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or considering the implementation of, ... June 27, 2011 TRB Transportation Research E-Circular E-C154: Development of Warranty Programs for Hot-Mix Asphalt is a synopsis of current information for the development of warranty programs for hot-mix asphalt pavements. The opinions expressed in this e-circular are those of ... | {
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with the goal of... July 11, 2011 TRB Transportation Research Electronic Circular E-C149: 75 Years of the Fundamental Diagram for Traffic Flow Theory: Greenshields Symposium includes the papers that were presented at a July 2008 conference that explored traffic flow theory’s history, recent developments, technological ... | {
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parties as they contemplate and fund research in this area. April 12, 2010 TRB Transportation Research Circular E-C143: Modal Primer on Greenhouse Gas and Energy Issues for the Transportation Industry is primer designed to provide transportation decision makers with an inclusive, educated, and objective overview of the... | {
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environment, including climate change, alternative fuels, and sustainability. The circular focuses on the state of science, rather than on policy, and the authors of various portio... May 19, 2009 TRB’s Transportation Research Circular E-C137: Glossary of Highway Quality Assurance Terms contains terms of common usage a... | {
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ameliorated. Background papers prepared for the workshop are also included. The workshop included perspectives on the issue... December 03, 2008 Transportation Research Circular E-C131, Transportation Asset Management: Strategic Workshop for Department of Transportation Executives summarizes a workshop held in Washingt... | {
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Symposium on Snow and Ice Control (June 17–19, 2008), which were both held in Indianapolis, Indiana. The E-Circular includes pa... March 01, 2008 TRB’s Transportation Research Circular E-C125: Evolving Role of Statewide Transportation Planning in an Era of Regional Funding and Governance reports on a July 7-8, 2006 pee... | {
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as part of a May 2007 workshop in Washington, D.C., designed as a forum to exchange knowledge about successful strategies in the collection and the analysis of traffic data. August 19, 2007 TRB’s Transportation Research Circular E-C119: North American Freight Transportation Data Workshop summarizes sessions that took p... | {
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Opportunities in Radio Frequency Identification Transportation Applications summarizes a conference by the same title that was held October 17 to 18, 2006, in Washington, D.C. This conference focused on current and future research on radio frequency identification (RFID) technologies in transportation applications, whi... | {
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the raw material, and information as the processed, useful product supporting decisions. The report also examines the value of information in specific real decisions and outlines an ongoing process designed to help ensure that transpo... October 19, 2006 TRB’s Transportation Research Circular E-C108: Geospatial Informa... | {
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Material for the 21st Century examines some of the research innovations in the 20th century that led to advancements in concrete. The circular also explores what the future may hold as a result of the continuing advancements in high-performance durable concrete. August 21, 2006 TRB Transportation Research Circular E-C1... | {
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held in Charleston, South Carolina, July 16-20, 2006. The report includes papers on outsourcing, pavements, roadside, winter operations, bridges, maintenance managemen... July 12, 2006 TRB’s Transportation Research Circular E-C097, International Perspectives on Urban Street Design: Proceedings of the Context-Sensitive ... | {
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included in the session. The circular is intended as a... February 10, 2006 TRB’s Transportation Research Circular E-C092, Maintenance and Operations of Transportation Facilities: 2005 Strategic Vision explores the major trends that affect maintenance; cites current and emerging innovations in management systems, techn... | {
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in January 2005. Hot-mix asphalt (HMA) overlays can also be considered for application to an existing, deteriorated portland cement ... December 06, 2005 TRB Electronic Circular E-C086: Evaluation of Chemical Stabilizers: State-of-the-Practice Report is designed to provide information on practices that agencies have fo... | {
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to improve the Highway Capacity Manual (HCM). The proposed program was developed by the members of the TRB Highway Capacity and Quality of Service Committee with the assistance of feedback from users of the HCM. The program includes 38 research studies that wo... December 02, 2005 TRB Electronic Circular E-C080: Freigh... | {
