Brower Fixed Point Theorem

brower定理证明在数学中，布劳威尔不动点定理是拓扑学里一个非常重要的不动点定理，它可应用到有限维空间并构成了一般不动点定理的基石。

布劳威尔不动点定理得名于荷兰数学家鲁伊兹·布劳威尔（英语：L. E. J. Brouwer）。

布劳威尔不动点定理说明：对于一个拓扑空间中满足一定条件的连续函数f，存在一个点x0，使得f(x0) = x0。

布劳威尔不动点定理最简单的形式是对一个从某个圆盘D射到它自身的函数f。

而更为广义的定理则对于所有的从某个欧几里得空间的凸紧子集射到它自身的函数都成立。

●定理表述不动点定理（fixed-point theorem）：对应于一个定义于集合到其自身上的映射而言，所谓不动点，是指经过该映射保持“不变的”点。

不动点定理是用于判断一个函数是否存在不动点的定理。

常用的不动点定理有：(1)布劳威尔不动点定理(1910年)：若A⊂R(N维实数集合)且A为非空、紧凸集，f：A→A是一个从A到A的连续函数，则该函数f(·)有一个不动点，即存在x∈A，x=f(x)。

该定理常被用于证明竞争性均衡的存在性。

(2)角谷(kakutani)不动点定理(1941年)：若A⊂R且A为非空、紧凸集，f ：A→A是从A到A的一个上半连续对应，且f(x)⊂A对于任意x∈A是一个非空的凸集，则f(·)存在一个不动点。

不动点定理一般只给出解的存在性判断，至于如何求解，则需要用到20世纪60年代末斯卡夫(H.E.Scarf)提出的不动点算法。

因此，不动点定理常被用于解决经济模型中出现的存在性问题，例如多人非合作对策中均衡点的存在性等。

数学定义设(A，d)为完备的度量空间，f为从A到其自身中的李普希茨映射。

如果李普希茨比的级数λ(fn)收敛，则存在A的唯一的点a，在f下该点不动。

其次，对A的任一元素x0，由递推关系：定义的级数(xn)必收敛于a。

这一定理尤其适用于f为压缩映射的情况。

二分皿平板对峙法英文English:"The binary opposition method, also known as the dichotomy method in physics experiments, particularly in electrostatics, involves two flat plates facing each other. This experimental setup is primarily used to study the properties and behaviors of electric fields between charged plates. In a typical experiment, the two plates are charged with opposite types of charges, which creates a uniform electric field in the region between them. This setup provides a controlled environment to measure the strength and direction of electric fields, which is crucial for understanding fundamental electrostatic principles. It also allows for the exploration of other phenomena, such as the effect of dielectric materials inserted between the plates and the force experienced by charged particles placed in the field. The clarity and simplicity of the binary opposition method make it an indispensable tool in both educational and research settings for illustrating complex electrostatic interactions in a visually and conceptually accessible manner. By adjusting variables like the distance between the plates, the amount of charge on each plate,and the medium between the plates, researchers can investigate a range of electrostatic effects and principles. This method's practical applications extend to designing capacitors, where the understanding of electric fields between plates is crucial for optimizing performance. Moreover, this method fosters a deeper comprehension of the theoretical aspects of electric fields, aiding in the development of more advanced technologies and solutions in electrical engineering and related fields."中文翻译:"二分皿平板对峙法，也称为物理实验中的二分法，特别是在静电学中，涉及两个平板面对面放置。

