part (a)‚ the proof is similar. Solution: Give 5 points for each parts. Note‚ for both (a) and (b) two possible proofs are possible (see alternate proof in part (b)). It suffices for students to give the solution in terms of either products of binomials or factorials. It is fine for students to prove (b) first and apply it to get (a). (a) Consider how many choices we have for the positions of each of the letters‚ (1) there are (2) there are (3) there are (4) there are (5) there are (6) there
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At age fourteen‚ Blaise started to attend his father’s meetings. While there he met Girard Desargues and at age fifteen came to admire his work. In June of 1639‚ Blaise Pascal came in with a piece of paper holding most of his mind’s geometric theorems‚ including Pascal’s mystic hexagon. That winter‚ Pascal’s family moved once again‚ this time to Rouen‚ France. Etienne Pascal was given a tax collector job for Upper Normandy. Shortly after moving‚ Blaise published Essay on Conic Sections in February
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the field of optics and provided a law on light travel and made wrote a few papers about calculus well before Isaac Newton and Gottfried Leibniz were actually born.Fermat’s most important work was done in the development of modern number theory which was one of his favorite areas 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
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RES/341 Final Examination Study Guide This study guide will prepare you for the Final Examination you will complete in the final week. It contains practice questions‚ which are related to each week’s objectives. In addition‚ refer to each week’s readings and your student guide as study references for the Final Examination. Week One: Defining Research Process and Data Measurement Objective: Define business research and its purpose. 1. The main distinction between applied and basic business
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with the Pythagorean Theorem within trigonometry. However‚ some sources doubt that is was him who constructed the proof (Some attribute it to his students‚ or Baudhayana‚ who lived some 300 years earlier in India). Nonetheless‚ the effect of such‚ as with large portions of fundamental mathematics‚ is commonly felt today‚ with the theorem playing a large part in modern measurements and technological equipment‚ as well as being the base of a large portion of other areas and theorems in mathematics. But
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Eigen values and eigen vectors. Calculus: Functions of single variable‚ Limit‚ continuity and differentiability‚ Mean value theorems‚ Evaluation of definite and improper integrals‚ Partial derivatives‚ Total derivative‚ Maxima and minima‚ Gradient‚ Divergence and Curl‚ Vector identities‚ Directional derivatives‚ Line‚ Surface and Volume integrals‚ Stokes‚ Gauss and Green’s theorems. Differential equations: First order equations (linear and nonlinear)‚ Higher order linear differential equations with
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Uncertainty in Task Duration and Cost Estimates: Fusion of Probabilistic Forecasts and Deterministic Scheduling Downloaded from ascelibrary.org by GEORGE WASHINGTON UNIVERSITY on 04/08/14. Copyright ASCE. For personal use only; all rights reserved. Homayoun Khamooshi‚ Ph.D. 1; and Denis F. Cioffi‚ Ph.D. 2 Abstract: A model for project budgeting and scheduling with uncertainty is developed. The traditional critical-path method (CPM) misleads because there are few‚ if any‚ real-life deterministic
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Invertible matrix 1 Invertible matrix In linear algebra an n-by-n (square) matrix A is called invertible or nonsingular or nondegenerate‚ if there exists an n-by-n matrix B such that where I n denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case‚ then the matrix B is uniquely determined by A and is called the inverse of A‚ denoted by A −1 . It follows from the theory of matrices that if for finite square matrices A and B
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Coursework 2 Mathematical Finance Group 27 Q1. Hedging in Complete and Incomplete market Solution: Complete market Suppose we have m states. A complete market A is one with the marketed subspace Span(A.1‚A.2‚ ⋯‚ A.n) includes all possible payoffs over the m states‚ i.e.‚ if it contains all possible m-dimensional vectors. Incomplete market Suppose we have m states. An incomplete market corresponds to a market with fewer linear independent
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are Plimpton 322 (Babylonian mathematics c. 1900 BC)‚ the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem‚ which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans‚ who coined the term
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