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Evaluate 1. the double integral and y = 0. 1 2 y y R 1 2xy dx dy and R is bounded by y = x, y = 2 x (4y y 2 ) dy = e x 2 y 2 Answer: 0 2xy dx dy = 0 2 3 2. Evaluate the double integral 4}. 2 2 1 R 2 0 dx dy where R = {(x, y) : 1 x2 + y 2 Answer: 0 re r dr d = 2 1 e 4 + e 1 d = (e 1 e 4 ). 2 16 x2 y 2 dx dy where R is inside x2 +y 2 = 9 3. Evaluate the double integral and outside x2 + y 2 = 4 and between y = x and y = x with y 0. 3 3 /4 1 3 /4 (123/2 73/2 ) d = (123/2 73/2 ). Answer: r 16 r 2 dr d = 3 /4 6 2 /4 R 4. Evaluate the double integral (8 + 3x y 2 ) dx dy where R is bounded by R y = 2x 1, y = 2x + 5, y = 1 3x and y = 7 3x. 1 1 Answer: u = y + 3x and v = y 2x implies x = 5 (u v) and y = 5 (2u + 3v). 1 |J| = 5 5 1 3 1 5 7 1 8 + (u v) (2u + 3v)2 du dv = (1104 378v 54v 2 ) dv = 5 1 1 5 25 125 1 36 . 25 5. Compute the volume of the solid bounded by x + 2y + z = 8 and the coordinate planes, x = 0, y = 0, and z = 0. 4 8 2y 4 128 1 . Answer: (8 x 2y) dx dy = 64 32y (5 2y)2 + 4y 2 dy = 2 3 0 0 0 6. Compute the volume of the solid under z = 6 x2 y 2 and inside x2 + y 2 = 1. 2 1 2 11 11 Answer: (6 r 2 )r dr d = . d = 4 2 0 0 0 7. Find the surface area of portion that of the paraboloid z = 1 + x2 + y 2 that lies below the plane z = 5. 2 2 Answer: 1 + 4x2 + 4y 2 dx dy = r 1 + 4r 2 dr d = (173/2 1). 6 S 0 0 8. Determine the surface area of the portion of z = 4y + 3x2 between y = 2x, y = 0 and x = 2. 2 2x 0 Answer: 0 2 17 + 36x2 dy dx = 0 2 2x 17 + 36x2 dx = (161)3/2 (17)3/2 . 3 9. Determine the surface area of the surface de ned by x = u, y = v + 2, z = 2uv for 0 u 2 and 0 v 1. Answer: | 1, 0, 2v 0, 1, 2u | = 1 + 4u2 + 4v 2 . (Put integration on a calculator.) 1 int2 1 + 4u2 + 4v 2 du dv = 5.2335. 0 0 10. Evaluate the triple integral 0 2x+4 4+2x y by 2x + y + z = 4 and the coordinate planes, x = 0, y = 0 and z = 0. Answer: 0 2 0 Q 6xz 2 dV where Q is the tetrahedron bounded 2xz 3 dz dy dx = 2x+4 2 0 0 (128x + 192x2 96xy + 96x3 96x2 y + 24xy 2 + 16x4 24x3 y + 12x2 y 2 0 2 2xy 3 ) dy dx = (8x5 + 64x4 + 192x3 + 256x2 + 128x) dx = Q 256 . 15 11. Evaluate the triple integral and z = 4. 2 Answer: 0 0 2 4 r2 (x y) dV where Q is bounded by z = x2 + y 2 (r 2 cos r 2 sin ) dz dr d = 0. 12. Convert to polar coordinates and evaluate the integral 2 0 /2 /2 0 2 4 x2 4 x2 /2 /2 x2 dx dy. Answer: r 3 cos2 dr d = (2 + 2 cos (2 )) d = 2 .
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Pittsburgh >> M >> 0250 (Fall, 2009)
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Pittsburgh >> M >> 25 (Fall, 2009)
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Pittsburgh >> M >> 0250 (Fall, 2009)
5 (15/(4*pi*2+9)*(3*cos(2*pi*t)+sin(2*pi*t) 5*cos(2*pi*t) 4 3 2 1 0 -1 -2 -3 -4 -5 0 0.2 0.4 0.6 0.8 1 1 line 1 line 2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 ...