USER GUIDEPMP/PTP 450 SeriesCovers:PMP 450 AP / PMP 450 SM / PTP 450 BH / PMP 450d PMP 450i / PTP 450iPMP/PTP 450b Mid-Gain / 450b High-GainPMP 450mPMP 430AccuracyWhile reasonable efforts have been made to assure the accuracy of this document, Cambium Networks assumes no liability resulting from any inaccuracies or omissions in this document, or from use of the information obtained herein. Cambium reserves the right to make changes to any products described herein to improve reliability, function, or design, and reserves the right to revise this document and to make changes from time to time in content hereof with no obligation to notify any person of revisions or changes. Cambium does not assume any liability arising out of the application or use of any product, software, or circuit described herein; neither does it convey license under its patent rights or the rights of others. 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All Rights Reserved.ContentsContents (i)List of Figures (xiv)List of Tables (xx)About This User Guide (1)Contacting Cambium Networks (1)Purpose (2)Product notation conventions in document (2)Cross references (3)Feedback (3)Important regulatory information (4)Application software (4)USA specific information (4)Canada specific information (5)Renseignements specifiques au Canada (6)EU Declaration of Conformity (7)Specific expertise and training for professional installers (7)Ethernet networking skills (8)Lightning protection (8)Training (8)Problems and warranty (9)Reporting problems (9)Repair and service (9)Hardware warranty (9)Security advice (10)Warnings, cautions, and notes (11)Warnings (11)Cautions (11)Notes (11)Caring for the environment (12)In EU countries (12)In non-EU countries (12)Chapter 1:Product description................................................................................................... 1-1 Overview of the 450 Platform Family .......................................................................................... 1-2 Purpose........................................................................................................................................ 1-2 PMP 450m Series....................................................................................................................... 1-2 PMP/PTP 450i Series ................................................................................................................ 1-4 PMP/PTP 450b Series............................................................................................................... 1-8PMP/PTP 450 Series ................................................................................................................. 1-9 Supported interoperability for 450m/450i/450b/450 Series........................................ 1-12 Typical deployment................................................................................................................. 1-13 Product variants....................................................................................................................... 1-15 Wireless operation ......................................................................................................................... 1-16 Time division duplexing.......................................................................................................... 1-16 Encryption................................................................................................................................. 1-19 MIMO.......................................................................................................................................... 1-19 MU-MIMO................................................................................................................................... 1-19 System management ..................................................................................................................... 1-21 Management agent.................................................................................................................. 1-21 Web server................................................................................................................................ 1-21 Remote Authentication Dial-in User Service (RADIUS).................................................... 1-23 Network Time Protocol (NTP)............................................................................................... 1-23 cnMaestro™ ............................................................................................................................... 1-24 Wireless Manager (WM)......................................................................................................... 1-24 Radio recovery mode.............................................................................................................. 1-26 Chapter 2:System hardware ....................................................................................................... 2-1 System Components........................................................................................................................ 2-2 Point-to-Multipoint (PMP)........................................................................................................ 2-2 Backhaul (PTP)........................................................................................................................... 2-6 450 Platform Family interfaces............................................................................................... 2-8 ATEX/HAZLOC variants......................................................................................................... 2-18 Diagnostic LEDs....................................................................................................................... 2-19 Power supply options ............................................................................................................. 2-23 ODU mounting brackets & accessories ............................................................................... 2-32 Lightning protection ............................................................................................................... 2-32 ODU interfaces................................................................................................................................ 2-34 PMP 450m Series 5 GHz AP .................................................................................................. 2-34 PMP 450m Series 3GHz AP ................................................................................................... 2-35 PMP/PTP 450i.......................................................................................................................... 2-36 PMP/PTP 450b Mid-Gain SM................................................................................................. 2-38 PMP/PTP 450b High Gain SM ............................................................................................... 2-39 Cabling.............................................................................................................................................. 2-40 Ethernet standards and cable lengths................................................................................. 2-40 Outdoor copper Cat5e Ethernet cable................................................................................ 2-41 SFP module kits ....................................................................................................................... 2-42 Main Ethernet port................................................................................................................... 2-44 Aux port .................................................................................................................................... 2-44 Ethernet cable testing............................................................................................................. 2-48 Lightning protection unit (LPU) and grounding kit ................................................................. 2-50 DC LPU and Grounding Kit .................................................................................................... 2-51 Cable grounding kit................................................................................................................. 2-53Antennas and antenna cabling..................................................................................................... 2-54 Antenna requirements ............................................................................................................ 2-54 Supported external AP antennas.......................................................................................... 2-54 Supported external BH/SM antenna .................................................................................... 2-54 RF cable and connectors........................................................................................................ 2-55 Antenna accessories ............................................................................................................... 2-55 GPS synchronization...................................................................................................................... 2-56 GPS synchronization description.......................................................................................... 2-56 Universal GPS (UGPS)............................................................................................................ 2-56 CMM5 ......................................................................................................................................... 2-58 CMM5 Controller Module........................................................................................................ 2-61 CMM5 Injector Module............................................................................................................ 2-62 CMM5 Injector Compatibility Matrix..................................................................................... 2-62 CMM5 Specifications............................................................................................................... 2-63 CMM4 (Rack Mount) .............................................................................................................. 2-65 CMM4 (Cabinet with switch)............................................................................................... 2-68 CMM4 (Cabinet without switch)......................................................................................... 2-68 CMM3/CMMmicro .................................................................................................................... 2-69 Installing a GPS receiver................................................................................................................ 2-71 GPS receiver location.............................................................................................................. 2-71 Mounting the GPS receiver .................................................................................................... 2-72 Cabling the GPS Antenna....................................................................................................... 2-73 Installing and connecting the GPS LPU............................................................................... 2-73 Ordering the components............................................................................................................. 2-74 Chapter 3:System planning......................................................................................................... 3-1 Typical deployment.......................................................................................................................... 3-2 ODU with PoE interface to PSU .............................................................................................. 3-2 Site planning...................................................................................................................................... 3-7 Site selection for PMP/PTP radios.......................................................................................... 3-7 Power supply site selection ..................................................................................................... 3-8 Maximum cable lengths............................................................................................................ 3-8 Grounding and lightning protection....................................................................................... 3-8 ODU and external antenna location..................................................................................... 3-10 ODU ambient temperature limits.......................................................................................... 3-10 ODU wind loading ................................................................................................................... 3-11 Hazardous locations................................................................................................................ 3-16 Drop cable grounding points................................................................................................. 3-16 Lightning Protection Unit (LPU) location............................................................................ 3-17 Radio Frequency planning............................................................................................................ 3-18 Regulatory limits...................................................................................................................... 3-18 Conforming to the limits ........................................................................................................ 3-18 Available spectrum.................................................................................................................. 3-18 Analyzing the RF Environment ............................................................................................. 3-19Channel bandwidth ................................................................................................................. 3-19 Anticipating Reflection of Radio Waves ............................................................................. 3-19 Obstructions in the Fresnel Zone ......................................................................................... 3-20 Planning for co-location ......................................................................................................... 3-20 Multiple OFDM Access Point Clusters.................................................................................. 3-21 Considerations on back-to-back frequency reuse............................................................. 3-23 Link planning ................................................................................................................................... 3-28 Range and obstacles............................................................................................................... 3-28 Path loss .................................................................................................................................... 3-28 Calculating Link Loss .............................................................................................................. 3-29 Calculating Rx Signal Level.................................................................................................... 3-29 Calculating Fade Margin......................................................................................................... 3-30 Adaptive modulation .............................................................................................................. 3-30 Planning for connectorized units................................................................................................. 3-31 When to install connectorized units..................................................................................... 3-31 Choosing external antennas .................................................................................................. 3-31 Calculating RF cable length (5.8 GHz FCC only)............................................................... 3-31 Data network planning .................................................................................................................. 3-33 Understanding addresses....................................................................................................... 3-33 Dynamic or static addressing................................................................................................ 3-33 DNS Client................................................................................................................................. 3-34 Network Address Translation (NAT).................................................................................... 3-34 Developing an IP addressing scheme .................................................................................. 3-35 Address Resolution Protocol................................................................................................. 3-36 Allocating subnets................................................................................................................... 3-36 Selecting non-routable IP addresses ................................................................................... 3-36 Translation bridging................................................................................................................ 3-37 Engineering VLANs ................................................................................................................. 3-37 Network management planning .................................................................................................. 3-41 Planning for SNMP operation ................................................................................................ 3-41 Enabling SNMP......................................................................................................................... 3-41 Security planning............................................................................................................................ 3-42 Isolating AP/BHM from the Internet .................................................................................... 3-42 Encrypting radio transmissions............................................................................................. 3-42 Planning for HTTPS operation............................................................................................... 3-43 Planning for SNMPv3 operation............................................................................................ 3-43 Managing module access by passwords ............................................................................. 3-44 Planning for RADIUS operation............................................................................................. 3-45 Filtering protocols and ports................................................................................................. 3-45 Encrypting downlink broadcasts .......................................................................................... 3-49 Isolating SMs in PMP ............................................................................................................... 3-49 Filtering management through Ethernet............................................................................. 3-49 Allowing management from only specified IP addresses ................................................ 3-50Configuring management IP by DHCP ................................................................................ 3-50 Controlling PPPoE PADI Downlink Forwarding ................................................................. 3-51 Remote AP Deployment................................................................................................................ 3-52 Remote AP (RAP) Performance ........................................................................................... 3-53 Example Use Case for RF Obstructions............................................................................... 3-53 Example Use Case for Passing Sync .................................................................................... 3-54 Physical Connections Involving the Remote AP................................................................ 3-55 Passing Sync signal ................................................................................................................. 3-57 Wiring to Extend Network Sync ........................................................................................... 3-60 Chapter 4:Legal and regulatory information........................................................................... 4-1 Cambium Networks end user license agreement....................................................................... 4-2 Definitions ................................................................................................................................... 4-2 Acceptance of this agreement................................................................................................ 4-2 Grant of license .......................................................................................................................... 4-2 Conditions of use....................................................................................................................... 4-3 Title and restrictions ................................................................................................................. 4-4 Confidentiality............................................................................................................................ 4-4 Right to use Cambium’s name................................................................................................. 4-5 Transfer ....................................................................................................................................... 4-5 Updates ....................................................................................................................................... 4-5 Maintenance................................................................................................................................ 4-5 Disclaimer.................................................................................................................................... 4-6 Limitation of liability.................................................................................................................. 4-6 U.S. government ........................................................................................................................ 4-6 Term of license........................................................................................................................... 4-7 Governing law ............................................................................................................................ 4-7 Assignment................................................................................................................................. 4-7 Survival of provisions................................................................................................................ 4-7 Entire agreement....................................................................................................................... 4-7 Third party software ................................................................................................................. 4-7 Compliance with safety standards .............................................................................................. 4-22 Electrical safety compliance.................................................................................................. 4-22 Electromagnetic compatibility (EMC) compliance............................................................ 4-22 Human exposure to radio frequency energy...................................................................... 4-23 Hazardous location compliance............................................................................................ 4-37 Compliance with radio regulations.............................................................................................. 4-39 Type approvals......................................................................................................................... 4-40 Brazil specific information...................................................................................................... 4-41 Australia Notification .............................................................................................................. 4-41 Regulatory Requirements for CEPT Member States () ........................... 4-41 Chapter 5:Preparing for installation.......................................................................................... 5-1 Safety.................................................................................................................................................. 5-2 Hazardous locations.................................................................................................................. 5-2。

