Instructions: I expect you to work on the problems by yourself. You can refer to any textbook (or the technical literature in general), but not confer with any person. 1. Short Questions (a) How many bits are needed to specify a selection of k objects from n objects? (n and k are assumed to be known and the selection of k objects is unordered.) (b) Either prove that the following code is unique...

Introduction 1 1. Continuous functions 2 2. Measures and σ­algebras 10 3. Measureability of functions 16 4. Integration 19 5. Hilbert space 30 6. Test functions 34 7. Tempered distributions 42 8. Convolution and density 47 9. Fourier inversion 58 10. Sobolev embedding 63 11. Differential operators. 67 12. Cone support and wavefront set 83 13. Homogeneous distributions 96 14. Wave equation 97 15...

Lognormal price dynamics and passage to the continuum limit. After a brief recap of our pricing formula, this section introduces the lognormal model of stock price dynamics, and explains how it can be approximated using binomial trees. Then we use these binomial trees to price contingent claims. The Black-Scholes analysis is obtained in the limit δt → 0. As usual, Baxter–Rennie captures the cen...

Problem 3. True or False, and Justify (12 parts) [84 points] Circle T or F for each of the following statements to indicate whether the statement is true or false, respectively. If the statement is correct, briefly state why. If the statement is wrong, explain why. The more content you provide in your justification, the higher your grade, but be brief. Your justification is worth more points th...

BACKGROUND The widespread use of organophosphorus (OP) pesticides has led to frequent exposure in adults and children. Because such exposure may cause adverse health effects, particularly in children, the sources and patterns of exposure need to be studied further. OBJECTIVES We assessed young urban/suburban children's longitudinal exposure to OP pesticides in the Children's Pesticide Exposur...

chickpea fall planting may increase seed yield in comparison with spring planting due to the increased duration of growing period and increased water use efficiency. the most important problem with fall planting is the danger of occurrence of plant death and decreased seed yield quantitatively and qualitatively due to the cold stress. the major strategy in breeding program is development of gen...

 Background: The phenomenon of dust is a serious environmental problem in dry and semi dry regions. It has a destructive effect for the residents of such regions. Over two-thirds of Iran is in areas of dry and semi-dry climate conditions. Methods: In this research, dust fall measurements were taken in 41 stations across Yazd city using a marble dust collector (MDCO) located in diffe...