SOME relationships among events happening in
|
|
- Elmer Ray
- 6 years ago
- Views:
Transcription
1 Possible nd Impossible Vector Clock Sets Estebn Meneses nd Frncisco J. Torres-Rojs Abstrct It is well known tht vector clocks cpture perfectly the cuslity reltionship mong events in distributed system. However, there re some interesting properties of vector clocks tht re still to be explored. In prticulr, we re interested in discovering whether there is n efficient procedure for deciding if given set of vector clocks is contined in some distributed history. We cll this the possible vector clock set problem. Index Terms Distributed Computing, Logicl Clocks, Vector Clocks. I. INTRODUCTION SOME reltionships mong events hppening in rel life cn be deduced just by compring the times when they hppened. Typiclly, if event occurred before event b we cn sy tht is potentilly cuse for b. The sme hppens with events in distributed system. The generl problem here is to find mechnism such tht, given two rbitrry events nd b, it cn be estblished which the cuslity reltionship between them is or, if it were the cse, conclude tht both events re concurrent [6]. Vector clocks [3], [5] define technique to determine precisely the cuslity reltionship mong events in distributed system, including the concurrent cse. Given distributed system with N sites, ech site keeps N entry vector of integers where the j-th entry ccounts for the quntity of events occurred in site j tht re known to this site. When event hppens t certin site, it is timestmped with the current vector clock of the site. There re few, but precise, rules for updting locl clocks when messge in sent or received. By compring the vector clocks ssigned to ny two rbitrry events, it cn be precisely determined the cuslity reltion between them. Besides this well known property of vector clocks, there re still Estebn Meneses (estebn.meneses@predisoft.com), Predisoft nd Centro de Investigción en Computción e Informátic Avnzd (CIenCIA), Cost Ric. Frncisco J. Torres-Rojs (torres@ic-itcr.c.cr), Cost Ric Institute of Technology nd Centro de Investigción en Computción e Informátic Avnzd (CIenCIA), Cost Ric. number of interesting questions bout vector clocks tht re worthy of exploring [8]. In prticulr, we propose the possible (or impossible) vector clock set problem, where, given n rbitrry, finite set of vector clocks, we re to find distributed history tht contins events timestmped with the vector clocks of the set. As we will show, there exist impossible sets, for which there re no distributed history tht contins them. In Section II, some bsic concepts bout vector clocks re reviewed. The problem of possible sets of vector clocks is presented in Section III. Some properties of these logicl clocks, useful for the problem t hnd, re explined in Section IV. A preliminry pproch to finding distributed history tht contins given set of vector clocks is explored in Section V, nd Section VI gives n exmple of the proposed technique. Finlly, conclusions nd future work re listed in Section VII. II. VECTOR CLOCKS Let s ssume distributed system with N sites, where ll the communictions re mde through messge exchnge. There re three kinds of events: internl, send nd receive. The locl history H i for site i is the totl-ordered sequence of events H i =e i1 e i2... tht re executed t site i. The globl or distributed history H for the distributed system is the prtilly ordered set of events occurring t every site in the system. Leslie Lmport [4] proposed the concept of logicl clocks, i.e., mpping between events in distributed history H nd integer numbers tht cn be used to detect some of the cusl reltionships between events. Although they re very esy to implement, Lmport clocks cnnot cpture ll the cuslity informtion. For exmple, concurrency between events is poorly detected. These clocks re consistent with cuslity, but they do not chrcterize it [6]. Vector clocks [3], [5], [2], [6] consist of mpping between events in distributed history H nd