Lecture VI:Existence of Nash equilibriumMarkus M.M¨o biusFebruary26,2008•Osborne,chapter4•Gibbons,sections1.3.B1Nash’s Existence TheoremWhen we introduced the notion of Nash equilibrium the idea was to come up with a solution concept which is stronger than IDSDS.Today we show that NE is not too strong in the sense that it guarantees the existence of at least one mixed Nash equilibrium in most games(for sure in allfinite games). This is reassuring because it tells that there is at least one way to play most games.1Let’s start by stating the main theorem we will prove:Theorem1(Nash Existence)Everyfinite strategic-form game has a mixed-strategy Nash equilibrium.Many game theorists therefore regard the set of NE for this reason as the lower bound for the set of reasonably solution concept.A lot of research has gone into refining the notion of NE in order to retain the existence result but get more precise predictions in games with multiple equilibria(such as coordination games).However,we have already discussed games which are solvable by IDSDS and hence have a unique Nash equilibrium as well(for example,the two thirds of the average game),but subjects in an experiment will not follow those equilibrium prescription.Therefore,if we want to describe and predict 1Note,that a pure Nash equilibrium is a(degenerate)mixed equilibrium,too.1the behavior of real-world people rather than come up with an explanation of how they should play a game,then the notion of NE and even even IDSDS can be too restricting.Behavioral game theory has tried to weaken the joint assumptions of rationality and common knowledge in order to come up with better theories of how real people play real games.Anyone interested should take David Laibson’s course next year.Despite these reservation about Nash equilibrium it is still a very useful benchmark and a starting point for any game analysis.In the following we will go through three proofs of the Existence Theorem using various levels of mathematical sophistication:•existence in2×2games using elementary techniques•existence in2×2games using afixed point approach•general existence theorem infinite gamesYou are only required to understand the simplest approach.The rest is for the intellectually curious.2Nash Existence in2×2GamesLet us consider the simple2×2game which we discussed in the previousequilibria:lecture on mixed Nash2Let’s find the best-response of player 2to player 1playing strategy α:u 2(L,αU +(1−α)D )=2−αu 2(R,αU +(1−α)D )=1+3α(1)Therefore,player 2will strictly prefer strategy L iff2−α>1+3αwhich implies α<14.The best-response correspondence of player 2is therefore:BR 2(α)=⎧⎨⎩1if α<14[0,1]if α=140if α>14(2)We can similarly find the best-response correspondence of player 1:BR 1(β)=⎧⎨⎩0if β<23[0,1]if β=231if β>23(3)We draw both best-response correspondences in a single graph (the graph is in color -so looking at it on the computer screen might help you):We immediately see,that both correspondences intersect in the single point α=14and β=23which is therefore the unique (mixed)Nash equilibrium of the game.3What’s useful about this approach is that it generalizes to a proof that any two by two game has at least one Nash equilibriu,i.e.its two best response correspondences have to intersect in at least one point.An informal argument runs as follows:1.The best response correspondence for player2maps eachαinto atleast oneβ.The graph of the correspondence connects the left and right side of the square[0,1]×[0,1].This connection is continuous -the only discontinuity could happen when player2’s best response switches from L to R or vice versa at someα∗.But at this switching point player2has to be exactly indifferent between both strategies-hence the graph has the value BR2(α∗)=[0,1]at this point and there cannot be a discontinuity.Note,that this is precisely why we need mixed strategies-with pure strategies the BR graph would generally be discontinuous at some point.2.By an analogous argument the BR graph of player1connects the upperand lower side of the square[0,1]×[0,1].3.Two lines which connect the left/right side and the upper/lower sideof the square respectively have to intersect in at least one point.Hence each2by2game has a mixed Nash equilibrium.3Nash Existence in2×2Games using Fixed Point ArgumentThere is a different way to prove existence of NE on2×2games.The advantage of this new approach is that it generalizes easily to generalfinite games.Consider any strategy profile(αU+(1−α)D,βL+(1−β)R)represented by the point(α,β)inside the square[0,1]×[0,1].Now imagine the following: player1assumes that player2follows strategyβand player2assumes that player1follows strategyα.What should they do?They should play their BR to their beliefs-i.e.player1should play BR1(β)and player2should play BR2(α).So we can imagine that the strategy profile(α,β)is mapped onto(BR1(β),BR2(α)).This would describe the actual play of both players if their beliefs would be summarizes by(α,β).We can therefore define a4giant correspondence BR:[0,1]×[0,1]→[0,1]×[0,1]in the following way:BR(α,β)=BR1(β)×BR2(α)(4) The followingfigure illustrates the properties of the combined best-response map BR:The neat fact about BR is that the Nash equilibriaσ∗are precisely the fixed points of BR,i.e.σ∗∈BR(σ∗).In other words,if players have beliefs σ∗thenσ∗should also be a best response by them.The next lemma followsdirectly from the definition of mixed Nash equilibrium:Lemma1A mixed strategy profileσ∗is a Nash equilibrium if and only if it is afixed point of the BR correspondence,i.e.σ∗∈BR(σ∗).We therefore look precisely for thefixed points of the correspondence BR which maps the square[0,1]×[0,1]onto itself.There is well developed mathematical theory for these types of maps which we utilize to prove Nash existence(i.e.that BR has at least onefixed point).3.1Kakutani’s Fixed Point TheoremThe key result we need is Kakutani’sfixed point theorem.You might have used Brower’sfixed point theorem in some mathematics class.This is not5sufficient for proving the existence of nash equilibria because it only applies to functions but not to correspondences.Theorem2Kakutani A correspondence r:X→X has afixed point x∈X such that x∈r(x)if1.X is a compact,convex and non-empty subset of n.2.r(x)is non-empty for all x.3.r(x)is convex for all x.4.r has a closed graph.There are a few concepts in this definition which have to be defined: Convex Set:A set A⊆ n is convex if for any two points x,y∈A the straight line connecting these two points lies inside the set as well.Formally,λx+(1−λ)y∈A for allλ∈[0,1].Closed Set:A set A⊆ n is closed if for any converging sequence {x n}∞n=1with x n→x∗as n→∞we have x∗∈A.Closed intervals such as[0,1]are closed sets but open or half-open intervals are not.For example(0,1]cannot be closed because the sequence1n converges to0which is not inthe set.Compact Set:A set A⊆ n is compact if it is both closed and bounded.For example,the set[0,1]is compact but the set[0,∞)is only closed but unbounded,and hence not compact.Graph:The graph of a correspondence r:X→Y is the set{(x,y)|y∈r(x)}. If r is a real function the graph is simply the plot of the function.Closed Graph:A correspondence has a closed graph if the graph of the correspondence is a closed set.Formally,this implies that for a sequence of point on the graph{(x n,y n)}∞n=1such that x n→x∗and y n→y∗as n→∞we have y∗∈r(x∗).2It is useful to understand exactly why we need each of the conditions in Kakutani’sfixed point theorem to be fulfilled.We discuss the conditions by looking correspondences on the real line,i.e.r: → .In this case,afixed point simply lies on the intersection between the graph of the correspondenceand the diagonal y=x.Hence Kakutani’sfixed point theorem tells