Polygons with primitive normal and additive theories

Previously, primitive normal, primitive connected, and additive theories of S-polygons were studied. In particular, it was proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid. The present paper is a continuation of this research. Here, Spolygons with primitive normal, additive, and antiadditive theories are described in the language of a primitive equivalence structure. It is shown that the class of all S-polygons is antiadditive only for a linearly ordered monoid S, that is, this class is antiadditive iff it is primitive normal.     

Testing primitivity on partial words

Primitive words, or strings over a finite alphabet that cannot be written as a power of another string, play an important role in numerous research areas including formal language theory, coding theory, and combinatorics on words. Testing whether or not a word is primitive can be done in linear time in the length of the word. Indeed, a word is primitive if and only if it is not an inside factor of its square. In this paper, we describe a linear time algorithm to test primitivity on partial words which are strings that may contain a number of “do not know” symbols. Our algorithm is based on the combinatorial result that under some condition, a partial word is primitive if and only if it is not compatible with an inside factor of its square. The concept of special, related to commutativity on partial words, is foundational in the design of our algorithm. A World Wide Web server interface at http://www.uncg.edu/mat/primitive/ has been established for automated use of the program.     

Marcus contextual languages consisting of primitive words

In this paper we prove that the language of all primitive (strongly primitive) words over a nontrivial alphabet can be generated by certain types of Marcus contextual grammars.     

A self-dual language for graph theory

A symbolic language is developed around the primitive notions of edge and spanning tree, with each spanning tree being interpreted as a binary relation on edges. A sense is considered in which the properties expressible in this language are precisely the self-dual properties of graph theory.     

Solid codes and disjunctive domains

In this paper, a solid code is introduced and disjunctive domain is defined as a language by which one can determined whether a given language is disjunctive or not. Some properties of solid codes were investigated. We show that a solid code consists of only primitive words. A characterization of disjunctive domains is obtained. The author would like to thank the referee for his valuable suggestions     

On list primitive recursion and the complexity of computinginf

We present a primitive recursive programinf_with_lists computing the minimum of two natural numbersn andp (written in unary notation) and using primitive recursion on lists. This program has at first sight the required property of visiting simultaneously its inputs, so it is a counterexample to a theorem showing that such a program cannot be written in the language of primitive recursion on natural numbers, in the more general framework of primitive recursion on term algebras. However, its complexity is at leastinf(n,p)² so it does not implement the algorithm we have in mind to computeinf(n,p).     

Optimal control of the primitive equations of the ocean with state constraints

We investigate in this article Pontryagin’s maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang (2002) [7] for the three-dimensional Navier-Stokes (NS) equations. The main difference between this work and Wang (2002) [7] is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Mathematical Aspect of Optimal Control Finite Element Method for Navier-Stokes Problems

This study deals with the theoretical basis of optimal control methods in primitive variable formulation and penalty function formulations of Navier-Stokes problems. Numerical examples demonstrating application are provided.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

μ-Reducible Languages

A word w is said to be a primitive word if it cannot be expressed as a power of any other word. A language L consisting of non-empty words is called -reducible if there exists a non-empty word w such that Lw contains only finitely many powers of each primitive word. We show that every regular component, context-free component, local language and every regular language containing no primitive words are -reducible. Languages which are not -reducible are investigated and characterized. We show that every code is -reducible. But there are 2-codes which are not -reducible. The -annihilator of a language L is the set of all non-empty words w such that Lw contains only finitely many powers of each primitive word. This paper also concerns the properties of the -annihilators of languages. The -annihilators of 2-codes and some other languages are investigated and characterized in this paper. The results provide an outline of the relationship between the catenation of languages and the powers of primitive words.     

