de balanço e equações constitutivas. b. What is mathematical modelling? V. Dabbagian. c. Computational engeneering – Introduction to numerical methods, M Schafer . d. Métodos numéricos para engenharia, Chapra & Canale - Equações de balanço. e. Philosophy of Science: Models in Science, Kristina Rolin. f. Stanford Encyclopedia of Philosophy g. UML – Unified Modelling Language. h. O site plus.math.org e o NRICH. i. Modelagem matemática em biologia. j. Outros livros de modelagem. Tópicos abordados:
models, differential equations, game theoretic models, recurrence relations, an algorithm for calculation of a sequence of related states (e.g. equilibrium states) and possibly even more forms. The governing equations of a mathematical model describes how the unknown variables (i.e. the dependent variables) will change. The change of variables w.r.t. time may be explicit (i.e. a governing equation includes derivative with respect to time) or implicit (e.g. a governing equation has velocity or flux as unknown variable or an algorithm). The classic governing equations in continuum mechanics are •Balance of mass •Balance of (linear) momentum •Balance of angular momentum •Balance of energy •Balance of entropy
conservation equations in physics. A governing equation may also take the form of a flux equation like the diffusion equation or the heat conduction equation. In these cases the flux itself is a variable describing change of the unknown variable or property (e.g. mole concentration or internal energy or temperature). A governing equation may also be an approximation and adaption of the above basic equations to the situation or model in question. A governing equation may also be derived directly from experimental results and therefore be an empiric equation. A governing equation may also be an equation describing the state of the system, and thus actually be a constitutive equation that has "stepped up the ranks" because the model in question was not meant to include a time- dependent term in the equation. This is the case for a model of a petroleum processing plant. Results from one thermodynamic equilibrium calculation are input data to the next equilibrium calculation together with some new state parameters and so on. In this case the algorithm and sequence of input data form a chain of actions, or calculations, that describes change of states from the first state (based solely on input data) to the last state that finally comes out of the calculation sequence.
equations Hele-Shaw flow Plate theory Kirchhoff–Love plate theory or Bending of Kirchhoff-Love plates Mindlin–Reissner plate theory or Bending of thick Mindlin plates or Bending of Reissner-Stein cantilever plates Vortex shedding Annular fin Astronautics Finite volume method for unsteady flow Acoustic theory Precipitation hardening Kelvin's circulation theorem Kernel function for solving integral equation of surface radiation exchanges Nonlinear acoustics Large eddy simulation Föppl–von Kármán equations Timoshenko beam theory
A theory is a collection of statements that can have a formal representation as an axiomatic system. We can separate the logical structure of the theory from its empirical content. (The “syntax” is the study of the rules that determine how sentences are formed.) Statements are given an interpretation (and empirical content) by means of “correspondence rules.” Example: Formal representation: For every x it is the case that if H(x), then M(x). Correspondence rules: H(x)= x is a human being; M(x)= x is mortal. Empirical content: All human beings are mortal. Theories of theories: part I
collection of models and theoretical hypotheses. Models are non-linguistic entities. A theoretical hypothesis is a statement asserting some sort of relationship between a model and a class of real systems in the world. Giere, Ronald. 1988. Explaining science: A cognitive approach. The University of Chicago Press. Suppe, Frederick. 1989. The semantic conception of theories and scientific realism. University of Illinois Press. Van Fraassen, Bas. 1980. The scientific image. Oxford University Press. Theories of theories: part II
where is the period of time for one complete oscillation, and is the length of the pendulum. The model is “constructed” so that the equation describes it truthfully (Giere 1988, 79). Model: an example
in the earth-moon system are similar to those of a two-particle Newtonian gravitational model (in some respects and to some degree) (Giere 1988, 81). “To believe a theory is to believe that one of its models correctly represents the world” (Van Fraassen 1980, 47). Theoretical hypothesis: an example
do not represent any real system in the world. • Fundamental laws (such as F=ma) are abstract, and they can relate to a real system only via models that are representations of concrete phenomena in the world. • Models are constructed so that fundamental laws can feature in them. Such laws do not apply directly to a real system in the world. Hence, the fundamental laws “lie.” • A “simulacrum” account of explanation: Models do explain well even though they are not literally true of any real system in the world. • Cartwright, Nancy. 1983. How the Laws of Physics Lie. Oxford: • Clarendon Press.