Pittsburgh >> M >> 25 (Fall, 2009)
5 (15/(4*pi*2+9)*(3*cos(2*pi*t)+sin(2*pi*t) 5*cos(2*pi*t) 4 3 2 1 0 -1 -2 -3 -4 -5 0 0.2 0.4 0.6 0.8 1 1 line 1 line 2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 ...
Pittsburgh >> M >> 0250 (Fall, 2009)
s x s d i u v q P s d ~jo)~}otj)\'`o)~}obd i h h h x s v tq h h `tbPB x u v ry)sc u v q i hw d h h m h i Pjlh8zj\'ojl}ot)yl`~\' s B~)ojl}ot)h s x tri s q i h h ...
Pittsburgh >> M >> 25 (Fall, 2009)
s x s d i u v q P s d ~jo)~}otj)\'`o)~}obd i h h h x s v tq h h `tbPB x u v ry)sc u v q i hw d h h m h i Pjlh8zj\'ojl}ot)yl`~\' s B~)ojl}ot)h s x tri s q i h h ...
Pittsburgh >> MATH >> 1270 (Fall, 2008)
Series solutions 1 Regular series We will study the solutions to certain second order linear differential equations. We will start with equations that are analytic at t = 0 and then turn to those that have singularities. Recall that a function is ...
Pittsburgh >> MATH >> 1470 (Fall, 2008)
Answer to First Midterm for Math 1470 October 1, 2008 1. Answer the following questions: (a) (6 points) Is the PDE uxy + sin (x + y) u + x3 = 0 linear or nonlinear? Why? Solution: It is linear and nonhomogeneous. The corresponding homogeneous equatio...
Pittsburgh >> MATH >> 0230 (Fall, 2008)
1. Evaluate the given integral (a) 3xe-x dx Answer: 3 2 - e-x + c 2 2 (b) 3 x ln x dx Answer: 4 2x3/2 ln x - x3/2 + c 3 (c) x2 x+5 dx +x-2 - ln |x + 2| + 2 ln |x - 1| + c Answer: (d) x sin (x) dx Answer: x 1 - cos (x) + 2 sin (x) + c (e) ...
Pittsburgh >> MATH >> 1470 (Fall, 2008)
Answer to Sample Problems for Second Midterm, Math 1470 November 12, 2008 1. Find the Fourier sine series of f (x) = 1 on [0, ]. Answer: Bn = 2 1 sin nxdx 0 2 cos nx = n 0 n 2 (1 (1) ) = n 0 if n is even, = 4 if n is odd. n k=1 Hence 1 n o...
Pittsburgh >> MATH >> 0220 (Fall, 2008)
HW-3 Name 1. Determine the given limit: x2 3x + 2 lim x2 x2 |x2 3x + 2| lim x2 x2 lim x+22 x2 x2 25x 1 lim x0 x ex 1 lim x0 x eh+2 (2h 1) lim h0 h x1 lim ln x cos (x h0 lim 4 1+h 4 h h0 lim (3 + h)2 9 h mg 2. The velocity, v, ...
Pittsburgh >> MATH >> 0220 (Fall, 2008)
Dierentiate the following 1. f (x) = 4x2 2x + 1 + 2. y = x2 (x + 2 sin 3x)3 3. g(x) = 5 2x + 2 x5 4. y(x) = x + cos x 3x tan (2x) 1 3 3 4+ x x 2 x 5. f (x) = ln (x2 + 1) 6. y(x) = x2 ln (x + 2 sin 3x) 7. y = 1 x2 8. y = 2e0.5x sin (0.25x) 9....
Pittsburgh >> SOC >> 0150 (Fall, 2008)
Notes for Thursday, January 19th Soc 0150: Social Theory w/ Dr. Lisa Brush (Notes by Sam Goldberg with some supplementary material from Dr. Brush) *Dr. Brush acknowledged our anxiety over the text and future exams: She suggested grappling with the t...
BU >> GG >> 101 (Fall, 2009)
Natural Environments: The Atmosphere GG 101 Spring 2004 Boston University Myneni L31: Projections of Future Climate Change Apr-12-04 (1 of 13) Further Reading: Detailed Notes Posted on Class Web Sites Outline - global climate models - global me...
BU >> GG >> 101 (Fall, 2009)
Natural Environments: The Atmosphere GG 101 Spring 2004 Boston University Myneni Lecture 11:Temperature-Regimes-02 Feb-06-04 (1 of 14) Further Reading: Chapter 05 of the text book Outline - continental vs. marine regimes - temperature structure...