us that 2If the correspondence is a function then the closed graph requirement is equivalent to assuming that the function is continuous.It’s easy to see that a continuous function hasa closed graph.For the reverse,you’ll need Baire’s category theorem.6a correspondence r:[0,1]→[0,1]which fulfills the conditions above always intersects with the diagonal.3.1.1Kakutani Condition I:X is compact,convex and non-empty. Assume X is not compact because it is not closed-for example X=(0,1). Now consider the correspondence r(x)=x2which maps X into X.However, it has nofixed point.Now consider X non-compact because it is unbounded such as X=[0,∞)and consider the correspondence r(x)=1+x which maps X into X but has again nofixed point.If X is empty there is clearly nofixed point.For convexity of X look atthe example X=[0,13]∪[23,1]which is not convex because the set has a hole.Now consider the following correspondence(seefigure below):r(x)=34if x∈[0,13]14if x∈[23,1](5)This correspondence maps X into X but has nofixed point again.From now on we focus on correspondences r:[0,1]→[0,1]-note that[0,1] is closed and bounded and hence compact,and is also convex.73.1.2Kakutani Condition II:r (x )is non-empty.If r (x )could be empty we could define a correspondence r :[0,1]→[0,1]such as the following:r (x )=⎧⎨⎩34if x ∈[0,13]∅if x ∈[13,23]14if x ∈[23,1](6)As before,this correspondence has no fixed point because of the hole in the middle.3.1.3Kakutani Condition III:r (x )is convex.If r (x )is not convex,then the graph does not have to have a fixed point as the following example of a correspondence r :[0,1]→[0,1]shows:r (x )=⎧⎨⎩1if x <12 0,13 ∪ 23,1 if x =120if x >12(7)The graph is non-convex because r (12)is not convex.It also does not have a fixed point.83.1.4Kakutani Condition IV:r(x)has a closed graph.This condition ensures that the graph cannot have holes.Consider the follow-ing correspondence r:[0,1]→[0,1]which fulfills all conditions of Kakutaniexcept(4):r(x)=⎧⎨⎩12if x<1214,12if x=1214if x>12(8)Note,that r(12)is the convex set14,12but that this set is not closed.Hencethe graph is not closed.For example,consider the sequence x n=12andy n=12−1n+2for n≥1.Clearly,we have y n∈r(x n).However,x n→x∗=12and y n→y∗=12but y∗/∈r(x∗).Hence the graph is not closed.3.2Applying KakutaniWe now apply Kakutani to prove that2×2games have a Nash equilibrium, i.e.the giant best-response correspondence BR has afixed point.We denote the strategies of player1with U and D and the strategies of player2with L and R.9We have to check(a)that BR is a map from some compact and convex set X into itself,and(b)conditions(1)to(4)of Kakutani.•First note,that BR:[0,1]×[0,1]→[0,1]×[0,1].The square X= [0,1]×[0,1]is convex and compact because it is bounded and closed.•Now check condition(2)of Kakutani-BR(σ)is non-empty.This is true if BR1(σ2)and BR2(σ1)are non-empty.Let’s prove it for BR1-the proof for BR2is analogous.Player1will get the following payoffu1,β(α)from playing strategyαif the other player playsβ:u1,β(α)=αβu1(U,L)+α(1−β)u1(U,R)++(1−α)βu1(D,L)+(1−α)(1−β)u1(D,R)(9) The function u1,βis continuous inα.We also know thatα∈[0,1]which is a closed interval.Therefore,we know that the continuous function u1,βreaches its maximum over that interval(standard min-max result from real analysis-continuous functions reach their minimum and max-imum over closed intervals).Hence there is at least one best response α∗which maximizes player1’s payoff.•Condition(3)requires that if player1has tow best responsesα∗1U+ (1−α∗1)D andα∗2U+(1−α∗2)D to player2playingβL+(1−β)R then the strategy where player1chooses U with probabilityλα∗1+(1−λ)α∗2 for some0<λ<1is also a best response(i.e.BR1(β)is convex).But since both theα1and theα2strategy are best responses of player 1to the sameβstrategy of player2they also have to provide the same payoffs to player1.But this implies that if player1plays strategyα1 with probabilityλandα2with probability1−λshe will get exactly the same payoffas well.Hence the strategy where she plays U with probabilityλα∗1+(1−λ)α∗2is also a best response and her best response set BR1(β)is convex.•Thefinal condition(4)requires that BR has a closed graph.To show this consider a sequenceσn=(αn,βn)of(mixed)strategy profiles and ˜σn=(˜αn,˜βn)∈BR(σn).Both sequences are assumed to converge to σ∗=(α∗,β∗)and˜σ∗=(˜α∗,˜β∗),respectively.We now want to show that˜σ∈BR(σ)to prove that BR has a closed graph.We know that for player1,for example,we haveu1(˜αn,βn)≥u1(α ,βn)10for anyα ∈[0,1].Note,that the utility function is continuous in both arguments because it is linear inαandβ.Therefore,we can take the limit on both sides while preserving the inequality sign:u1(˜α∗,β∗)≥u2(α ,β)for allα ∈[0,1].This shows that˜α∗∈BR1(β)and therefore˜σ∗∈BR(σ∗).Hence the graph of the BR correspondence is closed.Therefore,all four Kakutani conditions apply and the giant best-response correspondence BR has afixed point,and each2×2game has a Nash equilibrium.4Nash Existence Proof for General Finite CaseUsing thefixed point method it is now relatively easy to extend the proof for the2×2case to generalfinite games.The biggest difference is that we cannot represent a mixed strategy any longer with a single number such asα. If player1has three pure strategies A1,A2and A3,for example,then his set of mixed strategies is represented by two probabilities-for example,(α1,α2) which are the probabilities that A1and A2are chosen.The set of admissible α1andα2is described by:Σ1={(α1,α2)|0≤α1,α2≤1andα1+α2≤1}(10) The definition of the set of mixed strategies can be straightforwardly ex-tended to games where player1has a strategy set consisting of n pure strategies A1,..,A n.Then we need n−1probabilitiesα1,..,αn−1such that:Σ1={(α1,..,αn−1)|0≤α1,..,αn−1≤1andα1+..+αn−1≤1}(11) So instead of representing strategies on the unit interval[0,1]we have to represent as elements of the simplexΣ1.Lemma2The setΣ1is compact and convex.Proof:It is clearly convex-if you mix between two mixed strategies you get another mixed strategy.The set is also compact because it is bounded (all|αi|≤1)and closed.To see closedness take a sequence(αi1,..,αi n−1)of elements ofΣ1which converges to(α∗1,..).Then we haveα∗i≥0and n−1α∗i≤1because the limit preserves weak inequalities.QED i=111We can now check that all conditions of Kakutani are fulfilled in the gen-eralfinite case.Checking them is almost1-1identical to checking Kakutani’s condition for2×2games.Condition1:The individual mixed strategy setsΣi are clearly non-empty because every player has at least one strategy.SinceΣi is compact Σ=Σ1×...×ΣI is also compact.Hence the BR correspondence BR:Σ→Σacts on a compact and convex non-empty set.Condition2:For each player i we can calculate his utiltiy u i,σ−i (σi)forσi∈Σi.SinceΣi is compact and u i,σ−i is continuous the set of payoffs is alsocompact and hence has a maximum.Therefore,BR i(Σi)is non-empty.Condition3:Assume thatσ1i andσ2i are both BR of player i toσ−i. Both strategies have to give player i equal payoffs then and any linear com-bination of these two strategies has to be a BR for player i,too.Condition4:Assume thatσn is a sequence of strategy profiles and ˜σn∈BR(σn).Both sequences converge toσ∗and˜σ∗,respectively.We know that for each player i we haveu i˜σn i,σn−i≥u iσ i,σn−ifor allσ i∈Σi.Note,that the utility function is continuous in both arguments because it is linear.3Therefore,we can take the limit on both sides while preserving the inequality sign:u i˜σ∗i,σ∗−i≥u iσ i,σ∗−ifor allσ i∈Σi.This shows that˜σ∗i∈BR iσ∗−iand therefore˜σ∗∈BR(σ∗).Hence the graph of the BR correspondence is closed.So Kakutani’s theorem applies and the giant best-response map BR has afixed point.3It is crucial here that the set of pure strategies isfinite.12。

