# Recurrence. 1 Definitions and main statements

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def. Defnton 1.1 A state S s sad to be recurrent f P {X n = for some n 1 X 0 = } = 1. In words: S s recurrent f the probablty that the MC startng from wll return to s 1. A state s sad to be transent f t s non-recurrent. In other words, s sad to be transent f the probablty that the Markov chan startng from wll never to return to s strctly postve. As usual, we would lke to be able to tell whether or not a MC s recurrent n terms of propertes of transton probabltes of ths chan. Remember that def j = P {X n = j X 0 = } and that these probabltes can be found, at least n prncple, n terms of P, namely ( j ) = Pn. Theorem 1.2 A state S s recurrent f and only f =. (1) Defnton 1.3 We say that the MC X n s recurrent f all states of ths MC are recurrent. Theorem 1.2 wll be proved n the next secton. But we start wth two examples of concrete Markov chans for whch ths theorem allows one to establsh recurrence or transence (whchever s approprate). Example 1. Let X n be a fnte regular MC. Then ths chan s recurrent. Indeed, snce regular chans have the property that lm n k = w > 0, t follows that p(n) j =. A dfferent and, n a sense, much more natural proof of ths statement wll be gven n Secton 3. Example 2. To defne a smple one-dmensonal random walk magne a partcle movng along a one-dmensonal lattce Z = {... 2, 1, 0, 1, 2,...}. The partcle s observed at equdstant tme moments n = 0, 1, 2... Let X n be the coordnate of the partcle at tme n. If at tme n the partcle s at ste k, then at tme n + 1 t wll 1

2 jump to the rght wth probablty p or to the left wth probablty q def = 1 p and the of length of the jump s one; by defnton, each next move of the partcle does not depend on the hstory of ts movements (or, one could say, on ts past postons). It s the clear that the sequence X n forms a MC wth transton probabltes P {X n+1 = k + 1 X n = k} = p, P {X n+1 = k 1 X n = k} = q. We shall suppose that X 0 = 0. Remark. Note that ths MC has an nfnte state space - the set of all nteger numbers. In the context of the theory of random walks, t s common to call ths space state the one-dmensonal lattce. Statement. The state 0 s recurrent f p = q = 1 and s transent f p q. 2 To see ths, note frst that we can compute the probabltes of return to 0 explctly. Namely, the frst and easy remark s that p (2n+1) 00 = 0 because the random walk has to make an even number of steps n order to return to the startng pont. Next, p (2n) 00 = ( 2n n )p n q n (prove ths or see the explanaton gven n the lecture). The Statement wll follow f we show that the seres ( ) 2n j p n q n n converges when p q and dverges when p = q = 0.5. The case p q s smple because the rato test can be appled. Namely, t s easy to check that ( 2n+2 ) p n+1 q n+1 lm n n+1 2(2n + 1)pq ( 2n ) = lm = 4pq < 1 f p q. pn q n n n n + 1 (We use here the nequalty p(1 p) < 1 f p 0.5). Hence the seres converges and 4 the random walk s transent. The rato test does not work when p = q = 0.5 because then 4pq = 1. We shall prove that n ths case the seres dverges usng the famous Strlng s formula: n! 2πnn n e n. You can check that ths formula mples ( ) p (2n) 2n 00 = p n q n = (2n)! n (n!) 2 pn q n (4pq)n. πn When p = q = 0.5 we obtan p (2n) 00 1 πn. The seres thus dverges whch, accordng to the above theorem, mples the recurrence of the random walk. Remark. We say that a n b n f lm n a n bn = 1. 2 Proof of Theorem 1.2 In order to prove the theorem we need to ntroduce the followng random varable: R = the tme of the frst return to. (2) 2