BU >> CS >> 320 (Fall, 2009)
Modularizing Programs On Lists With Higher-Order Functions Modularity is an essential ingredient of successful software design. 1 All programming languages include features that help programmers to modularize their programs. A modular program solves ...
BU >> CS >> 320 (Fall, 2009)
3 Modularizing Programs On Trees With Higher-Order Functions The discussion in this chapter continues that in Chapter 2 and presupposes a careful reading of the earlier one. We explore the benets of modularizing programs on user-dened datatypes, now ...
BU >> CS >> 320 (Fall, 2009)
CS 320 Appendix 04 3 Sept 2008 Indentation, Naming Conventions, and Coding Conventions Reading For layout and identation conventions in general, read [HCFP, pp 47-48]. For naming conventions for identiers program (or value) variables, type vari...
BU >> CS >> 320 (Fall, 2009)
5 Modularizing Programs With Lazy Evaluation Lazy evaluation means that an argument to a function will be evaluated only if that arguments value is needed by the function in order to continue its computation. Extended to structured arguments, such as...
BU >> CS >> 320 (Fall, 2009)
Computer Science 320 (Fall, 2008) Concepts of Programming Languages Problem Set 8: Compilation and Monads Out: Tuesday, December 2, 2008 Due: Friday, December 12, 2008 In this assignment, you will write a compiler for a subset of Mini-Haskell. You ...
BU >> CS >> 320 (Fall, 2009)
Computer Science 320 (Fall Term, 2007) Concepts of Programming Languages Solutions for End-of-Term Examination Monday, 17 December 2007 Problem 1. (Several Multiple-Choice Questions) For each question, there may be more than one correct answer. For ...
BU >> CS >> 320 (Fall, 2009)
C OMPUTER S CIENCE 320 (FALL T ERM , 2005) C ONCEPTS OF P ROGRAMMING L ANGUAGES Solutions for Mid-Term Examination T HURSDAY, O CTOBER 20, 2005 Problem 1. Higher-Order Functions. Part (a) [4 points] Dene a function curry that takes as input a func...
BU >> CS >> 320 (Fall, 2009)
C OMPUTER S CIENCE 320 (FALL T ERM , 2005) C ONCEPTS OF P ROGRAMMING L ANGUAGES Problem Set 6: A Small Interpreter (Part II) O UT: T HURSDAY, O CTOBER 13, 2005 D UE : 11:59 PM ON T HURSDAY, O CTOBER 20, 2005 There are 5 problems in this set, each w...
BU >> CS >> 320 (Fall, 2009)
Computer Science 320 (Fall Term, 2005) Concepts of Programming Languages Problem Set 9: Mutable Storage in SML Out: November 10, 2005 Due: 11:59 PM on Thursday, November 17, 2005 There are 5 problems in this set, each worth as indicated, for a total...
BU >> CS >> 320 (Fall, 2009)
Computer Science 320 (Spring Term, 2005) Concepts of Programming Languages Problem Set 10: A Type System for Mini-ML Out: Thursday, November 17, 2005 Due: 11:59 PM on Tuesday,November 29, 2005 There are 5 problems in this set, each worth as indicate...
BU >> CS >> 320 (Fall, 2009)
Computer Science 320 (Fall Term, 2005) Concepts of Programming Languages Problem Set 12: Continuations Out: Tuesday, December 6, 2005 Due: 11:59 PM on Tuesday, December 13, 2005 There are 4 problems in this set, each worth as indicated, for a total ...
BU >> CS >> 320 (Fall, 2009)
Computer Science 320 (Fall Term, 2005) Concepts of Programming Languages Problem Set 8: Datatypes and Structures Out: Thursday, November 3, 2005 Due: 11:59 PM on Thursday, November 10, 2005 There are 6 problems in this set, each worth as marked, for...
BU >> CS >> 320 (Fall, 2009)
CS320 Handout 12 Side Effects, Aliasing, Managing Share State Assaf Kfoury 26 September 2005 Typical Scheme programs are mostly written in a pure functional style. But imperative features have their place in programming, as many computational tasks a...
BU >> DCC >> 2 (Fall, 2009)
2001Q2 Attachment #1 VIRMED Health Insurance Company IMPLEMENTATION OF THE GRANT PROGRAM Intermediate Technical Report Development and Implementation of a New GP Payment Model in MHI Environment in Tula Region. Development of Technical Guidelines...