2 integer vectors. Ech site i keeps its own vector clock V i with N entries, where N ccounts for the number of sites in the system. Entry V i [i] mintins the locl clock of site i, while V i [j] keeps trck of the ctivity t site j, from the point of view of site i. Ech time n event occurs t site i, its locl clock ticks nd vector clock is ssocited with tht event. Also, every messge sent in the system is piggybcked with the vector clock corresponding to the send event. Site i updtes its vector clock obeying the following rules: V i [j] = 0,0 j N 1 - Initil vlue. Ech time n internl event occurs: V i [i] = V i [i]+ (typiclly =1) When messge with timestmp T is received: V i [j] = mx(v i [j],t[j]),0 j N 1 V i [i] = V i [i]+ Let H i. We sy tht hs timestmp V(), where V() is the vlue of V i t the instnt when ws executed. Vector clocks re compred following these rules: v = w v[j] = w[j],0 j N 1 v w v[j] w[j],0 j N 1 v < w v w nd j such tht v[j] < w[j] v w k such tht v[k] < w[k] nd j such tht v[j] > w[j] Mttern showed in [5] tht there is n isomorphism between timestmps obtined from vector clock when events re executed, nd the cuslity reltionship mong events in H. Thus, vector clocks stisfy the Strong Clock Condition, where nd b H: 1) = b V() = V(b) 2) b V() < V(b) 3) b V() V(b) Given two vector clocks v nd w, only one of four cses might hppen: v = w, v w, w v or v w. Following the comprisons defined bove nd using the fct tht the reltionship genertes prtil order, Hsse digrm is obtined when drwing ll those reltionships [7]. Vector clocks hve severl interesting properties, some of which hve been studied in [2], [8], [9]. Definition 1: If v nd w re two vector clocks, we sy tht t is the mximum of them, denoted s mximum(v,w), if these conditions hold: v t nd w t It doesn t exist vector clock z such tht v z w z z t The definition for minimum is nlogous to the one given for mximum. It cn be esily verified tht given vector clocks v nd w, t is mximum(v,w) iff t[k] = mx(v[k],w[k]), 0 k N 1, nd tht u is minimum(v,w) iff u[k] = min(v[k],w[k]), 0 k N 1. The next two lemms re needed for the min result of next section: Lemm 1: No other site hs more updted informtion bout the ctivity of Site i (i.e., lrger vlue in its i-th entry) thn Site i itself. Proof: From the rules defined previously, it is evident tht site i is the only one tht increments V i [i], nd the vlue of this entry in ny other site is either zero, or the vlue sent (directly or indirectly) by site i, which in turn could hve incremented it since then. Lemm 2: Let,b H. In prticulr, let H k, i.e., ws executed t site k. Then: b V()[k] V(b)[k]. Proof: This is corollry of the isomorphism between vector clocks nd events in the distributed history, nd the comprison of vector clocks defined previously. Since occurred t site k nd it is cuslly before b, then, when b occurs, the timestmp of site k is t lest the one hd, nd vice vers. This result is usully known s the Simple Strong Clock Condition. III. THE POSSIBLE VECTOR CLOCK SET PROBLEM Not every rbitrry set of vector clocks is vlid or possible, in the sense tht it might be impossible to find distributed history H tht contins subset of events such tht their vector clocks correspond to the ones in the given set. In other words, certin combintions of vector clocks cn not coexist in the sme distributed history. Theorem 1 (Deserted Zone): Let 1, 2,..., n be n events in H ssocited with the n vector clocks v 1,v 2,...,v n, respectively. Also, let the n events be concurrent with ech other. If t=mximum(v 1,v 2,...,v n ), then, it cnnot exist n event z H such tht v i V(z) t, for ny i. Proof: By contrdiction. Let s ssume tht there exists n event z H such tht v i V(z) t for some i. Without loss of generlity, let i be
3 1, nd let z hppen on site k, i.e., z H k. Now, using Lemm 1, we hve v 1 [k] < V(z)[k]. Given tht V(z) t, then v 1 [k] < V(z)[k] t[k] = mx(v 1 [k],v 2 [k],...,v n [k]), which mens definitively tht V(z)[k] v p [k], for some p {2,3,...,n}. In virtue of Lemm 2 nd the isomorphism between vector clocks nd events, V(z)[k] v p [k] implies tht z p, but by hypothesis 1 z, nd, therefore, 1 p, which contrdicts the fct tht 1 nd p re concurrent. Hence, z cnnot exist. Let s consider the cse with n=2, where 1 nd 2 re two concurrent events with vector clocks v 1 nd v 2, respectively. If t=mximum(v 1, v 2 ), then v 1, v 2 nd t define kind of deserted zone, where no event might occur. Hence, Theorem 1 ffirms tht there cnnot be ny event z in the sme distributed history tht contins concurrent events 1 nd 2, with its vector clock lying in the deserted zone induced by 