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7–8, 2004, peer exchange that focused on the expanding role of asset management in planning and operations as a comprehensive approach to managing agency resources and transportation systems. The peer exchange was designed to gather additional information about the state of the... September 01, 2005 TRB Electronic Circ... | {
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TRB's E-Circular 70 - Optimizing the Dissemination and Implementation of Research Results: A Summary of Workshop and Midyear Meeting Activities examines the issues and challenges associated with optimizing the dissemination and implementation of research results. This TRB E-Circular also documents the problem explorati... | {
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the TRB’s 83rd Annual Meeting, January 11–15, 2004, and emphasizes the important security duties transportation professional’ play in protecting against and reacting to security events. May 07, 2004 TRB’s Transportation Research Circular E-C064 -- Data Requirements in Transportation Reauthorization Legislation: What Is... | {
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Pittsburgh, Pennsylvania, in 2000-2002. The objective of this workshop series was to provide a forum for the exchange of new ideas and developments in the field of accelerated construction. January 13, 2004 TRB Transportation Research Circular E-C058 includes the proceedings of the 9th National Light Rail Transit Confe... | {
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Bridge Management and Hybrid Knowledge Representation in Bridge Management Systems. This circular is a supplement to Circular E-C049 ( . Papers in both Circulars were presented at the 9th Internati... July 18, 2003 TRB Transportation Research Circular EC052: Maintenance Management 2003 contains papers presented at the ... | {
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budget, total costs, and fleet upgrade factors. November 14, 2002 TRB Transportation Research Circular E-C046: Using Spatial Data, Tools, and Technologies to Improve Program Delivery chronicles a peer exchange that focused on optimum use of spatial data, tools, and information to facilitate decision making and to deliv... | {
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Federal Governments is the proceedings from a conference held on October 18–21, 2001 in Albuquerque, New Mexico that focused on the complexity of broad transportation issues of importance for Native American nations. June 10, 2002 TRB Transportation Research Circular E-C038: Standards for Testing, Evaluating, and Locat... | {
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Title: URL Source: Markdown Content: # Lecture notes: Studying distributed systems – Reliable Broadcast M2 MOSIG: Large-Scale Data Management and Distributed Systems Thomas Ropars 2024 This lecture studies the implementation of reliable broadcast primitives 1. Considering a crash-stop failure model, it studies broadca... | {
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the required safety and liveness properties of the algorithm. Then we propose an implementation that satisfies these properties. > 1Acknowledgments: This lecture is strongly inspired from Chapter 3 of the book of Cachin, Guerraoui, and Rodrigues 1For all algorithms, we consider a crash-stop failure model for processes... | {
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the same set of messages. The goal of the (regular) Reliable Broadcast primitive 2 is to ensure this property. 4.1 Specification The reliable broadcast abstraction has the following properties: • Integrity: Each process delivers message m at most once, and only if was broadcasted by some process. • Validity: If a corre... | {
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deliver this message: The agreement property only applies to correct processes. > 2From this point on, when we simply say “ reliable broadcast ”, we are referring to the regular reliable broadcast abstraction. > 3The fact that this agreement property is a liveness property can be counter-intuitive since if message mis ... | {
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delivered m because that process crashed afterwards. We want to implement a broadcast primitive with a stronger agreement property, which is called Uniform Agreement . We find the need for such a uniform property in many other abstractions. 5.1 Specification The uniform reliable broadcast abstraction has the following ... | {
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all [s, m] ∈ pending: > 66 tryDeliver([s,m]) > 68 Function canDeliver(m): > 69 return (correct ⊆ ack[m]) # we receive m from all correct processes > 71 Function tryDeliver([s, m]): > 72 if canDeliver(m) and m /∈ delivered: > 73 delivered = delivered ∪ m > 74 Trigger urb.deliver([s, m]) Figure 3: Implementation of Unifo... | {
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process delivers message m at most once, and only if was broadcasted by some process. • Validity: If a correct process broadcasts a message m, then every correct process eventually delivers m. • Agreement: If a message m is delivered by some correct process, then m is eventually delivered by every correct process. • FI... | {