BU >> BI >> 505 (Fall, 2009)
BI505 Lecture2 Recapitulationary theories Prior to the theory of evolution by natural selection, and even after this theory became widely accepted, humanity has tended to measure nature on a single progressive scale. This tendency is a tremendous ha...
BU >> BI >> 505 (Fall, 2009)
EVOLUTION bDepartm...
BU >> BI >> 505 (Fall, 2009)
EVOLUTION & DEVELOPMENT 2:2, 114124 (2000) Adaptation and constraint in the evolution of Drosophila melanogaster wing shape A. S. Gilchrist,a,* R. B. R. Azevedo,b L. Partridge,a and P. OHigginsc a Department of Biology, Galton Laboratory, Universi...
BU >> BI >> 505 (Fall, 2009)
EVOLUTION & DEVELOPMENT 1:2, 9099 (1999) Developmental constraints on the mode of reproduction in the facultatively parthenogenetic cockroach Nauphoeta cinerea Laura S. Corley, Jill R. Blankenship, Allen J. Moore, and Patricia J. Moore* School of B...
BU >> BI >> 505 (Fall, 2009)
EVOLUTION & DEVELOPMENT 2:6, 340347 (2000) Molluscan engrailed expression, serial organization, and shell evolution David K. Jacobs,a* Charles G. Wray,a,1 Cathy J. Wedeen,b Richard Kostriken,b Rob DeSalle,c Joseph L. Staton,a,2 Ruth D. Gates,a and ...
BU >> BI >> 505 (Fall, 2009)
EVOLUTION & DEVELOPMENT 1:2, 7789 (1999) Chelicerate Hox genes and the homology of arthropod segments Arhat Abzhanov, Aleksandar Popadic, and Thomas C. Kaufman* Howard Hughes Medical Institute, Department of Biology, Indiana University, Bloomington...
BU >> BI >> 301 (Fall, 2009)
Caminalcules Phylogeny BI 301 / Laboratory 2 by John R. Finnerty The Caminalcules The Caminalcules are imaginary taxa with known phylogenetic relationships that were developed to test and illustrate principles of phylogenetic analysis. The Caminalcu...
BU >> CUMM >> 005 (Fall, 2009)
Dev Genes Evol (2007) 217:6372 DOI 10.1007/s00427-006-0111-6 SEQUENCE CORNER Rel homology domain-containing transcription factors in the cnidarian Nematostella vectensis James C. Sullivan Thomas D. Gilmore & John R. Finner...
BU >> BI >> 301 (Fall, 2009)
BI 301 Invertebrate Zoology Final Exam December 14, 2001 Name: 1. Metazoan Phylogeny. Draw a phylogeny of the Metazoa including 13 different phyla.List the common name or one representative taxon for each phylum. Indictate the Eumetazoa, the Bilate...
BU >> BI >> 505 (Fall, 2009)
The following presentation was given by Kelly Bonner in BI 505, Evolution and Development, on February 27, 2002. Kelly has graciously allowed me to use her presentation for illustrative purposes. My comments and suggestions are in red. In a couple ...
BU >> BI >> 505 (Fall, 2009)
Drosophila Axial Patterning /Segmentation Gene Cascade MATERNAL EFFECT GENES (e.g., anterior: bicoid caudal) GAP GENES (e.g., hunchback, knirps) PAIR-RULE GENES (even-skipped, fushi-tarazu) SEGEMENT POLARITY GENES (e.g...
BU >> CUMM >> 005 (Fall, 2009)
BOSTON UNIVERSITY FACULTY OF GRADUATE STUDIES PROGRAM OF THE FINAL THESIS SEMINAR FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN BIOLOGY of ADAM MICHAEL REITZEL B.A., Illinois Wesleyan University, 1999 M.S., University of Florida, 2002 Thursday, August 9...
BU >> BI >> 301 (Fall, 2009)
BI 301 Invertebrate Zoology Midterm 2 November 20, 2001 Name: 1. Assume the ancestral mollusc had the body plan pictured below. Fill in the character state matrix for the molluscs on the next page (20 pts). Describe the niches of the major classes o...
BU >> CUMM >> 005 (Fall, 2009)
Nucleic Acids Research, 2006, Vol. 34, Database issue D495D499 doi:10.1093/nar/gkj020 StellaBase: The Nematostella vectensis Genomics Database James C. Sullivan1, Joseph F. Ryan2,3, James A. Watson2, Jeramy Webb1, James C. Mullikin3, Daniel Rokhsar4...