1 nd 2. Notice, however, tht there could exist mny integer vectors which hve this property. Nevertheless, none of them cn be ssocited to ny event in the sme distributed history. From the result of Theorem 1, we relize the existence of certin impossible sets of vector clocks. For instnce, the set {<100, 0, 100>, <0, 100, 0>, <50, 100, 0>} is impossible. Thus, if presented with set of vector clocks, it mkes sense to sk ourselves whether or not this set of vector clocks is possible subset of timestmps seen in distributed history. This question is wht we clled the possible vector clock set problem. In its more generl form, this problem cn be stted s: Given set of vector timestmps, decide whether or not there is distributed history contining them. There is not informtion vilble bout the site where ech vector clock is supposed to hve occurred. There could be version of the problem where the sites corresponding to certin vector clocks in the set re known, while other vector clocks re free to be ssigned to ny site: Given set of vector timestmps, some of them ssigned to specific sites, decide whether or not there is distributed history contining them ll. Fig. 1. <0,0,1> <1,0,0> <2,0,1> <0,1,0> <1,2,0> <1,3,2> <0,0,2> Underlying distributed history for set A Finlly, the, pprently, softest version of the problem pins down ll the vector clocks to known sites: Given set of vector timestmps, ech one ssigned to specific site, decide whether or not there is distributed history contining them. As n exmple of the ltest sttement of the problem, consider the vector clock set A over distributed system with 3 sites: A = { [<2,0,1>, site 1], [<0,1,0>, site 2], [<1,3,2>, site 2], [<0,0,2>, site 3] } To demonstrte tht A is possible set, we just hve to show distributed history H which contins every element of A s timestmp. In this cse, A is possible, becuse there re multiple distributed histories tht stisfy the requirement. Figure 1 shows n exmple of such distributed history (the timestmps from A re written in bold). On the other hnd, the vector clock set B is impossible: B = { [<2,0,1>, site 1], [<0,1,0>, site 2], [<0,1,1>, site 2], [<1,3,2>, site 2], [<0,0,2>, site 3] } To prove tht B is n impossible set, consider the concurrent vector clocks v 1 =< 0,1,0 > nd v 2 =< 2,0,1 >, both in B. Let s sy tht these vector clocks correspond to events 1 nd 2. Following Theorem 1, if we compute the mximum between v 1 nd v 2, we obtin t=< 2,1,1 >. Then, there cn not exist n event z in the sme distributed history s 1 nd 2, such tht its ssocited vector clock V(z)
4 < 0,1,0 > Fig. 2. < 0,1,1 > z t < 2,1,1 > 1 2 Impossible set due to deserted zone theorem < 2,0,1 > hs the property: v 1 < V(z) < t or v 2 < V(z) < t. However, vector clock < 0, 1, 1 > in B hppens to be between < 0,1,0 > nd t=< 2,1,1 >, which renders set B s impossible. Figure 2 shows grphic representtion of this prticulr sitution. IV. INTERMEDIATE RESULTS In this section, we consider certin properties of vector clocks nd some impossibility results, which re useful to explore the problem of possible vector clock sets. Lemm 3: Let timestmp v correspond to event inh i. Then, for everyj i nd v[j] 0, thev[j]-th event of H j is send. Proof: Left to the reder. Thus, if every timestmp in set S is ssocited with some site, then, just by checking these timestmps, necessry set of send events in the distributed history cn be deduced. However, this resultnt group of send events might not be sufficient for building the underlying distributed history. Let v j nd v k be two timestmps corresponding to events hppening in sites j nd k, respectively. Also, let v j [i] = v k [i] = x (i j nd i k). As Lemm 3 estblished, the x-th event in Site i must be send. In order to construct timestmps v j nd v k, this send must rech, directly or indirectly, sites j nd k. Since our model does not llow multicsting, the x-th event in Site i must be directed to, let s sy, Site j, nd from there to Site k with n extr send. We cll this kind of send deferred send. Lemm 4: Given vector clock set S where ech timestmp is ssocited with some site, if two timestmps (occurring in different sites) coincide in their i-th entry nd neither of them occurred in Site i, deferred send is required in the underlying distributed history. Proof: It is strightforwrd