Basics on Epipolar Geometry

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petrel-属性建模

Nugget: Degree of dissimilarity at zero • Vertical
distance.
Basic Statistics
Example of Experimental Variogram calculation Semi-variance for 1 Lag distance Semi-variance for 2 lag distance
- Determine Layer thickness - Determine directions/degree of Anisotropy - Determine correlation/connectedness of facies data
Used as Quality Control to compare data before and after modeling process
Variogram Map
Good for visualizing anisotropy and its direction.
Sample Variogram
Good for finding Major and minor Range horizont
Variogram Map – Theory
EXERCISE A WELL with a string of porosity values in depth steps of 1m: 3, 5, 7, 6, 4, 1, 1, 4. Calculate the variogram values for lags 1, 2, 3, and 4 m respectively. Plot the variogram. Is there a pattern?

Epipolar geometry

Rectification
• Rotate the left camera so epipole goes to infinity along the horizontal axis • Apply the same rotation to the right camera • Rotate the right camera by R • Adjust the scale
Fundamental matrix
• Encodes information of the intrinsic and extrinisic parameters • F is of rank 2, since S has rank 2 (R and M and M’ have full rank) • Has 7 degrees of freedom There are 9 elements, but scaling is not significant and det F = 0
Triangulation
ap’ ray through C’ and p’, bRp + T ray though C and p expressed in right coordinate system
ap'bRp c( p'Rp) T
R Rr RlT T Tr RTl
3D Reconstruction
• Stereo: we know the viewing geometry (extrinsic parameters) and the intrinsic parameters: Find correspondences exploiting epipolar geometry, then reconstruct • Structure from motion (with calibrated cameras): Find correspondences, then estimate extrinsic parameters (rotation and direction of translation), then reconstruct. • Uncalibrated cameras: Find correspondences, Compute projection matrices (up to a projective transformation), then reconstruct up to a projective transformation.

Modular Robotics PowerCube系列产品说明书

PGElectrical · Principle of Function · Universal Gripper1044Modular RoboticsModular-Standardized interfaces for mechatronics and control for rapid and simple assembly without complicated designs-Cube geometry with diverse possibilities for creating individual solutions from the modular systemIntegrated-The control and power electronics are fully integrated in the modules for minimal space requirements and interfering contours-Single-cable technology combines data transmission and the power supply for minimal assembly and start-up costs Intelligent-Integrated high-end microcontroller for rapid data processing -Decentralized control system for digital signal processing -Universal communication interfaces for rapid incorporation in existing servo-controlled conceptsYour advantages and benefitsThe modules of the PowerCube series provide the basis for flexible combinatorics in automation. Complex systems and multiple-axis robot structures with several degrees of freedom can be achieved with minimum time and expenditure spent on design and programming.Module overviewThe innovative technology of the PowerCube modules already forms the basis of numerous applications in the fields of measuring and testing systems, laboratory automation, service robotics and flexiblerobot technology.PGServo-electric2-Finger Parallel Gripper PRServo-electric Rotary Actuators PWServo-electricRotary Pan Tilt ActuatorsPSMServo-motors with integrated position controlPDUServo-positioning motor with precision gearsPLSServo-electric Linear Axes withball-and-screw spindle drivePG·Universal Gripper1045Method of actuationThe PowerCube modules work completely independently. The master control system is only required for generating the sequential program and sending it step by step to the connected modules. Therefore, only the current sequential command is ever stored in the modules, and the subsequent command is stored in the buffer. The current, rotational speed and positioning are controlled in the module itself. Likewise, functions such as temperature and limit monitoring are performed in the module itself. Real-time capability is not absolutely essential for the master control or bus system. For the communication over Bus-System the SMP - SCHUNK Motion Protocol - is used. This enables you to create industrial bus networks,and ensures easy integration in control systems.Control version AB Hardware Control with PLC (S7)Control with PC Interface Profibus DP CAN bus / RS-232SoftwareWindows (from Windows 98) operating systemLINUX operating systemDevelopment platforms MC-Demo Operating Software PowerCube (LabView, Diadem)with Online documentation, standard softwaregsd-file, programming examples(gsd file, programming examples)on requeston requestIncluded with the ''Mechatronik DVD'' (ID 9949633): Assembly and Operating Manual with manufacturer's declaration, MCDemo software and description and gsd-file for S7 use.1234567889ᕃ24VDC / 48VDC power supply provided by the customerᕄControl system provided by the customer (see control versions A, B and C)ᕅPAE 130 TB terminal block for connecting the voltage supply, the communication and the hybrid cable (Option for easy connection)ᕆPDU servo-motorᕇLinear axis with PLS ball-and-screw spindle drive and PSM servo-motorᕈHybrid cable (single-cable technology) for connecting the PowerCube modules (voltage supply and communication). Not recommended for the use in Profibus applications ᕉPW Servo-electric Rotary Pan Tilt Actuator ᕊPG Servo-electric 2-Finger Parallel Gripper ᕋPR Servo-electric Rotary ActuatorPG· Universal Gripper1046Size 70Weight 1.4 kg Gripping force up to 200 N Stroke per finger 35 mm Workpiece weight1 kgApplication exampleDouble rotary gripper module for loading and unloading of sensitive componentsPG 70 Servo-electric 2-Finger Parallel Gripper PR 70 Servo-electric Rotary ActuatorPGUniversal Gripper1047Gripping force control in the range of 30 - 200 N for the delicate gripping of sensitive workpieces Long stroke of 70 mm for flexible workpiece handlingFully integrated control and power electronics for creating a decentralized control systemVersatile actuation optionsfor simple integration in existing servo-controlled concepts via Profibus-DP, CAN bus or RS-232Standard connecting elements and uniform servo-controlled conceptfor extensive combinatorics with other PowerCube modules (see explanation of the PowerCube system)Single-cable technology for data transmission and power supplyfor low assembly and start-up costsServo-electric 2-finger parallel gripper with highly precise gripping force control and long strokeUniversal GripperArea of applicationUniversal, ultra-flexible gripper for great part variety and sensitive components in clean working environmentsYour advantages and benefitsGeneral information on the seriesWorking principle Ball screw driveHousing materialAluminum alloy, hard-anodized Base jaw materialAluminum alloy, hard-anodized ActuationServo-electric, by brushless DC servo-motorWarranty 24 monthsScope of deliveryGuide centering sleeves and ‘’Mechatronik DVD’’ (contains an Assembly and Operating Manual with manufacturer’s declarartion and MC-Demo software withdescription)PG· Universal Gripper1048Control electronicsintegrated control and power electronics for controlling the servo-motorEncoderfor gripper positioning and position evaluationDrivebrushless DC servo-motorGear mechanismtransfers power from the servo-motor to the drive spindleSpindletransforms the rotational movement into the linear movement of the base jaw Humidity protection cap link to the customer’s systemThe brushless servo-motor drives the ball screw by means of the gear mechanism.The rotational movement is transformed into the linear movement of the base jaw by base jaws mounted on the spindles.Function descriptionThe PG gripper is electrically actuated by the fully integrated control and power electronics. In this way, the module does not require any additional external control units.A varied range of interfaces, such as Profibus-DP, CAN-Bus or RS-232 are available as methods of communication. For the communication over Bus-System the SMP - SCHUNK Motion Protocol - is used. This enables you to create industrial bus networks, and ensures easy integration in control systems.If you wish to create combined systems (e.g. a rotary gripper module), various other modules from the Mechatronik-Portfolio are at your disposal.Electrical actuationSectional diagramPGUniversal Gripper1049Gripping forceis the arithmetic total of the gripping force applied to each base jaw at distance P (see illustration), measured from the upper edge of the gripper.Finger lengthis measured from the upper edge of the gripper housing in the direction of the main axis.Repeat accuracyis defined as the spread of the limit position after 100 consecutive strokes.Workpiece weightThe recommended workpiece weight is calculated for a force-type connection with a coefficient of friction of 0.1 and a safety factor of 2 against slippage of theworkpiece on acceleration due to gravity g. Considerably heavier workpiece weights are permitted with form-fit gripping.Closing and opening timesClosing and opening times are purely the times that the base jaws or fingers are in motion. Control or PLC reaction times are not included in the above times and must be taken into consideration when determining cycle times.General information on the seriesCentering sleevesElectrical accessories PAE terminal blockPAM standardconnecting elementsAccessoriesHybrid cableFor the exact size of the required accessories, availability of this size and the designation and ID, please refer to the additional views at the end of the size in question. You will find more detailed information on our accessory range in the …Accessories“ catalog section.PG 70· Universal Gripper1050Technical dataFinger loadMoments and forces apply per base jaw and may occur simultaneously. M y may arise in addition to the moment generated by the gripping force itself. If the max.permitted finger weight is exceeded, it is imperative to throttle the air pressure so that the jaw movement occurs without any hitting or bouncing. Service life may bereduced.Gripping force, I.D. grippingDescriptionPG 70Mechanical gripper operating data ID 0306090Stroke per finger [mm]35.0Constant gripping force (100 % continuous duty)[N]200.0Max. gripping force [N]200.0Min. gripping force [N]30.0Weight [kg] 1.4Recommended workpiece weight [kg] 1.0Closing time [s] 1.1Opening time [s] 1.1Max. permitted finger length [mm]140.0IP class20Min. ambient temperature [°C] 5.0Max. ambient temperature [°C]55.0Repeat accuracy [mm]0.05Positioning accuracy [mm]on request Max. velocity [mm/s]82.0Max. acceleration [mm/s 2]328.0Electrical operating data for gripper Terminal voltage [V]24.0Nominal power current [A] 1.8Maximum current [A] 6.5Resolution [µm] 1.0Controller operating data Integrated electronics Yes Voltage supply [VDC]24.0Nominal power current [A]0.5Sensor system EncoderInterfaceI/O, RS 232, CAN-Bus, Profibus DPPG 70Universal Gripper1051ᕃ24 VDC power supply provided by thecustomerᕄControl (PLC or similar) provided bythe customerᕅPAE 130 TB terminal block(ID No. 0307725) for connecting the power supply, the communication and the hybrid cableᕆHybrid cable for connecting thePowerCube modulesMain viewsThe drawing shows the gripper in the basic version with closed jaws, the dimensions do not include the options described below.ᕃGripper connection ᕄFinger connectionᕓᕗM16x1.5 for cable glandActuation DescriptionID Length PowerCube Hybrid cable, coiled 03077530.3 m PowerCube Hybrid cable, coiled03077540.5 mPowerCube Hybrid cable, straight (per meter)9941120The ‘Hybrid cable’ is recommended for the use in CAN-Bus- or RS232-systems. For Profibus applications we recommend to use a separate standardized Profibus cable for the communication.You can find further cables in the …Accessories“ catalog section.Interconnecting cablePG 70· Universal Gripper1052Special lengths on requestRight-angle standard element for connecting size 70 PowerCube modulesSpecial lengths on requestConical standard element for connecting size 70 and 90 PowerCube modulesSpecial lengths on requestStraight standard element for connecting size 70 PowerCube modules Right-angle connecting elements Description ID DimensionsPAM 120030782090°/70.5x98Conical connecting elements Description ID DimensionsPAM 110030781090x90/45/70x70 mm PAM 111030781190x90/90/70x70 mmStraight connecting elements Description ID DimensionsPAM 100030780070x70/35/70x70 mm PAM 101030780170x70/70/70x70 mmMechanical accessoriesYou can find more detailed information and individual parts of the above-mentioned accessories in the …Accessories“ catalog section.。