BU >> BI >> 301 (Fall, 2009)
BI 301 Invertebrate Zoology Midterm Exam 1 October 30, 2001 Name: 1. A. Phylogeny On the next page is the phylogeny for the animal kingdom that I presented during lecture 3. Provide the appropriate name for each of the phyla (e.g., Choradata), sub-...
BU >> BI >> 505 (Fall, 2009)
Drosophila Hox Cluster (HOM-C) (inter-homeobox distances: 70-100 kilobases) ANTP-C Lab [cuticle] pb z2 z1 bcd [ama] Dfd Scr ftz Antp BX-C Ubx abdA AbdB eve 1 2 3 4 5 6 7 8 8 9-13 Anterior Spatial Colinearity: A9 Mx Ma T1 T3 A2 A4 A6 A8 A7...
BU >> BI >> 505 (Fall, 2009)
Isolation of Hox Genes by Degenerate PCR Helix 1 Primers: Helix 2 Helix 3/4 RRRKRTAYTRYQLLELEKEFHFNRYLTRRRRIELAHSLNLTERQVKIWFQNRRMKWKKEN Reverse translation Reverse translation + reverse complement Primers: Degenerate Sequences of ELEKEF & WFQNRR...
BU >> BI >> 301 (Fall, 2009)
BI301 Midterm October 25, 2005 Record all answers on the answer sheet. Write your name and BU ID# on the answer sheet. Note: For some of the multiple choice questions, you are instructed to supply all correct answers. For others, you supply only o...
BU >> BI >> 505 (Fall, 2009)
Drosophila Axial Patterning /Segmentation Gene Cascade MATERNAL EFFECT GENES (e.g., anterior: bicoid caudal) GAP GENES (e.g., hunchback, knirps) PAIR-RULE GENES (even-skipped, fushi-tarazu) SEGEMENT POLARITY GENES (e.g...
BU >> CUMM >> 005 (Fall, 2009)
Genome Informatics 17(1): 219229 (2006) 219 A High Percentage of Introns in Human Genes Were Present Early in Animal Evolution: Evidence from the Basal Metazoan Nematostella vectensis James C. Sullivan busully@bu.edu Adam M. Reitzel reitzel@bu.edu...
BU >> BI >> 301 (Fall, 2009)
BI 301 Invertebrate Zoology Final Exam December 15, 2003 Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 TOTAL / 30 points / 39 / 28 / 30 / 25 / 14 / 25 / 191 Name: 1. Metazoan Phylogeny. Draw a phylogeny of the Metaz...
BU >> BI >> 301 (Fall, 2009)
Parazoa Non-Bilateria Metazoa Eumetazoa Bilateria Porifera Coelenterata Platyhelminthes Nematoda Deuterostomia Protostomia DIPLOBLASTIC ACOELOMATE PSEUDOCOEL. EUCOELOMATE COELOM PSEUDOCOELOM MESODERM Classic View (e.g., Hyman, 1940) Par...
BU >> BI >> 505 (Fall, 2009)
BI505 Lecture3 Recapitulation, part II 7. Recapitulation Needs a New Explanation, a Causal Explanation. The fact of evolution forced recapitulationists to recast their theory because it implied that the resemblance of the embryonic stages of higher f...
BU >> BI >> 505 (Fall, 2009)
BI 505 Lecture 18 Developmental Developmental Genetics I Finnerty In the previous lecture, we considered developmental genes in a theoretical context. Specifically, we asked: What properties of a developmental gene define its role in development, a...
BU >> BI >> 505 (Fall, 2009)
BI 505 Lecture 19 Drosophila genes II [Based on Arthur, Chapter 6 & Gilbert Chapter 14] The choice of cell fate in Drosophila is a two-step process: Finnerty 1) Specification: cell fate is still flexible and is can be altered by environmental cues....
BU >> BI >> 505 (Fall, 2009)
BI505 Lecture 1 Pre-evolutionary thinking on development Evolutionary developmental biology is one of the newest sub-disciplines of biology. Twenty years ago, there was almost no one studying how development evolves. Slide: Van Valen Quote Developmen...
BU >> BI >> 505 (Fall, 2009)
BI 505 Evolution and Development Midterm exam. February 23, 2001 The exam comprises 8 questions. Please take the time to verify that your copy of the exam has all 8 questions. You will have 1 hour. Promptly at 3:00 p.m., I must collect all exams. The...
