...9. Recurrent and Transient States 9.1 Definitions (n) 9.2 Relations between fi and pii 9.3 Limiting Theorems for Generating Functions 9.4 Applications to Markov Chains (n) 9.5 Relations Between fij and pij 9.6 Periodic Processes 9.7 Closed Sets 9.8 Decomposition Theorem 9.9 Remarks on Finite Chains 9.10 Perron-Frobenius Theorem 9.11 Determining Recurrence and Transience when Number of States is Infinite 9.12 Revisiting Statistical Equilibrium 9.13 Appendix. Limit Theorems for Generating Functions 304 9.1 Definitions Define (n) fii = P {Xn = i, X1 = i, . . . , Xn−1 = i|X0 = i} = Probability of first recurrence to i is at the nth step. ∞ (n) fi = fii = fii = Prob. of recurrence to i. n=1 Def. A state i is recurrent if fi = 1. Def. A state i is transient if fi < 1. Define Ti = Time for first visit to i given X0 = 1. This is the same as Time to first visit to i given Xk = i. (Time homogeneous) ∞ mi = E(Ti |X0 = i) = (n) nfii = mean time for recurrence n=1 (n) Note: fii = P {Ti = n|X0 = i} 305 Similarly we can define (n) fij = P {Xn = j, X1 = j, . . . , Xn−1 = j|X0 = i} = Prob. of reaching state j for first time in n steps starting from X0 = i. fij = ∞ n=1 (n) fij = Prob. of ever reaching j starting from i. Consider fii = fi = prob. of ever returning to i. If fi < 1, 1 − fi = prob. of never returning to i. i.e. 1 − fi = P {Ti = ∞|X0 = i} fi = P {Ti < ∞|X0 = i} 306 TH. If N is no. of visits to i|X0 = i ⇒ E(N |X0 = i) = 1/(1 − fi ) Proof: E(N |X0 = i) = E[N |Ti = ∞, X0...
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...A Laboratory Information Management System(LIMS) is computer software that is used in the laboratory for the management of samples, laboratory users, instruments, standards and other laboratory functions such as invoicing, plate management, and work flow automation. A LIMS, a Laboratory Information System (LIS) or Process Development Execution System (PDES) perform similar functions. The primary difference is that LIMS are generally targeted toward environmental, research or commercial analysis, such as pharmaceutical or petrochemical, and LIS are targeted toward the clinical market (hospitals and other clinical labs). PDES normally cover a wider scope of functionalities including, for example, virtual manufacturing techniques, while they usually don't integrated with the laboratory equipment. Today's trend is to move the whole process of information gathering, decision making, calculation, review and release out into the workplace and away from the office. The goal is to create a seamless organization where: Instruments used are integrated in the lab network; receive instructions and worklists from the LIMS and return finished results including raw data back to a central repository where the LIMS can update relevant information to external systems such as a Manufacturing Execution System or Enterprise Resource Planning application. Lab personnel will perform calculations, documentation and review results using online information from connected instruments, reference...
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...Infinity : We say lim f ( x ) = L if we Precise Definition : We say lim f ( x ) = L if x ®a for every e > 0 there is a d > 0 such that whenever 0 < x - a < d then f ( x ) - L < e . “Working” Definition : We say lim f ( x ) = L if we can make f ( x ) as close to L as we want by taking x sufficiently close to a (on either side of a) without letting x = a . Right hand limit : lim+ f ( x ) = L . This has x ®a x ®a can make f ( x ) as close to L as we want by taking x large enough and positive. There is a similar definition for lim f ( x ) = L x ®-¥ x ®¥ except we require x large and negative. Infinite Limit : We say lim f ( x ) = ¥ if we can make f ( x ) arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting x = a . There is a similar definition for lim f ( x ) = -¥ except we make f ( x ) arbitrarily large and negative. x ®a x ®a the same definition as the limit except it requires x > a . Left hand limit : lim- f ( x ) = L . This has the x ®a same definition as the limit except it requires x 0 and sgn ( a ) = -1 if a < 0 . 1. lim e x = ¥ & x®¥ x ®¥ x®- ¥ lim e x = 0 x ®0 - 5. n even : lim x n = ¥ x ®± ¥ 2. lim ln ( x ) = ¥ 3. If r > 0 then lim x ®¥ & lim ln ( x ) = - ¥ 6. n odd : lim x n = ¥ & lim x n = -¥ x ®¥ x ®- ¥ b =0 xr 4. If r > 0 and x r is real for negative x b then lim r = 0 x ®-¥ x 7. n even : lim a x + L + b x + c = sgn ( a ) ¥ n x ®± ¥ 8. n odd : lim a x n + L + b...
