Please answer the following questions in your blue book, one question per blue book. No credit will be awarded for answers without explanations. Calculators are allowed, but no points will be taken o¤ if you are unable to simplify a mathematical expression. Point totals are provided next to each question; the suggested number of minutes to spend on each question roughly corresponds to the point...

Please answer your questions in the order specified. Write legibly, and use proper sentences; an unreadable answer is a wrong answer. . . Answer questions concisely, but with sufficient detail and precision. Always explain your answers. Please leave space in the margin for correction marks. Use a separate piece of scratch paper for draft answers, private computations or remarks. You are not all...

Over recent years, computers have been making their way into the classroom and lecture theatre. Overhead projectors, blackboards, and whiteboards have been joined or even displaced by smartboards and computer based multimedia presentations. Students with laptops are a common sight, and courses have their lecture notes on the web. Students are taking courses in programming, web-site design, comp...

Basic constructions: The completeness and compactness theorems. The Löwenheim-Skolem theorem. Omitting types. Ultraproducts. Indiscernible sequences. Nice properties: Saturation, homogeneity, and the monster model. Elimination of imaginaries and Meq. Quantifier elimination. Model completeness. Countably categorical theories. Stable theories: Characterizations of a stable formula. Stable, supers...

Measure spaces, measurable functions and positive measures, integration, convergence*. Examples. 1) fn(x) = n−1χ(0,n)(x), 2) fn(x) = χ(n,n+1)(x), 3) fn(x) = nχ[0,1/n](x), 4) L1 convergent but oscillating. Definition. pointwise convergence, uniform convergence, and convergence in Lp . Definition. We say that { fn} of measurable complex-valued functions on (X ,M ,μ) is Cauchy in measure if for ev...

Richard Neville: Over the lecture period [semester 1] three laboratory components was undertaken. The results from my labs [lab1RN and Lab3RN] showed the students had a moderately good understanding of a subset of concepts presented during the lectures [prior to these laboratories]. The exam results enforced this view – with a good spread of results. The exam results show the module’s learning ...

a) What are the possible values for the rank of A? Why? Solution By the Rank-Nullity Theorem 0 ≤ rank (A) ≤ min{6, 4} = 4. b) What are the possible values for the dimension of the kernel of A? Why? Solution Since dim(image(A)) ≤ 4, by the Rank-Nullity Theorem 2 ≤ dim ker(A) ≤ 6. c) Suppose the rank of A is as large as possible. What is the dimension of ker(A)? Explain. Solution Since then dim(i...

background: score worksheets have been extracted for phd candidates in epidemiology, microbiology and parasitology for the academic year of 1386-87. the main objective in this study is the impact and importance of different subjects on the discrimination results of admission. methods: extracted scores from worksheets are analyzed by minitab15 statistical toolbox. the regression values of differ...

BACKGROUND Analysis of aberrant hypermethylation in stool DNA might provide a novel strategy for noninvasive detection of colorectal cancer. AIMS To explore the feasibility of detecting hypermethylation in Spastic paraplegia-20 promoter as a stool-based DNA marker for detection of colorectal cancer. METHODS We collected 96 tissue and stool samples from patients with colorectal cancer and 30...