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Title: URL Source: Markdown Content: # 3. Reliable Broadcast He said: “I could have been someone”; She replied: “So could anyone.” (The Pogues) This chapter covers broadcast communication abstractions. These are used to dis-seminate information among a set of processes and differ according to the reliability of the di... | {
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reliable . Reliability in this context usually means that, under some assumptions (which are, by the way, often not completely understood by most system design-ers), messages exchanged between the two processes are not lost or duplicated, and are delivered in the order in which they were sent. Typical implementations o... | {
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last message sent while others do not. This may lead to an inconsistent view of the system state by different group members. When the sender of a message exhibits arbitrary-faulty behavior, assuring that the recipients deliver one and the same message is an even bigger challenge. The broadcast abstractions in this book... | {
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With best-effort broadcast, the burden of ensuring reliability is only on the sender. Therefore, the remaining processes do not have to be concerned with enforcing the reliability of received messages. On the other hand, no delivery guarantees are offered in case the sender fails. Best-effort broadcast is characterized... | {
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simply consists of sending the message to every process in the system using perfect point-to-point links, as illustrated by Fig. 3.1 (in the figure, white arrowheads represent request/indication events at the module interface and black arrowheads represent message exchanges). The algorithm works because the properties ... | {
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a reliable broadcast algorithm ensure that the correct processes agree on the set of messages they deliver, even when the senders of these messages crash during the transmission. It should be noted that a sender may crash before being able to transmit the message, in which case no process will del-iver it. The specific... | {
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best-effort broadcast abstraction described in the previous section, as well as the perfect failure detector abstraction P introduced earlier. Algorithm 3.2: Lazy Reliable Broadcast > Implements: > ReliableBroadcast, instance rb . > Uses: > BestEffortBroadcast, instance beb ;PerfectFailureDetector, instance P. > upon e... | {
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, denoting the set of processes that have not been detected to crash by P. Our algorithm is said to be lazy in the sense that it retransmits a message only if the original sender has been detected to have crashed. The variable from is an array of sets, indexed by the processes in Π, in which every entry s contains the ... | {
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from the fact that every process relays every message that it rb -delivers when it detects the sender, and from the use of a perfect failure detector. Performance. If the initial sender does not crash then the algorithm requires a single communication step and O(N ) messages to rb -deliver a message to all processes. O... | {
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phase is exactly what guarantees the agreement property of reliable broadcast. The resulting algorithm (Algorithm 3.3) is called “Eager Reliable Broadcast.” The algorithm assumes a fail-silent model and does not use any failure detec-tor: it relies only on the best-effort broadcast primitive described in Sect. 3.2. In ... | {
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in both reliable broadcast algorithms that we presented (eager and lazy). It is thus pos-sible that no other process, including correct ones, ever rb -delivers that message. There are cases where such behavior causes problems because even a process that rb -delivers a message and later crashes may bring the application... | {
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| p, m 〉: Delivers a message m broadcast by process p. > Properties: URB1–URB3: Same as properties RB1–RB3 in (regular) reliable broadcast (Mod-ule 3.2). > URB4: Uniform agreement: If a message mis delivered by some process (whether correct or faulty), then mis eventually delivered by every correct process. > rb−delive... | {
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> upon event 〈beb , Deliver | p, [D ATA , s, m ] 〉 do > ack [m]:= ack [m]∪ { p}; > if (s, m )∈pending then > pending := pending ∪ { (s, m )}; > trigger 〈beb , Broadcast | [D ATA , s, m ] 〉; > upon event 〈 P , Crash | p 〉 do > correct := correct \ { p}; > function candeliver( m) returns Boolean is return (correct ⊆ ack... | {
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representation. Note that the last upon statement of the algorithm is triggered by an internal event defined on the state of the algorithm. Correctness. The validity property follows from the completeness property of the failure detector and from the validity property of the underlying best-effort broad-cast. The no du... | {