1R e l e a s e d a t e : 2018-10-22 10:47D a t e o f i s s u e : 2018-10-22262163_e n g .x mlGermany: +49 621 776 4411Pepperl+Fuchs Group Refer to “General Notes Relating to Pepperl+Fuchs Product Information”.USA: +1 330 486 0001Singapore: +65 6779 9091 ************************.com************************.com************************.com Technical dataGeneral specifications Passage speed v ≤ 6 m/s Measuring range max. 10000 m Light type Integrated LED lightning (red)Read distance 100 mm Depth of focus ± 40 mm Reading field 60 mm x 40 mm Ambient light limit 100000 Lux Resolution ± 0.1 mm Nominal ratingsCamera Type CMOS , Global shutterProcessor Clock pulse frequency 600 MHz Speed of computation 4800 MIPS Functional safety related parametersMTTF d 103 a Mission Time (T M ) 51 a Diagnostic Coverage (DC) 0 %Indicators/operating means LED indication 7 LEDs (communication, alignment aid, status information)Electrical specifications Operating voltage U B 15 ... 30 V DC , PELV No-load supply current I 0max. 400 mA Power consumption P 0 6 W InterfaceInterface type 100 BASE-TX Protocol EtherNet/IP Transfer rate 100 MBit/s Interface 2Interface type USB Service InputInput type 1 funtion input0-level: -U B or unwired1-level: +8 V ... +U B , programmableInput impedance ≥ 27 k ΩOutputOutput type 1 to 3 switch outputs , programmable , short-circuitprotectedSwitching voltage Operating voltage Switching current 150 mA each output Standard conformity Emitted interference EN 61000-6-4:2007+A1:2011 Noise immunity EN 61000-6-2:2005 Shock resistance EN 60068-2-27:2009Vibration resistance EN 60068-2-6:2008Ambient conditions Operating temperature 0 ... 60 °C (32 ... 140 °F) , -20 ... 60 °C (-4 ... 140 °F)(noncondensing; prevent icing on the lens!)Storage temperature -20 ... 85 °C (-4 ... 185 °F)Relative humidity 90 % , noncondensing Mechanical specifications Connection type 8-pin, M12x1 connector, standard (supply+IO)4-pin, M12x1 socket, D-coded (LAN) 4-pin, M12x1 socket, D-coded (LAN)Housing width 70 mm Housing height 70 mm Housing depth 50 mm Degree of protection IP67MaterialHousing PC/ABS Mass approx. 200 g Approvals and certificates UL approvalcULus Listed, General Purpose, Class 2 Power Source,Type 1 enclosureCCC approvalCCC approval / marking not required for products rated ≤36 VModel numberPCV100-F200-B25-V1D-6011Read head for incident light positioning systemFeatures•Non-contact positioning on Data Matrix code tape •Mechanically rugged: no wearing parts, long operating life, maintenance-free •High resolution and precisepositioning, especially for facilities with curves and switch points as well as inclines and declines.•T ravel ranges up to 10 km, in X and Y direction •Integrated switch •EtherNet/IPDiagrammsXZYCoordinates2Germany: +49 621 776 4411Pepperl+Fuchs Group Refer to “General Notes Relating to Pepperl+Fuchs Product Information”.USA: +1 330 486 0001Singapore: +65 6779 9091 ************************.com************************.com************************.com3R e l e a s e d a t e : 2018-10-22 10:47D a t e o f i s s u e : 2018-10-22262163_e n g .x mlGermany: +49 621 776 4411Pepperl+Fuchs Group Refer to “General Notes Relating to Pepperl+Fuchs Product Information”.USA: +1 330 486 0001Singapore: +65 6779 9091 ************************.com************************.com************************.com Alignment guide for PCV100-* read head PCV-MB1Mounting bracket for PCV* read head V19-G-ABG-PG9-FEFemale connector, M12, 8-pin, shielded, field attachable V19-G-ABG-PG9Female connector, M12, 8-pin, shielded, field attachable PCV-SC12AGrounding clip for PCV system V19-G-2M-PUR-ABGFemale cordset, M12, 8-pin, shielded, PUR cableV19-G-10M-PUR-ABGFemale cordset, M12, 8-pin, shielded, PUR cableV19-G-5M-PUR-ABGFemale cordset, M12, 8-pin, shielded, PUR cableV1SD-G-5M-PUR-ABG-V45-G Connection cable, M12 to RJ-45, PUR cable 4-pin, CAT5eV1SD-G-30M-PUR-ABG-V45-G Connection cable, M12 to RJ-45, PUR cable 4-pin, CAT5eV1SD-G-2M-PUR-ABG-V45-G Connection cable, M12 to RJ-45, PUR cable 4-pin, CAT5eV1SD-G-10M-PUR-ABG-V45-G Connection cable, M12 to RJ-45, PUR cable 4-pin, CAT5e Vision ConfiguratorOperating software for camera-based sensorsPCV-KBL-V19-STR-USB USB cable unit with power supplyAccessoriesmust be provided such that the depth of field of the reading head is not closed during operation.All reading heads can be optimally customized by parameterization for specific requirements.Displays and ControlsThe reading head allows visual function check and fast diagnosis with 6 indicator LEDs. The reading head has 2 buttons on the reverse of the device to activate the alignment aid and pa-rameterization mode.LEDsAlignment aid for the Y and Z coordinatesThe activation of the alignment aid is only possible within 10 minutes of switching on the read-ing head. The switchover from normal operation to “alignment aid operating mode is via button 1 on the reverse of the reading head.•Press the button 1 for longer than 2 s. LED3 flashes green for a recognized code band.LED3 flashes red for an unrecognized code band.•Z coordinate: If the distance of the camera to the code band too small, the yellow LED5lights up. If the distance of the camera to the code band too large, the yellow LED5 lights up. Within the target range, the yellow LED5 flashes at the same time as the green LED3. •Y coordinate: If the optical axis of the camera is too deep in relation to the middle of the code band, the yellow LED4 lights up. If the optical axis is too high, the yellow LED4 extin-guishes. Within the target range, the yellow LED4 flashes at the same time as the green LED3.• A short press on button 1 ends the alignment aid and the reading head changes to normal operation.LED Color Label Meaning1green BUS LINKCommunication status 2yellow BUS ACTIVITY Data transfer3red / green PWR / ADJSYSERR / NO CODECode recognized / not recognized, Error 4yellow OUT1/ADJ Y Output 1, Alignment aid Y 5yellow OUT2/ADJ Z Output 2, Alignment aid Z 6red/green/yellow INTERNALDIAGNOSTICInternal diagnostics。

