term; in the aspect of design rules, modularization may restrict the search space of the incremental innovation within each module, but under some conditions the option value of modularity may offset or even exceed the restriction effect of design rules.