THICK NON-CROSSING PATHS

Installation and Maintenance ManualCompact Vacuum UnitSeries ZBThis manual contains essential information for the protection of users andothers from possible injury and/or equipment damage.∙Read this manual before using the product, to ensure correct handling,and read the manuals of related apparatus before use.∙Keep this manual in a safe place for future reference.∙ These instructions indicate the level of potential hazard by label of“Caution”, “Warning” or “Danger”, followed by important safety informationwhich must be carefully followed.∙To ensure safety of personnel and equipment the safety instructions inthis manual and the product catalogue must be observed, along with otherrelevant safety practices.CautionIndicates a hazard with a low level of risk, which ifnot avoided, could result in minor or moderateinjury.WarningIndicates a hazard with a medium level of risk,which if not avoided, could result in death orserious injury.DangerIndicates a hazard with a high level of risk, whichif not avoided, will result in death or seriousinjury.∙ The compatibility of pneumatic equipment is the responsibility of theperson who designs the pneumatic system or decides its specifications.Since the products specified here can be used in various operatingconditions, their compatibility with the specific pneumatic system must bebased on specifications or after analysis and/or tests to meet specificrequirements.∙Only trained personnel should operate pneumatically operatedmachinery and equipment.Compressed air can be dangerous if an operator is unfamiliar with it.Assembly, handling or repair of pneumatic systems should be performedby trained and experienced personnel.∙Do not service machinery/equipment or attempt to removecomponents until safety is confirmed.1) Inspection and maintenance of machinery/equipment should only beperformed after confirmation of safe locked-out control positions.2) When equipment is to be removed, confirm the safety process asmentioned above. Switch off air and electrical supplies and exhaust allresidual compressed air in the system.3) Before machinery/equipment is re-started, ensure all safety measuresto prevent sudden movement of cylinders etc. (Supply air into the systemgradually to create back pressure, i.e. incorporate a soft-start valve).∙Do not use this product outside of the specifications. Contact SMCif it is to be used in any of the following conditions:1) Conditions and environments beyond the given specifications, or if theproduct is to be used outdoors.2) Installations in conjunction with atomic energy, railway, air navigation,vehicles, medical equipment, food and beverage, recreation equipment,emergency stop circuits, press applications, or safety equipment.3) An application which has the possibility of having negative effects onpeople, property, or animals, requiring special safety analysis.∙Ensure that the air supply system is filtered to 5 microns.2 SpecificationsRefer to the operation manual for this product.3 Installation3.1 Installation∙Do not install the product unless the safety instructions have been readand understood.Note the following points when mounting and installing the product.■ Common Precautions for Mounting and Installation1) It is necessary to perform maintenance and replacement of the suctionfilter regularly to maintain the proper operation of the ejector and vacuumpump system. Ensure sufficient space for maintenance work wheninstalling the product.2) The filter case of this product is integrated with the vacuum piping.Secure sufficient space and some length of the tube with the piping(tubes) on the vacuum side so that the case can be removed.3) Do not fix the piping on the vacuum side such that a load is alwaysapplied to the filter case in a bending or pulling direction. This candamage the body and/or the filter case.4) If the ejector (silencer exhaust specification) is operated in a dustyenvironment or if there is dust on the surface of the workpieces, it cancause clogging of the silencing material as well as the suction filter dueto dust being sucked in. Secure space necessary for the maintenancecheck and replacement of the silencer when the ejector performancedecreases.5) Keep the ambient temperature of the product between -5 and 50o C. Inenvironments such as inside a panel where heat radiation efficiency ispoor, the ambient temperature will rise due to the heat generation of thecoil of the solenoid valve, causing malfunction.6) When handling the product, do not lift it by the lead wires or cables ofthe solenoid valve, pressure sensor or pressure switch for vacuum.Otherwise, it can cause vacuum leakage or broken wire or damage tothe product.■Mounting and Installation of Single Unit Ejectors1) The tightening torque for mounting the product to the wall should bebetween 0.075 and 0.096Nm. Using excessive torque may causedamage to the body. (The width of the product is 10mm.)2) Do not block the exhaust port of the ejector. The ejector of the singleunit specification has only one exhaust port on one side. If the ejector ismounted with the exhaust port facing a wall, secure a space of at least1mm between the product and the wall using a spacer, shim orequivalent.(3) Secure the space for connecting piping on the supply side wheninstalling the product.Mounting with a bracket for single unit (Width of the bracket: 1mm)3 Installation (cont.)Mounting on the wall and the port released to the atmosphere at the bottomPart No. of the bracket for single unit: ZB1-BK1-A (Provided with 2mounting screws M2x14 with washer and 2 hexagon nuts M2.)Recommended tube fittings for the set-up shown above: KQ2H04-M5N,KQ2L04-M5N, KQ2W04-M5N.■Mounting and Installation of an Ejector of Manifold SpecificationWhen mounting the manifold base, it is recommended to mount a spaceron the filter case side in order to make it easier to perform maintenanceservice of the filter element. (Width of the manifold base mounting hole:11.6mm)3.2 Environment∙Do not use in an environment where corrosive gases, chemicals, saltwater or steam are present.∙Do not use in an explosive atmosphere.∙Do not expose to direct sunlight. Use a suitable protective cover.∙Do not install in a location subject to vibration or impact. Check the productspecifications.∙Do not mount in a location exposed to radiant heat.3.3 Piping■Piping for Air Pressure Supply and Vacuum Pressure Supply1) Preparation before pipingBefore piping, perform air blow (flushing) or cleaning to remove anycutting chips, cutting oil, dust, etc. from the piping.2) Wrapping of pipe tapeWhen installing piping or a tube fitting into a port, prevent cuttingchips and sealant material from getting inside the product.If a sealant tape is used,leave 1 thread exposedat the end of threads.3) When connecting tubing, consider factors such as changes in thetubing length due to pressure, and allow a sufficient margin.Otherwise, it can damage the fitting and cause the tube to come off.Refer to Fittings & Tubing Precautions from 1 to 4 shown in BestPneumatics 6 on SMC’s website (URL ) forthe recommended piping conditions.3 Installation (cont.)■Piping to the Manifold Base1) For the PV port of the manifold base, use a tube fitting whose maximumbore size of the outside dimension is smaller than 12mm.Otherwise, the exterior of the fitting will interfere with the manifold baseinstallation face.Recommended tube fittings:KQ2S06-01□S, KQ2S04-01□S2) Follow the tightening instructions shown below for each thread.1/8 (PV port) : 7 to 9NmTightening torque is 3 to 5N as a guide.M5 (PV, PD port): After tightening by hand, increase the tightening byabout 1/6 turn with a tightening tool.Tightening torque is 1 to 1.5N as a guide.3) When mounting or removing the tube fitting,etc. to the manifold base,hold the manifold base hold the manifold base with a spanner.If the ejector/vacuum pump system is held, it may cause air leakage ordamage to the product.■Piping to the Vacuum (V) Port1) Allow a sufficient margin of tube length when piping, in order to preventtwisting, tensile, moment loads, vibration or impact being applied to thetubes and fittings.This can cause damage to the tube fittings and crushing, bursting ordisconnection of tubing.2) Piping to the product is assumed to be static piping.If the tube moves, it may become worn, elongated or torn due to tensileforces, or disconnected from the fitting. Ensure the tube is in a staticcondition at all times before using.3) Prevent the connected tube from being rotated.If the fittings are used in this way, the fitting may be broken.4) Do not lift the product by holding the piping after the tube is connected tothe vacuum (V) port.Otherwise, the filter case and/or the One-touch tube fitting will bedamaged.4 Settings■Manual OverrideVacuum for the ejector or the vacuum pump system is generated orreleased by manual operation.Use the manual override after confirming that there is no danger.When operating the locking type with a screwdriver, turn it gently using awatchmaker’s screwdriver. (Torque: Less than 0.1Nm)Sealant tapeLeave 1 threadexposed.Wrap this way.Non-locking push type (Tool required)It is turned ON by pressing the manual override allthe way in the direction indicated by the arrow (↓),and it is turned OFF by releasing it.11.65 Maintenance■Construction of ZB seriesManifold / With pressure sensorSingle unit / With vacuum pressure switch■ComponentsThe components from (7) to (15) are available as service parts.■Implement the maintenance and check shown below in order to use the ejector and the vacuum system safely and in an appropriate way for a long period of time.1) Maintenance should be performed according to the procedure indicated in the Operation Manual.Improper handling can cause damage and malfunction of equipment and machinery.2) Maintenance workCompressed air can be dangerous when handled incorrectly. Therefore, in addition to observing the product specifications, replacement of elements and other maintenance activities should be performed by personnel with sufficient knowledge and experience pertaining to pneumatic equipment. 3) DrainingRemove condensate from air filters and mist separators regularly. If the collected drainage is drained to the downstream side, it can stick inside of the product, causing operation failure and failure to reach the specified vacuum pressure.4) Replace the filter element built into the ejector and the vacuum pump system and the silencer regularly. (Refer to the replacement procedure below.)It is recommended to replace the filter element and the silencer when the pressure drop reaches 5kPa as a guideline. The replacement cycle varies depending on the operating conditions, operating environment and supply air quality.However, if there is a vacuum pressure drop and/or delay in the vacuum (adsorption) response time which causes problem with the settings during operation, stop the operation of the product and replace the element regardless of the above mentioned replacement guideline. 