Quadrature for meshless Nitsche's method

In this article, we study effect of numerical integration on Galerkin meshless method (GMM), applied to approximate solutions of elliptic partial differential equations with essential boundary conditions (EBC). It is well‐known that it is difficult to impose the EBC on the standard approximation space used in GMM. We have used the Nitsche's approach, which was introduced in context of finite element method, to impose the EBC. We refer to this approach as the meshless Nitsche's method (MNM). We require that the numerical integration rule satisfies (a) a “discrete Green's identity” on polynomial spaces, and (b) a “conforming condition” involving the additional integration terms introduced by the Nitsche's approach. Based on such numerical integration rules, we have obtained a convergence result for MNM with numerical integration, where the shape functions reproduce polynomials of degree k ≥ 1. Though we have presented the analysis for the nonsymmetric MNM, the analysis could be extended to the symmetric MNM similarly. Numerical results have been presented to illuminate the theoretical results and to demonstrate the efficiency of the algorithms.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 265–288, 2014     

HOW RAPID SHOULD EMISSION REDUCTION BE? A GAME‐THEORETIC APPROACH

Abstract In this paper, we search for multistage realization of international environmental agreements. To analyze countries' incentives and the results of their interactions, we mathematically represent players' strategic preferences and apply a game‐theoretic approach to make predictions about their outcomes. The initial decision on emissions reduction is determined by the Stackelberg equilibrium concept. We generalize Barrett's static “emission” model to a dynamic framework and answer the question “how rapid should the emission reduction be?” It appears that sharper abatement is desirable in the early term, which is similar to the conclusion of the Stern review. Numerical example demonstrates that abatement dynamics of the coalition and the free‐rider differ when discounting of the future payoffs increases. We show that without incentives from external organizations or governments, such pollution reduction path can actually lead to a decline in the agreement's membership size.     

Prototype Proofs in Type Theory

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ‐calculus and act as “proof‐schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type‐quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.     

On Martos' and Chaehes-Coopeb's approach vis-a-vis

In this paper a linear fractional programming problem is studied in presence of “singular-points”. It is proved that “singular points”, if present, exist at an extreme point of S: = {x ? R ⁿ | Ax = b, x ≧0}

It is also shown that a “singular point” is adjacent to an optimal point of S and a characterization of a non-basic vector is obtained, whose entry into the optimal basis in Martos' approach yields the “singular point”.     

Hannah's Prior Knowledge About Chemicals: A Case Study of One Fourth‐Grade Child

Chemical principles are taught in elementary education across much of the United States because the National Science Education Standards include concepts about the nature of matter, states of matter, and changes in matter among other science concepts within the first to fifth grade levels. “Chemicals” is a word related to the nature of matter that is used not only in formal instruction, but also in everyday conversations. Children's prior knowledge about chemicals gained from everyday experiences will influence how they learn about chemical principles. The research described herein reveals insights into one child's conceptual structure related to the word “chemical,” which includes how she uses the word both inside and outside of school. Hannah was purposefully chosen for this case study because she exemplified “children's science.” Her understanding of chemicals as cleaners, in foods, and used for a purpose were primarily gained from everyday experiences. The implications of these findings are discussed with respect to both future research and elementary science education.     

Ethnic Preferences and Residential Segregation: Theoretical Explorations Beyond Detroit

We are strongly supportive of Fossett's theoretical approach and modeling methodology, which uses computational methods to perform thought experiments that generate compelling insights into the enigma of persistent residential segregation in the U.S. We also agree with his theoretical results, which challenge the prevailing view among demographers that institutional discrimination is the essential cause. However, we think he did not go far enough. Fossett limited his analysis to a narrow region of the parameter space that corresponded to conditions observed in one city at one time. This precludes generalization to other times and places and exploration of theoretically motivated “what if” scenarios that trespass beyond the Detroit city limits. When we extended the parameter space, we noticed two interesting results. First, Fossett's “paradox of weak minority preferences” requires qualification. Disproportionate in-group preferences among minorities are indeed segregation-promoting, not integration-promoting, but they generally have less impact on segregation than the in-group preferences of the majority. Second, not only are exclusionary practices and institutional discrimination not necessary for segregation (as Fossett demonstrates), we show that in certain regions of the parameter space they are not even sufficient.     

A generalization of chordal graphs

In a 3-connected planar triangulation, every circuit of length ≥ 4 divides the rest of the edges into two nontrivial parts (inside and outside) which are “separated” by the circuit. Neil Robertson asked to what extent triangulations are characterized by this property, and conjectured an answer. In this paper we prove his conjecture, that if G is simple and 3-connected and every circuit of length ≥ 4 has at least two “bridges,” then G may be built up by “clique-sums” starting from complete graphs and planar triangulations. This is a generalization of Dirac's theorem about chordal graphs.     

Understanding preservation theorems,II

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non‐meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak ^ωω ‐bounding property, and the property “the set of reals of the ground model has positive outer measure” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Expanded understanding of student conceptions of engineers: Validation of the modified draw-an-engineer test (mDAET) scoring rubric

With this research, we sought to expand the pool of engineering education assessment instruments. We drew upon earlier efforts to define an expanded protocol for prompting children's drawings of engineers; and evaluate the reliability of a new scoring rubric for the modified Draw-an-Engineer Test (mDAET). An earlier paper introduced our ground theory approach in gathering theoretical propositions derived from children's drawings, explanations of engineers (n = 940) to develop the mDAET Scoring Rubric. Here we evaluate the reliability of the mDAET, a component of the larger mDAET rubric validation process. In the current study, eight participants scored twelve sets (including anchor sets) of drawings (n = 36) and, scores across drawings were evaluated by percentage match with the known-score and interrater reliability. Kappa values for exact matches across eight raters met the threshold for “substantial” agreement and matches that were exact or adjacent indicated “almost perfect” agreement. Notably, 92% of the possible matches among common anchor items were either exact or adjacent matches. These results indicate adequate inter-rater reliability evidence of the mDAET scoring rubric and demonstrate the feasibility of largescale administration that may no longer require individual child interviews.     