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... |bn − b| < b 2 then 3b b < bn < 2 2 2 b2 b so eventually bn > 2 . If b > 0 then eventually > 1 bbn so 2 |bn − b| 1 1 |bn − b| > = | − | > 0. b2 |bbn | bn b Hence by the sandwich theorem an a bn → b . c) 1 bn → 1 b and then if we let an → a then by the product rule an × 1 bn →a× 1 b and hence n+1 Ratio Lemma: If | aan | ≤ l where 0 ≤ l < 1 then an → 0. n+1 Want to prove that if | aan | ≤ l where 0 ≤ l < 1 then 0 < an+1 < ln a1 We will use proof by induction. First for n = 1, |a2 | ≤ l|a1 | which is true by the assumption on the sequence. Now assume that 0 ≤ |an+1 | ≤ ln |a1 |. Then for the induction step since |an+2 | < l|an+1 | then |an+2 | ≤ ln+1 |an+1 |. So since 0 ≤ l < 1, ln → 0 and then by the sandwich rule an → 0 d) By definition lim sup an = lim sup am . n→∞ n→∞ m≥n Fix > 0 then as limn→∞ bn = b > 0 then ∃N s.t. for n ≥ N |bn − b| < . So for n ≥ N sup am bm ≥ sup am (b − ) = (b − ) sup am . m≥n m≥n m≥n Taking the limit as n → ∞ lim sup an bn ≥ (b − ) lim sup an . n→∞ n→∞ Since this is true ∀ > 0 we have that lim sup an bn ≥ b lim sup an . n→∞ n→∞...
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...Pre-2000, ATLAM relied on customized single-user system bought from a vendor where the academy’s desktop computers (PCs) were solely used to enter accounting entries which is very poor state and far behind its other compatriots in Europe and Japan in term of IT infrastructure. However, improvements has been done in Information Technology since it has been privatized by MICT Berhad and ATLAM had overhauled its Information Technology (IT) in year 2000.gain The system had not been functioned to produce financial statement. After knowing the weakness, ATLAM has two alternatives which is ACCPAC and Petra’s group-wide SAP. ATLAM has been asked to upgrade its accounting system with the Petra group-wide SAP system. Zulkifli as ATLAM’s finance manager need to critically assess the risks associated with the decision on whether to or not to implement SAP in ATLAM. Zulkifli has arranged a meeting to discuss the new system upgrade. He had invited Sani (Project Manager), Gopal (User Project Manager), Lim (User Representative) and Kamal (Functional Analyst) to discuss about implementing SAP in ATLAM. As it shown in the case, the first meeting have not come to a mutual agreement on whether to implement SAP in ATLAM as Lim has pointed out that implement ACCPAC is a cheaper option compared to SAP that would be incurred a huge cost for ATLAM as it will becomes a financial constraint and burden to ATLAM. In second meeting, Lim stated that the SAP is impractical in ATLAM as the the company don’t even...
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...denominator equal to 0: (x − 4)2 = 0 =⇒ x = 4. So x = 4 is a potential vertical asymptote. Because of our simplification, this should be a vertical asymptote, but we should look at lim f (x) to be sure. But x→4 216 . Already this is enough to say that x = 4 is a vertical asymptote. If lim f (x) → x→4 0 you want to be extra careful, though, you can look at the left and right limits. It should be noted that x = 0 is not a vertical asymptote, even though as we originally wrote the function, x = 0 is a zero of the denominator. (b) The line y = b is a horizontal asymptote of a function f (x) if lim f (x) = b or x→∞ x→−∞ lim f (x) = b. Technically we should consider both limits, but because it is a well- known fact that horizontal asymptotes of rational functions are completely determined 3x4 + 3x3 − 6x2 by only one of these limits. So we need only look at lim f (x) = lim 3 = x→∞ x→∞ x − 8x2 + 16x 3 6 1 + 3x − 3x2 3x4 3x4 lim = lim Remember that the later (rather nasty 8 16 x→∞ x→∞ x3 x3 1 − x + x2 looking, and even more nasty to encode) fraction disappears because its limit is 1. But 3x4 = lim 3x = ∞. Therefore, f does not have any horizontal asympthen lim x→∞ x→∞ x3 totes. (c) lim f (x) → 0 which is indeterminate. So we simplify f as we did in part a and get x→0 0 3x(x + 2)(x − 1) 0 lim f (x) = lim = = 0. x→0 x→0 (x − 4)2 16 (d) Remember that a function can potentially change sign at its points of discontinuity or its zeroes. We can get a rational functions points of discontinuity...