5) Operation in an environment where there is a lot of dust in the airThe processing capacity of the filter element built into the product may be insufficient. It is recommended to use SMC's air suction filter (ZFA, ZFB, ZFC series) in order to avoid problems beforehand. 6) Check before and after the maintenance workWhen the product is to be removed, turn off the power supply, and be sure to cut off the supply pressure and exhaust the compressed air. Confirm that the air is released to atmosphere.When mounting the product after the maintenance work, supplycompressed air, connect to the power, check if it functions properly and have a leakage inspection. Especially for the latching type supply valve, be sure to check that the supply valve is OFF in the initial condition because it is possible that it is ON due to vibration.7) Do not disassemble or modify the product, other than the replacement parts specified in this manual.■Spare parts listNo. Description [Application]ModelRemarks (7) Supply valve[Generates vacuum.] ZB1-VQ110U-□□□ N.C. ZB1-VQ110L-□□□ Latching ZB1-VQ120U-□□□ N.C. (8) Release valve [Releases vacuum] ZB1-VQ110-□□□ N.C. (9) V-port Assembly [For vacuum port]ZB1-VPN3-□-A (10)Exchange the One-touch tube fittings with the port plugs KJ □□-C1(11)Filter element[For suction filter]ZB1-FE3-A(12) Silencer [For silencer] ZB1-SE1-A (13) Pressure sensor assemblyZB1-PS □-A (14)Pressure switch assembly forvacuumZB1-ZS □□□□-A(15) Manifold base assemblyZZB □-□□□ ■Replacement Procedure for Filter Element- Hold the V port assembly with your fingers, and turn it45 degrees in the counter-clockwise direction and pull it out.For the straight type One-touch tube fitting, it can be removed by using a hexagon wrench (width across flats: 2).- Remove the filter element from the removed filter case, and mount a new filter element securely to the back of the case. (See Fig. to the right) - Confirm that the filter case gasket is not displaced and that it has no foreign matter stuck to it.- Insert the V port assembly into the ejector/vacuum pump system (Fig. to the right), press it slightly and turn it for approximately 45 degrees in the clockwise direction until it stops. (See Fig. to the right)(Mount the V port assembly in the direction specified in the figure. If the convex side is mounted downward, it will interfere with the floor when the element is mounted on its bottom surface, causing breakage of the filter case and the element.)■Replacement procedure for silencer*- Turn the body upside down. Apply a watchmaker’s screw driver or your finger to the notch, and slide the silencer cover in the direction indicated by the △ mark.- When it clicks, the hook is disconnected. Put your Pry up and remove part A, cover.- Remove the silencer by using a watchmaker’s screw driver.- Insert a new silencer, and mount the cover by the reverse procedure of the disassembly procedure for reassembly.(When replacing the silencer, the metal diffuser can be seen. This part is important to the function. Do not touch or apply force to the metal diffuser when replacing the silencer.)* For vacuum pump system, the silencer is not built in.■Replacement Procedure for Solenoid Valves (supply valve, release valve)-This product has a “supply valve” for generating vacuum and a “release valve” for breaking vacuum. Follow the procedure below to replace the solenoid valves after the product has been used for a long period of time or malfunctions.1) Remove the mounting screw of the solenoid valve. 2) Remove the solenoid valve.3) Before mounting the replacement solenoid valve, check that it has no dust or scratches on the mounting surface. Be certain that the gasket and filter element R of the supply valve are properly mounted as well. (Filter element R is installed in the release valve only.)4) Tighten the mounting screw of the solenoid valve to the specified torque below.Appropriate tightening torque (Nm) 0.054 to 0.08- When replacing the solenoid valves, the valve body will come off if boththe supply valve and the release valve are removed at the same time.Removal and mounting of the solenoid valves should be done one at atime to prevent parts from dropping and foreign matter from entering.* Function of the filter element R: When the supply valve is switched OFF from ON, atmospheric pressure flows from the vent port into the spaceinside the valve where there is “vacuum pressure”. Filter element R is afilter mounted in the flow path. It prevents the dust in the operating environment from entering inside the solenoid valve.Manifold Products■Increasing and Decreasing the Number of Manifold Stations- When decreasing the number of manifold stations, order the manifoldbase (a) exclusive for the required number of stations. When increasing the number of stations, order the required number of single units of the body type 3 valve (b). Refer to Model Indication and How to Order for the part numbers for placing an order. The part number for the manifold base is different depending on whether pressure sensor/ vacuum pressure switch are mountable or not.- When mounting each station, check that all the gaskets are in place and tighten the screws to the specified torque. If the tightening torque is exceeded, the body can be broken.- For the manifold with pressure sensor/vacuum pressure switch, order themanifold base (a) for the required number of stations. When increasing the number of stations, order the required number of single unit of the body type 3 valve (b) and the required number of either the pressure sensor assembly (c) or the vacuum pressures switch assembly (d).- In this case, the pressure sensor (c) /vacuum pressure switch (d) is tightened together with the single unit of the product (b). (Refer to the figure on the right.)- When mounting the pressure sensor/ vacuum pressure switch, be sure to check that the O-ring on the mounting surface of the manifold base is mounted properly and that the O-ring is not displaced from the mounting groove. If the O-ring is not mounted properly, it can cause vacuum pressure leakage.Locking push type (Tool required) <Latching type> - Turn the manual override to the left and line up the arrow () with 0 to return it to the RESET state (flow from A toP). (It is set to RESET state when shipped.)SET RESETNo. Item Main partsmaterial Remarks(1) Valve bodyassembly Resin/HNBR Solenoid valve mounting part(2) Needle assembly Resin/ Brass/ NBR For adjusting release flow, with lock nut retaining mechanism(3) Body ResinBodies for ejector and for pump system both available. (4)Nozzle Aluminum For vacuum pump system: Spacer (5) Diffuser AluminumFor vacuum pump system: No diffuser(6) Silencer cover Resin △ markSilencer coverConcave Part A SilencerMounting screw Valve bodySupply valve Filter element R *)Gasket Gasket(a)Appropriate tighteningtorque (Nm) 0.075 to 0.096(b)(a) (b)(d) (c) Vacuum pressure release port (Pressure sensor/vacuum pressureswitch can be mounted)Locking type (Tool required) <Semi-standard>- Turn the manual override to the left and line up () manual override.Note) For the locking type manual override, be sure to release the lock before starting the normal operation.Filter case■Special transparent filter case made of nylonDo not use in an environment where chemicals such as alcohol are present and where they could stick to the filter case.Vacuum Break Flow Adjusting Needle ■Vacuum break flow characteristicsThe graph on the right shows the flow characteristics with various supply pressures when the vacuum break flow adjustment needle is opened from the fully close state “n” turns.However, the flow characteristics shown in this graph are represent values of the single unit of the product.The flow at the absorption part may vary depending on the piping conditions to the vacuum (V) port, circuit etc.The flow characteristics and the number of rotations of the needle vary due to the range of the specifications of the product.■ This product has a needle retaining mechanism.The needle stops rotating when it reaches the rotation stop position. It may damage the product if the needle is rotated past its stop position. ■Do not tighten the needle any more after it reaches the fully closed position (fully Clockwise).The fully closed position is when the end of the needle touches the resin hole. If it is tightened any more after the needle reaches the position where it stops, the resin part will be deformed, causing breakage. ■Do not tighten the handle with tools such as pliers. This can result in breakage due to idle turning.Exhaust from Ejector■Avoid back pressure being applied to the exhaust air of the ejector. The exhaust resistance should be as small as possible to obtain the full ejector performance.There should be no shield around the exhaust port for the silencer exhaust specification. For the port exhaust specification, the back pressureincrease should be 0.005MPa (5kPa) at maximum, as exhaust resistance is generated with some piping bore sizes and piping lengths. For tube ID 4, as a guideline, it is recommended to make the piping length 1000mm at maximum, although it varies depending on the condition of the equipment at the end.For the silencer exhaust specification, the silencer will gradually getclogged if dust in the operating environment is sucked in or if the supply air is not clean enough. If the silencer is clogged, back pressure is applied to the ejector exhaust which results in a reduction in the vacuum pressure and the adsorption flow rate.7 ContactsAUSTRIA (43) 2262 62280-0 LATVIA(371) 781 77 00 BELGIUM (32) 3 355 1464 LITHUANIA(370) 5 264 8126 BULGARIA (359) 2 974 4492 NETHERLANDS (31) 20 531 8888 CZECH REP. (420) 541 424 611 NORWAY (47) 67 12 90 20 DENMARK (45) 7025 2900 POLAND (48) 22 211 9600 ESTONIA (372) 651 0370 PORTUGAL (351) 21 471 1880 FINLAND (358) 207 513513 ROMANIA (40) 21 320 5111 FRANCE (33) 1 6476 1000 SLOVAKIA (421) 2 444 56725 GERMANY (49) 6103 4020 SLOVENIA (386) 73 885 412 GREECE (30) 210 271 7265 SPAIN (34) 945 184 100 HUNGARY (36) 23 511 390 SWEDEN(46) 8 603 1200 IRELAND (353) 1 403 9000 SWITZERLAND (41) 52 396 3131 ITALY(39) 02 92711UNITED KINGDOM(44) 1908 563888URL : http// (Global) http// (Europe) Specifications are subject to change without prior notice from the manufacturer. © 2012 SMC Corporation All Rights Reserved.。