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failure detector. Algorithm 3.5, called “Majority-Ack Uniform Reliable Broadcast,” is similar to Algorithm 3.4 (“All-Ack Uniform Reliable Broadcast”) in the fail-silent model, except that processes do not wait until all correct processes have seen a message, but only until a majority quorum has seen and retransmitted t... | {
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uniform agreement , and let q be any process that urb -delivers m. To do so, q must have beb -delivered m from a majority of the processes. Because of the assumption of a correct majority, at least one correct process must have beb -broadcast m. Hence, all correct processes eventually beb -deliver m by the validity pro... | {
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property of best-effort broadcast is not ensured. Stubborn broadcast is the first broadcast abstraction in the fail-recovery model considered in this chapter (more will be introduced in the next two sections). As the discussion of logged perfect links in Chap. 2 has shown, communication abstrac-tions in the fail-recove... | {
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exchanges at least N messages. Of course, the stubborn links may retransmit the same message several times and, in practice, an optimization mechanism is needed to acknowledge the messages and stop the retransmission. # 3.6 Logged Best-Effort Broadcast This section and the next one consider broadcast abstractions in th... | {
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log-delivered more than once. > LBEB3: No creation: If a process log-delivers a message mwith sender s, then m > was previously broadcast by process s. has no memory of m, because the delivery of m occurred asynchronously and could not be anticipated. There should be some way for process q to find out about m upon reco... | {
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Implements: > LoggedBestEffortBroadcast, instance lbeb . > Uses: > StubbornPointToPointLinks, instance sl . > upon event 〈lbeb ,Init 〉do > delivered := ∅;store (delivered ); > upon event 〈lbeb ,Recovery 〉do retrieve (delivered ); > trigger 〈lbeb ,Deliver |delivered 〉; > upon event 〈lbeb ,Broadcast |m〉do forall q∈Πdo tr... | {
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stubborn links 90 3 Reliable Broadcast may retransmit the same message several times and, in practice, an optimization mechanism is needed to acknowledge the messages and stop the retransmission. Additionally, the algorithm requires a log operation for each delivered message. # 3.7 Logged Uniform Reliable Broadcast In ... | {
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lurb ,Init 〉 do delivered := ∅; pending := ∅; forall m do ack [m] := ∅;store( pending , delivered ); upon event 〈 lurb ,Recovery 〉 do retrieve( pending , delivered ); trigger 〈 lurb ,Deliver | delivered 〉; forall (s, m ) ∈ pending do trigger 〈 sbeb ,Broadcast | [D ATA , s, m ] 〉; upon event 〈 lurb ,Broadcast | m 〉 do p... | {
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and this will ensure the retransmission to all correct processes. Correctness. Consider the agreement property and assume that some correct pro-cess p log-delivers a message m. To do so, a majority of the processes must have 92 3 Reliable Broadcast retransmitted the message. As we assume a majority of the processes is ... | {
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for large-scale systems or systems exposed to attacks. As we will see, it is often possible to build scalable probabilistic algorithms that exploit randomization and provide good reliability guarantees. Moreover, the abstractions considered in this book can almost never be mapped to physical systems in real deployments... | {
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messages and to collect acknowledgments, for instance, by arranging the processes in a binary tree, as illustrated in Fig. 3.5b. Hierarchies can reduce the load of each process but increase the latency of the com-munication protocol. Additionally, hierarchies need to be reconfigured when faults occur (which may not be ... | {
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broadcasts a message m, the probability that every correct process eventually delivers mis at least 1−ε. > PB2: No duplication: No message is delivered more than once. > PB3: No creation: If a process delivers a message mwith sender s, then mwas previously broadcast by process s. 3.8.3 Specification The probabilistic b... | {
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processes. It returns k random samples chosen from Π \ { self } according to the uniform distribution without replacement. The 3.8 Probabilistic Broadcast 95 Algorithm 3.9: Eager Probabilistic Broadcast > Implements: > ProbabilisticBroadcast, instance pb . > Uses: > FairLossPointToPointLinks, instance fll . > upon even... | {
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amount of redundancy in the message exchanges: any given process may receive the same message many times. A three-round execution of the algorithm with fanout three is illustrated in Fig. 3.6 for a system consisting of nine processes. However, increasing the fanout is costly. The higher the fanout, the higher the load ... | {
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derive a simple estimate of the probability that a particular correct pro-cess delivers a message. Suppose that the underlying fair-loss links deliver every message sent by the first infected correct process (i.e., the original sender) but no further message; in other words, only the sender disseminates the broadcast m... | {