3 The probablty mass functon of R shall be denoted f (n), n 1. Obvously, f (n) = P {R = n X 0 = } = P {X n =, X k for k = 1,..., n 1 X 0 = }. (3) Clearly f (1) p (k) = p. All other probabltes f (n) because of the followng Lemma 2.1 For n 1 can be calculated recursvely terms of = f (1) p (n 1) + f (2) Indeed, f (4) holds then f (n) p (n 2) = (f (1) p (n 1) + f (2) f (n 1) p (1) + f (n) p (n 2) n f (n 1) p (1) ) n 1 p(n) f (k) p (n k). (4) f (k) p (n k) and hence we can calculate f (2) usng f (1), then we can calculate f (3) usng f (1), and so on. f (2) Proof of Lemma 2.1. The event {X n = } can take place only together wth one of mutually exclusve events {R = k}, k = 1, 2,..., n. Hence, by the Total Probablty Law, But P {X n = X 0 = } = = n P {R = k X 0 = }P {X n = R = k, X 0 = } n f (k) P {X n = R = k, X 0 = }. P {X n = R = k, X 0 = } =P {X n = X k =, X l when 1 l k 1, X 0 = } =P {X n = X k = }} = p (n k), where the last equalty follows from the Markov property. The Lemma s proved. Next, set β def = =1 f (n). Obvously, β = P {X n = for at least one n 1 X 0 = } P { MC startng at would return to }. We can now say that the state S s recurrent f and only f β = 1 and Theorem 1.2 can be stated as follows: Theorem 2.2 and = f and only f β = 1 (5) 3

4 or, equvalently, < f and only f β < 1 (6) Proof of Theorem 2.2. Suppose frst that U def = p(n) < ; we shall now prove that then β < 1. To ths end, let us wrte equatons (4) as follows: p (1) =f (1) p (2) =f (1) p (1) + f (2) p (3) =f (1) p (2) + f (2) p (1) + f (3). =f (1) p (n 1) + f (2) p (n 2) f (n 1) p (1) + f (n) (7) Defne U N def = N p(n). Let us add up the frst N equatons n (7). We obtan U N = f (1) (1 + U N 1 ) + f (2) (1 + U N 2 ) f (N 1) (1 + U 1 ) + f (N). (8) Snce lm N U N = U <, we can pass to the lmt n (8) and obtan U = f (1) (1 + U) + f (2) (1 + U) f (n) (1 + U) +... = β (1 + U). (9) Hence β = U 1 + U < 1 (10) (whch means that the random walk s transent). Let us now consder the case U = whch by defnton means that lm N U N =. We shall agan make use of (8). Namely, let us replace all the factors of the form 1 + U n n the rght hand sde of (8) by 1 + U N. Snce 1 + U n 1 + U N for all n N, we obtan: U N f (1) (1 + U N ) + f (2) (1 + U N ) f (N 1) (1 + U N ) + f (N) (1 + U N ) =(f (1) + f (2) f (N) )(1 + U N ) β (1 + U N ), (11) where the last step s due to the fact that β = and therefore β lm N β U N 1 + U N U N 1 + U N = lm N f (n). Hence 1 1 U N + 1 = 1. But β 1 because ths number s probablty of some event. Hence β = 1 whch means that the random walk s recurrent. Theorem 2.2 s proved and thus also Theorem 1.2 s proved.. 4

5 3 Recurrence and communcaton classes Let, as n secton 2, X n be a MC wth a state space S (S may be an nfnte set). Defnton 3.1 We say that two states, j S ntercommuncate f there are k 1 and l 1 such that p (k) j > 0 and p (l) j > 0. In other words the states and j ntercommuncate f the two probabltes, to reach j startng from and to reach startng from j, are strctly postve. We shall rght j nstead of sayng ntercommuncates wth j. By conventon,. Note next that f j and j k then k (explan ths statement!). Defnton 3.2 We say that a subset C S forms a communcaton class f any, j C ntercommuncate wth each other. Each state S belongs to exactly one communcaton class, namely the one formed by all states from S whch ntercommuncate wth (and therefore also ntercommuncate wth each other). Hence the state space can be parttoned nto (non-ntersectng) communcaton classes. If we call them C 1, C 2,... then S = C 1 C 2... C k, wth C C j = f j. Remarks. 1. If S s a fnte set then of course the number of communcaton classes s fnte. If S s an nfnte set the number of communcaton classes can be nfnte too; However, we shall not consder such chans. 2. A communcaton class may conssts of just one state; for nstance, ths happens f s an absorbng state or f cannot be reached from any other state. 3. In the lterature communcaton classes are often called equvalence classes. They ndeed are equvalence classes wth respect to the ntercommuncaton property. Let us frst consder the smplest case, k = 1, and thus S = C 1. Snce all states ntercommuncate wth each other ths smply means that the Markov chan s rreducble. (Remember the defnton of an rreducble Markov chan!) Next, let k = 2, that s S = C 1 C 2. Then there are two further possbltes: (a) No state from from C 2 can be reached from any state from C 1 and vse versa, no state from C 1 can be reached from any state from C 2. (b) No state from C 2 can be reached from any state from C 1, but some sates from C 1 can be reached from some states from C 2 (formally, there s a thrd possblty whch n fact s the same as the second one; what s t?). Exercses. 1. Consder Markov chans wth transton matrces and (12)