第30卷第3期2010年6月国际病理科学与临床杂志 http ://www.g jb.l netIn ternati onal Journal ofPat h ol ogy and C li n i calM ed ici ne Vo.l 30 N o .3J un . 2010收稿日期:2010-05-02 修回日期:2010-05-20作者简介:赵渊源,七年制学生,主要从事细胞信号转导的研究。

通信作者:李春艳,E m ai:l li chunyan915@yahoo W nt 与Notc h 信号通路的串话与肿瘤发生、发展的关系赵渊源1 综述 李春艳2 审校(中国医科大学 1.2007级七年制;2.实验技术中心三部,沈阳110001)[摘要] 肿瘤是危害人类健康与生命的常见疾病之一,其发病机制至今尚未完全探明。

W nt 与N otch 信号通路在生物体的生长发育中起着重要的调控作用,同时也都被证实与肿瘤相关。

两条通路之间存在着许多直接或间接的串话。

在肿瘤的发生发展中,二者间的相互作用取决于不同的组织背景,既可以表现为相互协同,也可以表现为相互拮抗。

W nt 与N o tch 信号通路的串话与肿瘤的关系,在基础与临床肿瘤研究中都具有重大意义。

本文对两条信号通路的组成、作用机制、相互作用,以及其串话与肺癌、皮肤癌、乳腺癌、白血病、黑色素瘤等肿瘤疾病间的关系进行了阐述与分析。

[关键词]W nt ; N o tch ; 串话; 肿瘤do:i 10.3969/.j issn .1673 2588.2010.03.016Cross talk betw een W nt and Notch signali ng path wayand its role i n the genesis and devel op m ent of tu m orsZHAO Yuanyuan 1,LI Chunyan2(1.Grad e 2007,S e ven year Syst e m;2.Th ird De part m e n t of Labora tory T e chnolo gy C e n ter ,Ch i na M edical Universit y ,Shenyang 110001,Ch i na )[Abstract] Tum or is a co mm on disease that da m ages the health and life of hum an beings .The pathogenesis o f tum or is stillnot fully understood .BothW nt and Notch signaling pathw ays play an i m portant ro le i n regulati n g the gro w t h and deve l o p m ent of o r gan is m s and have been proved to be i n vo l v ed i n neoplas m s .There ex ist d irect or i n d irect cross ta l k s bet w een these 2pathways .I n the genesis and devel opm ent of tumo rs ,the i n teracti o n bet w een W n t and Notch show s either synergetic o r antagonistic effect depend i n g on the type o f tissues .C larification o f t h e cr oss tal k s bet w een W nt and N otch si g nali n g path w ays and its ro le i n tum orgenesis is of great si g nificance in bo th basic and clinical neop l a s m stud i e s .This article rev ie wed the essential co m ponents ofW nt and No tch si g na li n g pathw ays ,the cross ta l k s bet w een these t w o pathw ays and the corre lation w it h the developm ent of t h e t u m ors i n cl u ding lung cancer ,sk i n cancer ,breast cancer ,leuke m ia ,and m elano m a .[Key w ords] W n;t N otch; cross talk ; tumo r[Int J Pathol C linM ed ,2010,30(3):0255 05]W nt 与No tch 信号通路是两条在多细胞动物中广泛存在且高度保守的信号转导通路。

介绍濒危动物短语英语作文1. Wow, have you heard about the critically endangered Sumatran tiger? It's such a majestic creature, with its distinctive orange coat and black stripes. It's a shamethat there are only around 400 of them left in the wild. We really need to do something to protect their habitat and stop illegal hunting.2. Did you know that the vaquita, a small porpoise found in the Gulf of California, is on the brink of extinction? There are less than 10 of them left in the world! It's heartbreaking to think that such a beautiful marine mammal could disappear forever. We should raise awareness about the importance of sustainable fishing practices to save these precious creatures.3. The Javan rhinoceros is one of the rarest animals on Earth, with only about 60 individuals remaining. They are known for their single horn and thick, armor-like skin.It's devastating to think that they could be wiped out dueto poaching and habitat loss. We must take immediate action to protect their last remaining habitats and crack down on illegal wildlife trade.4. Have you heard of the pangolin? It's the most trafficked mammal in the world, with its scales being highly sought after in traditional medicine. These scaly creatures are facing extinction due to illegal hunting and habitat destruction. It's crucial that we raise awareness about the importance of protecting these unique animals and put an end to the illegal wildlife trade.5. The black rhinoceros is critically endangered, with only around 5,000 individuals left in the wild. They are often targeted by poachers for their horns, which are falsely believed to have medicinal properties. It's heartbreaking to see such magnificent creatures being pushed to the brink of extinction. We need to strengthen anti-poaching efforts and educate people about the importance of conservation.6. The African elephant, one of the largest landanimals, is also under threat. They are hunted for their ivory tusks, leading to a significant decline in their population. It's tragic to think that future generations might not get to witness the beauty and intelligence of these gentle giants. We must enforce stricter laws against poaching and support conservation efforts to protect these iconic animals.7. The orangutan, found in the rainforests of Borneo and Sumatra, is critically endangered due to deforestation and illegal pet trade. These intelligent primates are losing their homes at an alarming rate, and their population has been drastically reduced. It's crucial that we work towards sustainable palm oil production and fight against the illegal wildlife trade to save these incredible creatures.8. The Hawksbill sea turtle is facing numerous threats, including habitat loss and illegal hunting for their beautiful shells. These graceful creatures play a vitalrole in maintaining the health of coral reefs. It's disheartening to see their population decline rapidly. Weneed to protect their nesting beaches, promote sustainable fishing practices, and educate the public about the importance of marine conservation.9. The Amur leopard, found in Russia and China, is one of the rarest big cats in the world, with less than 100 individuals remaining. They are threatened by habitat loss and poaching. It's devastating to think that we might lose this magnificent species forever. We need to establish protected areas and enforce strict laws against poaching to ensure their survival.10. The Sumatran orangutan, a close relative of the Bornean orangutan, is critically endangered due to habitat destruction caused by palm oil plantations. Theseintelligent and gentle creatures are losing their homes at an alarming rate. It's essential that we support sustainable palm oil production and raise awareness about the devastating impact of deforestation on wildlife.。