given the previous discussion. Although the im of deferred send is to provide time informtion indirectly, this functionlity could be fulfilled by one of the regulr sends derived from Lemm 3. Thus, it could be the cse in n underlying distributed history, tht some deferred send is collpsed with nother send event. The following lemm proves the existence of impossible vector clock sets with just one element: Lemm 5: The vector clock set S={< 1,1,1 >} is impossible. Proof: Let s suppose, without loss of generlity, tht timestmp < 1,1,1 > occurred t Site 1. So, entry 1 is justified by being the locl stmp, which mens tht this vector clock corresponds to the first event of this site. Lemm 3 indictes tht the first event of both nd must be send events. Notice tht the first event of must be receive. Assume, for instnce, tht it receives messge from. Then, its third entry cn not be 1, since there is no time for such informtion to be trnsmitted. Note tht deferred send is not sufficient either, becuse if sends messge to, then the first event of should be receive event, being unble to send the required informtion to. Theorem 2: Consider distributed system with N sites. Let S = {v 1,v 2,...,v k } be set with k N vector clocks concurrent to ech other nd let t = mximum(v 1,v 2,...,v k ). There cnnot be v i (1 i k) such tht v i [j] < t[j] for ll j = 1,2,...,N. Proof: By contrdiction. Assume tht such v i exists, nd tht it occurred t Site p. So, by ssumption, v i [p] < t[p]. Since t is the mximum of the vector set, there must exist some v j, such tht v j [p] = t[p]. The only wy to explin this vlue of entry v j [p] is vi chin of messges strting with send event t Site p. Let V() be the vector clock of. Clerly, V() < v j, becuse event cuslly precedes the event with timestmpv j. Now, nd the event with timestmp v i come from the sme site p, nd since v i [p] < V()[p], we hve v i < V() < v j, which contrdicts the premise tht ll vector clocks re concurrent.
5 V. AN EXHAUSTIVE APPROACH By using n exhustive strtegy, it cn be possible to decide whether or not there is n underlying distributed history for some vector clock set. We nlyze solution for the softest version, where ech timestmp is ssocited with some specific site. Strtegies for the other two versions of the problem cn be obtined by extending this one. We re looking for set A of pirs of events of the form < Site : send,site : receive >, where send nd receive indicte the reltive positions of these events in ech locl history. In fct, ny distributed history cn be chrcterized by the number of sites, the number of events in ech site nd set A. Thus, the lgorithm bsiclly tests series of lterntives for the underlying history, until set A tht produces ll required timestmps in the distributed history is obtined. Given vector clock set S ={v 1,v 2,...,v m } with m timestmps, ech one ssocited to some site, the following steps will find n underlying distributed history: Step 1: The number of sites N is equl to the length of every timestmp in S. Step 2: Let t = mximum(v 1,v 2,...,v m ), then t[i] is the number of events in site i, for i=1,2,...,n. Step 3: Obtin ll the necessry send events set R, using Lemm 3. Step 4: Obtin ll the deferred send events set D, using Lemm 4. Step 5: Associte every element of R nd D with pproprite receive events, computing set A If timestmp v ppers in Site i, it is known tht it ws the v[i]-th event in this site. Let s cll this event. Now, we hve to justify ll the other entries in v. Lemms 3 nd 4 estblish the quntity of send events, being deferred or not, tht must rech site i in order to justify timestmp v for event. Definition 2: The receive block of event includes the preceding events of in the sme site nd itself, which cn be used to receive informtion for justify timestmp of event. Figure 3 shows the receive blocks for two different events nd b, occurring in the sme site i, nd whose timestmps must be justified. Notice tht if these timestmps shre the j-th entry (with j i), then j-th must not be justified for in the receive Site i Fig. 3. Fig. 4. b Receive blocks for events b nd, respectively Originl distributed history <4,5,3> <1,6,3> <3,3,5> <3,7,7 <6,5,6> block for. This entry, on the contrry, hs to be kept nd not overridden by messge with lrge j-th entry in its timestmp. Finlly, we hve