Tangent and Normal Cones to Real Surfaces

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Computational and mathematical approaches to societal transitions

After an introduction of the theoretical framework and concepts of transition studies, this article gives an overview of how structural change in social systems has been studied from various disciplinary perspectives. This overview first leads to the conclusion that computational and mathematical approaches and their practical form, modeling, up till now, have been almost absent in the research and theorizing of structural change or transitions in social systems. Second, this review of the social science literature suggests numerous theoretical constructs relevant for transition modeling. Relevant concepts include the conceptualization of the micro-to-macro link, the importance of explaining both stability and change, quantitative and qualitative definitions of structural change, the use of dichotomies, synchronic and diachronic reasoning in explaining structural change, definitions of basic patterns of social change, the conceptualization of resistance to change and intentional and normative aspects of social change. This article employs these theoretical concepts to describe and discuss the models presented in this special issue in order to develop an understanding of what exactly entails a computational or mathematical approach to societal transitions.

ℓ-norm and its Application to Learning Theory

We investigate connections between an important parameter in the theory of Banach spaces called the -norm, and two properties of classes of functions which are essential in Learning Theory – the uniform law of large numbers and the Vapnik–Chervonenkis (VC) dimension. We show that if the -norm of a set of functions is bounded in some sense, then the set satisfies the uniform law of large numbers. Applying this result, we show that if X is a Banach space which has a nontrivial type, then the unit ball of its dual satisfies the uniform law of large numbers. Next, we estimate the -norm of a set of {0,1}-functions in terms of its VC dimension. Finally, we present a `Gelfand number' like estimate of certain classes of functions. We use this estimate to formulate a learning rule, which may be used to approximate functions from the unit balls of several Banach spaces.     

Symmetric and quasi-symmetric functions associated to polymatroids

To every subspace arrangement X we will associate symmetric functions ℘[X] and ℋ[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant ℋ[X] specializes to the Tutte polynomial . Billera, Jia and Reiner recently introduced a quasi-symmetric function ℱ[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function for polymatroids which has this property as well. Moreover, specializes to ℘[X], ℋ[X], and ℱ[X]. The author is partially supported by the NSF, grant DMS 0349019.     

Measurement of Returns to Scale and Damages to Scale for DEA-based operational and environmental assessment: How to manage desirable (good) and undesirable (bad) outputs?

Environmental assessment is increasingly important in preventing various types of pollutions. Data Envelopment Analysis (DEA) has been long used as an operational performance measure, but we have insufficiently explored the use of DEA for environmental assessment. This study explores a new use of DEA for the environmental assessment in which outputs are classified into desirable (good) and undesirable (bad) outputs. Such an output separation is important in the DEA-based environmental assessment. This study extends the use of DEA to the measurement of both Returns to Scale (RTS) for desirable outputs and Damages to Scale (DTS) for undesirable outputs. A Range-Adjusted Measure (RAM) is used as a DEA model for this study because the non-radial model can easily combine the two types of outputs in a unified treatment. All the mathematical features regarding the RAM-based RTS/DTS measurement are first discussed from the operational and environmental performance in a separate treatment. Then, this study combines the two performance measures as a unified measure. The RAM-based RTS/DTS is mathematically explored from the unified measure for operational and environmental performance.     

Paul Painlevé and his contribution to science

The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlevé transcendents. The contribution of Paul Painlevé to the study of algebraic nonintegrability of the N-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.     

α-Returns to scale and multi-output production technologies

This contribution proposes a specification of strictly increasing and decreasing returns to scale in multi-output technologies. Along this line a notion of α-returns to scale is derived from that of homogeneous multi-output technology. For a large class of technologies we establish necessary and sufficient conditions characterizing strictly increasing and strictly decreasing returns to scale to scale. Furthermore, a relationship between input, output and graph distance functions is established. These connections lead naturally to a link between the various Malmquist indexes and the Chavas–Cox productivity index. Finally, we show that these concepts can be implemented in a DEA context using a piecewise homogeneous constant elasticity substitution–transformation model due to [Färe, R., Grosskopf, S., Njinkeu, D., 1988b. On piecewise reference technologies. Management Science 34, 1507–1511].     

Critical Point Theorems and Applications to Differential Equations

This paper contains a generalization of the well-known Palais-Smale and Cerami compactness conditions. The compactness condition introduced is used to prove some general existence theorems for critical points. Some applications are given to differential equations.     