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1 , the infections in round r + 1 spread about as fast as if one process would have infected the others during additional Ir rounds. Summing this up over all R rounds, we obtain our second estimate: the probability of some correct process being infected after R rounds is about E2 = 1 − (1 − γ)∑ R−1 > r=0 Ir . The two e... | {
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∅;stored := ∅; > procedure gossip (msg ) is forall t ∈ picktargets (k) do trigger 〈 fll ,Send | t, msg 〉; > upon event 〈pb , Broadcast | m 〉 do > lsn := lsn + 1 ; > trigger 〈upb , Broadcast | [D ATA , self , m, lsn ] 〉; > upon event 〈upb , Deliver | p, [D ATA , s, m, sn ] 〉 do if random ([0 , 1]) > α then > stored := s... | {
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processes, say, half of them, obtains the message after the first phase. The primitive could typically be implemented on top of fair-loss links (as the “Eager Probabilistic Broadcast” algorithm) and should work efficiently, that is, not cause an excessive amount of redundant message transmissions. Algorithm 3.10–3.11, ... | {
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sequence numbers associated with messages. The array variable next contains an entry for every process p with the sequence num-ber of the next message to be pb -delivered from sender p. The process detects that it has missed one or more messages from p when the process receives a message from p with a larger sequence n... | {
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through retransmissions or delayed messages from s)and that should be processed; a more complete implementation would deliver these messages and remove them from pending . Correctness. The no creation and no duplication properties follow from the under-lying point-to-point links and the use of sequence numbers. The pro... | {
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in the link to the LAN). Similarly, the retransmission procedure, instead of being completely random, may search first for a copy in the local LAN and only afterward at more distant processes. # 3.9 FIFO and Causal Broadcast So far, we have not considered any ordering guarantee among messages delivered by different pro... | {
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Title: URL Source: Markdown Content: # Some problems I have proposed # Warut Suksompong 1. Asian Pacific Mathematics Olympiad 2022, Problem 3 Find all positive integers k r .3. Crux Mathematicorum, September 2021, Problem 4669 For a given positive integer n, a 4 n × 4n table is partitioned into 16 n2 unit squares, ea... | {
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paths in a diverse collection of paths from A to B. Similarly, let NBA denote the maximal number of paths in a diverse collection of paths from B to A.Prove that the equality NAB = NBA holds if and only if the number of roads going out from A is the same as the number of roads going out from B.5. Asian Pacific Mathemat... | {
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∈ C such that a(r, c ) ≤ a(r, c ′) for all r ∈ R.A saddle pair ( R, C ) is called a minimal pair if for each saddle pair ( R′, C ′) with R′ ⊆ R and C′ ⊆ C, we have R′ = R and C′ = C.Prove that any two minimal pairs contain the same number of rows. 8. Asian Pacific Mathematics Olympiad 2020, Problem 3 Determine all posi... | {
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b1 ≥ · · · ≥ bn ≥ 0 be integers such that 1. a1 + · · · + ai ≥ b1 + · · · + bi for all i = 1 , . . . , n − 1; 2. a1 + · · · + an = b1 + · · · + bn.Assume that there are n boxes, with box i containing ai balls. In each move, Alice is allowed to take two boxes with an unequal number of balls, and move one ball from the b... | {
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with at most one outgoing road. Prove that she can always find out by asking at most 4 n questions. 15. Asian Pacific Mathematics Olympiad 2019, Problem 1 Let Z+ be the set of positive integers. Determine all functions f : Z+ → Z+ such that a2 + f (a)f (b) is divisible by f (a) + b for all positive integers a and b.16.... | {
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given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of n.19. International Mathematical Olympiad 2017 Shortlist, Problem N3 Determine all integers n ≥ 2 with the following property: for any integers a1, a 2, . . . , a n whose sum is not ... | {
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coefficients such that for any positive integer n ≥ 2016, the integer P (n) is positive and S(P (n)) = P (S(n)) . 23. Asian Pacific Mathematics Olympiad 2016, Problem 3 Let AB and AC be two distinct rays not lying on the same line, and let ω be a circle with center O that is tangent to ray AC at E and ray AB at F . Let... | {
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2015, Problem 4 (with Pakawut Jiradilok) Let n be a positive integer. Consider 2 n distinct lines on the plane, no two of which are parallel. Of the 2 n lines, n are colored blue, the other n are colored red. Let B be the set of all points on the plane that lie on at least one blue line, and R the set of all points on ... | {