Lecture7

(PowerPoint ****)(simplified version from Wilson)
Slides from the text book (From the CD included in Atkins ,**)
Operators and Expectation Values
Re view of average calculations
Hˆ ψ(x) = Eψ(x) ; Fˆ ψ(x) ≠ kψ(x)
In this case the meassurement of F will give one of the eigenvalues of F Fξi = ki ξi
The average value from a large number
the average value < F > is also called an expectation value
Operators and Expectation Values New postulate 5.
Consider the special case where ψ(x) is a simultanious eigenfunction to Hˆ and Fˆ
Consider a large number N of identical boxes with identical particles all described by the same wavefunction Ψ(x, t) :
Let us for each system at the same time meassure the property F
of meassurements will be
<F

ASQ CQE (考试题目与答案)

Study Questions and Answers This material is protected under United States copyright laws. It may not be reproduced, transmitted ordistributed in any form without the express written consent of Carl Nocera.Copyright © 2004 by Carl D. Nocera. All rights reserved.Produced in the United States of America.QREVIEW STUDY QUESTIONS and ANSWERSBasic Quality Concepts 1Basic Probability 7Statistics 15 Statistical Inference 23Sampling 27 Control Charts 31Reliability 35 Regression and Correlation 37Cost of Quality 39Design of Experiments 43Metrology and Calibration 47Answer Sheet 49Basic Quality Concepts 1 QREVIEW STUDY QUESTIONSINTRODUCTIONThe following questions include key concepts which are representative of the CQE examination. Each question should be worked out as completely as possible before looking at the answer. Some questions have been taken from previously published ASQ CQE exams.BASIC QUALITY CONCEPTS1. In many companies, what is generally the weakest link in the quality auditingprogram?a) Audit reportingb) Follow-up of corrective action implementationc) Scheduling of auditsd) Inadequate audit checklists2. A quality control program is considered to bea) a collection of quality control procedures and guidelines.b) a step by step list of all quality control check points.c) a summary of company quality control policies.d) a system of activities to provide quality of products and services.3. The "quality function" of a company is best described asa) the degree to which the company product conforms to a design or specification.b) that collection of activities through which "fitness for use" is achieved.c) the degree to which a class or category of product possesses satisfaction forpeople generally.d) All of the above.4. In preparing a product quality policy for your company, you should do all of the followingexcepta) specify the means by which quality performance is measured.b) develop criteria for identifying risk situations and specify whose approval isrequired when there are known risks.c) include procedural matters and functional responsibilities.d) state quality goals.2 QReview Study Questions5. What natural phenomenon created the necessity to control product and process quality?a) Gravityb) Variationc) Human Errord) Management6. The three basic elements of a quality system area) Quality Management, Purchasing and Document Controlb) Quality Management, Quality Control and Quality Assurancec) SPC, Inspection and Quality Assuranced) Quality Control, Quality Costs and Control Charts7. What are the two basic categories of quality?a) Design and Conformance Qualityb) Good and Bad Qualityc) Defective and Non-Defective Qualityd) Attribute and Variable Quality8. The Law of Large Numbers states thata) individual occurrences are predictable and group occurrences are unpredictable.b) group data always follows a normal pattern.c) individual occurrences are unpredictable and group occurrences are predictable.d) the standard deviation of group data will always be greater than ten.9. Statistical quality control is best described asa) keeping product characteristics within certain bounds.b) calculating the mean and standard deviation.c) the study of the characteristics of a product or process, with the help of numbers,to make them behave the way we want them to behave.d) the implementation of ISO 9000.10. Which of the following is the most important element in Statistical Quality Control?a) The Feedback Loopb) Make Operationc) Inspectiond) Quality of Incoming MaterialBasic Quality Concepts 311. When measurements are accurate and precise,a) the data are distributed randomly throughout the entire range.b) the data are clustered closely around the central value.c) minimum variation will exist.d) the data are normally distributed.12. All of the following are included in a quality system excepta) document control.b) corrective action.c) management responsibility.d) employee salaries.13. Which of the following best describes a statistical distribution?a) A model that shows how data are distributed over a range of measurements.b) An Analysis of Variance table.c) A sampling plan.d) A graph that contains data plotted on a normal curve.14. Which of the following are two types of data used in statistical quality control?a) Design and Conformance Datab) Precise and Accurate Datac) Variables and Attributes Datad) Mean and Variance Data15. The primary reason for evaluating and maintaining surveillance over a supplier's qualityprogram is toa) perform product inspection at source.b) eliminate incoming inspection cost.c) motivate suppliers to improve quality.d) make sure the supplier's quality program is functioning effectively.16. Which one of the following are ISO 9001 requirements?a) Process Flow Chartb) Quality Manualc) Operations Manuald) TQM Program4 QReview Study Questions17. Which of the following does not generate product-quality characteristics?a) Designerb) Inspectorc) Machinistd) Equipment engineer18. Incoming material inspection is based most directly ona) design requirements.b) purchase order requirements.c) manufacturing requirements.d) customer use of the end product.19. The acronym ISO meansa) International Standards Organization.b) Internal Service Organization.c) equal.d) third party auditing organization.20. Products should be subjected to tests which are designed toa) demonstrate advertised performance.b) demonstrate basic function at minimum testing cost.c) approximate the conditions to be experienced in customer's application.d) assure that specifications are met under laboratory conditions.21. The advantage of a written procedure isa) it provides flexibility in dealing with problems.b) unusual conditions are handled better.c) it is a perpetual coordination device.d) coordination with other departments is not required.22. In spite of the Quality Engineer's best efforts, situations may develop in which hisdecision is overruled. The most appropriate action would be toa) resign from the position based upon convictions.b) report findings to an outside source such as a regulatory agency or the press.c) document findings, report them to superiors and move on the next assignment.d) discuss findings with co-workers in order to gain support, thereby forcing action.Basic Quality Concepts 523. If a test data does not support a Quality Engineer's expectations, the best thing to do isa) adjust the data to support expectations if it is only slightly off.b) draw the expected conclusion omitting the data not supporting it.c) re-evaluate the expectations of the test based upon the data.d) report the data and expected conclusion with no reference to one another.24. In case of conflict between contract specifications and shop practice,a) arbitration is necessary.b) the customer is always right.c) good judgment should be exercised.d) contract specifications normally apply.25. A quality audit program should begin witha) a study of the quality documentation system.b) an evaluation of the work being performed.c) a report listing findings, the action taken and recommendations.d) a charter of policy, objectives and procedures.26. Selection of auditors shall ensure thata) auditors are completely independent from the organization being audited.b) management is aware of the audit activities.c) auditors do not audit their own work.d) auditors are trained.27. Analysis of data on all product returns is important becausea) failure rates change with length of product usage.b) changes in design and in customer use are often well reflected.c) immediate feedback and analysis of product performance becomes available.d) All of the above.28. All of the following are considerations when a total quality management (TQM) programis implemented excepta) the use of statistical tools and techniques.b) a program of continuous quality improvement.c) the manager responsible for product quality.d) total involvement from management to production associates.6 QReview Study Questions29. According to Juran, all of the following are widespread errors in perception that have ledmany managers astray excepta) the work force is mainly responsible for the company's quality problems.b) workers could do quality work but they lack the motivation to do so.c) quality will get top priority if upper management so decrees.d) return on investment is everything.30. An essential technique in making training programs effective is toa) set group goals.b) have training classes which teach skills and knowledge required.c) feed back to the employee meaningful measures of his performance.d) post results of performance before and after the training program.31. An engineer has the job of providing a written plan of quality related tasks to hismanager, including a detailed timeline, for the following year. Which of the following tools should be used?a) Histogramb) Flow Chartc) Gantt Chartd) Frequency DistributionBasic Probability 7 BASIC PROBABILITY32. The time it takes to answer a technical support line has a continuous uniform distributionover an interval from 17 to 20 minutes. All of the following are true excepta) P(x = 18.5) = 1/2b) P(x ≤ 20) = 1c) P(17 ≤ x ≤ 18) = 1/3d) P(x ≥ 17) = 133. For two events, A and B, which one of the following is a true probabilitystatement?a) P(A or B) = P(A) + P(B) if A and B are independentb) P(A or B) = P(A) + P(B) if A and B are mutually exclusivec) P(A and B) = P(A) x P(B) if A and B are mutually exclusived) P(A or B) = P(A) x P(B) if A and B are independent34. What is the probability of getting a head or a tail in 1 toss of a coin?a) 1/16b) 1/4c) 1/2d) 135. What is the probability of getting a head and a tail in 2 tosses of a fair coin? And, what isthe probability of getting a head and a tail, in that order, in 2 tosses of a fair coin?a) 1/2, 1/2b) 1/4, 1/4c) 1/2, 1/4d) 1/4, 1/236. A coin is tossed 10 times. The first 9 tosses come up heads. What is the probability thatthe 10th toss will come up heads?a) 1/512b) 1/256c) 1/32d) 1/237. What is the probability of obtaining exactly 2 heads in 4 tosses of a fair coin?a) 1/4b) 3/8c) 1/2d) 1/68 QReview Study Questions38. What is the probability of getting a 3 when rolling a single die? ( A die is one of a pair ofdice)a) 1/5b) 3/5c) 1/6d) 1/339. What is the probability of getting an odd number when rolling a pair of dice? (Spots onthe two dice sum to odd number)a) 1/4b) 1/2c) 1/3d) 3/1040. What is the probability of obtaining a sum of 7 when rolling a pair of dice?a) 1/5b) 3/5c) 1/6d) 1/3Use the following information to answer questions 41, 42 and 43. The probability is 1/2 that Bob will pass the CQE exam, 1/3 that Amy will pass and 3/4 that Jon will pass.41. What is the probability that Bob, Amy and Jon will all pass the exam?a) 1/8b) 4/9c) 4/11d) 1/342. What is the probability that neither Bob, Amy nor Jon will pass the exam?a) 1/9b) 7/8c) 1/12d) 2/343. What is the probability that only one of the three will pass the exam?a) 1/4b) 1/3c) 3/4d) 3/8Basic Probability 9 44. Four people shoot at a target and the probability that each will hit the target is 1/2 (50%).What is the probability that the target will be hit?a) 1/16b) 15/16c) 1/2d) 1/445. A committee of 5 people is chosen at random from a room that contains 4 men and 6women. What is the probability that the committee is composed of 2 men and 3 women?a) 1/2b) 10/21c) 5/21d) 1/346. A vendor is trying to sell you a box of 50 fuses that contains exactly 5 defective fuses.You select 2 fuses from the box for testing. If both are good you will buy the entire box. If one or both are defective, you will not buy the box. What is the probability that you will buy the box?a) .7533b) .8082c) .9769d) .853147. What is the probability of winning the Super Lotto? (Winning = getting all 6 numbers outof 47)a) 1/10,737,573b) 1/7,731,052,560c) 1/3,457,296d) 1/12,966,82148. A box contains 12 connectors, 9 good and 3 defective. What is the probability ofobtaining exactly 2 good and 1 defective connector in drawing 3 parts from the boxwithout replacement?a) .4219b) .4909c) .5022d) .691510 QReview Study Questions49. A box contains 12 connectors, 9 good ones and 3 defective ones. What is the probabilityof obtaining exactly 2 good and 1 defective connector in drawing 3 parts from the box with replacement?a) .4219b) .4909c) .5022d) .691550. You