See Statesman, 266e. 1) 0 is a natural number. 2) Whenever n is a
... A Digression on the status of the principle of proof by induction Is this principle a purely logical one—on a par with modus ponens, say? The answer is an unequivocal “it all depends.” If we ask what justifies a logical principle, the answer commonly given is in terms of meaning. In the case of modu ...
... A Digression on the status of the principle of proof by induction Is this principle a purely logical one—on a par with modus ponens, say? The answer is an unequivocal “it all depends.” If we ask what justifies a logical principle, the answer commonly given is in terms of meaning. In the case of modu ...
M a th Ec ono mics - the Department of Economics
... The B.A. in Economics - Mathematical Economics Track provides rigorous training in economics combined with coursework in mathematics. The major provides excellent training for analytical jobs in the private and public sector. It also provides strong preparation for graduate school in economics. ...
... The B.A. in Economics - Mathematical Economics Track provides rigorous training in economics combined with coursework in mathematics. The major provides excellent training for analytical jobs in the private and public sector. It also provides strong preparation for graduate school in economics. ...
Extremal axioms: logical, mathematical and cognitive aspects
... clearly visible that the axiom cannot be formulated in the object language. It is a minimal axiom. There exist some other formulations of the axiom of restriction, notably those given by Roman Suszko (the axiom of canonicity) and John Myhill. The axioms of restriction in set theory became criticized ...
... clearly visible that the axiom cannot be formulated in the object language. It is a minimal axiom. There exist some other formulations of the axiom of restriction, notably those given by Roman Suszko (the axiom of canonicity) and John Myhill. The axioms of restriction in set theory became criticized ...
THEORY AND MODELS IN ECOLOGY: A DIFFERENT
... Their most successful model (the 'cascade model') is only marginally more complex, but does an astonishing job of duplicating both the qualitative and quantitative patterns of observed webs. In the cascade model, the species are ordered; each species can be potentially fed on by all species higher i ...
... Their most successful model (the 'cascade model') is only marginally more complex, but does an astonishing job of duplicating both the qualitative and quantitative patterns of observed webs. In the cascade model, the species are ordered; each species can be potentially fed on by all species higher i ...
Why proofs by mathematical induction are
... is like many typical scientific explanations. For example, Kepler’s laws of planetary motion are explained by being deduced from Newton’s laws of motion and gravity, that the sun and planets are roughly spherical hunks of matter, that the sun contains nearly all of the solar system’s matter, and so ...
... is like many typical scientific explanations. For example, Kepler’s laws of planetary motion are explained by being deduced from Newton’s laws of motion and gravity, that the sun and planets are roughly spherical hunks of matter, that the sun contains nearly all of the solar system’s matter, and so ...
1 THE NATURE OF MATHEMATICS
... the various topics under study. Of course, not all mathematical topics are immediately useful and some have historically been developed for their own sake with their applications coming later. The first task, though, will be to see some of the ways in which mathematics is developed and used; in part ...
... the various topics under study. Of course, not all mathematical topics are immediately useful and some have historically been developed for their own sake with their applications coming later. The first task, though, will be to see some of the ways in which mathematics is developed and used; in part ...
Functional Modeling - Discussion groups
... the first part, the Black-Box-Part, the questions `Which information is for which component?’ and ‘Which information is necessary for another part of the process?’ can be answered. The inner structure of the used components is not answered yet. This is an exercise for the second part, the White-Box- ...
... the first part, the Black-Box-Part, the questions `Which information is for which component?’ and ‘Which information is necessary for another part of the process?’ can be answered. The inner structure of the used components is not answered yet. This is an exercise for the second part, the White-Box- ...
Teaching Cryptologic Mathematics
... afford other curricular courses. When preparing Cryptologic Mathematics lessons two major questions should be answered: what to teach and how to teach it. In order to answer the first question, one of the factors the teacher has to take into account is the preliminary mathematical knowledge of the s ...
