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Experimental Mathematics: Recent Developments and Future Outlook
 CECM PREPRINT 99:143] FFL J.M. BORWEIN AND P.B. BORWEIN, &QUOT;CHALLENGES FOR MATHEMATICAL COMPUTING,&QUOT; COMPUTING IN SCIENCE & ENGINEERING, 2001. [CECM PREPRINT 01:160
, 2000
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Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
Understanding Open Source Software Evolution
 Applying, Breaking, and Rethinking the Laws of Software Evolution
, 2003
"... This chapter examines the evolution of open source software and how their evolutionary patterns compare to prior studies of software evolution of proprietary (or closed source) software. Free or open source software (F/OSS) development focuses attention to systems like the GNU/Linux operating system ..."
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This chapter examines the evolution of open source software and how their evolutionary patterns compare to prior studies of software evolution of proprietary (or closed source) software. Free or open source software (F/OSS) development focuses attention to systems like the GNU/Linux operating system, Apache Web server, and Mozilla Web browser,
Formal proof—theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
Simulation modeling in organizational and management research
 ACADEMY OF MANAGEMENT REVIEW
, 2007
"... Simulation modeling provides a powerful methodology for advancing theory and research on complex behaviors and systems, yet it has been embraced more slowly in management than in some associated social science disciplines. We suspect that part of the reason is that simulation methods are not well un ..."
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Simulation modeling provides a powerful methodology for advancing theory and research on complex behaviors and systems, yet it has been embraced more slowly in management than in some associated social science disciplines. We suspect that part of the reason is that simulation methods are not well understood. We therefore aim to promote understanding of simulation methodology and to develop an appreciation of its potential contributions to management theory by describing the nature of simulations, its attractions, and its special problems, as well as some uses of computational modeling in management research.
Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus
 FOUNDATIONS OF SCIENCE
, 2012
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Variations by complexity theorists on three themes of Euler, . . .
 COMPUTATIONAL COMPLEXITY
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 15 (3 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways.
A lambda calculus for real analysis
, 2005
"... Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoni ..."
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Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoning looks remarkably like a sanitised form of that in classical topology. This paper is an introduction to ASD for the general mathematician, and applies it to elementary real analysis. It culminates in the Intermediate Value Theorem, i.e. the solution of equations fx = 0 for continuous f: R → R. As is well known from both numerical and constructive considerations, the equation cannot be solved if f “hovers ” near 0, whilst tangential solutions will never be found. In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of “overtness”. The zeroes are captured, not as a set, but by highertype operators � and ♦ that remain (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than sets of points leads to
Paradox and theorizing within the resourcebased view
 Academy of Management Review
, 2006
"... By working with and through the paradoxes present in the resourcebased view (RBV) of strategic management, scholars can advance understanding concerning the contradictions and tensions inherent in creating and sustaining superior firm performance. We identify and discuss various RBV paradoxes, il ..."
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By working with and through the paradoxes present in the resourcebased view (RBV) of strategic management, scholars can advance understanding concerning the contradictions and tensions inherent in creating and sustaining superior firm performance. We identify and discuss various RBV paradoxes, illustrating how paradoxical thinking can enhance theorizing and open up new vistas for knowing and understanding. Finally, we discuss the utility of the paradoxical perspective in furthering RBV scholarship. Paradox, “the dynamic tensions of juxtaposed opposites ” (Rosen, 1994: xvii), underpins much organization and management scholarship and practice. The use of paradox can promote divergent or “oppositional ” thinking (Cameron, 1986;
Qualitative Reasoning beyond the Physics Domain: The Density Dependence Theory of Organizational Ecology
 Proceedings of QR95
, 1995
"... Abstract: Qualitative reasoning is traditionally associated with the domain of physics, although the domain of application is, in fact, much broader. This paper investigates the application of qualitative reasoning beyond the domain of physics. It presents a case study of application in the social s ..."
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Abstract: Qualitative reasoning is traditionally associated with the domain of physics, although the domain of application is, in fact, much broader. This paper investigates the application of qualitative reasoning beyond the domain of physics. It presents a case study of application in the social sciences: the density dependence theory of organizational ecology. It discusses how the different nature of soft science domains will complicate the process of model building. Furthermore, it shows that the “model building ” process can also help making theoretically important decisions, and, as a result, have an impact on the original theory. This will require a shift in focus from the “model simulation ” process towards the “model building ” process. 1