Afixed-point theorem for multi-valued functionswith an application tomultilattice-based logic programmingJ.Medina ,M.Ojeda-Aciego ,and J.Ruiz-Calvi˜n oDept.Matem´a tica Aplicada.Universidad de M´a lagaEmail:{jmedina,aciego,jorgerucal}@ctima.uma.esAbstract.This paper presents a computability theorem forfixed pointsof multi-valued functions defined on multilattices,which is later used inorder to obtain conditions which ensure that the immediate consequenceoperator computes minimal models of multilattice-based logic programsin at mostωiterations.1IntroductionFollowing the trend of generalising the structure of the underlying set of truth-values for fuzzy logic programming,multilattice-based logic programs were in-troduced in[7]as an extended framework for fuzzy logic programming,in which the underlying set of truth-values for the propositional variables is considered to have a more relaxed structure than that of a complete lattice.Thefirst definition of multilattices,to the best of our knowledge,was in-troduced in[1],although,much later,other authors proposed slightly different approaches[4,6].The crucial point in which a complete multilattice differs from a complete lattice is that a given subset does not necessarily have a least upper bound(resp.greatest lower bound)but some minimal(resp.maximal)ones.As far as we know,thefirst paper which used multilattices in the context of fuzzy logic programming was[7],which was later extended in[8].In these papers, the meaning of programs was defined by means of afixed point semantics;and the non-existence of suprema in general but,instead,a set of minimal upper bounds,suggested the possibility of developing a non-deterministicfixed point theory in the form of a multi-valued immediate consequences operator.Essentially,the results presented in those papers were the existence of mini-mal models below any model of a program,and that any minimal model can be attained by a suitable version of the iteration of the immediate consequence op-erator;but some other problems remained open,such as the constructive nature of minimal models or the reachability of minimal models after at most countably many iterations.Partially supported by Spanish project TIC2003-09001-C02-01Partially supported by Spanish project TIN2006-15455-C03-01Partially supported by Andalusian project P06-FQM-02049The aim of this paper is precisely to present conditions which ensure that min-imal models for multilattice-based logic programs can be reached by a“bounded”iteration of the immediate consequences operator,in the sense thatfixed points are attained after no more thanωiterations.Obviously,the main theoretical problem can be stated in the general framework of multi-valued functions on a multilattice.Some existence results in this line can be found in[2,3,5,9,10],but they worked with complete lattices instead of multilattices.The structure of the paper is as follows:in Section2,some preliminary def-initions and results are presented;later,in Section3,we introduce the main contribution of the paper,namely,reachability results for minimalfixed points of multi-valued functions on a multilattice;then,in Section4,these results are instantiated to the particular case of the immediate consequences operator of multilattice-based logic programs;the paperfinishes with some conclusions and prospects for future work.2PreliminariesIn order to make this paper self-contained,we provide in this section the basic notions of the theory of multilattices,together with a result which will be used later.For further explanations,the reader can see[7,8].Definition1.A complete multilattice is a partially ordered set, M, ,such that for every subset X⊆M,the set of upper(resp.lower)bounds of X has minimal(resp.maximal)elements,which are called multi-suprema(resp.multi-infima).The sets of multi-suprema and multi-infima of a set X are denoted by multisup(X)and multinf(X).It is straightforward to note that these sets consist of pairwise incomparable elements(also called antichains).An upper bound of a set X needs not be greater than any minimal up-per bound(multi-supremum);such a condition(and its dual,concerning lower bounds and multi-infima)has to be explicitly required.This condition is called coherence,and is formally introduced in the following definition,where we use the Egli-Milner ordering,i.e.,X EM Y if and only if for every y∈Y there exists x∈X such that x y and for every x∈X there exists y∈Y such that x y.Definition2.A complete multilattice M is said to be coherent if the following pair of inequations hold for all X⊆M:LB(X) EM multinf(X)multisup(X) EM UB(X)where LB(X)and UB(X)denote,respectively,the sets of lower bounds and upper bounds of the set X.Coherence together with the non-existence of infinite antichains(so that the sets multisup(X)and multinf(X)are alwaysfinite)have been shown to be useful conditions when working with multilattices.Under these hypotheses,the follow-ing important result was obtained in[7]:Lemma1.Let M be a coherent complete multilattice without infinite antichains, then any chain1in M has a supremum and an infimum.3Reaching Fixed Points for Multi-valued Functions on MultilatticesIn order to proceed to the study of existence and reachability of minimalfixed points for multi-valued functions,we need some preliminary definitions.Definition3.Given a poset P,by a multi-valued function we mean a function f:P−→2P(we do not require that f(x)=∅for every x∈P).We say that x∈P is afixed point of f if and only if x∈f(x).The adaptation of the definition of isotonicity and inflation for multi-valued functions is closely related to the ordering that we consider on the set2M of subsets of M.We will consider the Smyth ordering among sets,and we will write X S Y if and only if for every y∈Y there exists x∈X such that x y. Definition4.Let f:P−→2P be a multi-valued function on a poset P:–We say that f is isotone if and only if for all x,y∈P we have that x y implies f(x) S f(y).–We say that f is inflationary if and only if{x} S f(x)for every x∈P.As our intended application is focused on multilattice-based logic programs, we can assume the existence of minimalfixed points for a given multi-valued function on a multilattice(since in[7]the existence of minimalfixed points was proved for the T P operator).Regarding reachability of afixed point,it is worth to rely on the so-called orbits[5]:Definition5.Let f:M−→2M be a multi-valued function an orbit of f is a transfinite sequence(x i)i∈I of elements x i∈M where the cardinality of M is less than the cardinality of I(|M|<|I|)and:x0=⊥x i+1∈f(x i)xα∈multisup{x i|i<α},for limit ordinalsαNote the following straightforward consequences of the definition:1A chain X is a totally ordered subset.Sometimes,for convenience,a chain will be denoted as an indexed set{x i}i∈I.1.In an orbit,we have f(x i)=∅for every i∈I.2.As f(x i)is a nonempty set,there might be many possible choices for x i+1,so we might have many possible orbits.3.If(x i)i∈I is an orbit of f and there exists k∈I such that x k=x k+1,thenx k is afixed point of f.Providing sufficient conditions for the existence of such orbits,we ensure the existence offixed points.Note that the condition f( )=∅directly implies the existence of afixed point,namely .4.Any increasing orbit eventually reaches afixed point(this follows from theinequality|M|<|I|).This holds because every transfinite increasing sequence is eventually sta-tionary,and an ordinalαsuch that xα=xα+1∈f(xα)is afixed point.Under the assumption of f being non-empty and inflationary,the existence of increasing orbits can be guaranteed;the proof is roughly sketched below: The orbit can be constructed for any successor ordinalαby using the inequal-ity{xα} S f(xα),which follows by inflation,since any element xα+1∈f(xα) satisfies xα xα+1.The definition for limit ordinals,directly implies that it is greater than any of its predecessors.As a side result,note that when reaching a limit ordinal,under the assump-tion of f being inflationary,the initial segment is actually a chain;therefore,by Lemma1it has only one multi-supremum(the supremum of the chain);this fact will be used later in Propositions1and2.Regarding minimalfixed points,the following result shows conditions under which any minimalfixed point is attained by means of an orbit:Proposition1.Let f:M−→2M be inflationary and isotone,then for any