Introduction to Formal Methods Chapter 04: CTL Model Checking Roberto Sebastiani DISI‚ Università di Trento‚ Italy – rseba@disi.unitn.it URL: http://disi.unitn.it/~rseba/DIDATTICA/fm2012/ teaching assistant: Silvia Tomasi – silvia.tomasi@disi.unitn.it CDLM in Informatica‚ academic year 2011-2012 last update: April 26‚ 2012 Copyright notice: some material (text‚ figures) displayed in these slides is courtesy of M. Benerecetti‚ A. Cimatti‚ P. Pandya‚ M. Pistore‚ M. Roveri‚ and S.Tonetta‚ who detain
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in math. He is best remembered for his number theory‚ in particular for Fermat’s Last Theorem. This theorem states that: xn + yn = zn has no non-zero integer solutions for x‚ y and z when n is greater than 2. Fermat almost certainly wrote the marginal note around 1630‚ when he first studied Diophantus’s Arithmetic. It may well be that Fermat realized that his prove was wrong‚ however‚ since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians
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Mathematicians have long been intrigued by Pierre Fermat’s famous assertion that Ax + Bx = Cx is impossible (as stipulated) and the remark written in the margin of his book that he had a demonstration or "proof". This became known as Fermat’s Last Theorem (FLT) despite the lack of a proof. Andrew Wiles proved the relationship in 1994‚ though everyone agrees that Fermat’s proof could not possibly have been the proof discovered by Wiles. Number theorists remain divided when speculating over whether or
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ordinates of curved lines‚ which is analogous to that of the then unknown differential calculus‚ and his research into number theory. He made notable contributions to analytic geometry‚ probability‚ and optics. He is best known for Fermat ’s Last Theorem‚ which he described in a note at the margin of a copy of Diophantus ’ Arithmetica. Pierre de Fermat was the most brilliant mathematician of his era and‚ along with Déscartes‚ one of the most influential. Although mathematics was just
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polygon with 17 sides could be drawn using just a compass and a straight edge. The first theorem he proved was‚ “The Fundamental Theorem of Algebra.” This theorem says that every algebraic equation has at least one root or solution. Later he published a book called‚ “Disquisitions Arithmetic” which he published in 1801. It showed the study of the number theory and also provided proof for‚ “The Fundamental Theorem of Algebra.” Also he calculated the orbit of asteroid Ceres‚ and was able to predict its
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of symmetry‚ the centroid will lie somewhere along the line of symmetry. Perpendicular Axis Theorem • The moment of inertia (MI) of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis. • That means the Moment of Inertia Iz = Ix+Iy Parallel Axis Theorem • The moment of area of an object about any axis parallel to the centroidal axis is the sum of
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205 How could you use Descartes’ rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial as well as find the number of possible positive and negative real roots to a polynomial? | Descartes rule is really helpful because it eliminates the long list of possible rational roots and you can tell how many positives or negatives roots you will have. Fundamental Theorem of Algebra finds the maximum number of zeros which includes real and complex numbers
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didn’t leave much time for his love of math. Since math was just his hobby‚ he never wanted any of his work to be published. When he did publish his work‚ it was always anonymously. Fermat would state theorems‚ but always neglected the proofs. For example‚ his most famous work‚ ‘Fermat’s Last Theorem‚’ didn’t include a proof until when Andrew J. Wiles provided the first in 1993. He made many contributions in the field of mathematics. For example‚ he is considered as one of the ‘fathers’ of analytic
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CALCULUS Calculus is the study of change which focuses on limits‚ functions‚ derivaties‚ integrals‚ and infinite series. There are two main branches of calculus: differential calculus and integral calculus‚ which are connected by the fundamental theorem of calculus. It was discovered by two different men in the seventeenth century. Gottfried Wilhelm Leibniz – a self taught German mathematician – and Isaac Newton - an English scientist - both developed calculus in the 1680s. Calculus is used in a
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MODIGLIANI-MILLER THEOREM Courtney A. Hopley February‚ 2003 Economics Major Despite the fact that the Tax Reform Act of 1986 was the most sweeping tax reform effort in recent US history‚ critics are concerned that the act could have worsened the distortion of corporate financing decisions by failing to address the unequal treatment of debt and equity finance. Two conflicting theories‚ the traditional theory of corporate finance and the Modigliani-Miller Theorem‚ make different predictions
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