have been asked to sample a lot of 300 units from a vendor whose past quality hasbeen about 2% defective. A sample of 40 pieces is drawn from the lot and you have been told to reject the lot if you find two or more parts defective. What is the probability of rejecting the lot?a) 0.953b) 0.809c) 0.191d) 0.047Use the following information to answer questions 51 and 52. A company produces capacitors by a process that normally yields 5% defective product. A sample of 4 capacitors is selected.51. What is the probability that all 4 capacitors are good?a) .9790b) .9213c) .8617d) .814552. What is the probability that all 4 capacitors are defective?a) .1383b) .1855c) .0000258d) .00000625Use the following information to answer questions 53, 54, 55 and 56. A company makes ball bearings that are found to be 10% defective in the long run. A sample of 10 bearings is selected.53. What is the probability that 0 bearings will be defective?a) .3487b) .3874c) .4126d) .1110Basic Probability 1154. What is the probability of obtaining exactly 1 defective bearing?a) .3487b) .3874c) .4126d) .257455. What is the probability of obtaining exactly 3 defective bearings?a) .0574b) .4448c) .7361d) .156256. What is the probability of obtaining more than 1 defective bearing?a) .3874b) .4126c) .2639d) .228557. How many defective connectors would be expected in a sample of 200 parts if theprocess averages 2% defective?a) 1b) 2c) 4d) 758. What is the probability of obtaining exactly 2 defective connectors in a sample of 6 partsif the process averages 2% defective?a) .0135b) .0055c) .0009d) .000159. All of the following are probabilistic events excepta) the number rolled in a game of dice.b) the number of defects in a random sample.c) the acceleration of an apple when it drops from a tree.d) the number of games played in the world series.12 QReview Study QuestionsUse the following information to answer problems 60, 61 and 62. A company produces integrated circuits (chips) by a process that normally yields 2000 ppm defective product for electrical test requirements (ppm = defective parts per million). A sample of 5 chips is selected and tested.60. What is the probability that all 5 chips are good?a) .9900b) .9603c) .9213d) .856361. What is the probability that 1 or more chips are defective?a) .0051b) .0009c) .0269d) .010062. What is the probability that more than 1 chip is defective?a) .01931b) .00510c) .00008d) .01000Use the following information to answer problems 63, 64 and 65. A capability study was made to determine the defective rate of 28AZ transistors. The study showed the rate to be 5000 ppm. Ten of the transistors were shipped to a customer.63. What is the probability that the shipment contains no defective transistors?a) .9511b) .9066c) .8512d) .921364. What is the probability that the shipment contains exactly 1 defective transistor?a) .0001b) .0478c) .1048d) .1165Basic Probability 1365. What is the probability that the shipment contains 2 or more defective transistors?a) .0001b) .0478c) .0237d) .0011Use the following information to answer problems 66, 67 and 68. A circuit board operation yields 2 defects per board on the average. A sample of 1 board is selected at random.66. What is the probability of finding exactly 2 defects on the selected board?a) .3522b) .2706c) .1550d) .029567. What is the probability of finding less than 2 defects on the selected board?a) .4060b) .6352c) .3522d) .384968. What is the probability of finding more than 2 defects on the selected board?a) .4060b) .2706c) .3522d) .3235Use the following information to answer problems 69, 70 and 71. In manufacturing material for automobile seats it was found that each 100-foot roll contained, on average, 2 defects (flaws). A sample of 1 roll is selected at random from the process.69. What is the probability that the selected roll contains 0 defects?a) .1353b) .2707c) .8647d) .729314 QReview Study Questions70. What is the probability that the selected roll contains exactly 1 defect?a) .1353b) .7293c) .8647d) .270671. What is the probability that the selected roll contains more than 1 defect?a) .3233b) .5941c) .7293d) .8647Use the following information to answer problems 72 and 73. A firm that makes T-shirt decals has determined that their process yields, on average, 3 defects per day. Fifty decals are inspected each day.72. What is the probability of finding exactly 2 defective decals in any given day? (Assumeone defect per defective decal.)a) .7361b) .1494c) .2240d) .074673. What is the probability of buying a decal that contains more than 1 defect?a) .0005b) .0042c) .0001d) .001774. A parts dealer buys parts from a warehouse. Parts are made by either Company A orCompany B but are not identified as to which company produces them. One company produces all parts in one shipment or lot. On the average, we know:Company A produces 2.5% defective parts.Company B produces 5.0% defective parts.The warehouse states that 70% of parts will come from Company A and 30% from Company B. If the dealer selects 4 parts at random from a lot and finds 1 defective part, what is the probability that the lot was produced by Company A?a) .4422b) .5580c) .6915d) .3085Statistics 15 STATISTICS75. What is the expected value of the random variable x for the following data?x f(x)12 0.210 0.514 0.120 0.2a) 13.6b) 14.0c) 12.8d) 14.576. In the standard normal table, what value of z has 5% of the area in the tail beyond it?a) 1.960b) 1.645c) 2.576d) 1.28277. Which distribution should be used to determine a confidence interval when σ is notknown and the sample size is 10?a) zb) tc) Fd) χ278. Which of the following methods should be used to test 6 population means for statisticalsignificance?a) Chi Square Testb) Analysis of Variancec) F Testd) Duncan's Multiple Range Test79. A sample size of 120 is taken from a process and is represented graphically on ahistogram. What is the appropriate number of histogram cells to use?a) 1 - 8b) 9 - 20c) 21 - 35d) 12016 QReview Study Questions80. Which of the following conditions makes it possible for a process to produce a largenumber of defective units while it is in statistical control?a) When the specification limits are not set correctly.b) When the process capability is wider than the tolerance.c) When unknown external forces affect the process.d) When the sample size, from which the reject data is found, is too small.81. For the normal probability distribution, which of the following is true about therelationship among the median, mean and mode?a) They are all equal to the same value.b) The mean and mode have the same value but the median is different.c) Each has a value different from the other two.d) The mean and median are the same but the mode is different.82. All of the following statistical techniques can be used to determine the effectiveness of asupplier improvement program excepta) Pareto analysis.b) x bar and R charts.c) a PERT chart.d) a flow chart.83. A sample of n observations has a mean x and a standard deviation s > 0. If a singleobservation, which equals the value of the sample mean x, is removed from the sample, which of the following is true?a) x and s both changeb) x and s remain the samec) x remains the same but s increasesd) x remains the same but s decreases84. The factory installed brake linings for a certain kind of car have a mean lifetime of 60,000miles with a 6,000 mile standard deviation. A sample of 100 cars has been selected for testing. What is the standard error of x? (Assume that the finite population correction may be ignored.)a) 60 milesb) 6000 milesc) 600 milesd) 6100 milesStatistics 17 Use the following information to answer problems 85 - 90. A sample of 7 rivets was taken from a shipment of 1000 rivets and the length was measured. The following data are obtained:Sample Number Length (inches)1 3.12 3.13 3.24 3.75 3.66 3.77 3.185. What is the mean length of the rivets?a) 3.20 inchesb) 3.36 inchesc) 4.00 inchesd) 3.65 inches86. What is the standard deviation of the length of the rivets (estimate of population standarddeviation)?a) 0.27 inchesb) 2.16 inchesc) 0.29 inchesd) 2.00 inchesNote: In the following 4 problems, the sample sizes are less than 30 and the t statistics should be used to solve the problems. Analyses of this type usually involve sample sizes of 30 or greater. Handle the problems just as if the sample sizes were greater than 30 and use the z statistics.87. What percentage of rivets have lengths less than 2.80 inches?a) 2.69%b) 5.00%c) 1.22%d) 3.23%18 QReview Study Questions88. What percentage of rivets have lengths greater than 3.65 inches?a) 17.1%b) 14.2%c) 15.9%d) 7.10%89. What percentage of rivets have lengths between 3.1 inches and 3.9 inches?a) 89.37%b) 78.45%c) 52.25%d) 99.75%90. In the shipment of 1000 rivets, how many good parts will we find if a good part is definedas having a minimum of 3 inches and a maximum of 4 inches?a) 999b) 967c) 912d) 878The following information is used to answer problems 91 - 95. Data are taken from a manufacturing process that produces optical glass. The sample size is 5 parts and the characteristic measured is the diameter of the plates.(mm)SampleNumber Diameter1 302 313 294 335 3491. What is the mean diameter of the optical glass?a) 31.4 mmb) 29.0 mmc) 31.0 mmd) 34.0 mmStatistics 1992. What is the standard deviation of the population?a) 1.00 mmb) 2.07 mmc) 2.22 mmd) 1.22 mm93. The specifications for the glass plates are 30.5 ± 2 mm. What percentage of parts madeby this company will not meet specifications?a) 32.5%b) 5.00%c) 35.0%d) 37.9%94. What percentage of parts will be less than 29.5 mm?a) 17.9%b) 7.21%c) 15.9%d) 24.3%95. What percentage of parts will be greater than 33 mm?a) 78.5%b) 24.3%c) 15.9%d) 22.1%96. The Zoglen Corporation markets a product, which is a blend of 3 ingredients (A, B, C). Ifthe individual tolerances for the weight of the 3 ingredients are as shown, what should the tolerance be for the net weight of the product?A: 40.5 ± 2.236 grams, B: 30.4 ± 2.000 grams, C: 18.1 ± 1.732 gramsa) 89.0 ± 2.443 gramsb) 89.0 ± 3.464 gramsc) 89.0 ± 5.968 gramsd) 89.0 ± 4.732 grams20 QReview Study Questions97. A random sample of size n is to be taken from a large population that has a standarddeviation of 1 inch. The sample size is determined so that there will be a 95% chance that the sample average will be within ±0.1 inch of the true mean. Which of the following values is nearest to the required sample size?a) 385b) 200c) 100d) 4098. All of the following conditions must be met for the process capability to be within thespecification limits excepta) C pk≥ 1.0b) C p≥ 1.0c) C p = C pkd) a stable process.99. A value on the abscissa of the t distribution is 1.093. What is the area to the right of thisvalue if the sample size is 11?a) 0.30b) 0.15c) 0.05d) 0.10100. The spread of individual observations from a normal process capability distribution may be expressed numerically asa) 6R/d2b) 2 x A2Rc) R/d2d) D4R101. What percentage of data will normally fall within a process capability?a) 99.00%b) 99.73%c) 1.00%d) 0.27%Statistics 21 Use the following information to answer problems 102 - 105. A winding machine wraps wire around a metal core to make small transformers. The design engineers have determined that the nominal number of windings are to be 10,060 with a minimum of 10,025 and a maximum of 10,095. A sample of 300 transformers was selected in a three month period and the wire was unwrapped on each part to determine the number of windings. The results were:x = 10,052 windings and s = 10 windings102. What is the process capability?a) 10020 - 10100b) 10052 - 10020c) 10022 - 10082d) 10020 - 10060103. Compute the value of C pa) 30.0b) 10.0c) 0.67d) 1.17104. Compute the value of C pka) 0.90b) 0.72c) 3.22d) 2.67105. What is the expected percent defective?a) 1.00%b) 0.35%c) 2.13%d) 0.49%22 QReview Study Questions Blank PageStatistical Inference 23 STATISTICAL INFERENCE106. Which of the following cannot be a null hypothesis?a) The population means are equal.b) p = 0.5c) The sample means are equal.d) The difference in the population means is 3.85.107. In a sampling distribution which of the following represents the critical region?a) αb) βc) 1 - βd) 1 - α108. In a hypothesis test which of the following represents the acceptance region?a) αb) βc) 1 - βd) 1 - α109. The Chi Square distribution isa) a distribution of averages.b) a distribution of variances.c) a distribution of standard deviations.d) a distribution of frequencies.110. Which of the following is a number derived from sample data that describes the data in some useful way?a) constantb) statisticc) parameterd) critical value111. A null hypothesis assumes that a process is producing no more than the maximum allowable rate of defective items. What does the type II error conclude about theprocess?a) It is producing too many defectives when it actually isn't.b) It is not producing too many defectives when it actually is.c) It is not producing too many defectives when it is not.d) It is producing too many defectives when it is.。