... afford other curricular courses. When preparing Cryptologic Mathematics lessons two major questions should be answered: what to teach and how to teach it. In order to answer the first question, one of the factors the teacher has to take into account is the preliminary mathematical knowledge of the s ...
Americana, American History Mathematics
... This book is an integrated work published in two volumes. The first volume treats the basic Markov process and its variants; the second, semi-Markov and decision processes. Its intent is to equip readers to formulate, analyze, and evaluate simple and advanced Markov models of systems, ranging from g ...
... This book is an integrated work published in two volumes. The first volume treats the basic Markov process and its variants; the second, semi-Markov and decision processes. Its intent is to equip readers to formulate, analyze, and evaluate simple and advanced Markov models of systems, ranging from g ...
A variation of Miquel`s theorem and its generalization
... is determined by the intersection of the perpendicular bisectors of XM and YM, the same spiral similarity by ∠XME that maps P1 to P will map XM to EM with scale factor EM/XM, YM to HM and P1M to PM with the result that right triangle P1PM is similar to XEM and YHM. In the same way for the other poin ...
... is determined by the intersection of the perpendicular bisectors of XM and YM, the same spiral similarity by ∠XME that maps P1 to P will map XM to EM with scale factor EM/XM, YM to HM and P1M to PM with the result that right triangle P1PM is similar to XEM and YHM. In the same way for the other poin ...
Affect and Meta-Affect in Mathematical Problem Solving
... exchanging information with cognitive systems. Meta-affect is a construct central to our theory, standing in relation to affect much as metacognition stands in relation to cognition, and powerfully transforming individuals’ emotional feelings. Mathematical intimacy refers to deep, vulnerable emotion ...
... exchanging information with cognitive systems. Meta-affect is a construct central to our theory, standing in relation to affect much as metacognition stands in relation to cognition, and powerfully transforming individuals’ emotional feelings. Mathematical intimacy refers to deep, vulnerable emotion ...
Modeling Electrical and Thermal Conductivities of
... constant. Dashed lines represent the time in which the first roll-off was achieved in each case. The maximum difference was 6% between cases 1 and 4 at ≈220 s. This difference was only 3.5% at 6 minutes. The same comparison was made for cases 5 to 8 (not shown) and differences in the lesion short di ...
... constant. Dashed lines represent the time in which the first roll-off was achieved in each case. The maximum difference was 6% between cases 1 and 4 at ≈220 s. This difference was only 3.5% at 6 minutes. The same comparison was made for cases 5 to 8 (not shown) and differences in the lesion short di ...
High Cognitive Demand Tasks
... • Ask guiding questions instead of leading questions (or providing answers). – Be less helpful! ...
... • Ask guiding questions instead of leading questions (or providing answers). – Be less helpful! ...
PDF
... (d) Entries that include too many unexplained concepts, or too many terms that are undefined elsewhere on PlanetMath is not publishable. The said entry is deemed publishable only when sufficient background is given. This may mean that additional definitions defined, theorems listed (or proved), etc ...
... (d) Entries that include too many unexplained concepts, or too many terms that are undefined elsewhere on PlanetMath is not publishable. The said entry is deemed publishable only when sufficient background is given. This may mean that additional definitions defined, theorems listed (or proved), etc ...
INforM – Interactive Notebooks for Mathematics
... Keywords: calculations, equivalent, links, mental, mind map, SMART Ideas. Book 8 – Straight line graphs through the origin The materials support the development of the algebraic representation of straight-line graphs in coordinates in the form y = mx + c and the relationship between m and the gradie ...
... Keywords: calculations, equivalent, links, mental, mind map, SMART Ideas. Book 8 – Straight line graphs through the origin The materials support the development of the algebraic representation of straight-line graphs in coordinates in the form y = mx + c and the relationship between m and the gradie ...