Probabilities of Independent Choices Being Ordered

A simple proof of the following result of Baumert et al. is given. If A and B are chosen independently according to a probability distribution defined on subsets of an n element set, then the probability that A contains B is at least . It's shown to lead to a stronger conclusion, and to suggest definition of concepts of “partition number” and of “orthogonality of partitions into chains” as well as several conjectures.     

Cultural Influences on Children's Probabilistic Thinking

This study investigated selected cultural influences on probabilistic thinking of 11–12-year-old children in England. Language, beliefs and experience are shown to influence the children's “informal knowledge” of probability, i.e., the intuitive knowledge they bring to school and use in thinking about probabilistic situations presented in school. Some of the pupils' responses in interviews and in a questionnaire were consistent with the “outcome approach” and with the use of certain heuristics: “representativeness”; “availability”; “equiprobability.” A significant proportion of pupils revealed superstitions. Standard “random devices” such as dice were regarded by some children as subject to personal, religious or causative influences. In a comparison between two culturally-contrasted subgroups in the same school, we found significant differences in scores on probability tests, even when taking account of numerical and non-verbal ability. But the differences between the two groups could almost entirely be explained by differences in language ability.     

The algebraic theory approach for ordinary differential equations: Highly accurate finite differences

This article reports further developments of Herrera's algebraic theory approach to the numerical treatment of differential equations. A new solution procedure for ordinary differential equations is presented. Finite difference algorithms of 0(h^r), for arbitrary “r” are developed. The method consists in constructing local approximate solutions and using them to extract information about the sought solution. Only nodal information is derived. The local approximate solutions are constructed by collocation, using polynomials of degree G. When “n” collocation points are used at each subinterval, G = n + 1and the order of accuracy is 0(h^2n?1). The procedure here presented is very easy to implement. A program in which n can be chosen arbitrarily, was constructed and applied to selected examples.     

G ∞-Formality Theorem in Terms of Graphs and Associated Chevalley–Eilenberg–Harrison Cohomology #

In 1997, M. Kontsevich proved the L _∞-formality conjecture (which implies the existence of star-products for any Poisson manifold) using graphs. A year later, D. Tamarkin gave another proof of a more general conjecture (for G _∞-structures) using operads and cohomological methods. In this article, we show how Tamarkin's construction can be written using graphs. For that, we introduce a generalization of Kontsevich graphs on which we define a “Chevalley–Eilenberg–Harrison” complex. We show that this complex on graphs is related to the “Chevalley–Eilenberg–Harrison” complex for maps on polyvector fields, which is trivial and give Tamarkin's formality theorem as a consequence. This formality reduces to an L _∞-formality.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

Helping Students Come to Grips With the Meaning of Division

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by defining two meanings for division). However, in the author's classes many students had relatively low scores (ranging from 60–80%) when trying to explain simple operations. Reported in this paper are ways to supplement the text that force students to address the actual meaning of division by stressing the relation between a “whole” and a “package,” and connect that meaning with previously learned operational definitions for area and volume.     

A Distribution- Theoretic Approach to a Stability Problem of Orr

In a classic paper [1] of 1907, W. M'Farr Orr discovered, among other things, the “infinitesimal” instability of inviscid plane Couette flow. Surprisingly, although Orr's paper remains a standard reference in the field, later investigators [2, 3] have been able to call inviscid plane Couette flow stable without finding it necessary to controvert Orr's result. What has happened is that, at least in problems governed by linear (or linearized) equations with time-independent coefficients, the term “instability” has come to be identified with the presence of solutions exhibiting exponential time-growth. Orr found instability indeed: a class of solutions certain members of which grow in time by more than each preassigned factor. Unlike the exponential instabilities, however, Orr's solutions die away like 1/t after achieving their greatest growth. This ephemerality probably accounts for the discounting of Orr's result. Orr did not look into the general initial value problem. This is done in the sequel, with the result that the situation becomes clear. Under general disturbances, Couette flow turns out to be neither stable nor quasi-asymptotically stable*. The rate of growth depends on the smoothness of the initial data: classical solutions grow no faster than t, but sufficiently rough distribution-valued initial data leads to growth matching any power of t. Before presenting detailed results, we briefly review Orr's fundamental work on the problem.     

A duality theorem for graph embeddings

A generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. If K, L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs K?,L? in orientable surfaces S?, such that S? are branched covers of S and the restrictions of the branched coverings to K?,L? are wqc's of K, L. the theory is applied to obtain genus embeddings of composition graphs G[nK₁] from embeddings of “quotient” graphs G.     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009     

Automatic Optimal Regularization for Ill-Posed Problems with Stochastical Noise without Imposing Smoothness Assumptions

We consider the compact operator A : 𝒳 → 𝒴 for the separable Hilbert spaces 𝒳 and 𝒴. The problem Ax = y is called ill-posed when the singular values s_k , k = 1, 2, … of the operator A tend to zero. Classically one assumes that y is biased with “deterministic noise”; we will also consider “stochastic noise” where the noise element is a weak Gaussian random variable. There classical stopping rules (e.g. Morozov) do not work. We will show that both for the “deterministic noise” case as well for the “stochastical noise” case we can regularize in an (asymptotically almost) optimal way without knowledge of the smoothness of the solution using Lepskij's method. Furthermore the method also works for estimated error levels and error behavior. So we can assure regularization which is just dependent on measurements obtainable in reality, e.g. satellite problems. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)