precalculus知识点总结

precalculus知识点总结Precalculus is an essential branch of mathematics that serves as a bridge between algebra, geometry, and calculus. This subject is crucial for students preparing to undertake advanced courses in mathematics, physics, engineering, and other technical fields. In this precalculus knowledge summary, we will cover important topics such as functions, trigonometry, and analytic geometry.FunctionsOne of the fundamental concepts in precalculus is that of functions. A function is a relationship between two sets of numbers, where each input is associated with exactly one output. In other words, it assigns a unique value to each input. Functions can be represented in various forms, such as algebraic expressions, tables, graphs, and verbal descriptions.The most common types of functions encountered in precalculus include linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions. Each type of function has its own unique characteristics and properties. For example, linear functions have a constant rate of change, while quadratic functions have a parabolic shape.Functions can be manipulated by performing operations such as addition, subtraction, multiplication, division, composition, and inversion. These operations can be used to create new functions from existing ones, or to analyze the behavior of functions under different conditions.TrigonometryTrigonometry is the study of the relationships between the angles and sides of triangles. It plays a crucial role in precalculus and is essential for understanding periodic phenomena such as oscillations, waves, and circular motion.The primary trigonometric functions are sine, cosine, and tangent, which are defined in terms of the sides of a right-angled triangle. These functions have various properties, such as periodicity, amplitude, and phase shift, which are important for modeling and analyzing periodic phenomena.Trigonometric functions can also be extended to the entire real line using their geometric definitions. They exhibit various symmetries and periodic behaviors, which can be visualized using the unit circle or trigonometric graphs. Additionally, trigonometric identities and equations are essential tools for simplifying expressions, solving equations, and proving theorems.Analytic GeometryAnalytic geometry is a branch of mathematics that combines algebra and geometry. It deals with the use of algebraic techniques to study geometric shapes and their properties. Inprecalculus, this subject is primarily concerned with the study of conic sections, such as circles, ellipses, parabolas, and hyperbolas.The equations of conic sections can be derived using geometric constructions, or by using algebraic methods such as completing the square, factoring, and manipulating equations. These equations can then be used to describe the geometric properties of conic sections, such as their shape, size, orientation, and position.Furthermore, analytic geometry also involves the study of vectors and matrices, which are important tools for representing and manipulating geometric objects in higher dimensions. Vectors can be used to represent points, lines, and planes in space, while matrices can be used to perform transformations such as rotations, reflections, and scaling.Other TopicsIn addition to the core topics mentioned above, precalculus also covers other important concepts such as complex numbers, polar coordinates, sequences and series, and mathematical induction. Complex numbers are used to extend the real number system to include solutions to equations that have no real roots. They have applications in various fields such as electrical engineering, quantum mechanics, and signal processing.Polar coordinates provide an alternative way of describing points in the plane using radial distance and angular direction. They are particularly useful for representing periodic and circular motion, as well as for simplifying certain types of calculations in calculus.Sequences and series are ordered lists of numbers that have a specific pattern or rule. They can be finite or infinite, and their sums can be used to represent various types of mathematical and physical phenomena. For example, arithmetic sequences are used to model linear growth or decline, while geometric series are used to model exponential growth or decay.Finally, mathematical induction is a powerful method for proving statements about positive integers. It is based on the principle that if a certain property holds for a base case, and if it can be shown that it also holds for the next case, then it holds for all subsequent cases as well. This method is widely used in various areas of mathematics, such as number theory, combinatorics, and discrete mathematics.ConclusionIn conclusion, precalculus is a diverse and rich subject that covers a wide range of mathematical concepts and techniques. It provides students with the necessary foundation to tackle more advanced topics in calculus and beyond. By mastering the core topics of precalculus, students will be well-equipped to understand and apply advanced mathematical methods in various technical fields. Whether it be functions, trigonometry, analytic geometry, or any other topic, a solid understanding of precalculus is essential for success in higher mathematics.。