Odd Crossing Number Is Not Crossing NumberMichael J.Pelsmajer1,Marcus Schaefer2,and DanielˇStefankoviˇc31Department of Applied Mathematics,Illinois Institute of Technology,Chicago,IL60616pelsmajer@2Department of Computer Science,DePaul University,Chicago,IL60604mschaefer@3Department of Computer Science,University of Rochester,Rochester,NY14627, and Department of Computer Science,Comenius University,Bratislava,Slovakiastefanko@Abstract.The crossing number of a graph is the minimum number ofedge intersections in a plane drawing of a graph,where each intersectionis counted separately.If instead we count the number of pairs of edgesthat intersect an odd number of times,we obtain the odd crossing num-ber.We show that there is a graph for which these two concepts differ,answering a well-known open question on crossing numbers.To derivethe result we study drawings of maps(graphs with rotation systems).1A Confusion of Crossing NumbersIntuitively,the crossing number of a graph is the smallest number of edge cross-ings in any plane drawing of the graph.As it turns out,this definition leaves room for interpretation,depending on how we answer the questions:what is a drawing,what is a crossing,and how do we count crossings?The papers by Pach and T´o th[7]and Sz´e kely[9]discuss the historical development of various interpretations and,often implicit,definitions of the crossing number concept.A drawing D of a graph G is a mapping of the vertices and edges of G to the Euclidean plane,associating a distinct point with each vertex,and a simple plane curve with each edge such that the ends of an edge map to the endpoints of the corresponding curve.For simplicity,we also require that–a curve does not contain any endpoints of other curves in its interior,–two curves do not touch(that is,intersect without crossing),and–no more than two curves intersect in a point(other than at a shared end-point).In such a drawing the intersection of the interiors of two curves is called a crossing.Note that by the restrictions we placed on a drawing,crossings do not involve endpoints,and at most two curves can intersect in a crossing.We often identify a drawing with the graph it represents.For a drawing D of a graph G in the plane we define–cr(D)-the total number of crossings in D;–pcr(D)-the number of pairs of edges which cross at least once;and–ocr(D)-the number of pairs of edges which cross an odd number of times. Remark1.For any drawing D,we have ocr(D)≤pcr(D)≤cr(D).We let cr(G)=min cr(D),where the minimum is taken over all drawings D of G in the plane.We define ocr(G)and pcr(G)analogously.Remark2.For any graph G,we have ocr(G)≤pcr(G)≤cr(G).The question(first asked by Pach and T´o th[7])is whether the inequalities are actually equalities.4Pach[6]called this“perhaps the most exciting open problem in the area.”The only evidence for equality is an old theorem by Chojnacki, which was later rediscovered by Tutte—and the absence of any counterexamples. Theorem1(Chojnacki[4],Tutte[10]).If ocr(G)=0then cr(G)=0.5 In this paper we will construct a simple example of a graph with ocr(G)< pcr(G)=cr(G).We derive this example from studying what we call weighted maps on the annulus.Section2introduces the notion of weighted maps on arbi-trary surfaces and gives a counterexample to ocr(M)=pcr(M)for maps on the annulus.In Section3we continue the study of crossing numbers for weighted maps,proving in particular that cr(M)≤c n·ocr(M)for maps on a plane with n holes.One of the difficulties in dealing with the crossing number is that it is NP-complete[2].In Section4we show that the crossing number can be com-puted in polynomial time for maps on the annulus.Finally,in Section5we show how to translate the map counterexample from Section2into an infinite family of simple graphs for which ocr(G)<pcr(G).2Map Crossing NumbersA weighted map M is a2-manifold S and a set P={(a1,b1),...,(a m,b m)}of pairs of distinct points on∂S with positive weights w1,...,w m.A realization R of the map M=(S,P)is a set of m properly embedded arcsγ1,...,γm in S whereγi connects a i and b i.6Leti(γk,γ )w k w ,cr(R)=1≤k< ≤m[i(γk,γ )>0]w k w ,pcr(R)=1≤k< ≤m[i(γk,γ )≡1(mod2)]w k w ,ocr(R)=1≤k< ≤m4Doug West lists the problem on his page of open problems in graph theory[12].Dan Archdeacon even conjectured that equality holds[1].5In fact they proved something stronger,namely that in any drawing of a non-planar graph there are two non-adjacent edges crossing an odd number of times.Also see[8]. 6If we take a realization R of a map M,and contract each boundary component toa vertex,we obtain a drawing of a graph with a given rotation system[3].For ourpurposes,maps are a more visual way to look at graphs with a rotation system.where i(γ,γ )is the geometric intersection number ofγandγ and[x]is1if the condition x is true,and0otherwise.We define cr(M)=min cr(R),where the minimum is taken over all realiza-tions R of M.We define pcr(M)and ocr(M)analogously.Remark3.For every map M,ocr(M)≤pcr(M)≤cr(M).Conjecture1.For every map M,cr(M)=pcr(M).Lemma1.If Conjecture1is true then cr(G)=pcr(G)for every graph G. Proof.Let D be a drawing of G with minimal pair crossing number.Drill small holes at the vertices.We obtain a drawing R of a weighted map M.If Con-jecture1is true,there exists a drawing of M with the same crossing number. Collapse the holes to vertices to obtain a drawing D of G with cr(D )≤pcr(G).We can,however,separate the odd crossing number from the crossing number for weighted maps,even in the annulus(a disk with a hole).Fig.1.ocr<pcr.When analyzing crossing numbers of drawings on the annulus,we describe curves with respect to an initial drawing of the curve and a number of Dehn twists.Consider,for example,the four curves in the left part of -paring them to the corresponding curves in the right part,we see that the curves labeled c and d have not changed,but the curves labeled a and b have each un-dergone a single clockwise twist.Two curves are isotopic rel boundary if they can be obtained from each other by a continuous deformation which does not move the boundary∂M.Isotopy rel boundary is an equivalence relation,its equivalence classes are called isotopy classes.An isotopy class on annulus is determined by a properly embedded arc connecting the endpoints,together with the number of twists performed. Lemma2.Let a≤b≤c≤d be such that a+c≥d.For the weighted map M in Figure1we have cr(M)=pcr(M)=ac+bd and ocr(M)=bc+ad.Proof.The upper bounds follow from the drawings in Figure1,the left drawing for crossing and pair crossing number,the right drawing for odd crossing number; it remains to prove the two lower bounds.First,we claim thatpcr(M)≥ac+bd.Proof of the claim.Let R be a drawing of M minimizing pcr(R).We can apply twists so that the thick edge d is drawn as in the left part of Figure1.Letα,β,γbe the number of clockwise twists that are applied to arcs a,b,c in the left part of Figure1to obtain the drawing R.Then,pcr(R)=cd[γ=0]+bd[β=−1]+ad[α=0]+bc[β=γ]+ab[α=β]+ac[α=γ+1].(1) Ifγ=0then pcr(R)≥cd+ab because at least one of the lastfive conditions in (1)must be true;the lastfive terms contribute at least ab(since d≥c≥b≥a), and thefirst term contributes cd.Since d(c−b)≥a(c−b),cd+ab≥ac+bd, and the claim is proved in the case thatγ=0.Now assume thatγ=0.Equation(1)becomespcr(R)=bd[β=−1]+bc[β=0]+ad[α=0]+ac[α=1]+ab[α=β].(2) Ifβ=−1then pcr(R)≥bd+ac because eitherα=0orα=1.Since bd+ac≥bc+ad,the claim is proved in the case thatβ=−1.This leaves us with the case thatβ=−1.Equation(2)becomespcr(R)=bc+ad[α=0]+ac[α=1]+ab[α=−1].(3) The right-hand side of Equation(3)is minimized forα=0.In this case pcr(R)= bc+ac+ab≥ac+bd because we assume that a+c≥d.Second,we claim thatocr(M)≥bc+ad.Proof of the claim.Let R be a drawing of M minimizing ocr(R).Letα,β,γbe as in the previous claim.We haveocr(R)=cd[γ]2+bd[β+1]2+ad[α]2+bc[β+γ]2+ab[α+β]2+ac[α+γ+1]2,(4) where[x]2is0if x≡0(mod2),and1otherwise.Ifβ≡γ(mod2)then the claim clearly follows unlessγ=0,β=1,and α=0(all modulo2).In that case ocr(R)≥bc+ab+ac≥bc+ad.Hence,the claim is proved ifβ≡γ(mod2).Assume then thatβ≡γ(mod2).Equation(4)becomesocr(R)=cd[β]2+bd[β+1]2+ad[α]2+ab[α+β]2+ac[α+β+1]2.(5) Ifα≡1(mod2)then the claim clearly follows because either cd or bd contributes to the ocr.Thus we can assumeα≡0(mod2).Equation(5)becomesocr(R)=(cd+ab)[β]2+(bd+ac)[β+1]2.(6) For bothβ≡0(mod2)andβ≡1(mod2)we get ocr(R)≥bc+ad.This finishes the proof of the second claim.2We get a separation of pcr and ocr for maps with small integral weights.Corollary 1.There is a weighted map M on the annulus with edges of weight a =1,b =c =3,and d =4for which cr(M )=pcr(M )=15and ocr(M )=13.Optimizing the gap over the reals yields b =c =1,a =(√3−1)/2,andd =1+a ,giving us the following separation of pcr(M )and ocr(M ).Corollary 2.There exists a weighted map M on the annulus with ocr(M )≤√3/2pcr(M ).Conjecture 2.For every weighted map M on the annulus,ocr(M )≥√32pcr(M ).3Upper Bounds on Crossing NumbersIn Section 5we will transform the separation of ocr and pcr on maps into a separation on graphs.In particular,we will show that for every ε>0there is a graph G such that ocr(G )<(√3/2+ε)cr(G ).The gap,however,cannot be arbitrarily large,as Pach and T´o th showed.Theorem 2(Pach,T´o th [7]).Let G be a graph.Then cr(G )≤2(ocr(G ))2.7This result suggests the question whether the linear separation can be im-proved.We do not believe this to be possible:Conjecture 3.There is a c >0such that cr(G )<c ·ocr(G ).Using a graph redrawing idea from from [8](which investigates other appli-cations of that idea),we can show something weaker:Theorem 3.cr(M )≤ocr(M ) n +44 /5for weighted maps M on the plane with n holes,with strict inequality if n >1.As a special case of the theorem,we have that if M is a (weighted)map on the annulus (n =2)then cr(M )<3ocr(M ),which comes reasonably close to the √3/2lower bound from the previous section.The theorem shows that any counterexample to Conjecture 3cannot be constructed on a plane with a small,fixed number of holes.For reasons of space,we do not include the proof of the theorem.7In terms of pcr(G )better upper bounds on cr(G )are known [11,5].4Computing Crossing Numbers on the AnnulusLet M be a map on the annulus.We explained earlier that as far as crossing numbers are concerned we can describe a curve in the realization of M by a properly embedded arcγab connecting endpoints a and b on the inner and outer boundary of the annulus,and an integer k∈Z,counting the number of twists applied to the curveγab.Our goal is to compute the number of intersections between two arcs after applying a number of twists to each one of them.Since twists can be positive and negative and cancel each other out,we need to count crossings more carefully.Let us orient all arcs from the inner boundary to the outer boundary.Traveling along an arcα,a crossing withβcounts as+1ifβcrosses from right to left,and as−1if it crosses from left to right.Summing up these numbers over all crossings for two arcsαandβyieldsˆi(α,β),the algebraic crossing number ofαandβ.Tutte[10]introduced the notionacr(G)=minD{e,f}∈(E2)|ˆi(γe,γf)|,the algebraic crossing number of a graph,a notion that apparently has not drawn any attention since.Let D k(γ)denote the result of adding k twists to the curveγ.For two curves αandβconnecting the inner and outer boundary we have:ˆi(D k(α),D (β))=k− +ˆi(α,β).