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Potcharapol Suteparuk) Let ABC be an acute-angled triangle with orthocenter H, and let W be a point on the side BC , lying strictly between B and C. The points M and N are the feet of the altitudes from B and C, respectively. Denote by ω1 the circumcircle of BW N ,and let X be the point on ω1 such that W X is a diamete... | {
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can perform only finitely many such iterations. 33. International Mathematical Olympiad 2012 Shortlist, Problem N1 Call admissible a set A of integers that has the following property: If x, y ∈ A (possibly x = y) then x2 + kxy + y2 ∈ A for every integer k.Determine all pairs m, n of nonzero integers such that the only ... | {
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Title: Proof by Contrapositive: If n^2 is Even then n is Even URL Source: Markdown Content: Proof by Contrapositive: If n^2 is Even then n is Even - YouTube =============== • NaN / NaN Back ![Image 1 SA Skip navigation Search Search with your voice Sign in SA Proof by Contrapositive: If n^2 is Even then n is Even Sear... | {
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TikTok!]( Show less []( Proof by Contrapositive: If n^2 is Even then n is Even ====================================================== 38,934 views 38K views Jul 9, 2021 620 Share Download Download Save 40 Comments ----------- Sort comments Sort by Top comments Newest first !Image 11: Default profile photo]( Can't wait ... | {
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Reply ![Image 29]( · 1 reply ![Image 30]( · 1 reply ![Image 31]( ### [@Zero-ef4sc]( [5 months ago]( Why can't n^2 = 2 which disproves the statement? Since squareroot of 2 is irrational, it isn't even. Though, I'd assume that we're limiting ourselves to integers in which case it makes sense, but that's never stated. Sho... | {
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![Image 54]( ❤ by Wrath of Math Reply ![Image 55]( · 1 reply ![Image 56]( · 1 reply ![Image 57]( ### [@SeeTv.]( [3 years ago]( In this case couldn't you just prove the converse, i.e. "when n is even then n^2 is even" ? Show less Read more Like 1 Dislike ![Image 58]( ❤ by Wrath of Math Reply ![Image 59]( · 4 replies ![I... | {
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exclusive benefits: [![Image 78]( ◆ Donate on PayPal: [ Thanks to Robert Rennie, Barbara Sharrock, and Rolf Waefler for their generous support on Patreon! Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: [ Follow Wrath of Math on... ●... | {
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7:55 7:55 Now playing]( ### 1.2.1 Proof by Contradiction MIT OpenCourseWare MIT OpenCourseWare • • 116K views 8 years ago Nakia Rimmer (CalcCoach.com) • • 37K views 6 years ago]( ![Image 98 BriTheMathGuy BriTheMathGuy Verified • • 363K views 9 months ago]( [![Image 105]( 6:45 6:45 Now playing]( [### Directly Prove that... | {
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ago]( [![Image 106]( 18:46 18:46 Now playing]( [### Proof By Cases | 2 Examples Brain Gainz Brain Gainz • • 12K views 4 years ago]( Show more | {
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Title: How do I prove the statement "All even integers are divisible by 2." using proof by contradiction? : r/learnmath URL Source: Markdown Content: How do I prove the statement "All even integers are divisible by 2." using proof by contradiction? : r/learnmath =============== Skip to main content. * * * 386K Members... | {
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New * Controversial * Old * Q&A ![Image 6: u/theadamabrams avatar definition of "even integer"? Without knowing that, it's impossible to say what kind of proof to write. "An even integer is an integer that is divisible by two" is a perfectly valid definition, and in that case there is literally nothing to prove, but th... | {
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to prove anything using the method of contradiction, we start by assuming the opposite of what we are trying to prove. When we continue to progress with our assumption, we suddenly come at a contradiction that proves that our assumption is wrong and the fact that we made our assumption against is correct. So let's supp... | {
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to Reddit? Create your account and connect with a world of communities. المواصلة باستخدام Google المواصلة باستخدام Google Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreement. * * * 386K Members Online [### Proving that the sum of two even integers gives an even integer]( 1 upvot... | {
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good guys and odd look like bad guys .]( 105 upvotes ·63 comments * * * * How can I prove that no odd number is even?. * * * 386K Members Online ### How can I prove that no odd number is even?. * * * 386K Members Online [### Proof that the sum of consecutive numbers cannot be powers of 2?]( 35 upvotes ·55 comments * * ... | {
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online subscribers). * * * 386K Members Online ### The sum of two even integers is even, the sum of an even and an odd integer is odd, and the sum of two odd integers is even. What is the generalization of this statement to residue classes mod 3?. * * * 386K Members Online ### Why is it that a number whose digits add u... | {