Quantum Groups and Noncommutative Geometry

a r X i v :h e p -t h /0006167v 1 21 J u n 2000QUANTUM GROUPS AND NONCOMMUTATIVE GEOMETRYShahn MajidSchool of Mathematical Sciences,Queen Mary and Westﬁeld College University of London,Mile End Rd,London E14NS,UK November,1999Abstract Quantum groups emerged in the latter quarter of the 20th century as,on the one hand,a deep and natural generalisation of symmetry groups for certain integrable systems,and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain.Just as the last century saw the birth of classical geometry,so the present century sees at its end the birth of this quantum or noncommutative geometry,both as an elegant mathematical reality and in the form of the ﬁrst theoretical predictions for Planck-scale physics via ongoing astronomical measurements.Noncommutativity of spacetime,in particular,amounts to a postulated new force or physical eﬀect called cogravity.I Introduction Now that quantum groups and their associated quantum geometry have been around for more than a decade,it is surely time to take stock.Where did quantum groups come from,what have they achieved and where are they going?This article,which is addressed to non-specialists (but should also be interesting for experts)tries to answer this on two levels.First of all on the level of quantum groups themselves as mathematical tools and building blocks for physical models.And,equally importantly,quantum groups and their associated noncommutative geometry in terms of their overall signiﬁcance for mathematics and theoretical physics,i.e.,at a more conceptual level.Obviously this latter aspect will be very much my own perspective,which is that of a theoretical physicist who came to quantum groups a decade ago as a tool to unify quantum theory and gravity in an algebraic approach to Planck scale physics.This is in fact only one of the two main origins in physics of quantum groups;the other being integrable systems,which I will try to cover as well.Let me also say that noncommutative geometry has other approaches,notably the one of A.Connes coming out of operator theory.I will say something about this too,although,until recently,this has largely been a somewhat diﬀerent approach.We start with the conceptual signiﬁcance for theoretical physics.It seems clear to me that future generations looking back on the 20th century will regard the discovery of quantum mechanics in the 1920s,i.e.the idea to replace the coordinates x,p of classical mechanics bynoncommuting operators x,p,as one of its greatest achievements in our understanding of Nature, matched in its signiﬁcance only by the uniﬁcation of space and time as a theory of gravity.But whereas the latter was well-founded in the classical geometry of Newton,Gauss,Riemann and Poincar´e,quantum theory was something much more radical and mysterious.Exactly which variables in the classical theory should correspond to operators?They are local coordinates on phase space but how does the global geometry of the classical theory look in the quantum theory,what does it fully correspond to?The problem for most of this century was that the required mathematical structures to which the classical geometry might correspond had not been invented and such questions could not be answered.As I hope to convince the reader,quantum groups and their associated noncommutative geometry have led in the last decades of the20th century to theﬁrst deﬁnitive answers to this kind of question.There has in fact emerged a more or less systematic generalisation of geometry every bit as radical as the step from Euclidean to non-Euclidean,and powerful enough not to break down in the quantum domain.I do doubt very much that what we know today will be theﬁnal formulation,but it is a deﬁnitive step in a right and necessary direction and a turning point in the future development of mathematical and theoretical physics.For example,any attempt to build a theory of quantum gravity with classical starting point a smooth manifold–this includes loop-variable quantum gravity,string theory and quantum cosmology,is necessarily misguided except as some kind of eﬀective approximation:smooth manifolds should come out of the algebraic structure of the quantum theory and not be a starting point for the latter. There is no evidence that the real world is any kind of smooth continuum manifold except as a macroscopic approximation and every reason to think that it is fundamentally not.I therefore doubt that any one of the above could be a‘theory everything’until it becomes an entirely algebraic theory founded in noncommutative geometry of some kind or other.Of course,this is my personal view.At any rate,I do not think that the fundamental importance of noncommutative geometry can be overestimated.First of all,anyone who does quantum theory is doing noncommutative geometry whether wanting to admit it or not,namely noncommutative geometry of the phase space.Less obvious but also true,we will see in Section II that if the position space is curved then the momentum space is by itself intrinsically noncommutative.If one gets this far then it is also natural that the position space or spacetime by itself could be noncommutative,which would correspond to a curved or nonAbelian momentum group.This is one of the bolder predictions coming out of noncommutative geometry.It has the simple physical interpretation as what I call cogravity,i.e.curvature or‘gravity’in momentum space.As such it is independent of i.e. dual to curvature or gravity in spacetime and would appear as a quite diﬀerent and new physical eﬀect.Theoretically cogravity can,for example,be detected as energy-dependence of the speed8185909510000500981200yearpapersFigure 1:Growth of research papers on quantum groupsof light.Moreover,even if cogravity was very weak,of the order of a Planck-scale eﬀect,it could still in principle be detected by astronomical measurements at a cosmological level.Therefore,just in time for the new millennium,we have the possibility of an entirely new physical eﬀect in Nature coming from fresh and conceptually sound new mathematics .Where quantum groups precisely come into this is as follows.Just as Lie groups and their associated homogeneous spaces provided deﬁnitive examples of classical diﬀerential geometry even before Riemann formulated their intrinsic structure as a theory of manifolds,so quantum groups and their associated quantum homogeneous spaces,quantum planes etc.,provide large (i.e.inﬁnite)classes of examples of proven mathematical and physical worth and clear geomet-rical content on which to build and develop noncommutative diﬀerential geometry.They are noncommutative spaces in the sense that they have generators or ‘coordinates’like the non-commuting operators x ,p in quantum mechanics but with a much richer and more geometric algebraic structure than the Heisenberg or CCR algebra.In particular,I do not believe that one can build a theory of noncommutative diﬀerential geometry based on only one example such as the Heisenberg algebra or its variants (however fascinating)such as the much-studied noncommutative torus.One needs many more ‘sample points’in the form of natural and varied examples to obtain a valid general theory.By contrast,if one does a search of BIDS one ﬁnds,see Figure 1,vast numbers of papers in which the rich structure and applications of quantum groups are explored and justiﬁed in their own right (data complied from BIDS:published pa-pers since 1981with title or abstract containing ‘quantum group*’,‘Hopf alg*’,‘noncommutative geom*’,‘braided categ*’,‘braided group*’,‘braided Hopf*’.)This is the signiﬁcance of quantum groups.And of course something like them should be needed in a quantum world where there is no evidence for a classical space such as the underlying set of a Lie group.Finally,it turns out that noncommutative geometry,at least of the type that we shall de-scribe,is in many ways cleaner and more straightforward than the special commutative limit. One simply does not need to assume commutativity in most geometrical constructions,including diﬀerential calculus and gauge theory.The noncommutative version is often less inﬁnite,dif-ferentials are often more regularﬁnite-diﬀerences,etc.And noncommutative geometry(unlike classical geometry)can be specialised without eﬀort to discrete spaces or toﬁnite-dimensional algebras.It is simply a powerful and natural generalisation of geometry as we usually know it. So my overall summary and prediction for the next millennium from this point of view is:•All geometry will be noncommutative(or whatever comes beyond that),with conventional geometry merely a special case.•The discovery of quantum theory,its correspondence principle(and noncommutative ge-ometry is nothing more than the elaboration of that)will be considered one of the century’s greatest achievement in mathematical physics,commensurate with the discovery of clas-sical geometry by Newton some centuries before.•Quantum groups will be viewed as theﬁrst nontrivial class of examples and thereby point-ers to the correct structure of this noncommutative geometry.•Spacetime too(not only phase space)will be known to be noncommutative(cogravity will have been detected).