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...Task 1 Describe the procedure for storing scientific information in a laboratory management system. LIMS (laboratory information management system) is used in a laboratory for the management of samples, standards, instruments etc. The instruments are used in the laboratory network in order to be able to receive worklists from the LIMS so they are able to update any information that is relevant to other systems. An example of where LIMS is used is in industry as it maintains the information in the systems. Overall, LIMS have some special functions for example the registration of a sample, the storage of data and the equipment that is used in the analysis of the sample. The main feature of a LIMS is the sample management which involves tracking them. For instance if a sample container arrives the registration must be completed and the LIMS would track it and record its location. This could consist of printing off barcodes in order to identify the sample containers. LIMS could exchange and record electronic information. Along with all these functions of LIMS, it also manages the communication with your clients and the related documents. Also, all information is centralised. Some of the main types of information that would need to be stored on your lab’s LIMS would be COSHH records (control of substance hazardous to health) to ensure the awareness of health and safety, waste disposal to show how much waste is disposed and how much is actually produced, security because different...
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...ELEMENTARY MATHEMATICS III ICT MATHEMATICS TEAM UNIVERSITY OF JOS, NIGERIA October 13, 2009 ! Module 1: Functions 1 1.1 Objectives: In this module: We introduce with examples, the concept of a function and study some classes of functions. We discuss methods of nding the domain, range, zeros and singularities of some real-valued functions. We look at injective (one to one) and surjective (onto) functions and end the module by exploring the inverse of a function and the composition of functions. We stimulate students through classroom and other interactive activities to understand the concept of a function and then discover the di erences and similarities between the several types of functions introduced. 2 1.2 Learning Outcomes: At the end of this module students should be able to De ne a function; identify di erent types of functions and Identify relations which are not functions; Identify di erent classes of functions, nd the domain and range that makes a rule f (say), a function; Investigate the injectiveness, surjectiveness and the inverse of di erent functions; Establish the composition of two or more functions. 3 1.3 Learning Activities: Students should: 1 Explore notes and exercises, individually and in groups; Explore and use related materials on the Intranet, especially the e-granary and MIT open course ware; Explore and use related materials on the OLI Calculus course on the Internet(http://www.cmu.edu/OLI/courses/); Solve relevant questions...
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...laboratory information management system * M4:Explain the processes involved in storing information in a scientific workplace * D3: Discuss the advantages gained by keeping data and records on a laboratory management information system * Grading Criteria * P4:Describe the procedure for storing scientific information in a laboratory information management system * M4:Explain the processes involved in storing information in a scientific workplace * D3: Discuss the advantages gained by keeping data and records on a laboratory management information system * How Do I Do It? 1. For P4, learners must describe the procedures for storing scientific information in a laboratory information management system (LIMS). A prepared list of scientific data is provided below. Learners must decide which sets of information could be stored on a workplace record system. 2. For M4, learners must explain how scientific data and records are stored....
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...laboratory information management system * M4:Explain the processes involved in storing information in a scientific workplace * D3: Discuss the advantages gained by keeping data and records on a laboratory management information system * Grading Criteria * P4:Describe the procedure for storing scientific information in a laboratory information management system * M4:Explain the processes involved in storing information in a scientific workplace * D3: Discuss the advantages gained by keeping data and records on a laboratory management information system * How Do I Do It? 1. For P4, learners must describe the procedures for storing scientific information in a laboratory information management system (LIMS). A prepared list of scientific data is provided below. Learners must decide which sets of information could be stored on a workplace record system. 2. For M4, learners must explain how scientific data and records are stored....
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...Student Solutions Manual for SINGLE VARIABLE CALCULUS rS al e SEVENTH EDITION DANIEL ANDERSON University of Iowa Fo JEFFERY A. COLE Anoka-Ramsey Community College N ot DANIEL DRUCKER Wayne State University Australia . Brazil . Japan . Korea . Mexico . Singapore . Spain . United Kingdom . United States ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher except as may be permitted by the license terms below. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, ISBN-13: 978-0-8400-4949-0 ISBN-10: 0-8400-4949-8 Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd. e © 2012 Brooks/Cole, Cengage Learning ...
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...laboratory information management system * M4:Explain the processes involved in storing information in a scientific workplace * D3: Discuss the advantages gained by keeping data and records on a laboratory management information system * Grading Criteria * P4:Describe the procedure for storing scientific information in a laboratory information management system * M4:Explain the processes involved in storing information in a scientific workplace * D3: Discuss the advantages gained by keeping data and records on a laboratory management information system * How Do I Do It? 1. For P4, learners must describe the procedures for storing scientific information in a laboratory information management system (LIMS). A prepared list of scientific data is provided below. Learners must decide which sets of information could be stored on a workplace record system. 2. For M4, learners must explain how scientific data and records are stored....