Dynamics of dinosaurs Dynamics of dinosaurs Infectious disease
... involvement with biology, just as mathematics has already benefited and will continue to benefit from its historic involvement with physical problems. In classical times, physics, as first an applied then a basic science, stimulated enormous advances in mathematics. … “In the coming century, biology ...
... involvement with biology, just as mathematics has already benefited and will continue to benefit from its historic involvement with physical problems. In classical times, physics, as first an applied then a basic science, stimulated enormous advances in mathematics. … “In the coming century, biology ...
Dynamic Cost Allocation for Economic Lot Sizing Games
... once the allocated costs are collected from the participating retailers, there may be incentives for individual retailers or coalitions to defect later on in the planning horizon if they find themselves in an advantageous position. Finally, a static allocation ignores the typical rolling horizon app ...
... once the allocated costs are collected from the participating retailers, there may be incentives for individual retailers or coalitions to defect later on in the planning horizon if they find themselves in an advantageous position. Finally, a static allocation ignores the typical rolling horizon app ...
Economic Man as Model Man: Ideal Types, Idealization and
... whom economic motivations have a causal power but these do not lead us directly to the laws. This is not a model man for whom we can think of testing the outcomes of his behavior pattern by a counter-factual thought experiment, one in whom we can conceive that the character might be otherwise than i ...
... whom economic motivations have a causal power but these do not lead us directly to the laws. This is not a model man for whom we can think of testing the outcomes of his behavior pattern by a counter-factual thought experiment, one in whom we can conceive that the character might be otherwise than i ...
MatLab - Systems of Differential Equations
... After invoking pplane8 a window appears where you type in your differential equation, including the limits on your state variables. This will generate a new window showing the direction field for the phase portrait. With the mouse you can click at any point and have a solution trajectory be drawn. D ...
... After invoking pplane8 a window appears where you type in your differential equation, including the limits on your state variables. This will generate a new window showing the direction field for the phase portrait. With the mouse you can click at any point and have a solution trajectory be drawn. D ...
The algorithmic and dialectic aspects in proof and proving
... Through some examples we discuss the algorithmic and dialectic aspects in proof and proving, with a pedagogical implication on emphasizing both aspects which complement and supplement each other in this mathematical activity. In 1973 Peter Henrici coined the terms “algorithmic mathematics” and “dial ...
... Through some examples we discuss the algorithmic and dialectic aspects in proof and proving, with a pedagogical implication on emphasizing both aspects which complement and supplement each other in this mathematical activity. In 1973 Peter Henrici coined the terms “algorithmic mathematics” and “dial ...
Sergei Korchevoi - University of Exeter
... 2016a: 2). Bogardus (2012) even argues that we need to abandon the safety condition because of a risk that any knowledge is under threat: something could happen but didn't, therefore one's true belief is unsafe. I assume that the world where 'something could happen but didn't' and a world where 'som ...
... 2016a: 2). Bogardus (2012) even argues that we need to abandon the safety condition because of a risk that any knowledge is under threat: something could happen but didn't, therefore one's true belief is unsafe. I assume that the world where 'something could happen but didn't' and a world where 'som ...
Mathematics - Hanover College
... properties and applications of vectors, matrices, and linear transformations. Prerequisite: 121 or permission. Mat 231. Differential Equations. Survey of basic techniques for describing dynamical systems by means of equations involving derivatives of functions, and of methods for finding functions w ...
... properties and applications of vectors, matrices, and linear transformations. Prerequisite: 121 or permission. Mat 231. Differential Equations. Survey of basic techniques for describing dynamical systems by means of equations involving derivatives of functions, and of methods for finding functions w ...
Mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, the applied methods refer to those beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. An advantage claimed for the approach is its allowing formulation of theoretical relationships with rigor, generality, and simplicity.It is argued that mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics. Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.Broad applications include: optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing comparative statics as to a change from one equilibrium to another induced by a change in one or more factors dynamic analysis, tracing changes in an economic system over time, for example from economic growth.Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics.This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.