(7) Note that i(α,β)=|ˆi(α,β)|for any two curvesα,βon the annulus.Letπbe a permutation of[n].A map Mπcorresponding toπis constructed as follows.Choose n+1points on each of the two boundaries and number them 0,1,...,n in the clockwise order.Let a i be the vertex numbered i on the outer boundary and b i be the vertex numberedπi on the inner boundary,i=1,...,n. We ask a i to be connected to b i in Mπ.We will encode a drawing R of Mπby a sequence of n integers x1,...,x n as follows.Fix a curveβconnecting the a0and b0and chooseγi be such that i(β,γi)=0(for all i).We will connect a i,b i with the arc D x i(γi)in R.Note that for i<j,ˆi(γi,γj)=[πi>πj]and henceˆi(D x i(γi),D x j(γj))=x i−x j+[πi>πj].We haveacr(Mπ)=cr(Mπ)=mini<jx i−x j+[πi>πj]w i w j:x i∈Z,i∈[n]},(8)pcr(Mπ)=mini<j [x i−x j+[πi>πj]=0]w i w j:x i∈Z,i∈[n],(9)ocr(Mπ)=mini<j [x i−x j+[πi>πj]≡0(mod2)]w i w j:x i∈Z,i∈[n].(10)Consider the relaxation of the integer program for cr(Mπ):cr (Mπ)=mini<jx i−x j+[πi>πj]w i w j:x i∈R,i∈[n].(11)Since(11)is a relaxation of(8),we have cr (Mπ)≤cr(Mπ).The following lemma shows that cr (Mπ)=cr(Mπ).Lemma3.Let n be a positive integer.Let b ij∈Z and let a ij∈R be non-negative,1≤i<j≤n.Thenmini<j a ijx i−x j+b ij:x i∈R,i∈[n]has an optimal solution with x i∈Z,i∈[n].Proof.Let x∗be an optimal solution which satisfies the maximum number of x i−x j+b ij=0,1≤i<j≤n.Without loss of generality,we can assume x∗1=0.Let G be a graph on vertex set[n]with an edge between vertices i,j if x∗i−x∗j+b ij=0.Note that if i,j are connected by an edge and one of x∗i,x∗j is an integer then both x∗i and x∗j are integers.It is then enough to show that G is connected.Suppose that G is not connected.There exists non-empty A V(G)such that there are no edges between A and V(G)−A.LetχA be the characteristic vector of the set A,that is,(χA)i=[i∈A].Let f(λ)be the value of the objective function on x=x∗+λ·χA.Let I be the interval on which the signs of the x i−x j+b ij,1≤i<j≤n are the same as for x∗.Then I is not the entire line(otherwise G would be connected).Since f(λ)is linear on I and an open neighborhood of0belongs to I we conclude that f is constant on I.Choosing x=x∗+λχA forλan endpoint of I gives an optimal solution satisfying more x i−x j+b ij=0,1≤i<j≤n,a contradiction.Theorem4.The crossing number of maps on the annulus can be computed in polynomial time.Proof.Note that cr (Mπ)is computed by the following linear program Lπ:mini<j y ij w i w jy ij≥x i−x j+[πi>πj],1≤i<j≤ny ij≥−x i+x j−[πi>πj],1≤i<j≤n.Question1.Let M be a map on the annulus.Can ocr(M)be computed in polynomial time?Conjecture4.For any map M on the annulus cr(M)=pcr(M).5Separating Crossing Numbers of GraphsWe modify the map from Lemma2to obtain a graph G separating ocr(G)and pcr(G).The graph G will have integral weights on edges.From G we can get an unweighted graph G with ocr(G )=ocr(G)and pcr(G )=pcr(G)by replacing an edge of weight w by w parallel edges of weight1(this does not change any of the crossing numbers).If needed we can get rid of parallel edges by subdividing edges,which does not change any of the crossing numbers.We start with the map M from Lemma2with the following integral weights:a= √3−12m,b=c=m,d=√3+12m,where m∈N will be chosen later.We replace each pair(a i,b i)of M by w i pairs(a i,1,b i,1),...,(a i,wi ,b i,wi)where the a i,j(b i,j)occur on∂S in clockwise order in a small interval around of a i(b i).We can argue that all the curves corresponding to(a i,b i)can be routed in parallel in an optimal drawing,and,therefore,the resulting map N with unit weights will have the same crossing numbers as M.We then replace the boundaries of the annulus by cycles(using one vertex for each a i,j and b i,j),obtaining a graph G.We assign weight W=1+cr(N)to the edges in the cycles.This ensures that in a drawing of G minimizing any of the crossing numbers the boundary cycles are embedded without any intersections. This means that a drawing of G minimizing any of the crossing numbers looks very much like the drawing of a map on the annulus.With one subtle difference: one of the boundaries mayflip.Given the map N on the annulus,theflipped map N is obtained byflipping the order of the points on one of the boundaries.In other words,there are essentially two different ways of embedding the two boundary cycles of G on the sphere without intersections depending on the relative orientation of the boundaries.In one of the cases the drawing D of G gives a drawing of N,in the other case it gives a drawing of theflipped map N .Fortunately,in theflipped case the group of edges corresponding to the weighted edge from a i to b i must intersect often with each other(as illustrated in Figure2).Fig.2.The insideflipped.Now we know thatocr(G )≤ocr(N )(since every drawing of N is a drawing of G )≤w 1w 3+w 2w 4(by Lemma 2)≤32m 2(by the choice of weights).We will presently prove the following estimate on the flipped map.Lemma 4.ocr(N )≥2m 2−4m .With that estimate and our discussion of flipped maps,we havecr(G )=min {cr(N ),cr(N )}≥min {cr(N ),ocr(N )}(since ocr ≤cr)≥min {√3m 2−2m,2m 2−4m }(choice of w ,and Lemma 4).By making m sufficiently large,we can make the ratio of ocr(G )and cr(G )arbitrarily close to √3/2.Theorem 5.For any ε>0there is a graph G such thatocr(G )<(√3/2+ε)cr(G ).The proof of Lemma 4will require the following estimate.Lemma 5.Let 0≤a 1≤a 2≤···≤a n be such that a n ≤a 1+···+a n −1.Thenmax |y i |≤a i ⎛⎝n i =1y i 2−2n i =1y 2i ⎞⎠= n i =1a i 2−2n i =1a 2i .Proof of Lemma 4Let w 1=a,w 2=b,w 3=d,w 4=c (with a,b,c,d as in the definition of N ).In any drawing of N each group of the edges split into two classes,those with an even number of twists and those with an odd num-ber of twists (two twists make the same contribution to ocr(M )as no twists).Consequently,we can estimate ocr(N )as follows.ocr(N )=min0≤k i≤w i ⎛⎝4i=1k i2+4i=1w i−k i2+i=jk i(w j−k j)⎞⎠≥−124i=1w i+min0≤x i≤w i⎛⎝4i=1x2i2+4i=1(w i−x i)22+i=jx i(w j−x j)⎞⎠=−124i=1w i+144i=1w i2+min|y i|≤w i/2⎛⎝24i=1y2i−4i=1y i2⎞⎠≥124i=1w2i−124i=1w i(using Lemma5)≥12⎛⎝√3+12m−12+2m2+√3−12m−12−4m⎞⎠≥2m2−4m.(12)The equality between the second and third line can be verified by substituting y i=x i−w i/2.2 Proof of Lemma5Let y1,...,y n achieve the maximum value.Replacing the y i by|y i|does not decrease the objective function.Without loss of generality,we can assume0≤y1≤y2≤···≤y n.Note that y i<y j then y i=a i(otherwise increasing y i byεand decreasing y j byεincreases the objective function for smallε).Let k be the largest i such that y i=a i.Let k=0if no such i exists.We have y i=a i for i≤k and y k+1=···=y n.If k=n we are done.Letf(t)=ki=1a i+(n−k)t2−2ki=1a2i+(n−k)t2.We havef (t)=2(n−k)ki=1a i+(n−k−2)t.Note that for t<a k+1we have f (t)>0and hence the only optimal choice is t=a k+1.Hence y k+1=a k+1,a contradiction with our choice of k.2 6ConclusionThe relationship between the different crossing numbers remains mysterious, and we have already mentioned several open questions and conjectures.Here wewant to revive a questionfirst asked by Tutte(in slightly different form).Recall the definition of the algebraic crossing number from Section4:acr(G)=minD{e,f}∈(E2)|ˆi(γe,γf)|,whereγe is a curve representing edge e in a drawing D of G.It is clear thatacr(G)≤cr(G).Does equality hold?References1.Dan Archdeacon.Problems in topological graph theory.http://www.emba/∼archdeac/problems/altcross.html(accessed April7th,2005).2.Michael R.Garey and David S.Johnson.Crossing number is NP-complete.SIAMJournal on Algebraic and Discrete Methods,4(3):312–316,1983.3.Jonathan L.Gross and Thomas W.Tucker.Topological graph theory.Dover Pub-lications Inc.,Mineola,NY,2001.Reprint of the1987original.4.Chaim Chojnacki(Haim Hanani).¨Uber wesentlich unpl¨a ttbare Kurven im drei-dimensionalen Raume.Fundamenta Mathematicae,23:135–142,1934.5.Petr Kolman and Jiˇr´ıMatouˇs ek.Crossing number,pair-crossing number,andbin.Theory Ser.B,92(1):99–113,2004.6.J´a nos Pach.Crossing numbers.In Discrete and computational geometry(Tokyo,1998),volume1763of Lecture Notes in Comput.Sci.,pages267–273.Springer, Berlin,2000.7.J´a nos Pach and G´e za T´o th.Which crossing number is it anyway?bin.Theory Ser.B,80(2):225–246,2000.8.Michael J.Pelsmajer,Marcus Schaefer,and DanielˇStefankoviˇc.Removing evencrossings.Manuscript,April2005.9.L´a szl´o A.Sz´e kely.A successful concept for measuring non-planarity of graphs:the crossing number.Discrete Math.,276(1-3):331–352,2004.6th International Conference on Graph Theory.10.W.T.Tutte.Toward a theory of crossing binatorial Theory,8:45–53,1970.11.Pavel Valtr.On the pair-crossing number.Manuscript.12.Douglas West.Open problems-graph theory and combinatorics.http://www.math/∼west/openp/(accessed April7th,2005).。

