LECTURE NOTES ON MATHEMATICAL INDUCTION PETE L. CLARK Contents 1. Introduction 2. The (Pedagogically) First Induction Proof 3. The (Historically) First(?) Induction Proof 4. Closed Form Identities 5. More on Power Sums 6. Inequalities 7. Extending binary properties to n-ary properties 8. Miscellany 9. The Principle of Strong/Complete Induction 10. Solving Homogeneous Linear Recurrences 11. The Well-Ordering Principle 12. Upward-Downward Induction 13. The Fundamental Theorem of Arithmetic
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Partial Fractions A way of "breaking apart" fractions with polynomials in them. What are Partial Fractions? We can do this directly: Like this (read Using Rational Expressions to learn more): 2 + 3 = 2·(x+1) + (x-2)·3 x-2 x+1 (x-2)(x+1) Which can then be simplified to: = 2x+2 + 3x-6 = 5x-4 x2+x-2x-2 x2-x-2 ... but how do we go in the opposite direction? That is what we discover here: How to find the "parts" that make the single fraction (the "partial fractions")
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the size n becomes large (you may use the file splinemat.m)? 6. A parabolic runout spline is the interpolating function you get by changing the condition f ′′ (x1 ) = f ′′ (xn ) = 0 to the condition that p1 (x) and pn−1 (x) should be quadratic polynomials (that is‚ a1 = an−1 = 0). Modify the file splinemat.m so that it computes the matrix relevant to this modified problem. Call the modified file splinematpr.m. (Hand in a a print-out of the modified file and an explanation of your changes.) Use your
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of purchase(A) for each customer 2. Calculated the likelihood of churn(B) after last transaction Both the model was based on a regression frame work where generalized additive model (GAM) was used to identify non linear patterns in the data and polynomial
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Have you ever ridden on a rollercoaster and felt your heart drop as you were going downhill? Have you asked yourself how getting these feelings were possible? The answer is math. You may ask what math has to do with rollercoasters. Math is the reason for everything and anything that has to do with rollercoasters. Without math‚ it would be impossible to even be able to create one. To build a rollercoaster you need to be able to use numbers when talking about the costs‚ taking measurements‚ calculating
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CHAPTER 3 Response surface methodology 3.1 Introduction Response surface methodology (RSM) is a collection of mathematical and statistical techniques for empirical model building. By careful design of experiments‚ the objective is to optimize a response (output variable) which is influenced by several independent variables (input variables). An experiment is a series of tests‚ called runs‚ in which changes are made in the input variables in order to identify the reasons for changes in
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Quality guarantees required l Limited time 2 Approach l l Constraint Based Reasoning – Distributed Constraint Optimization Problem (DCOP) Adopt algorithm – First-ever distributed‚ asynchronous‚ optimal algorithm for DCOP – Efficient‚ polynomial-space l Bounded error approximation – Principled solution-quality/time-to-solution tradeoffs 3 Constraint Representation Why constraints for multiagent systems? l Constraints are natural‚ general‚ simple – Many successful applications
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3.2 CP-ASBE Algorithm In the CP-ABE scheme‚ all the attributes belong to one monolithic set and there is no concept of sets involved. Hence the main challenge in its implementation is to prevent user collusion‚ which means different users combinign their attributes to satisfy a policy. This is acheived in CP-ABE by binding together attribute keys with a random unique nunber assigned to each user. In case of CP-ASBE‚ one must also prevent combining attributes belonging to different sets. This is
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Modeling nature Task 1 Theory construction and Model-building skills (chapter 1-3) Scientific research is a process that is designed to extend our understandings and to determine if they are correct or useful. Science strives to operate according to a more rigorous set of rules. Realism: reality exists independent of any human presence‚ there is an external world comprised of objects that follow a myriad of natural facts and laws. --> mirror world. It is possible to make claims regarding
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quantum-mechanical phenomena‚ such as superposition andentanglement‚ to perform operations on data.[1] Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits (bits)‚ each of which is always in one of two definite states (0 or 1)‚ quantum computation uses qubits (quantum bits)‚ which can be in superpositionsof states. A theoretical model is the quantum Turing machine‚ also known as the universal quantum computer
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