whether the asserted relationship between a model and a real system holds. • The relationship between a model and a real system cannot be one of truth or falsity since neither is a linguistic entity (Giere 1988, 80). Theoretical hypothesis
adequate theory of science.” (Giere 1988, 81). • The relation between a model and a real system is isomorphism (sameness of structure) (Van Fraassen 1980, 46). • The relation between a model and a real system is similarity (with respect to some aspects and to some degree) (Giere 1988, 81). ”Truths” in science?
specified theoretical model and a designated real system. But since anything is similar to anything else in some way or other, the claim of similarity must be limited … to a specified set of respects and degrees” (Giere 1988, 93). Scientific representations as “maps”
facilitates access to that phenomenon” (Bailer-Jones 2009, 1). “Facilitating access” involves focusing on specific aspects of the phenomenon, disregarding others. As a result, models are partial representations. Bailer-Jones, Daniela. 2009. Scientific models in philosophy of science. The University of Pittsburgh Press. Models as partial representations
is only a model”). Mature theories were thought to render models redundant (Bailer-Jones 2009, 82). • In the rational reconstruction of scientific knowledge, theories play a central role, not models (92). • Models can play a central role in the context of discovery but not in the context of justification (93). Models in logical empiricism
postulate non-observable entities that are difficult to construct out of observations (e.g., electromagnetic waves). • A model-based conception of theories provides an explanation: such entities are part of models. Models in logical empiricism
statements), such as “exemplars” and “concrete puzzle solutions.” These are claimed to play a central role in the learning of scientific practice. • Yet, if models are understood as “exemplars” or “concrete puzzle solutions,” it is not clear how they relate to theories (understood as statements). • Kuhn acknowledges that visual representations play a significant role in scientific knowledge: a change of paradigm is described as a ”gestalt shift.” Models in Kuhn’s paradigms
stand in a relation of analogy to nature. • Positive analogy refers to those aspects that two things (e.g., billiard balls and molecules) are known to have in common. Negative analogy refers to those aspects that are known to be different. • Neutral analogy refers to those aspects for which commonality or difference is yet to be established. These aspects are interesting because they allow scientists to make predictions (e.g., knowledge of the mechanics of billiard balls can be used to make predictions about the expected behavior of gases). • Hesse, Mary. 1966. Models and analogies in science. University of Notre Dame Press. Analogies in science
system (Bailer- Jones 2009, 56 and 74). • Analogies are often instrumental in scientific discovery, in the formulation of new hypotheses, and in the process of constructing new models (61). • Analogies are often used for the purpose of illustration in science instruction (62). • Analogies are used to transport mathematical methods from one domain to another (73). Analogies are not models
language is common in scientific theories (even though many metaphors in science are “dead”): e.g., electric field, light wave, excited state, chemical bonds, black hole, brain as a computer, critical mass in the social sciences. • Keller, Evelyn Fox. 1985. Reflections on Gender and Science. Yale University Press. • Schiebinger, Londa. 1993. Nature’s Body: Gender inthe Making of Modern Science. Beacon Press. Metaphors in science
is, they employ different external representational tools. • Whereas theories aim to be general, models aim to match specific empirical situations well. • Morgan, Mary, and Morrison, Margaret (eds.). 1999. Models as Mediators. Cambridge University Press. Models as mediators between theories and the world
It can account for the role of unobservable entities in scientific theories: Such entities are part of models. • It can account for idealization and approximation in scientific theories: Models are partial representations of some phenomena in the world. • It can account for the persistence of analogies and metaphors in scientific theories: Analogies and metaphors are not models but the use of analogies and metaphors in science is a spin-off of models (Bailer-Jones 2009, 117). Today’s message
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