BU >> BI >> 505 (Fall, 2009)
BI 505 Lecture 30 Developmental Evolution within Populations [Primary source: Arthur, Origin of Animal Body Plans, chapters 8 & 9] Finnerty Most of the studies we have discussed in the last 2 months come from the field of developmental genetics. Th...
BU >> BI >> 505 (Fall, 2009)
JOURNAL OF EXPERIMENTAL ZOOLOGY (MOL DEV EVOL) 285:1926 (1999) Why Do Almost All Mammals Have Seven Cervical Vertebrae? Developmental Constraints, Hox Genes, and Cancer FRIETSON GALIS* Institute for Evolutionary and Ecological Sciences, University o...
BU >> BI >> 505 (Fall, 2009)
BI 505 Lecture 9 Micro- versus Macromutations Today, we begin to lay the foundation for addressing a question that is one of the most important questions addressed by the field of Evolutionary Developmental biology. Is Macroevolution the sum of many ...
BU >> BI >> 505 (Fall, 2009)
BI 505 Lecture 34 Developmental Plasticity 2 Slide: Genetic assimilation in ostrich callosities Finnerty What does this example suggest about the genetic architecture underlying this kind of change? It seems as if there was an original cue for skin...
BU >> BI >> 505 (Fall, 2009)
http:/genomebiology.com/2000/2/1/research/0001.1 Research Expression profiles during honeybee caste determination Jay D Evans* and Diana E Wheeler Addresses: *Bee Research Lab, USDA-ARS, Beltsville, MD 20705, USA. VDepartment of Entomology, Univers...
BU >> BI >> 505 (Fall, 2009)
BI 505 Lecture 26 Evolution of Paired Appendages 3 Finnerty [based on Gilbert, Developmental Biology, 5e, chapter 18 and Gerhardt Evolution, chapter 10] In the past two lectures, we have covered the developmental mecha...
BU >> BI >> 505 (Fall, 2009)
Crucible of Creation, Simon Conway Morris,1998 Crucible of Creation, Simon Conway Morris,1998 Crucible of Creation, Simon Conway Morris,1998 Finnerty Figure Y. Lower Cambrian Trace Fossils (Possible Passive Flow Network) McMenamin & McMenamin, 1...
BU >> BI >> 505 (Fall, 2009)
EvoDevo Glossary Format of text ALL CAPS lower case Blue Source Gould (1977) Ontogeny and Phylogeny Ridley (1996) Evolution, 2d edition. Carroll et al., (2001) From DNA to Diversity ACCELERATION A speeding up of development in ontogeny (relative to ...
BU >> BI >> 505 (Fall, 2009)
Proc. Natl. Acad. Sci. USA Vol. 95, pp. 23552360, March 1998 Evolution Modification of expression and cis-regulation of Hoxc8 in the evolution of diverged axial morphology HEINZ-GEORG BELTING*, COODUVALLI S. SHASHIKANT*, Contributed by Frank H. Rudd...
BU >> EC >> 381 (Fall, 2009)
EC381/MN308 Probability and Some Statistics Yannis Paschalidis yannisp@bu.edu, http:/ionia.bu.edu/ Lecture 17 - Outline 1. Estimation theory a. ML and MAP estimation b. MMSE estimation c. LLSE estimation Dept. of Manufacturing Engineering Dept. of ...
BU >> EC >> 381 (Fall, 2009)
EC381/MN308 Probability and Some Statistics Yannis Paschalidis yannisp@bu.edu, http:/ionia.bu.edu/ Lecture 13 - Outline 1. Pairs of Random Variables Jointly Gaussian random variables Covariance matrices Dept. of Manufacturing Engineering Dept. of...
BU >> EC >> 381 (Fall, 2009)
1/28/2008 EC381/MN308 Probability and Some Statistics Yannis Paschalidis yannisp@bu.edu, http:/ionia.bu.edu/ 1. Set theory. Lecture 2 - Outline 2. Three key concepts: (, F, ( F P) = (S E P) (S, E, = (Sample space, Event space, Probability measure)...
Pittsburgh >> LTL >> 13 (Fall, 2009)
7 Representation and Basic Transformations This chapter presents techniques for representing points, lines and objects in graphical systems. The objective of the representation scheme is to allow the image created from the points, lines and objects t...
Pittsburgh >> LTL >> 13 (Fall, 2009)
4 Low Resolution Graphics Although the trend in computer graphics systems has been toward higher and higher levels of resolution, there exist many applications areas where low resolution graphics provide a satisfactory and effective low cost display ...
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