Package‘crossrun’October12,2022Version0.1.1Title Joint Distribution of Number of Crossings and Longest RunDescription Joint distribution of number of crossings and thelongest run in a series of independent Bernoulli trials.Thecomputations uses an iterative procedure where computationsare based on results from shorter series.The procedureconditions on the start value and partitions by furtherconditioning on the position of thefirst crossing(or none).Depends R(>=3.5)License GPL-3URL https:///ToreWentzel-Larsen/crossrunEncoding UTF-8LazyData trueImports Rmpfr(>=0.7-1)RoxygenNote7.1.2Suggests knitr,rmarkdownVignetteBuilder knitrNeedsCompilation noAuthor Tore Wentzel-Larsen[aut,cre],Jacob Anhøj[aut]Maintainer Tore Wentzel-Larsen<****************************>Repository CRANDate/Publication2022-04-1307:32:33UTCR topics documented:boxprobt (2)clshift (3)crossrunauto (3)crossrunbin (4)12boxprobtcrossrunchange (5)crossrunem (6)crossrunemcont (7)crossrunshift (8)crossrunsymm (8)cumsumm (9)cumsummcol (10)exactbin (10)joint100.6 (11)joint100symm (11)joint14.6 (12)joint14em (12)joint14symm (13)joint60.6 (14)joint60em (14)joint60symm (15)simclbin (15)simclem (16)Index17 boxprobt Box Cumulative SumsDescriptionA box cumulative sum is defined as the cumulative sum over a lower left rectangle.This functionis primarily for use when the components are point probabilities for the number of crossings C and the longest run L,then component(c,l)in the result is the probability P(C≥c,L≤l).Usageboxprobt(mtrx)Argumentsmtrx mpfr arrayValuempfr arrayclshift3 Examplesnill<-Rmpfr::mpfr(0,120)one<-Rmpfr::mpfr(1,120)two<-Rmpfr::mpfr(2,120)contents<-c(one,nill,nill,one,one,one,two,two,two)mtrx3<-Rmpfr::mpfr2array(contents,dim=c(3,3))print(mtrx3)print(boxprobt(mtrx3))clshift Number of Crossings and Longest RunDescriptionAuxiliary function for simclbin,computing the number of crossings(type=0)or longest run(type=2) in a sequence of independent normal observations.Crossings and runs are related to whether the observations are above a shift.Usageclshift(seri,shift=0,type=0)Argumentsseri numeric;seri a sequence of random drawsshift numeric;shift for the observatoobstype numeric;0number of crossings,1longest runValuenumber of crossings or longest run,numericcrossrunauto Joint Distribution for Crossings and Runs,autocorrelated SequenceDescriptionJoint probability distribution for the number of crossings C and the longest run L in a sequence of n autocorrelated Bernoulli observations with success probability p.To enhance precision,results are stored in mpfr arrays and the probabilities are multiplied by m n−1for a multiplier m.4crossrunbinUsagecrossrunauto(nmax=100,prob=0.5,changeprob=0.5,mult=2,prec=120,printn=FALSE)Argumentsnmax max sequence length.prob success probability p.changeprob unrestricted change probability.If p≥0.5,probability of changing to success, if not probability of changing to failure.mult multiplier for joint probabilities.prec mpft precision.printn logical for progress output.Valuelist of joint probabilities.Examples#p=0.6,independencecr10.6<-crossrunbin(nmax=10,prob=0.6,printn=TRUE)cra10.6<-crossrunauto(nmax=10,prob=0.6,changeprob=.6,printn=TRUE)Rmpfr::asNumeric(cr10.6$pt[[10]])Rmpfr::asNumeric(cr10.6$pt[[10]])Rmpfr::asNumeric(cr10.6$pt[[10]])-Rmpfr::asNumeric(cra10.6$pt[[10]])#equal#p=0.6,some dependencecr10.6<-crossrunbin(nmax=10,prob=0.6,printn=TRUE)cra10.6.u.5<-crossrunauto(nmax=10,prob=0.6,changeprob=.5,printn=TRUE)round(Rmpfr::asNumeric(cr10.6$pt[[10]]),1)round(Rmpfr::asNumeric(cra10.6.u.5$pt[[10]]),1)#not the samecrossrunbin Joint Distribution for Crossings and RunsDescriptionJoint probability distribution for the number of crossings C and the longest run L in a sequence of n independent Bernoulli observations with success probability p.To enhance precision,results are stored in mpfr arrays and the probabilities are multiplied by m n−1for a multiplier m.crossrunchange5Usagecrossrunbin(nmax=100,prob=0.5,mult=2,prec=120,printn=FALSE) Argumentsnmax max sequence length.prob success probability.mult multiplier for joint probabilities.prec mpft precision.printn logical for progress output.Valuelist of joint probabilities.Examplescrb10.6<-crossrunbin(nmax=10,prob=.6,printn=TRUE)print(crb10.6$pt[[10]])crossrunchange Joint Distribution for Crossings and Runs,Varying Success Probabil-ity.DescriptionJoint probability distribution for the number of crossings C and the longest run L in a sequence of n independent Bernoulli observations with p ossibly varying success probability.To enhance preci-sion,results are stored in mpfr arrays and the probabilities are multiplied by m n−1for a multiplier m.Usagecrossrunchange(nmax=100,prob=rep(0.5,100),mult=2,prec=120,printn=FALSE)Argumentsnmax max sequence length.prob success probabilities.mult multiplier for joint probabilities.prec mpft precision.printn logical for progress output.6crossrunem Valuelist pt of joint probabilities.Cumulative probabilities qt within each row are also included.Further, mostly for code checking,lists pat and qat conditional on starting with a success,and pbt and qbt conditional of starting with a failure,are included.Examplesprob10<-c(rep(.5,5),rep(.7,5))crchange10<-crossrunchange(nmax=10,prob=prob10,printn=TRUE)print(crchange10$pt[[10]])crossrunem Joint Distribution for Crossings and Runs Using the Empirical Me-dian.DescriptionJoint probability distribution for the number of crossings C and the longest run L in a sequence of n Bernoulli observations where the number of successes isfixed at m,m between0and n.Forfixed n,the joint distribution is computed for all m,this makes the computation demanding in terms of time and storage requirements.The joint distribution is computed separately for sequences where thefirst observation is,or is not,a success.The results are mainly intended for use when n is even and m=n/2,but computation in this case requires that all distributions are computed previously for all m,for all shorter sequences(lower n).In the case of even n and m=n/2,the distributions for sequences starting or not with a success are identical,and only the distribution among sequences starting with a success is used.In that case,this may be interpreted as the joint distribution for sequences around the empirical median.Usagecrossrunem(nmax=100,prec=120,printn=FALSE)Argumentsnmax max sequence length.prec mpft precision.printn logical for progress output.Valuenfi,number of sequences with m successes,starting with a success,and nfn,number of sequences with m successes,not starting with a success.Three-dimensional Rmpfr arrays for each n up to nmax,with dimensions n(C=0to n-1),n(L=1to n)and n+1(m=0to n).For n even and m=n/2, only nfi,and the part corresponding to C=1to n-1and L=1and m=n/2is non-zero and should be used.crossrunemcont7 Examplescrem14<-crossrunem(nmax=14,printn=TRUE)Rmpfr::asNumeric(crem14$nfi[[14]][,,"m=7"])#subsets of size7=14/2#restricted to possible values of C and LRmpfr::asNumeric(crem14$nfi[[14]][-1,1:7,"m=7"])#same as stored data joint14emRmpfr::asNumeric(crem14$nfn[[14]][-1,1:7,"m=7"])#the same#subsets of sizes different from14/2#size4,first observation includedRmpfr::asNumeric(crem14$nfi[[14]][,,"m=4"])#size14-4=10,first observation not includedRmpfr::asNumeric(crem14$nfn[[14]][,,"m=10"])#the samecrossrunemcont Continuation of an existing sequence of joint probabilities for cross-ings and longest run,based on the empirical median.DescriptionContinuation of an existing sequence of the number of crossings C and the longest run L in a sequence of n independent continuous observations classified as above or below the empirical me-dian.To enhance precision,results are stored in mpfr arrays and the probabilities are multiplied by choose(n,m)/2where m=n/2,even n assumed.The probabilities are integers in this representation.Usagecrossrunemcont(emstart,n1=61,nmax=100,prec=120,printn=FALSE)Argumentsemstart existing sequencen1sequence length for thefirst new case addedcnmax max sequence length.prec mpft precision.printn logical for including progress output.Valuenfi,number of sequences with m successes,starting with a success,and nfn,number of sequences with m successes,not starting with a success.8crossrunsymm crossrunshift wrapper for crossrunbin,success probability=pnorm(shift).Descriptionwrapper for crossrunbin,success probability=pnorm(shift).Usagecrossrunshift(nmax=100,shift=0,mult=2,prec=120,printn=FALSE)Argumentsnmax max sequence length.shift mean of normal distribution.mult multiplier for joint probabilities.prec mpft precision.printn logical for progress output.Valuelist pt of joint probabilities.Cumulative probabilities qt within each row are also included.Further, mostly for code checking,lists pat and qat conditional on starting with a success,and pbt and qbt conditional of starting with a failure,are included.Examplescrs15<-crossrunshift(nmax=15,printn=TRUE)print(crs15$pt[[15]])crossrunsymm Joint Probabilities for Crossings and Runs,Symmetric CaseDescriptionJoint probability distribution for the number of crossings C and the longest run L in a sequence of n independent Bernoulli observations with success probability p.To enhance precision,results are stored in mpfr arrays and the probabilities are multiplied by m n−1for a multiplier m.This is for the symmetric case with success probability0.5,in which the multiplied probabilities are integers for the default value2of the multiplier.Usagecrossrunsymm(nmax=100,mult=2,prec=120,printn=FALSE)cumsumm9 