Janet’s approach to presentations and resolutions for polynomials and linear pdes

Janets algorithm to create normal forms for systems of linear pdes is outlined and used as a tool to construct resolutions for finitely generated modules over polynomial rings over fields as well as over rings of linear differential operators with coefficients in a differential field. The main result is that a Janet basis for a module allows to read off a Janet basis for the syzygy module. Two concepts are introduced: The generalized Hilbert series allowing to read off a basis (over the ground field) of the modules, once the Janet basis is constructed, and the Janet graph, containing all the relevant information connected to the Janet basis. In the context of pdes, the generalized Hilbert series enumerates the free Taylor coefficients for power series solutions. Rather than presenting Janets algorithm as a powerful computational tool competing successfully with more commonly known Gröbner basis techniques, it is used here to prove theoretical results.Received: 6 September 2004     

Hardy and Bellman Transformations in Spaces Close to L ∞ and L 1

The replacement of coefficients of a trigonometric series by their arithmetic averages gives rise to the Hardy operator. The Bellman operator is its adjoint. The spaces _Lp with p[1,) are invariant under the Hardy transformation. This result was proved by Hardy. On the other hand, the space L is not invariant under the Hardy transformation, and the space _L1 is not invariant under the Bellman transformation. Golubov proved that the space BMO is not invariant under the Hardy transformation and is not invariant under the Bellman operator. In the present paper, the exact ``shift' of the domain under the action of these operators is described for certain Orlicz, Lorenz, and Marcinkiewicz spaces and the spaces BMO and . For the Hardy operator, this shift occurs if the domain is close to L , and for the Bellman operator the same happens if the domain is close to L ₁. Bibliography: 15 titles.     

From Oscillatory Integrals and Sublevel Sets to Polynomial Congruences and Character Sums

We present a slight extension of a classical lemma of Hensel and give various applications to polynomial congruences and character sums; in particular, we give a new proof of a classical result of Hua on complete exponential sums.     

Erratum to: A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis

Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.     

Local Lipschitz-α Mappings and Applications to Sublinear Expectations

The aim of this paper is to establish a series of important properties of local Lipschitz-α mappings from a subset of a normed space into a normed space. These mappings include Lipschitz operators, Lipschitz-α operators and local Lipschitz functions. Some applications to the theory of sublinear expectation spaces are given.     

Orderly Algorithm to Enumerate Central Groupoids and Their Graphs

A graph has the unique path property UPPn if there is a unique path of length n between any ordered pair of nodes. This paper reiterates Royle and MacKay＇s technique for constructing orderly algorithms. We wish to use this technique to enumerate all UPP2 graphs of small orders 3^2 and 4^2. We attempt to use the direct graph formalism and find that the algorithm is inefficient. We introduce a generalised problem and derive algebraic and combinatoric structures with appropriate structure. Then we are able to design an orderly algorithm to determine all UPP2 graphs of order 3^2, which runs fast enough. We hope to be able to determine the UPP2 graphs of order 4^2 in the near future.     

Stable-Range Approach to Short Wave and Khokhlov-Zabolotskaya Equations

Short wave equations were introduced in connection with the nonlinear reflection of weak shock waves. They also relate to the modulation of a gas-fluid mixture. Khokhlov-Zabolotskaya equation is used to describe the propagation of a diffraction sound beam in a nonlinear medium. We give a new algebraic method of solving these equations by using certain finite-dimensional stable range of the nonlinear terms and obtain large families of new explicit exact solutions parameterized by several functions for them. These parameter functions enable one to find the solutions of some related practical models and boundary value problems. Research supported by China NSF 10871193.     

Safety controls and applications to the Dubins’ car

In this paper we consider systems with cooperative controls. Such a model turns out to be efficient in those cases where one has to deal with different goals, as optimality and safety issues, at the same time. On the other hand, the drawback of this approach is the presence of discontinuities of the various controls. To deal with this problem, we give a general strategy, which is based on the concept of Krasowskii solution to discontinuous ODEs. Such a strategy is then applied to the safety driving of the Dubins car.     

An effective version of the Bombieri–Vinogradov theorem, and applications to Chen’s theorem and to sums of primes and powers of two

In this paper we give an effective version of the Bombieri–Vinogradov theorem of Bombieri, Friedlander, and Iwaniec. As applications, we refine Chen’s theorem on primes with difference h for h ≤ x ^1/56, with a consequence on integers of the form p + 2 ^k .     

Floer–Novikov homology and applications to Lagrangian submanifolds

We use a non-Hamiltonian version of Lagrangian Floer homology to prove that an exact Lagrangian submanifold in the cotangent bundle of the 3-torus T ³ must be diffeomorphic to T ³. This improves a previous result of Fukaya, Seidel and Smith.