•At some point a future Einstein will combine the then-standard noncommutative geomet-rical ideas with some deep philosophical ideas and explain something really fundamental about our physical reality.In the fun spirit of this article,I will not be above putting down my own thoughts on this last point.These have to do with what I have called for the last decade the Principle of representation-theoretic self-duality[1].In eﬀect,it amounts to extending the ideas of Born reciprocity,Mach’s principle and Fourier theory to the quantum domain.Roughly speaking, quantum gravity should be recast as gravity and cogravity both present and dual to each other and with Einstein’s equation appearing as a self-duality condition.The longer-term philosophical implications are a Kantian or Hegelian view of the nature of physical reality,which I propose in Section V as a new foundation for next millennium.We now turn to another fundamental side of quantum groups,which is at the heart of their other origin in physics,namely as generalised symmetry groups in exactly solvable lattice models. It leads to diverse applications ranging from knot theory to representation theory to Poisson geometry,all areas that quantum groups have revolutionised.What is really going on here in my opinion is not so much the noncommutative geometry of quantum groups themselves as a diﬀerent kind of noncommutativity or braid statistics which certain quantum groups induce onany objects of which they are a symmetry.The latter is what I have called‘noncommutativity of the second kind’or outer noncommutativity since it not so much a noncommutativity of one algebra as a noncommutative modiﬁcation of the exchange law or tensor product of any two independent algebras or systems.It is the notion of independence which is really being deformed here.Recall that the other great‘isation’idea in mathematical physics in this century(after ‘quantisation’)was‘superisation’,where everything is Z2-graded and this grading enters into how two independent systems are interchanged.Physics traditionally has a division into bosonic or force particles and fermionic or matter particles according to this grading and exchange behaviour.So certain quantum groups lead to a generalisation of that as braided geometry[2]or a process of braidiﬁcation.These quantum groups typically have a parameter q and its meaning is a generalisation of the−1for supersymmetry.This in turn leads to a profound generalisation of conventional(including super)mathematics in the form of a new concept of algebra wherin one‘wires up’algebraic operations much as the wiring in a computer,i.e.outputs of one into inputs of another.Only,this time,the under or over crossings are nontrivial(and generally distinct)operations depending on q.These are the so-called‘R-matrices’.Afterwards one has the luxury of both viewing q in this way or expanding it around1in terms of a multiple of Planck’s constant and calling it a formal‘quantisation’–q-deformation actually uniﬁes both ‘isation’processes.For example,Lorentz-invariance,by the time it is q-deformed[3],induces braid statistics even when particles are initially bosonic.In summary,•The notion of symmetry or automorphism group is an artifact of classical geometry and ina quantum world should naturally be generalised to something more like a quantum groupsymmetry.•Quantum symmetry groups induce braid statistics on the systems on which they act.In particular,the notion of bose-fermi statistics or the division into force and matter particles is an artifact of classical geometry.•Quantisation and the departure from bosonic statistics are two limits of the same phe-nomenon of braided geometry.Again,there are plenty of concrete models in solid state physics already known with quantum group symmetry.The symmetry is useful and can be viewed(albeit with hindsight)as the origin of the exact solvability of these models.These two points of view,the noncommutative geometrical and the generalised symmetry, are to date the two main sources of quantum groups.One has correspondingly two mainﬂavours or types of quantum groups which really allowed the theory to take oﬀ.Both were introduced at the mid1980s although the latter have been more extensively studied in terms of applicationsto date.They include the deformationsU q(g)(1)of the enveloping algebra U(g)of every complex semisimple Lie algebra g[4][5].These have as many generators as the usual ones of the Lie algebra but modiﬁed relations and,additionally, a structure called the‘coproduct’.The general class here is that of quasitriangular quantum groups.They arose as generalised symmetries in certain lattice models but are also visible in the continuum limit quantumﬁeld theories(such as the Wess-Zumino-Novikov-Witten model on the Lie group G with Lie algebra g).The coordinate algebras of these quantum groups are further quantum groups C q[G]deforming the commutative algebra of coordinate functions on G.There is again a coproduct,this time expressing the group law or matrix multiplication.Meanwhile, the type coming out of Planck scale physics[6]are the bicrossproduct quantum groupsC[M]◮⊳U(g)(2)associated to the factorisation of a Lie group X into Lie subgroups,X=GM.Here the in-gredients are the conventional enveloping algebra U(g)and the commutative coordinate algebra C[M].The factorisation is encoded in an action and coaction of one on the other to make a semidirect product and coproduct◮⊳.These quantum arose at about the same time but quite independently of the U q(g),as the quantum algebras of observables of certain quantum spaces. Namely it turns out that G acts on the set M(and vice-versa)and the quantisation of those orbits are these quantum groups.This means that they are literally noncommutative phase spaces of honest quantum systems.In particular,every complex semisimple g has an associated complexiﬁcation and its Lie group factorises G C=GG⋆(the classical Iwasawa decomposition) so there is an exampleC[G⋆]◮⊳U(g)(3)built from just the same data as for U q(g).In fact the Iwasawa decomposition can be understood in Poisson-Lie terms with g⋆the classical‘Yang-Baxter dual’of g.In spite of this,there is,even after a decade of development,no direct connection between the two quantum groups:gւց(4)U q(g)←?→C[G⋆]◮⊳U(g).They are both‘exponentiations’of the same classical data but apparently of completely diﬀerent type(this remains a mystery to date.)Figure2:The landscape of noncommutative geometry todayAssociated to these twoﬂavours of quantum groups there are corresponding homogeneous spaces such as quantum spheres,quantum spacetimes,etc.Thus,of theﬁrst type there is a q-Minkowski space introduced in[7]as a q-Lorentz covariant algebra,and independently about a year later in[8]as2×2braided hermitian matrices.It is characterised by[x i,t]=0,[x i,x j]=0.(5) Meanwhile,of the second type there is a noncommutativeλ-Minkowski space with[x i,t]=λx i,[x i,x j]=0(6)which is the one that provides theﬁrst known predictions testable by astronomical measurements (by gamma-ray bursts of cosmological origin[9]).This kind of algebra was proposed as spacetime in[10]and in the4-dimensional case it was shown in[11]to be covariant under a Poincar´e quantum group of bicrossproduct form.These are clearly in sharp contrast.There are of course many more objects than these.q-spheres,q-planes etc.In Section IV we turn to the notion of‘quantum manifold’that is emerging from all these examples.Riemann was able to formulate the notion of Riemannian manifold as a way to capture known examples like spheres and tori but broad enough to formulate general equations for the intrinsic structure of space itself(or after Einstein,space-time).We are at a similar point now and what this ‘quantum groups approach to noncommutative geometry’is is more or less taking shape.It has the same degree of‘ﬂabbiness’as Riemannian geometry(it is not tied to speciﬁc integrable systems etc.)while at the same time it includes the‘zoo’of already known naturally occurring examples,mostly linked to quantum groups.Such things as Ricci tensor and Einstein’s equation are not yet understood from this approach,however,so I would not say it is the last word.This approach is in fairly sharp contrast to‘traditional’noncommutative geometry as it was done before the emergence of quantum groups.That theory was developed by mathematiciansand mathematical physicists also coming from quantum mechanics but being concerned more with topological completions and Hilbert spaces.Certainly a beautiful theory of von-Neumann and C∗algebras emerged as an analogue of point-set topology.Some general methods such as cyclic cohomology were also developed in the1970s,with remarkable applications throughout mathematics[12].However,for concrete examples with actual noncommutative diﬀerential geometry one usually turned either to an actual manifold as input datum or to the Weyl algebra (or noncommutative torus)deﬁned by relationsvu=e2πıθuv.