Argumentsnmax;max sequence length.mult;multiplier for joint probabilities.Default2.prec;mpft precision.printn;logical for including progress output.Valuept,list of joint probabilities,multiplied with m n−1.In addition cumulative probabilities qt within each row are also included.Examplescrs10<-crossrunsymm(nmax=10,printn=TRUE)cumsumm Row-wise Cumulative SumsDescriptionRow-wise Cumulative Sums in mpfr Array.Usagecumsumm(mtrx)Argumentsmtrx mpfr two-dimensional array.Valuempfr array with row-wise cumulative sums,same dimension as the original array.Examplesnill<-Rmpfr::mpfr(0,120)one<-Rmpfr::mpfr(1,120)two<-Rmpfr::mpfr(2,120)contents<-c(one,nill,nill,one,one,one,two,two,two)mtrx3<-Rmpfr::mpfr2array(contents,dim=c(3,3))print(mtrx3)print(cumsumm(mtrx3))10exactbin cumsummcol Column-Wise Cumulative SumsDescriptionColumn-wise cumulative sums in mpfr array.Usagecumsummcol(mtrx)Argumentsmtrx mpfr two-dimensional array.Valuempfr array with column-wise cumulative sums,same dimension as the original array.Examplesnill<-Rmpfr::mpfr(0,120)one<-Rmpfr::mpfr(1,120)two<-Rmpfr::mpfr(2,120)contents<-c(one,nill,nill,one,one,one,two,two,two)mtrx3<-Rmpfr::mpfr2array(contents,dim=c(3,3))print(mtrx3)print(cumsummcol(mtrx3))exactbin Exact Joint Probabilities for Low nDescriptionExact joint probabilities,for low n,of the number of crossings C and the longest run L in n inde-pendent Bernoulli observations with success probability p.Probabilites are multiplied by2n−1. Usageexactbin(n,p=0.5,prec=120)Argumentsn number,length of seqience,at most6.p success probability.prec precision in mpfr calculations.Default120.joint100.611Valuempfr arrayExamplesexactbin(n=6)exactbin(n=5,p=0.6)joint100.6Joint probabilities,n=100,success probability0.6DescriptionThe joint probabilities of the number C og crossings(0,...99)and the longest run L(1,...,100)ina series of n=100independent Bernoulli observations for success probability0.6.The probabilitiesare stored in the"times"representations,multiplied by2100−1.Only the joint distributions for n=15, 60,100and success probabilities0.5and0.6are included in the package to avoid excessive storage, but many more cases may be generated by the function crossrunbin.Usagejoint100.6Formatmatrix,100rows and100columnsSourcegenerated by the function crossrunbin and transformed from an Rmpfr array to a matrixjoint100symm Joint probabilities,n=100,symmetric caseDescriptionThe joint probabilities of the number C og crossings(0,...99)and the longest run L(1,...,100)ina series of n=100independent Bernoulli observations for the symmetric case(success probability0.5).The probabilities are stored in the"times"representations,multiplied by2100−1and are inte-gers in the symmetric case.Only the joint distributions for n=15,60,100and success probabilities0.5and0.6are included in the package to avoid excessive storage,but many more cases may begenerated by the function crossrunsymm.Usagejoint100symm12joint14emFormatmatrix,100rows and100columnsSourcegenerated by the function crossrunsymm and transformed from an Rmpfr array to a matrixjoint14.6Joint probabilities,n=14,success probability0.6DescriptionThe joint probabilities of the number C og crossings(0,...13)and the longest run L(1,...,14)in a series of n=14independent Bernoulli observations for success probability0.6.The probabilities are stored in the"times"representations,multiplied by214−1=8192.Only the joint distributions for n=14,60,100and success probabilities0.5and0.6are included in the package to avoid excessive storage,but many more cases may be generated by the function crossrunbin.Usagejoint14.6Formatmatrix,14rows and14columnsSourcegenerated by the function crossrunbin and transformed from an Rmpfr array to a matrixjoint14em Joint probabilities,n=14,around the empirical medianDescriptionJoint probabilities of the number C of crossings(1,...13)and the longest run L(1,...,17)in a series of n=60Bernoulli observations around its empirical median.The probabilities are stored in the"times"representations,multiplied by(60by30)/2,the number of constellations starting above the median,and are integers.About the empirical median there is at least one crossing,and the longest run cannot exceed14/2=7.Only the joint distributions for n=14,60are included in the package to avoid excessive storage,but many more cases may be generated by the function ’crossrunem.Since these computations are demanding in terms of storage and computation time, they are at present not performed for n much above60.joint14symm13 Usagejoint14emFormatmatrix,13rows and7columnsSourcegenerated by the function crossrunsymm and transformed from an Rmpfr array to a matrixjoint14symm Joint probabilities,n=14,symmetric caseDescriptionJoint probabilities of the number C of crossings(0,...13)and the longest run L(1,...,14)in a series of n=14independent Bernoulli observations for the symmetric case(success probability0.5).The probabilities are stored in the"times"representations,multiplied by214−1=8192and are integers in the symmetric case.Only the joint distributions for n=14,60,100and success probabilities0.5and0.6are included in the package to avoid excessive storage,but many more cases may begenerated by the function crossrunsymm.Usagejoint14symmFormatmatrix,14rows and14columnsSourcegenerated by the function crossrunsymm and transformed from an Rmpfr array to a matrix14joint60em joint60.6Joint probabilities,60,success probability0.6DescriptionThe joint probabilities of the number C og crossings(0,...59)and the longest run L(1,...,60)ina series of n=60independent Bernoulli observations for success probability0.6.The probabilitiesare stored in the"times"representations,multiplied by260−1.Only the joint distributions for n=15, 60,100and success probabilities0.5and0.6are included in the package to avoid excessive storage, but many more cases are generated in the script crossrun1.R.Usagejoint60.6Formatmatrix,60rows and60columnsSourcegenerated by the function crossrunbin and transformed from an Rmpfr array to a matrixjoint60em Joint probabilities,n=60,around the empirical medianDescriptionJoint probabilities of the number C of crossings(1,...59)and the longest run L(1,...,30)in a series of n=14Bernoulli observations around its empirical median.The probabilities are stored in the"times"representations,multiplied by(14by7)/2=1716,the number of constellations starting above the median,and are integers.About the empirical median there is at least one crossing,and the longest runcannot exceed60/2=30.Only the joint distributions for n=14,60are included in the package to avoid excessive storage,but many more cases may be generated by the function ’crossrunem.Since these computations are demanding in terms of storage and computation time, they are at present not performed for n much above60.’#’Usagejoint60emFormatmatrix,59rows and30columnsSourcegenerated by the function crossrunem and transformed from an Rmpfr array to a matrixjoint60symm15 joint60symm Joint probabilities,n=60,symmetric caseDescriptionThe joint probabilities of the number C og crossings(0,...59)and the longest run L(1,...,60)in a series of n=60independent Bernoulli observations for the symmetric case(success probability0.5).The probabilities are stored in the"times"representations,multiplied by260−1and are integers in the symmetric case.Only the joint distributions for n=15,60,100and success probabilities0.5and0.6are included in the package to avoid excessive storage,but many more cases may be generatedby the function crossrunsymm.Usagejoint60symmFormatmatrix,60rows and60columnsSourcegenerated by the function crossrunsymm and transformed from an Rmpfr array to a matrixsimclbin Simulation of Independent Bernoulli ObservationsDescriptionSimulation of a sequence of independent Bernoulli Observations.To reduce the amount of random draws,each simulation is based on a sequence of standard normal variables,and whether each observation is above a shift defined by the binomial probabilities assumed.Usagesimclbin(nser=100,nsim=1e+05,probs=c(0.5,0.6,0.7,0.8,0.9))Argumentsnser length of sequence simulatednsim number of simulationsprobs binomial probabilites16simclemValuea data frame with the number of crossings and longest run for each probability.For instance thevariables nc0.5and lr0.5are the number of crossings and the longest run for success probability0.5.One row for each simulation.Examplescl30simbin<-simclbin(nser=30,nsim=100)mean(cl30simbin$nc0.5)#mean number of crossings,p=0.5mean(cl30simbin$lr0.9)#mean longest run,p=0.9simclem Check of joint probabilities by simulationsDescriptionSimulation of a sequence of n=2m observations around the median in the sequence.To be used for checking the results of crossrunem.Usagesimclem(m1=7,nsim=1e+05)Argumentsm1,half the sequence lengthnsim number of simulationsValuedata frame with cs,number of crossings and ls,longest run in the simulations.Examplessimclem14<-simclem(nsim=sum(joint14em))print(table(simclem14))#joint distributions in the simulationsprint(joint14em)#for comparisonIndex∗datasetsjoint100.6,11joint100symm,11joint14.6,12joint14em,12joint14symm,13joint60.6,14joint60em,14joint60symm,15boxprobt,2clshift,3crossrunauto,3crossrunbin,4crossrunchange,5crossrunem,6crossrunemcont,7crossrunshift,8crossrunsymm,8cumsumm,9cumsummcol,10exactbin,10joint100.6,11joint100symm,11joint14.6,12joint14em,12joint14symm,13joint60.6,14joint60em,14joint60symm,15simclbin,15simclem,1617。