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...ALGEBRAIC FORMULAE: 1.(a + b)2 = a2 + 2ab+ b2 . 2. (a - b)2 = a2 - 2ab+ b2 . 3.(a + b)3 = a3 + b3 + 3ab(a + b). 4. (a - b)3 = a3 - b3 - 3ab(a - b). 2 2 2 2 5.(a + b + c) = a + b + c +2ab+2bc +2ca. 6.(a + b + c)3 = a3 + b3 + c3+3a2b+3a2c + 3b2c +3b2a +3c2a +3c2a+6abc. 7.a2 - b2 = (a + b)(a – b ) . 8.a3 – b3 = (a – b) (a2 + ab + b2 ). 9.a3 + b3 = (a + b) (a2 - ab + b2 ). 10.(a + b)2 + (a - b)2 = 4ab. 11.(a + b)2 - (a - b)2 = 2(a2 + b2 ). 12.If a + b +c =0, then a3 + b3 + c3 = 3 abc . INDICES AND SURDS m n mn (ab)m = a m b m am 1. am an = am + n 2. = a m − n . 3. (a ) = a . 4. . an m am −m = 1 a 5. = . 6. a 0 = 1, a ≠ 0 . 7. a . 8. a x = a y ⇒ x = y m am b b 9. a x = b x ⇒ a = b 10. a ± 2 b = x ± y , where x + y = a and xy = b. S B SATHYANARAYANA M. Sc., M.I.E ., M Phil . 9481477536 2 LOGARITHMS a x = m ⇒ log m = x (a > 0 and a ≠ 1) a 1. loga mn = logm + logn. m 2. loga = logm – logn. n 3. loga mn = n logm. log a 4. logba = . log b 5. logaa = 1. 6. loga1 = 0. 1 7. logba = . log a b 8. loga1= 0. 9. log (m +n) ≠ logm +logn. 10. e logx = x. 11. logaax = x. PROGRESSIONS ARITHMETIC PROGRESSION a, a + d,...
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... 1.2. Functions 5 1.3. Composition and inverses of functions 7 1.4. Indexed sets 8 1.5. Relations 11 1.6. Countable and uncountable sets 14 Chapter 2. Numbers 21 2.1. Integers 22 2.2. Rational numbers 23 2.3. Real numbers: algebraic properties 25 2.4. Real numbers: ordering properties 26 2.5. The supremum and infimum 27 2.6. Real numbers: completeness 29 2.7. Properties of the supremum and infimum 31 Chapter 3. Sequences 35 3.1. The absolute value 35 3.2. Sequences 36 3.3. Convergence and limits 39 3.4. Properties of limits 43 3.5. Monotone sequences 45 3.6. The lim sup and lim inf 48 3.7. Cauchy sequences 54 3.8. Subsequences 55 iii iv Contents 3.9. The Bolzano-Weierstrass theorem Chapter 4. Series 4.1. Convergence of series 4.2. The Cauchy condition 4.3. Absolutely convergent series 4.4. The comparison test 4.5. * The Riemann ζ-function 4.6. The ratio and root tests 4.7. Alternating series 4.8. Rearrangements 4.9. The Cauchy product 4.10. * Double series...
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...The good Deed: The lili’s Dream. Narrator: One day, there was an old lady living in china, Her name was Old Mrs. Pan, and unfortunately, her son, Billy pan, brought her to new York because she is too old to live alone in their house and he is worried because there has a Local bullies in there. But when mrs. pan is in the new York, she felt uncomfortable and felt useless, A lot of things are different there in New York, the water are taste like metal while in china, the water taste naturally earth, the food are not fresh unlike in china, and there has a machine that there has no in china, she didn’t know what it is and how to use it. So she decided to die that to live like that, she never eat and did not want to outside of her room, she never bother to look at the window because there has a machine like car. Sophia, her son’s wife worried because of her, so she asked a help to her social worker friend who are Chinese too. Sophia: Mother, this is my social worker friend, her name was Miss lili yang, she came to see you. (Mrs.Pan is about to stand up but lili stops her) Lili yang: You must not rise to one suchyounger. Old mrs pan: You speak so much good in Chinese! Lili yang: My parents taught me. Old mrs pan: Have you ever been in our Village? Lili yang: That’s sorrow, I never been there, im here to listen to you to tell me. Sophia: excuse me, I must prepare the dinner. Old mrs pan: She is always busy. Anyway, I was seventeen when I...
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