(7)This in turn is basically the usual CCR or Heisenberg algebra[x,p]=ı (8)in exponentiated form.And at an algebraic level(i.e.until one considers the precise C∗-algebra completion)this is basically the usual algebra B(H)of operators on a Hilbert space as in quantum mechanics.Or at roots of unity it is M n(C)the algebra of n×n matrices.So at some level these are all basically one example.Unfortunately many of the tricks one can pull for this kind of example are special to it and not a foundation for noncommutative diﬀerential geometry of the type we need.For example,to do gauge theory Connes and M.Rieﬀel[13] used derivations for two independent vectorﬁelds on the torus.The formulation of‘vector ﬁeld’as a derivation of the coordinate algebra is what I would call the traditional approach to noncommutative geometry.For quantum groups such as C q[G]one simply does not have those derivations(rather,they are in general braided derivations).Similarly,in the traditional approach one deﬁnes a‘vector bundle’as aﬁnitely-generated projective module without any of the infrastructure of diﬀerential geometry such as a principal bundle to which the vector bundle might be associated,etc.All of that could not emerge until quantum groups arrived(one clearly should take a quantum group asﬁber).This is how the quantum groups approach diﬀers from the work of Connes,Rieﬀel,Madore and others.It is also worth noting that string theorists have recently woken up to the need for a noncommutative spacetime but,so far at least,have still considered only this‘traditional’Heisenberg-type algebra.In the last year or two there has been some success in merging these approaches,however;a trend surely to be continued. By now both approaches have a notion of‘noncommutative manifold’which appear somewhat diﬀerent but which have as point of contact the Dirac operator.Preliminaries.A full text on quantum groups is[14].To be self-contained we provide here a quick deﬁter on we will see many examples and various justiﬁcations for this concept. Thus,a quantum group or Hopf algebra is•A unital algebra H,1over theﬁeld C(say)•A coproduct∆:H→H⊗H and counitǫ:H→C forming a coalgebra,with∆,ǫalgebra homomorphisms.•An antipode S:H→H such that·(S⊗id)∆=1ǫ=·(id⊗S)∆.Here a coalgebra is just like an algebra but with the axioms written as maps and arrows on the maps reversed.Thus the coassociativity and counity axioms are(∆⊗id)∆=(id⊗∆)∆,(ǫ⊗id)∆=(id⊗ǫ)∆=id.(9)The antipode plays a role that generalises the concept of group inversion.Other than that the only new mathematical structure that the reader has to contend with is the coproduct∆and its associated counit.There are several ways of thinking about the meaning of this depending on our point of view.If the quantum group is like the enveloping algebra U(g)generated by a Lie algebra g,one should think of∆as providing the rule by which actions extend to tensor products.Thus,U(g)is trivially a Hopf algebra with∆ξ=ξ⊗1+1⊗ξ,∀ξ∈g,(10)which says that when a Lie algebra elementξacts on tensor products it does so byξin the ﬁrst factor and thenξin the second factor.Similarly it says that when a Lie algebra acts on an algebra it does so as a derivation.On the other hand,if the quantum group is like a coordinate algebra C[G]then∆expresses the group multiplication andǫthe group identity element e. Thus,if f∈C[G]the coalgebra is(∆f)(g,h)=f(gh),∀g,h∈Gǫf=f(e)(11)at least for suitable f(or with suitable topological completions).In other words it expresses the group product G×G→G by a map in the other direction in terms of coordinate algebras. From yet another point of view∆simply makes the dual H∗also into an algebra.So a Hopf algebra is basically an algebra such that H∗is also an algebra,in a compatible way,which makes the axioms‘self-dual’.For everyﬁnite-dimensional H there is a dual H∗.Similarly in the inﬁnite-dimensional case.It said that in the Roman empire,‘all roads led to Rome’.It is remarkable that several diﬀerent ideas for generalising groups all led to the same axioms.The axioms themselves wereﬁrst introduced(actually in a super context)by H.Hopf in1947in his study of group cohomology but the subject only came into its own in the mid1980s with the arrival from mathematical physics of the large classes of examples(as above)that are neither like U(g)nor like C[G],i.e.going truly beyond Lie theory or algebraic group theory.Acknowledgements.An announcement of this article appears in a short millennium article[15] and a version more focused on the meaning for Planck scale physics in[16].II Quantum groups and Planck scale physicsThis section covers quantum groups of the bicrossproduct type coming out of Planck-scale physics[6]and their associated noncommutative geometry.These are certainly less well-developed than the more familiar U q(g)in terms of their concrete applications;one does not have inter-esting knot invariants etc.On the other hand,these quantum groups have a clearer physical meaning as models of Planck scale physics and are also technically easier to construct.Therefore they are a good place to start.Obviously if we want to unify quantum theory and geometry then a necessaryﬁrst step is to cast both in the same language,which for us will be that of algebra.We have already mentioned that vectorﬁelds can be thought of classically as derivations of the algebra of functions on the manifold,and if one wants points they can be recovered as maximal ideals in the algebra, etc.This is the more of less standard idea of algebraic geometry dating from the late19th century and early on in the20th.It will certainly need to be modiﬁed before it works in the noncommutative case but it is a starting point.The algebraic structure on the quantum side will need more attention,however.II.A CogravityWe begin with some very general considerations.In fact there are fundamental reasons why one needs noncommutative geometry for any theory that pretends to be a fundamental one.Since gravity and quantum theory both work extremely well in their separate domains,this comment refers mainly to a theory that might hope to unify the two.As a matter of fact I believe that, through noncommutative geometry,this‘holy grail’of theoretical physics may now be in sight.Theﬁrst point is that we usually do not try to apply or extend our geometrical intuition to the quantum domain directly,since the mathematics for that has traditionally not been known. Thus,one usually considers quantisation as the result of a process applied to an underlying classical phase space,with all of the geometrical content there(as a Poisson manifold).But demanding any algebra such that its commutators to lowest order are some given Poisson bracket is clearly an illogical and ill-deﬁned process.It not only does not have a unique answer but also it depends on the coordinates chosen to map over the quantum operators.Almost always one takes the Poisson bracket in a canonical form and the quantisation is the usual CCR or canonical commutation relations algebra.Maybe this is the local picture but what of the global geometry of the classical phase space?Clearly all of these problems are putting the cart before the horse:the real world is to our best knowledge quantum so that should comeﬁrst.We should build models guided by the intrinsic(noncommutative)geometry at the level of noncommutative algebras and only at the end consider classical limits and classical geometry(and Poisson brackets)as emerging from a choice,where possible,of‘classical handles’in the quantum system.In more physical terms,classical observables should come out of quantum theory as some kind of limit and not really be the starting point;in quantum gravity,for example,classical geometry should appear as an idealisation of the expectation value of certain operators in certain states of the system.Likewise in string theory one starts with strings moving in classical spacetime, deﬁnes Lagrangians etc.and tries to quantise.Even in more algebraic approaches,such as axiomatic quantumﬁeld theory,one still assumes an underlying classical spacetime and classical Poincar´e group etc.,on which the operatorﬁelds live.Yet if the real world is quantum then phase space and hence probably spacetime itself should be‘fuzzy’and only approximately modeled by classical geometrical concepts.Why then should one take classical geometrical concepts inside the functional integral except other than as an eﬀective theory or approximate model tailored to the desired classical geometry that we hope to come out.This can be useful but it cannot possibly be the fundamental‘theory of everything’if it is built in such an illogical manner.There is simply no evidence for the assumption of nice smooth manifolds other than now-discredited classical mechanics.And in certain domains such as,but not only,in Planck scale physics or quantum gravity,it will certainly be unjustiﬁed even as an approximation.Next let us observe that any quantum system which contains a nonAbelian global symmetry group is already crying out for noncommutative geometry.This is in addition to the more obvious position-momentum noncommutativity of quantisation.The point is that if our quantum system has a nonAbelian Lie algebra symmetry,which is usually the case when the classical system does, then from among the quantum observables we should be able to realise the generators of this Lie algebra.That is,the algebra of observables A should contain the algebra generated by the Lie algebra,A⊇U(g).(12)Typically,A might be the semidirect product of a smaller part with external symmetry g by the action of U(g)(which means that in the bigger algebra the action of g is implemented by the commutator).This may soundﬁne but if the algebra A is supposed to be the quantum analogue of the‘functions on phase space’,then for part of it we should regard U(g)‘up side down’not as an enveloping algebra but as a noncommutative space with g the noncommutative coordinates.In other words,if we want to elucidate the geometrical content of the quantum algebra of observables then part of that will be to understand in what sense U(g)is a coordinate algebra,U(g)=C[?].(13)Here?cannot be an ordinary space because its supposed coordinate algebra U(g)is noncom-mutative.。