ll the pieces to solve the softest version of the problem. Notice tht the difference between versions of the problem is the fct tht site informtion is inexistent, incomplete or complete. When informtion bout sites is inexistent, this is prticulr cse of being incomplete. Then, if we hve n lgorithm for detecting possibility or impossibility when site informtion is complete, we cn iterte the lgorithm over the options tht re creted by ssocite the timestmps without site informtion with prticulr one. Those possibilities could be huge, but finite. VI. EXAMPLE: BUILDING UNDERLYING DISTRIBUTED HISTORY This section gives n exmple of the exhustive pproch introduced in the previous section. Suppose we re given set S of timestmps tken from the distributed history in Figure 4 (obviously, we do not know this history): S = { [<4,5,3>, site 1], [<6,5,6>, site 1], [<1,6,3>, site 2], [<3,3,5>, site 3], [<3,7,7>, site 3] } Let s cll the events ssocited with the previous timestmps,b,c,d nd e, respectively. Now, the lgorithm from section V will be pplied to verify if this set hs t lest one underlying or witness distributed history. First of ll, it is known tht N=3, so the system hve 3 sites where we re
6 to ccommodte ll the required send nd receive events. Secondly, the mximum for ll timestmps ins ist = mx(< 4,5,3 >,< 6,5,6 >,< 1,6,3 >,< 3,3,5 >,< 3,7,7 >) =< 6,7,7 >, which mens tht there re 6 events in, 7 in Site 2 nd 7 in. Using Lemm 3, the setris obtined. Thus, locl events {1,3} in must be sends. Also, events {3,5,7} in nd events {3,6} in. The next tble summrizes these results: Event Site Number Blocks s c s d,e s d s 4 2 5,b s e s 6 3 3,c s b This tble shows the nme of the event, the site where it occurred, the locl number of event nd finlly the blocks which it must rech. This lst column is very importnt, since the rison d être for ech send event is to rech some block nd thus provide some event with the required components for its timestmp. Intuitively, one could think tht if some send event hs two or more blocks in its lst column, then deferred send is required. However, this is not necessrily the cse. Events s 1,s 3,s 5 nd s 7 hve just one reching block. On the other side, events s 2 nd s 4 hve two reching blocks. Nevertheless, in ech cse, the two events occurred in the sme site, so it is sufficient for the send to rech the first of the two events. Moreover, it is necessry to void nother send event to rech these sites with more updted informtion. Then, for instnce,s 2 hve just to rech block d nd this locl timestmp will be propgted into block e. But, one must void ny other send event coming from nd occurring fter s 2. The lst rgument cnnot be pplied to s 6, so it must be considered inserting deferred send which reches block nd then block c or vice vers. Therefore, Lemm 4 hs been tken into ccount nd set D hs been built. Figure 5 depicts visul scheme of the lst sitution. It shows ll the events with timestmps in S nd the required send events. Note tht the deferred Fig. 5. Fig. 6. s 1 s 2 s 3 s 6 Scheme for building the witness distributed history Witness distributed history s 4 d b c s 7 s 5 e <4,5,3> <6,5, <1,6,3> <3,3,5> <3,7,7> send is not shown, becuse its event number must no be fixed priori. Finlly, pplying n exhustive ssignment for send events to its reching blocks nd testing if the required timestmps re generted, we verify the possibility of S. One of the witness distributed histories ppers in figure 6. Note tht it is different from the originl history of figure 4. In fct, there could exist mny underlying distributed histories for given set S. However this new history keeps some of the originl messges, which re depicted with solid lines. The new messges were drwn with dotted lines. As it cn be pprecited, the deferred send is locted in s the second event. Nevertheless, it is not necessry, becuse ll the informtion it ws supposed to provide, ws crried by send event s 4. VII. CONCLUSIONS AND FUTURE WORK Vector clocks hve been helpful tool in determining the cuslity reltions mong events in distributed system. Knowing their properties will ensure more relible time stmping system, s we cn check dditionl requirements. For instnce, if we hd n efficient procedure for determining if some set of vector clocks is possible or not, then we cn reject report of timestmps if they constitute n impossible or flsified set. We re interested in clssifying the process of deciding whether some vector clock set is possible or impossible. This problem is clerly soluble, t
