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The TestsversusProofs Conundrum
"... I n recent years, advances in formalproof systems and constraint solvers have enabled us to dream of a day when all the software we write can be proven correct. Software practitioners now must decide whether to persist in using testing to improve confidence in their code or to invest effort in le ..."
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in learning and adopting more formal approaches. How are they to choose? Well,... it depends. Proofs or Tests? Proofs are powerful. They’re mathematical constructs that guarantee that all possible executions of a program satisfy (under certain assump
Topic #4: Nonstandard Methods in the study of NavierStokes equations and in Math ematical Physics [Joint work with Brendan Enright.1] We discuss the solution of the nonhomogeneous stochastic NavierStokes equations: ρ
"... + ρ < u, ∇> u = ν∆u−∇p+ ρf(t, u) + ρg(t, u)dw dt div u = 0, u∂D = 0, ut=0 = u0(1) (2) ∂ρ ∂t + < u, ∇> ρ = 0, ρt=0 = ρ0 for the velocity field of a viscous incompressible fluid in a bounded domain D ⊂ Rd (d = 2, 3) with density ρ that is not constant. The term p is the pressure; f repr ..."
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, with g ≡ 1) was solved by Yashima [5]. We will outline the construction of solutions to the above equations with a general external force and multiplicative noise, using an extension of the Loeb space methods
Constructive Mathematics, Probability Theory, Interval Mathematics, and Fuzzy Mathematics Are Important
"... The recent death of Ray Moore, one of the fathers of interval mathematics, inspired these thoughts on why interval computations — and several related areas of study — are important, and what we can learn from the successes of these areas ’ founders and promoters. ..."
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The recent death of Ray Moore, one of the fathers of interval mathematics, inspired these thoughts on why interval computations — and several related areas of study — are important, and what we can learn from the successes of these areas ’ founders and promoters.
Nonconstructive proofs in combinatorics
 Proc. of the International Congress of Mathematicians, Kyoto
, 1990
"... One of the main reasons for the fast development of Combinatorics during the recent years is certainly the widely used application of combinatorial methods in the study and the development of efficient algorithms. It is therefore somewhat surprising that many results proved by applying some of the m ..."
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and their coloring have become in the last ten years part of the mathematical machinery commonly used in combinatorics. Many interesting examples appear in [12]. Some of the more recent results of this type deal with problems that are closely related to certain algorithmic problems. While the topological tools
HALVES, PIECES, AND TWOTHS: CONSTRUCTING REPRESENTATIONAL CONTEXTS IN TEACHING FRACTIONS
"... understanding, according to this view, entails making connections between informal understandings— —about mathematical ideas, quantitative and spatial patterns, and relationships——and more formal mathematical ideas. Connections must be forged among mathematical ideas (Fennema, Carpenter, and Peterso ..."
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Cited by 15 (1 self)
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measurement. "Knowing mathematics " includes knowing how to do mat ematics: "To know mathematics is to investigate and express relationships among patterns, to be able to discern patterns in complex and obscure contexts, to understand and transform relationships among patterns " (National
Constructing generalized mean functions using convex functions with regularity conditions
 SIAM J. Optim
"... Abstract. The generalized mean function has been widely used in convex analysis and mathematical programming. This paper studies a further generalization of such a function. A necessary and sufficient condition is obtained for the convexity of a generalized function. Additional sufficient condition ..."
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Abstract. The generalized mean function has been widely used in convex analysis and mathematical programming. This paper studies a further generalization of such a function. A necessary and sufficient condition is obtained for the convexity of a generalized function. Additional sufficient
Reified bayesian modelling and inference for physical systems (with discussion
 Journal of Statistical Planning and Inference
, 2009
"... We describe an approach, termed reified analysis, for linking the behaviour of mathematical models with inferences about the physical systems which the models represent. We describe the logical basis for the approach, based on coherent assessment of the implications of deficiencies in the mathemat ..."
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Cited by 38 (18 self)
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We describe an approach, termed reified analysis, for linking the behaviour of mathematical models with inferences about the physical systems which the models represent. We describe the logical basis for the approach, based on coherent assessment of the implications of deficiencies
Software for Evaluating Relevance of Steps in Algebraic Transformations? Rein Prank
"... Abstract. Students of our department solve algebraic exercises in mathematical logic in a computerized environment. They construct transformations step by step and the program checks the syntax, equivalence of expressions and completion of the task. With our current project, we add a program compo ..."
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Abstract. Students of our department solve algebraic exercises in mathematical logic in a computerized environment. They construct transformations step by step and the program checks the syntax, equivalence of expressions and completion of the task. With our current project, we add a program
Stochastics and tio: A
"... get simulation model. The importance of metamodel validation is emphasized by using the generalization of Raos test However, much of our knowledge of the world comes from lowresolution sources. Whenever possible, it is more suitable to construct a simple mathematical relationship that relates the ..."
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get simulation model. The importance of metamodel validation is emphasized by using the generalization of Raos test However, much of our knowledge of the world comes from lowresolution sources. Whenever possible, it is more suitable to construct a simple mathematical relationship that relates
Computational Techniques and Applications: CTAC95 1 World Scientic A MAPLE Package for Stochastic Differential Equations
"... The eects of intrinsic noise within physical phenomena is ignored when mathematical models of their behaviour are constructed using deterministic dierential equations. Such equations describe what is occurring on average to the system. However, more information is often required to obtain a full pi ..."
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The eects of intrinsic noise within physical phenomena is ignored when mathematical models of their behaviour are constructed using deterministic dierential equations. Such equations describe what is occurring on average to the system. However, more information is often required to obtain a full
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