6 What are the communcaton classes of these chans? Whch of the two cases (a) and (b) lsted above descrbe the behavor of each chan? 2. Suppose that S = C 1 C 2 C 3. What are the dfferent types of behavour, n terms of communcatons between the classes, whch one may observe? We shall now prove a smple but mportant theorem whch, as wll be seen, descrbes the recurrence propertes of communcatons classes. Theorem 3.3 If and j are ntercommuncatng states and s recurrent then j s recurrent. Proof. Snce and j ntercommuncate, there s a k such that p (k) j such that p (l) j > 0. But then p (k+n+l) jj p (l) j p(n) p (k) j. > 0 and and an l Ths nequalty holds because one of the possbltes to reach j startng from j n k + n + l transtons would be to frst reach n l transtons (wth probablty p (l) then to return to after n transtons (wth probablty ) and fnally to reach j from n k transtons (wth probablty p (k) j ). Hence jj p (k+n+l) jj p (l) j p(k) j because s recurrent and ths means that p(n) = j ), =. Theorem 3.3 s proved. Obvously, Theorem 3.3 mples that both recurrence and transence are propertes of a communcaton class: ether all states n the class are recurrent or all of them are transent. Examples. 1. Consder a fnte rreducble Markov chan. It follows from Theorem 3.3 that t s recurrent. Indeed, snce S t s a fnte set, there must be at least one state whch s vsted by the chan nfntely many tmes as n and hence s recurrent. But ths state ntercommuncates wth all other states. Hence the chan s recurrent. In partcular, every fnte regular Markov chan s recurrent (because regularty mples rreducblty). In example 1, Secton 1, the exstence of a more natural prof of recurrence was mentoned. So, here t s. 2. Consder the one-dmensonal random walk dscussed n Secton 1. Snce 0 ntercommuncates wth each state of the lattce, we conclude that all states are recurrent f p = q = 1 and are transent otherwse. 2 Exercse. For each of the two Marcov chans whose transton matrces are gven by (12) establsh whch states are recurrent and whch are transent? 6

### 1 Example 1: Axis-aligned rectangles

COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

### 8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

### BERNSTEIN POLYNOMIALS

On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

### Formula of Total Probability, Bayes Rule, and Applications

1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

### We are now ready to answer the question: What are the possible cardinalities for finite fields?

Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

### Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

### Support Vector Machines

Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

### Extending Probabilistic Dynamic Epistemic Logic

Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

### Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

### n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

### PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

### Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

### v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

### benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

### PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

### 1 Approximation Algorithms

CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

### Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz

### The OC Curve of Attribute Acceptance Plans

The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

### Solutions to the exam in SF2862, June 2009

Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodc-revew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r

### greatest common divisor

4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

### The eigenvalue derivatives of linear damped systems

Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

### What is Candidate Sampling

What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

### A Probabilistic Theory of Coherence

A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

### 8 Algorithm for Binary Searching in Trees

8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

### Interest Rate Fundamentals

Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

### Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

### Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

### 1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

### NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

### 6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

### Pricing Overage and Underage Penalties for Inventory with Continuous Replenishment and Compound Renewal Demand via Martingale Methods