7 lest using n exhustive strtegy. But, it would be importnt to verify if this problem is P or NPcomplete. Thus, future work is focused on discovering the nture of the possible set problem on vector clocks. It must be defined if it hs polynomil time solution or if it cn only be solved by n exhustive pproch. Besides, there re severl interesting properties nd theorems bout possible nd impossible sets tht deserve to be explored (e.g., closure properties, specil cses, geometric interprettions, etc.). REFERENCES [1] Ahmd, M. et l. Cusl memory: definitions, implementtion nd progrmming. Distributed Computing, [2] Bldoni, R. nd Rynl, M. Fundmentls of Distributed Systems: A Prcticl Tour of Vector Clocks Systems. IEEE Distributed Systems Online, Februry, [3] Fidge, C. Logicl Time in Distributed Computing Systems. Computer, Vol 24(8), August, [4] Lmport, L. Time, Clocks, nd the Ordering of Events in Distributed System. Communictions of the ACM, Vol 21(7): , July, [5] Mttern, F. Virtul Time nd Globl Sttes of Distributed Systems. Proceedings of the Interntionl Workshop on Prllel nd Distributed Algorithms, , [6] Schwrz, R. nd Mttern, F. Detecting Cusl Reltionships in Distributed Systems: In Serch of the Holy Gril. Distributed Computing, [7] Torres-Rojs, F. Prtilly Ordered Sets nd Logicl Clocks for Distributed Systems. Proceedings of Conferenci Ltinomericn en Informátic, [8] Torres-Rojs, F. nd Meneses, E. Alguns Propieddes Interesntes de los Relojes Vectoriles. Proceedings of Jornds Chilens de Computción, [9] Yng, Z. nd Mrslnd T.A. Globl Sttes nd Time in Distributed Systems. IEEE Computer Society Press, 1994.
Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationLocal orthogonality: a multipartite principle for (quantum) correlations
Locl orthogonlity: multiprtite principle for (quntum) correltions Antonio Acín ICREA Professor t ICFO-Institut de Ciencies Fotoniques, Brcelon Cusl Structure in Quntum Theory, Bensque, Spin, June 2013
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationStrong Bisimulation. Overview. References. Actions Labeled transition system Transition semantics Simulation Bisimulation
Strong Bisimultion Overview Actions Lbeled trnsition system Trnsition semntics Simultion Bisimultion References Robin Milner, Communiction nd Concurrency Robin Milner, Communicting nd Mobil Systems 32
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationLecture 09: Myhill-Nerode Theorem
CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationReinforcement Learning
Reinforcement Lerning Tom Mitchell, Mchine Lerning, chpter 13 Outline Introduction Comprison with inductive lerning Mrkov Decision Processes: the model Optiml policy: The tsk Q Lerning: Q function Algorithm
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationUniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is self-contined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationarxiv: v1 [quant-ph] 19 Dec 2017
Quntum supervlutionist ccount of the EPR prdox Arkdy Bolotin Ben-Gurion University of the Negev, Beersheb (Isrel) December, 17 rxiv:171.6746v1 [qunt-ph] 19 Dec 17 Abstrct In the pper, the EPR prdox is
More informationFrobenius numbers of generalized Fibonacci semigroups
Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC 29634-0975, USA gmtthe@clemson.edu Received:, Accepted:, Published:
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationGeneralized Fano and non-fano networks
Generlized Fno nd non-fno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationZero-Sum Magic Graphs and Their Null Sets
Zero-Sum Mgic Grphs nd Their Null Sets Ebrhim Slehi Deprtment of Mthemticl Sciences University of Nevd Ls Vegs Ls Vegs, NV 89154-4020. ebrhim.slehi@unlv.edu Abstrct For ny h N, grph G = (V, E) is sid to
More informationIntuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras
Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore-632014, TN, Indi
More information12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS
1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility
More informationVyacheslav Telnin. Search for New Numbers.