minimalfixed point there is an orbit converging to it.Proof.Let x be a minimalfixed point of f and let us prove that there is an increasing orbit(x i)i∈I satisfying x i x.We will build this orbit by transfinite induction:Trivially x0=⊥ x.If x i x,by isotonicity f(x i) S f(x).Then for x∈f(x)we can choose x i+1∈f(x i)such that x i+1 x and obviously x i x i+1by inflation.For a limit ordinalα,as stated above,xα=sup i<αx i;now,by induction we have that x i x for every i<α,hence xα x.The transfinite chain(x i)i∈I constructed this way is increasing,therefore there is an ordinalαsuch that xα=xα+1∈f(xα),so xαis afixed point and xα x but by minimality of thefixed point x,we have that x=xα.2 The usual way to approach the problem of reachability is to consider some kind of‘continuity’in our multi-valued functions,understanding continuity in the sense of preservation of suprema and infima.But it is obvious that we have to state formally what this preservation is meant,since in complete multilattices we only have for granted the existence of sets of multi-infima and sets of multi-suprema.This is just another reason to rely on coherent complete multilattices Mwithout infinite antichains so that,at least,we have the existence of suprema and infima of chains.Definition6.A multi-valued function f:M−→2M is said to be sup-preserving if and only if for every chain X=(x i)i∈I we have that:f(sup{x i|i∈I})={y|there are y i∈f(x i)s.t.y∈multisup{y i|i∈I}} Note that,abusing a bit the notation,the definition above can be rephrased in much more usual terms as f(sup X)=multisup(f(X))but we will not use it, since the intended interpretation of multisup(f(X))is by no means standard.Reachability of minimalfixed points is granted by assuming the extra condi-tion that our function f is sup-preserving,as shown in the following proposition. Proposition2.If a multi-valued function f is inflationary,isotone and sup-preserving,then at most countably many steps are necessary to reach a minimal fixed point(provided that some exists).Proof.Let x be a minimalfixed point and consider the approximating increasing orbit(x i)i∈I given by Proposition1.We will show that xωis afixed point of f and,therefore,xωequals x.As f is sup-preserving we have that f(xω)is the set{y|there are y i∈f(x i)s.t.y=multisup{y i|i<ω}} In order to prove that xωis afixed point,on the one hand,recall that we have,by definition,that xω=sup{x i|i<ω}.On the other hand,we will show that this construction can be also seen as a multi-supremum of a suitable sequence of elements y i∈f(x i).To do this we only have to recall that,by construction of the orbit,we know that x i+1∈f(x i),therefore for every0≤i<ωwe can consider y i=x i+1. Hence the element xωcan be seen as an element of f(xω).Thus,xωis afixed point of f and xω x and by minimality of x,we have that x=xω.24Application to fuzzy logic programs on a multilatticeIn this section we apply the previous results to the particular case of the im-mediate consequences operator for extended logic programs on a multilattice,as defined in[7,8].To begin with,we will assume the existence of a multilattice (coherent and without infinite antichains)M as the underlying set of truth-values,that is,our formulas will have certain degree of truth in M.In order to build our formulas,we will consider a set of computable n-ary isotone operators M n−→M which will be intended as our logical connectors.Finally,we will consider a setΠof propositional symbols as the basic blocks which will allow to build the set of formulas,by means of the connector functions.Now,we can recall the definition of the fuzzy logic programs based on a multilattice:Definition 7.A fuzzy logic program based on a multilattice M is a set P of rules of the form A ←B such that:1.A is a propositional symbol of Π,and2.B is a formula built from propositional symbols and elements of M by using isotone operators.Now we give the definition of interpretation and model of a program:Definition 8.1.An interpretation is a mapping I :Π−→M .2.We say that I satisfies a rule A ←B if and only if ˆI(B )≤I (A ),where ˆI is the homomorphic extension of I to the set of all formulae.3.An interpretation I is said to be a model of a program P iffall rules in P are satisfied by I .Example 1.Let us consider an example of a program on a multilattice.The program consists of the four rules below to the left,whereas the underlying multilattice is the six-element multilattice depicted below to the right:E ←AE ←B A ←a B ←b •⊥•bd d •a •d d d •c ¨¨¨¨r r r r • It is easy to check that the program does not have a least model but two minimal ones,I 1and I 2,given below:I 1(E )=cI 2(E )=d I 1(A )=aI 2(A )=aI 1(B )=b I 2(B )=b 2A fixed point semantics was given by means of the following consequences operator:Definition 9.Given a fuzzy logic program P based on a multilattice M ,an inter-pretation I and a propositional symbol A ;the immediate consequences operator is defined as follows:T P (I )(A )=multisup {I (A )}∪{ˆI (B )|A ←B ∈P }Note that,by the very definition,the immediate consequences operator is an inflationary multi-valued function defined on the set of interpretations of the program P ,which inherits the structure of multilattice.Moreover,models can be characterized in terms of fixed points of T P as follows:Proposition3(see[7]).An interpretation I is a model of a program if and only if I∈T P(I).Although not needed for the definition of either the syntax or the semantics of fuzzy logic programs,the requirement that M is a coherent multilattice without infinite antichains turns out to be essential for the existence of minimalfixed points,see[7].Hence,a straightforward application of Proposition2allows us to obtain the following result.Theorem1.If T P is sup-preserving,thenωsteps are sufficient to reach a min-imal model.5ConclusionsContinuing the study of computational properties of multilattices initiated in[7], we have presented a theoretical result regarding the attainability of minimal fixed points of multi-valued functions on a multilattice which,as an application, guarantees that minimal models of multilattice-based logic programs can be attained after at most countably many iterations of the immediate consequence operator.We recall that,in this paper,the existence of suchfixed points has been assumed because of the intended application in mind(that is,the existence of minimal models for multilattice-based logic programs was proved in[7]).As future work,this initial investigation onfixed points of multi-valued func-tions on a multilattice has to be completed with the study of sufficient conditions for the existence of(minimal)fixed points.Another interesting line of research,which turns out to be fundamental for the practical applicability of the presented result,is the study of conditions which guarantee that the immediate consequences operator is sup-preserving. References1.M.Benado.Les ensembles partiellement ordonn´e s et le th´e or`e me de raffinementde Schreier,II.Th´e orie des multistructures.Czechoslovak Mathematical Journal, 5(80):308–344,1955.2.V.d’Orey.Fixed point theorems for correspondences with values in a partially or-dered set and extended supermodular games.Journal of Mathematical Economics, 25:345–354,1996.3. F.Echenique.A short and constructive proof of Tarski’sfixed-point theorem.International Journal of Game Theory,2005.4. D.Hansen.An axiomatic characterization of multi-lattices.Discrete Mathematics,33(1):99–101,1981.5.M.A.Khamsi and D.Misane.Fixed point theorems in logic programming.Annalsof Mathematics and Artificial Intelligence,21:231–243,1997.6.J.Mart´ınez,G.Guti´e rrez,I.de Guzm´a n,and P.Cordero.Generalizations oflattices via non-deterministic operators.Discrete Mathematics,295:107–141,2005.7.J.Medina,M.Ojeda-Aciego,and J.Ruiz-Calvi˜n o.Multi-lattices as a basis forgeneralized fuzzy logic programming.In Proc.of WILF,volume3849of Lect.Notes in Artificial Intelligence,pages61–70,2006.8.J.Medina,M.Ojeda-Aciego,and J.Ruiz-Calvi˜n o.Fuzzy logic programming viamultilattices.Fuzzy Sets and Systems,158(6):674–688,2007.9. A.Stouti.A generalized Amman’sfixed point theorem and its application toNash equilibrium.Acta Mathematica Academiae Paedagogicae Ny´ıregyh´a ziensis, 21:107–112,2005.10.L.Zhou.The set of Nash equilibria of a supermodular game is a complete lattice.Games and economic behavior,7:295–300,1994.。