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r/learnmath]( yr. ago !Image 32: r/learnmath icon. * * * 386K Members Online ### Factoring: Even numbers are always divisible by 2, why aren't odd numbers always divisible by some fixed number besides 1? (that aren't primes). * * * 386K Members Online ### Can someone explain how subtracting integers work? have two answ... | {
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* * * * Help understanding endless primes proof from book. * * * 386K Members Online ### Help understanding endless primes proof from book = 1 work when proving sqrt(2) is irrational?]( 47: r/learnmath icon]( r/learnmath]( yr. ago !Image 48: r/learnmath icon. * * * 386K Members Online [### Why does the assumption gcd(a... | {
"page_id": null,
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0 to N in descending order.]( 6 comments * * * * If we erased all math, how different do you think it would eventually be?. * * * 386K Members Online ### If we erased all math, how different do you think it would eventually be?. * * * 386K Members Online ### How do I know how many times a number is divisible by 2 till ... | {
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* * Is -1 an odd number * [Amazing]( * [Animals & Pets]( * [Cringe & Facepalm]( * [Funny]( * [Interesting]( * [Memes]( * [Oddly Satisfying]( * [Reddit Meta]( * [Wholesome & Heartwarming]( * Games * [Action Games]( * [Adventure Games]( * [Esports]( * [Gaming Consoles & Gear]( * [Gaming News & Discussion]( * [Mobile Game... | {
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Title: URL Source: Markdown Content: Math 2001, January 14, 2020 # A very first proof Definition. An integer n is called even if it has the form n = 2 k for some integer k. Theorem. If n is an even integer, then n2 is an even integer. Proof. Suppose n is an even integer. Then n has the form n = 2 k for some integer k.... | {
"page_id": null,
"source": 7353,
"title": "from dpo"
} |
could be any even integer at all. Now, since it is even, it must be twice another integer. Let’s call the new one k. In other words, n is twice k, which we can write as n = 2 k.Next, the equation n = 2 k can be squared on both sides to yield another correct equation, namely: n2 = (2 k)2.We can rewrite the right-hand si... | {
"page_id": null,
"source": 7353,
"title": "from dpo"
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Title: Renaming URL Source: Markdown Content: **Note:** You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes and output y i, with the requirements: Termination Every nonfaulty process eventually decides. Uniqueness If p i ≠ p j then y i ≠ y j. Anonymity The code executed b... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
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of processes p π(1)..p π(n) with inputs x 1..x n in which p π(i) performs exactly the same operations as p i and obtains the same output y i. The last condition is like non-triviality for consensus: it excludes algorithms where p i just returns i in all executions. Typically we do not have to do much to prove anonymity... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
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induction on n. For n=1, we use 2 1-1=1 names; this is the base case. For larger n, suppose we use m names, and consider an execution in which one process p n runs to completion first. This consumes one name y n and leaves k names less than y n and m-k-1 names greater than y n. By setting all the inputs x i for i = 2 n... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
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a name yet. If a process proposes a name and finds that no other process has proposed the same name, it takes it; otherwise it chooses a new name by first computing its rank r among the active processes and then choosing the r-th unused name. Because the rank is at most n and there are at most n-1 names used by the oth... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
} |
and compute the same rank r for itself. Now look at the trying process i with the smallest original name, and suppose it has rank r. Let F = { z 1< z 2 ... } be the set of "free names" that are not proposed in a by any of the finished processes. Observe that no trying process j != i ever proposes a name in { z 1 ... z ... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
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same process, if it happens to run choose-name again). Now the bound on the number of names needed is 2k-1, where k is the maximum number of concurrently active processes. The algorithm above can be converted to a long-lived renaming algorithm by adding the following release-name procedure: procedure release-name(i) * ... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
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keep it—which means that with k processes, no process reaches row k+1 or column k+1. Here is the actual code for a splitter. We use two atomic registers: a register X, initialized to ⊥, that can hold original names, and a register Y, initialized to false, that holds a single flag bit (the register names are from the pa... | {
"page_id": null,
"source": 7357,
"title": "from dpo"
} |
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