Vycheslv Telnin Serch for New Numbers. 1 CHAPTER I 2 I.1 Introduction. In 1984, in the first issue for tht yer of the Science nd Life mgzine, I red the rticle "Non-Stndrd Anlysis" by V. Uspensky, in which
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationLet S be a numerical semigroup generated by a generalized arithmetic sequence,
Abstrct We give closed form for the ctenry degree of ny element in numericl monoid generted by generlized rithmetic sequence in embedding dimension three. While it is known in generl tht the lrgest nd
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationGenetic Programming. Outline. Evolutionary Strategies. Evolutionary strategies Genetic programming Summary
Outline Genetic Progrmming Evolutionry strtegies Genetic progrmming Summry Bsed on the mteril provided y Professor Michel Negnevitsky Evolutionry Strtegies An pproch simulting nturl evolution ws proposed
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationAdministrivia CSE 190: Reinforcement Learning: An Introduction
Administrivi CSE 190: Reinforcement Lerning: An Introduction Any emil sent to me bout the course should hve CSE 190 in the subject line! Chpter 4: Dynmic Progrmming Acknowledgment: A good number of these
More informationHow to simulate Turing machines by invertible one-dimensional cellular automata
How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More information1 Structural induction
Discrete Structures Prelim 2 smple questions Solutions CS2800 Questions selected for Spring 2018 1 Structurl induction 1. We define set S of functions from Z to Z inductively s follows: Rule 1. For ny
More informationThe Henstock-Kurzweil integral
fculteit Wiskunde en Ntuurwetenschppen The Henstock-Kurzweil integrl Bchelorthesis Mthemtics June 2014 Student: E. vn Dijk First supervisor: Dr. A.E. Sterk Second supervisor: Prof. dr. A. vn der Schft
More informationOn the free product of ordered groups
rxiv:703.0578v [mth.gr] 6 Mr 207 On the free product of ordered groups A. A. Vinogrdov One of the fundmentl questions of the theory of ordered groups is wht bstrct groups re orderble. E. P. Shimbirev [2]
More informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More information1.3 The Lemma of DuBois-Reymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationHomework Solution - Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More information19 Optimal behavior: Game theory
Intro. to Artificil Intelligence: Dle Schuurmns, Relu Ptrscu 1 19 Optiml behvior: Gme theory Adversril stte dynmics hve to ccount for worst cse Compute policy π : S A tht mximizes minimum rewrd Let S (,
More informationMonotonic Parallel and Orthogonal Routing for Single-Layer Ball Grid Array Packages
Monotonic Prllel nd Orthogonl Routing for Single-Lyer Bll Grid Arry Pckges Yoichi Tomiok Atsushi Tkhshi Deprtment of Communictions nd Integrted Systems, Tokyo Institute of Technology 2 12 1 S3 58 Ookym,
More informationRELATIONAL MODEL.
RELATIONAL MODEL Structure of Reltionl Dtbses Reltionl Algebr Tuple Reltionl Clculus Domin Reltionl Clculus Extended Reltionl-Algebr- Opertions Modifiction of the Dtbse Views EXAMPLE OF A RELATION BASIC
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More informationA BRIEF INTRODUCTION TO UNIFORM CONVERGENCE. In the study of Fourier series, several questions arise naturally, such as: c n e int
A BRIEF INTRODUCTION TO UNIFORM CONVERGENCE HANS RINGSTRÖM. Questions nd exmples In the study of Fourier series, severl questions rise nturlly, such s: () (2) re there conditions on c n, n Z, which ensure
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 22 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 22 September 2017 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationLecture 2: Fields, Formally
Mth 08 Lecture 2: Fields, Formlly Professor: Pdric Brtlett Week UCSB 203 In our first lecture, we studied R, the rel numbers. In prticulr, we exmined how the rel numbers intercted with the opertions of
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationMarkscheme May 2016 Mathematics Standard level Paper 1
M6/5/MATME/SP/ENG/TZ/XX/M Mrkscheme My 06 Mthemtics Stndrd level Pper 7 pges M6/5/MATME/SP/ENG/TZ/XX/M This mrkscheme is the property of the Interntionl Bcclurete nd must not be reproduced or distributed
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationIntegral points on the rational curve
Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin - Mrinette 750 W. Byshore Street Mrinette, WI 5443-453 Also: Konstntine Zeltor P.O. Box
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More information