Brouwer Fixed Point Theorem:A Proof forEconomics StudentsTakao Fujimoto∗Abstract This note is one of the efforts to present an easy and sim-ple proof of Brouwerfixed point theorem,which economics students can, hopefully,grasp both in terms of geometry and through its economic inter-pretation.In our proof,we use the implicit function theorem and Sard’s theorem.The latter is needed to utilize a global property.1IntroductionIn this note,we give a proof of Brouwerfixed point theorem([2]), based on the method in Kellogg,Li and Yorke([7]).Our idea con-sists in using a special set for a given map and a special boundary point to start with.Two theorems are used in our proof,i.e.,the im-plicit function theorem and Sard’s theorem([12]).(See Golubitsky and Guillemin([4])for a result as we use in this note.)Unfortu-nately,we still need Sard’s theorem,which seems to be somewhat beyond the mathematical knowledge of average economics students.In section2,we describe our proof,and in the following section 3,an economic interpretation of the proof is given.Thefinal section 4contains several remarks.∗Faculty of Economics,Fukuoka University12ProofLet f(x)be a map from a compact convex set X into itself.We assume that f(x)is twice continuously differentiable and that X is a set in the n-dimensional Euclidean space R n(n≥1)defined byX≡{x|x≥0,X n i=1x i≤1,x∈R n}.Then define a direct product set YY≡X×T,where T≡[0,∞).The symbol J f(x)denotes the Jacobian matrix of the map f(x).Now we begin our proof.First of all,as the hypothesis of math-ematical induction,we suppose that the theorem is true when the dimension is less than n.(When n=1,it is easy enough to show the existence of at least onefixed point.)Let us consider the set C≡{y|f(x)−t·x=0,y=(x,t)∈Y}.2(See Fig.1above and Fig.2below.)By the implicit function theo-rem,when the determinant|J f(x)−t·I|=0at a point(x,t),there exists a unique curve passing through the point in one of its neighborhoods.If the origin0is afixed point,the proof ends,and so,let us suppose not.Then,there is a point y0≡(x0,t0)in a neighborhood of the origin such that y0∈C, because of the inductive hypothesis.On the other hand,in the setB≡{x|x≥0,P n i=1x i=1,x∈R n},there is either afixed point or a point in C with t<1.(Thus,when the Jacobian|J f(x)|has no eigenvalue either in the interval[0,1)or(1,t0],we canfind out a curve in C which includes a point with t=1,i.e.,afixed point.)When there is nofixed point,we can construct a continuously differentiable map g(x)from X to its boundary∂X as the point where a line x−f(x)hits the boundary on the side of x.The Jacobian|J g(x)|surely vanishes,and yet its rank is(n−1)almost3everywhere because of Sard’s theorem.(Note that now n≥2.)We suppose without the loss of generality that the origin is one of the points where the rank of|J g(x)|is indeed(n−1).This means that any curve contained in C is locally unique as is shown in Kellogg, Li and Yorke([7]).So,starting from the initial point y0,we extend the curve which satisfies f(x)−t·x=0.This can be done by considering a solution of differential equation with respect to arc length s.½dx ds=(J f(x)−t·I)−1·x·dt ds when|J f(x)−t·I|=0,and(J f(x)−t·I)·dxds=0and dt=0when|J f(x)−t·I|=0.The above equation can be derived by differentiatingf(x)−t·x=0with respect to arc length s along the curve,that is,J f(x)dxds−dtdsx−tdxds=0.If there is nofixed point,this curve remains within a subset of C whose t∈[a,b],where1<a and b<t0,having an infinite length without crossing.(Note that there is a minorflaw in the proof of Theorem2.2in Kellogg,Li and Yorke([7,p.477]).)This leads to a contradiction because there is at least one accumulation point where the local uniqueness mentioned above is lost.Hence there should be afixed point.When f(x)is merely continuous,we can prove the theorem as is done in Howard([5]),which expounds Milnor([9])and Rogers ([11]).3Economic InterpretationWe can interpret our method of proof in a simple share game among (n+1)players.Let us suppose that in our economy there exists4one unit of a certain commodity which every individual player,num-bered from0to n,wishes to have.When a point x of X is given,it represents the shares p i’s of players:p0=1−P n i=1x i and p i=x i for i=1,...,n.Staying at the origin implies the whole commoditybelongs to player0.A given map f(x)stands for a rule of changes in players’shares.For a point x to be in C,it is required that after changes in shares by f(x),the ratios of shares among the players from1to n should remain the same as before,while the share of player0may increase or decrease.(See Fig.1above.) We know that in a neighborhood of the origin there is a point in C,provided that the origin is not afixed point.Starting from this point,we grope for successive neighboring points in C.As we continue the search,we reach afixed point,a special point where no changes are made after the transformation by f(x).This is because, otherwise,we would get into a path of an infinite length with no crossing point within a compact set,which yields a contradiction to the local uniqueness guaranteed by use of Sard’s theorem.4RemarksIt seems that the theorem wasfirst proved in1904by Bohl([1]) for the case of dimension3,and in the general case by Hadamard before or in1910,whose proof was published in an appendix of the book by Tannery([15]).It is reported that Hadamard was told about the theorem by Brouwer(Stuckless([14])).Thus,it may be more appropriate to call the theorem B2H theorem as astronomers do.(Stuckless([14])also mentions the contributions by Bolzano (1817)and Poincaré(1883)which are equivalent to Brouwerfixed point theorem.Then,more precisely,B3HP theorem.) Differentiability is useful to tame down possibly wild movements of continuous functions,and yet it is,in normal settings,powerful only to derive local properties.In order to prove Brouwerfixed point theorem,we need a gadget related to global properties:5(1)simplicial subdivisions and Sperner’s lemma([13])and KKM theorem([8]),(2)Sard’s theorem,or(3)volume integration as used by Kannai([6])and Howard([5]).As in Kellogg,Li and Yorke([7]),we can devise out a numerical procedure to compute afixed point.First note that(J f(x)−t·I)·dx=x·dtby the total differentiation of the equation f(x)−t·x=0.Thus, starting from the point y0,we proceed4x=(J f(x)−t·I)−1·hx when|J f(x)−t·I|=0,and4x=hv,where v is a real eigenvector of J f(x)with its eigenvalue being t,when|J f(x)−t·I|=0. Here,h is a small positive scalar of step-size.It is clear that near a fixed point,where t+1,this numerical procedue is quasi-Newton-Raphson method.Originally,the author tried to obtain a proof by extending the notion offixed point,and by using the result in Fujimoto([3]).This lead to the method in Kellogg,Li and Yorke([7]).When we define two sets asCG≡{y|f(x)−t·x≥0,t≥1,y=(x,t)∈Y},and CL≡{y|f(x)−t·x≤0,t≤1,y=(x,t)∈Y},these sets certainly have respective points in Y with t=1.If the two sets CG and CL intersect with each other,there exists afixed point.The reader is referred to Park and Jeong([10])for an interesting circular tour around Brouwerfixed point theorem.In this note,we have shown a proof of Brouwerfixed point theorem,using the implicit function theorem and Sard’s theorem, and thus twice continuous differentiability of the map.To be un-derstandable to economics students,it is desirable to establish the theorem without depending upon Sard’s theorem.I wish to have such proofs and present them in the near future.6References[1]Bohl,P.(1904):“über die Bewegung eines mechanischen Sys-tems in der Nähe einer Gleichgewichtslage”,Journal für die Reine und Angewandte Mathematik,127,pp179-276.[2]Brouwer,L. E.J.(1912):“Über Abbildung von Mannig-faltigkeiten”,Mathematische Annalen,71,pp.97-115.Submit-ted in July,1910.[3]Fujimoto,T.(1984):“An Extension of Tarski’s Fixed PointTheorem and Its Application to Isotone Complementary Prob-lems”,Mathematical Programming,28,pp.116-118.[4]Golubitsky,M.,Guillemin,V.(1973):Stable Mappings andTheir Singularities,Springer-Verlag,New York.[5]Howard,R.(2004):“The Milnor-Rogers Proof of the BrouwerFixed Point Theorem”,mimeo,University of South Carolina.[6]Kannai,Y.(1981):“An Elementary Proof of No-RetractionTheorem”,American Mathematical Monthly,88,pp.264-268.[7]Kellogg,R.B.,Li,T.Y.,Yorke,J.A.(1976):“A ConstructiveProof of the Brouwer Fixed Point Theorem and Computational Results”,SIAM Journal of Numerical Analysis,13,pp.473-483.[8]Knaster, B.,K.Kuratowski, C.,Mazurkiewicz,S.(1929):“Ein Beweis des Fixpunktsatzes für n-Dimensionale Simplex-e”,Fundamenta Mathematicae,14,pp.132-137.[9]Milnor,J.(1978):“Analytic Proofs of the‘Hairy Ball Theo-rem’and the Brouwer Fixed-Point Theorem”,American Math-ematical Monthly,85,pp.521-524.7[10]Park,S.,Jeong,K.S.(2001):“Fixed Point and Non-RetractTheorems–Classical Circular Tours”,Taiwanese Journal of Mathematics,5,pp.97-108.[11]Rogers,C.A.(1980):“A Less Strange Version of Milnor’sProof of Brouwer’s Fixed-Point Theorem”,American Mathe-matical Monthly,87,pp.525-527.[12]Sard,A.(1942):“The Measure of the Critical Points of Dif-ferentiable Maps”,Bulletin of the American Mathematical So-ciety,48,pp.883-890.[13]Sperner,E.(1928):“Neuer Beweis für der Invarianz der Di-mensionszahl und des Gebietes”,Abhandlungen aus dem Math-ematischen Seminar der Universität Hamburg,6,pp.265-272.[14]Stuckless,S.(2003):“Brouwer’s Fixed Point Theorem:Meth-ods of Proof and Generalizations”,M.Sc.Thesis,Simon Fraser University.[15]Tannery,J.(1910):Introductionàla Théorie des Fonctionsd’une Variable,2ème ed.,tome II,Hermann,Paris.8Below is given annual work summary, do not need friends can download after editor deleted Welcome to visit againXXXX annual work summaryDear every leader, colleagues:Look back end of XXXX, XXXX years of work, have the joy of success in your work, have a collaboration with colleagues, working hard, also have disappointed when encountered difficulties and setbacks. 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Some experiences in the past work, mentality is very important in the work, work to have passion, keep the smile of sunshine, can close the distance between people, easy to communicate with the customer. Do better in the daily work to communicate with customers and achieve customer satisfaction, excellent technical service every time, on behalf of the customer on our products much a understanding and trust.Fourth, we need to continue to learn professional knowledge, do practical grasp skilled operation. Over the past year, through continuous learning and fumble, studied the gas generation, collection and methods, gradually familiar with and master the company introduced the working principle, operation method of gas machine. 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