Prcng Overage and Underage Penaltes for Inventory wth Contnuous Replenshment and Compound Renewal emand va Martngale Methods RAF -Jun-3 - comments welcome, do not cte or dstrbute wthout permsson Junmn

### x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

### New bounds in Balog-Szemerédi-Gowers theorem

New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

### Embedding lattices in the Kleene degrees

F U N D A M E N T A MATHEMATICAE 62 (999) Embeddng lattces n the Kleene degrees by Hsato M u r a k (Nagoya) Abstract. Under ZFC+CH, we prove that some lattces whose cardnaltes do not exceed ℵ can be embedded

### Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

### Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

### An Alternative Way to Measure Private Equity Performance

An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

### QUANTUM MECHANICS, BRAS AND KETS

PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

### 5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)

5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a

### CALL ADMISSION CONTROL IN WIRELESS MULTIMEDIA NETWORKS

CALL ADMISSION CONTROL IN WIRELESS MULTIMEDIA NETWORKS Novella Bartoln 1, Imrch Chlamtac 2 1 Dpartmento d Informatca, Unverstà d Roma La Sapenza, Roma, Italy novella@ds.unroma1.t 2 Center for Advanced

### Using Series to Analyze Financial Situations: Present Value

2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

### Implied (risk neutral) probabilities, betting odds and prediction markets

Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

### Non-degenerate Hilbert Cubes in Random Sets

Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Non-degenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne

### + + + - - This circuit than can be reduced to a planar circuit

MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

### The University of Texas at Austin. Austin, Texas 78712. December 1987. Abstract. programs in which operations of dierent processes mayoverlap.

Atomc Semantcs of Nonatomc Programs James H. Anderson Mohamed G. Gouda Department of Computer Scences The Unversty of Texas at Austn Austn, Texas 78712 December 1987 Abstract We argue that t s possble,

### Loop Parallelization

- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

### The Power of Slightly More than One Sample in Randomized Load Balancing

The Power of Slghtly More than One Sample n Randomzed oad Balancng e Yng, R. Srkant and Xaohan Kang Abstract In many computng and networkng applcatons, arrvng tasks have to be routed to one of many servers,

### The complex inverse trigonometric and hyperbolic functions

Physcs 116A Wnter 010 The complex nerse trgonometrc and hyperbolc functons In these notes, we examne the nerse trgonometrc and hyperbolc functons, where the arguments of these functons can be complex numbers

### On Lockett pairs and Lockett conjecture for π-soluble Fitting classes

On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs

### The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

### z(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1

(4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at

### STATIONARY DISTRIBUTIONS OF THE BERNOULLI TYPE GALTON-WATSON BRANCHING PROCESS WITH IMMIGRATION

Communcatons on Stochastc Analyss Vol 5, o 3 (2 457-48 Serals Publcatons wwwseralsublcatonscom STATIOARY DISTRIBUTIOS OF THE BEROULLI TYPE GALTO-WATSO BRACHIG PROCESS WITH IMMIGRATIO YOSHIORI UCHIMURA

### 1 What is a conservation law?

MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2015 2016, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,

### REGULAR MULTILINEAR OPERATORS ON C(K) SPACES

REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of

### An Overview of Financial Mathematics

An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

### THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

### In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A

Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars

### Addendum to: Importing Skill-Biased Technology

Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

### A Bonus-Malus System as a Markov Set-Chain

A Bonus-alus System as a arov Set-Chan Famly and frst name: emec algorzata Organsaton: Warsaw School of Economcs Insttute of Econometrcs Al. epodleglosc 64 2-554 Warszawa Poland Telephone number: +48 6

### Generalizing the degree sequence problem

Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

### SCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar

SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,

### CHAPTER 7 VECTOR BUNDLES

CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U

### On Robust Network Planning

On Robust Network Plannng Al Tzghadam School of Electrcal and Computer Engneerng Unversty of Toronto, Toronto, Canada Emal: al.tzghadam@utoronto.ca Alberto Leon-Garca School of Electrcal and Computer Engneerng

### Stochastic epidemic models revisited: Analysis of some continuous performance measures

Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,

### Simple Interest Loans (Section 5.1) :

Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

### Calculation of Sampling Weights

Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

### Probabilities and Probabilistic Models

Probabltes and Probablstc Models Probablstc models A model means a system that smulates an obect under consderaton. A probablstc model s a model that produces dfferent outcomes wth dfferent probabltes

### Section 2 Introduction to Statistical Mechanics

Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.