Maximum Principle of Optimal Control of the Primitive Equations of the Ocean With State Constraint

We investigate in this paper Pontryagin's maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) for the three-dimensional Navier–Stokes (NS) equations. The main difference between this work and Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Fragments of HA based on \Sigma_1-induction

In the first part of this paper we investigate the intuitionistic version of (in the language of ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for and establishes a number of nice properties for , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: together with induction over arbitrary prenex formulas is -conservative over . In the second part of the article we study the relation of to (in the usual arithmetical language). The situation here is markedly different from the classical case in that and are mutually incomparable, while is significantly stronger than as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in whereas the provably recursive functions of are all majorized by polynomials over . 0 is unusual also in that it lacks closure under Markov's Rule . Received April 4, 1996     

The effects of meta-cognitive instruction on third and sixth graders solving word problems

This study examined the differential effects of a meta-cognitive instruction, called IMPROVE, on third and sixth graders’ solution of word problems. In particular, the study focused on the solution of two kinds of word problems: with consistent and with inconsistent language. Participants were 194 Israeli students who studied in third (N = 110) and sixth (N = 84) grades. All students were administered pre- and post-tests constructed of 16 word problems with consistent and inconsistent language. About half of the students within each grade level were exposed to IMPROVE and the others studied under a ‘traditional’ teaching method. The findings indicate that at both grade levels the IMPROVE students significantly outperformed their counterparts in the control group, but third graders benefited from IMPROVE more than sixth graders. In addition, the study indicates that the gap in achievement between IMPROVE and control groups was larger on word problems with inconsistent language compared to word problems with consistent language. The theoretical and practical implications of the study are discussed.     

循环矩阵本原指数的上界估计

本文给出了循环矩阵本原指数上界的新的估计及一种由级数较低的循环矩阵的本原指数估计级数较高的循环矩阵的本原指数的方法，解决了一类循环矩阵本原指数的计算问题．     

Optimizing a constrained convex polygonal annulus

In this paper we give solutions to several constrained polygon annulus placement problems for offset and scaled polygons, providing new efficient primitive operations for computational metrology and dimensional tolerancing. Given a convex polygon P and a planar point set S, the goal is to find the thinnest annulus region of P containing S. Depending on the application, there are several ways this problem can be constrained. In the variants that we address the size of the polygon defining the inner (respectively, outer) boundary of the annulus is fixed, and the annulus is minimized by minimizing (respectively, maximizing) the outer (respectively, inner) boundary. We also provide solutions to a related known problem: finding the smallest homothetic copy of a polygon containing a set of points. For all of these problems, we solve for the cases where smallest and largest are defined by either the offsetting or scaling of a polygon. We also provide some experimental results from implementations of several competing approaches to a primitive operation important to all the above variants: finding the intersection of n copies of a convex polygon.     

A counterexample to the reconstruction of <Emphasis Type="Italic">ω</Emphasis>-categorical structures from their endomorphism monoid

We present an example of two countable ω-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids—in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable ω-categorical structure in a finite relational language which can neither be reconstructed up to first-order biinterpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.     

Elimination of quantifiers for modules

Every first-order formula in the language ofR-modules (R an associative ring) is equivalent relative to the theory ofR-modules to a boolean combination of positive primitive formulas and ∀∃-sentence. Supported by Schweizerischer Nationalfonds.     

TACO: a toolkit for AMPL control optimization

We describe a set of extensions to the AMPL modeling language to conveniently model mixed-integer optimal control problems for ODE or DAE dynamic processes. These extensions are realized as AMPL user functions and suffixes and do not require intrusive changes to the AMPL language standard or implementation itself. We describe and provide TACO, a Toolkit for AMPL Control Optimization that reads AMPL stub.nl files and detects the structure of the optimal control problem. This toolkit is designed to facilitate the coupling of existing optimal control software packages to AMPL. We discuss requirements, capabilities, and the current implementation. Using the example of the multiple shooting code for optimal control MUSCOD-II, a direct and simultaneous method for DAE-constrained optimal control, we demonstrate how the problem information provided by the TACO toolkit is interfaced to the solver. In addition, we show how the MS-MINTOC algorithm for mixed-integer optimal control can be used to efficiently solve mixed-integer optimal control problems modeled in AMPL. We use the AMPL extensions to model three control problem examples and we discuss how those extensions affect the representation of optimal control problems. Solutions to these problems are obtained by using MUSCOD-II and MS-MINTOC inside the AMPL environment. A collection of further AMPL control models is provided on the web site http://mintoc.de. MUSCOD-II and MS-MINTOC have been made available on the NEOS Server for Optimization, using the TACO toolkit to enable input of AMPL models.