玩丛林穿越作文Imagine the feeling of trekking through a thick, mysterious jungle, surrounded by towering trees and exotic wildlife. 想象一下穿越茂密神秘的丛林时的感受，四周是高大的树木和奇异的野生动物。

The adventure of jungle trekking is an exhilarating experience that brings you closer to nature in its wildest form. 穿越丛林的冒险是一种令人兴奋的体验，让你更接近大自然最原始的形态。

Walking along rugged paths and crossing over gushing streams, the sense of freedom and connection with the environment is indescribable. 沿着崎岖的小径行走，穿过潺潺的溪流，那种自由感和与环境的联系感是无法言喻的。

The sights and sounds of the jungle are a symphony of nature's beauty – from the chirping of colorful birds to the rustling of leaves in the wind. 丛林的景色和声音是大自然美丽的交响乐——从五颜六色的鸟儿的鸣叫到树叶在风中的沙沙声。

However, it's not just about the physical journey through the jungle, but also about the mental and emotional transformation that takes place within. 然而，这不仅仅是关于穿越丛林的身体旅程，还包括内心和情感上的转变。

crossing the rubicon翻译"Crossing the Rubicon" 的翻译是"越过卢比孔河"。

造句（英文）：1. Julius Caesar's crossing of the Rubicon River was a bold and fateful decision that changed the course of history.2. Once you decide to cross the Rubicon, there's no turning back, and you have to face the consequences of your actions.3. Crossing the Rubicon was a metaphor for taking a significant and irreversible step in one's life or career.4. The general knew that crossing the Rubicon meant defying the Senate's orders and potentially sparking a civil war.5. The Rubicon River marked the boundary between Gaul and Italy, and crossing it with his army signified Caesar's intent to seize power.6. After crossing the Rubicon, Caesar famously said, "Alea iacta est," meaning "the die is cast."7. The phrase "crossing the Rubicon" is often used to describe a point of no return in any critical decision-making process.8. History is filled with examples of leaders who faced tough choices, but crossing the Rubicon required tremendous courage and determination.9. Crossing the Rubicon was a moment of great uncertainty, but it was also a defining moment that shaped the destiny of an empire.10. The consequences of crossing the Rubicon were profound, leading to a shift in the balance of power and the rise of an imperial rule.翻译（中文）：1. 凯撒大帝越过卢比孔河是一项大胆而决定性的举动，改变了历史的进程。