### On the Approximation Error of Mean-Field Models

On the Approxmaton Error of Mean-Feld Models Le Yng School of Electrcal, Computer and Energy Engneerng Arzona State Unversty Tempe, AZ 85287 le.yng.2@asu.edu ABSTRACT Mean-feld models have been used to

### Introduction to Statistical Physics (2SP)

Introducton to Statstcal Physcs (2SP) Rchard Sear March 5, 20 Contents What s the entropy (aka the uncertanty)? 2. One macroscopc state s the result of many many mcroscopc states.......... 2.2 States wth

### INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES.

INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introducton Degree theory studes mathematcal structures, whch arse from a formal noton

### An Interest-Oriented Network Evolution Mechanism for Online Communities

An Interest-Orented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne

### Supplementary material: Assessing the relevance of node features for network structure

Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada

### On some special nonlevel annuities and yield rates for annuities

On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

### Level Annuities with Payments Less Frequent than Each Interest Period

Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

### ErrorPropagation.nb 1. Error Propagation

ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

### Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

### Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

### Finite Math Chapter 10: Study Guide and Solution to Problems

Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

### OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected

Bandwdth Packng E. G. Coman, Jr. and A. L. Stolyar Bell Labs, Lucent Technologes Murray Hll, NJ 07974 fegc,stolyarg@research.bell-labs.com Abstract We model a server that allocates varyng amounts of bandwdth

### Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

### RELIABILITY, RISK AND AVAILABILITY ANLYSIS OF A CONTAINER GANTRY CRANE ABSTRACT

Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March RELIABILITY RIK AND AVAILABILITY ANLYI OF A CONTAINER GANTRY CRANE

### Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

### THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

### Efficient Project Portfolio as a tool for Enterprise Risk Management

Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

### LECTURE 2: CRYSTAL BASES

LECTURE 2: CRYSTAL BASES STEVEN SAM AND PETER TINGLEY Today I ll defne crystal bases, and dscuss ther basc propertes. Ths wll nclude the tensor product rule and the relatonshp between the crystals B(λ)

### Analysis of Energy-Conserving Access Protocols for Wireless Identification Networks

From the Proceedngs of Internatonal Conference on Telecommuncaton Systems (ITC-97), March 2-23, 1997. 1 Analyss of Energy-Conservng Access Protocols for Wreless Identfcaton etworks Imrch Chlamtac a, Chara

### J. Parallel Distrib. Comput.

J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

### General Auction Mechanism for Search Advertising

General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

### Stability, observer design and control of networks using Lyapunov methods

Stablty, observer desgn and control of networks usng Lyapunov methods von Lars Naujok Dssertaton zur Erlangung des Grades enes Doktors der Naturwssenschaften - Dr. rer. nat. - Vorgelegt m Fachberech 3

### In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

### Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

### Abstract. 260 Business Intelligence Journal July IDENTIFICATION OF DEMAND THROUGH STATISTICAL DISTRIBUTION MODELING FOR IMPROVED DEMAND FORECASTING

260 Busness Intellgence Journal July IDENTIFICATION OF DEMAND THROUGH STATISTICAL DISTRIBUTION MODELING FOR IMPROVED DEMAND FORECASTING Murphy Choy Mchelle L.F. Cheong School of Informaton Systems, Sngapore

### The material in this lecture covers the following in Atkins. 11.5 The informtion of a wavefunction (d) superpositions and expectation values

Lecture 7: Expectaton Values The materal n ths lecture covers the followng n Atkns. 11.5 The nformton of a wavefuncton (d) superpostons and expectaton values Lecture on-lne Expectaton Values (PDF) Expectaton

### 7.5. Present Value of an Annuity. Investigate

7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on