W Series科学计算器操作指南说明书

<W Series>C O N TEN TSRead Before Using12This calculator can operate in three different modes as follows.<Example>≈Read B efore Using ≈This operation g uide has been written based on the EL-531W , EL-509W , and EL-531W H models. Some functions described here are not featured on other models. In addition, key operations and symbols on the display may differ according to the model.•Mode = 0; normal mode for performing normal arithmetic and function calculations.[Normal mode]•Mode = 1; STAT-0 mode for performing 1-variable statisti-cal calculations.[STAT-0 mode]•Mode = 1; STAT-1–6 mode for performing 2-variable statistical calculations.[STAT-1–6 mode]W hen chang ing to the statistical sub-mode, press the corresponding number key afterperforming (LINE): Linear reg ression calculation(Q UAD): Q uadratic reg ression calculation (EX P):Exponential reg ression calculation(LO G): Log arithmic reg ression calculation (PW R): Power reg ression calculation (INV):Inverse reg ression calculation3For convenient and easy operation, this model can be used in one of four display modes.The selected display status is shown in the upper part of the display (Format Indicator).N ote: If more 0’s (zeros) than needed are displayed when the O N /C key is pressed, check whether or not the calculator is set to a Special Display Format.•Floating decimal point format (no symbol is displayed)Valid values beyond the maximum rang e are displayed in the form of a [10-dig it (mantissa) + 2-dig it (exponent)]•Fixed decimal point format (FIX is displayed)Displays the fractional part of the calculation result according to the specified number of decimal places.•Scientific notation (SC I is displa yed)Frequently used in science to handle extremely small or larg e numbers.•Eng ineering scientific notation (EN G is displayed)C onvenient for converting between different units.(specifies normal mode)<Example>Let’s compare the display result of[10000 8.1 =] in each display format.4. DI S P L AY F O R M AT A N DDE C I M A L S E T T I N G F U N C T I O N3. DI S P L AY P AT T E R NInitial display The actual display does not appear like this.This illustration is for explanatory purposes only.100008.1(normal mode)N ote: The calculator has two setting s for displaying a floating point number: N O RM 1 (default setting ) and N O RM 2. In each display setting , a number isautomatically displayed in scientific notation outside a preset rang e:• N O RM 1: 0.000000001 x 9999999999• N O RM 2: 0.01 x 9999999999(FIX mode TAB = 3)5.E X P O N E N T DI S P L AYThe distance from the earth to the sun is approx. 150,000,000 (1.5 x 108) km. Values such as this with many zeros are often used in scientific calculations, but entering the zeros one by one is a great deal of work and it’s easy to make mistakes.In such a case, the numerical values are divided into mantissa and exponent portions, displayed and calculated.<Example>W hat is the number of electronics flowing in a conductor whenthe electrical charg e across a g iven cross-section is 0.32 cou-lombs. (The charg e on a sing le electron = 1.6 x 10-19 coulombs).0.32191.645Ang ular values are converted from DEG to RAD to GRAD with each push of the DRG key . This function is used when doing calculations related to trig onometric functions or coordinate g eometry conversions.<Example>6. A N G U L A RU N I T(in DEG mode)••••••••O per ationof ang C heck to confirm 90 deg rees equaling π/2 radiansequaling 100 g rads. (π=3.14159...)90D egrees (DE G is shown at the top of the display)A commonly used unit of measure for ang les. The ang ular measure of a circle is expressed as 360°.R adians (R A D is shown at the top of the display)Radians are different than deg rees and express ang les based on the circumfer-ence of a circle. 180° is equivalent to π radians. Therefore, the ang ular mea-sure of a circle is 2π radians.G r ads (G R A D is shown at the top of the display)Grads are a unit of ang ular measure used in Europe, particularly in France. An ang le of 90 deg rees is equivalent to 100 g rads.6Turns the calculator on or clears the data. It also clears the contents of the calculator display and voids any calculator command; however, coeffi-cients in 3-variable linear equations and statistics, as well as values stored in the independent memory in normal mode, are not erased.Turns the calculator off.C lears all internal values, including coefficients in 3-variable linear equations andstatistics. Values stored in memory in normal mode are not erased.T hese arrow keys are useful for Multi-Line playback, which lets you scroll throug h calculation steps one by one. (refer to page 8)These keys are useful for editing equations. The key moves the cursor to the left, and theht. TheON/OFF, Entry Correction Keys≈Function and K ey Operation≈7Provided the earth is moving around the sun in a circular orbit,how many kilometers will it travel in a year?* The averag e distance between the earth and the sun being 1.496 x 108 km.C ircumference equals diameter x π; therefore,1.496 x 108 x 2 x π0 to 9Pressing π automatically enters the value for π (3.14159...).The constant π, used frequently in function calculations, is the ratio of thecircumference of a circle to its diameter.<Example>N umeric keys for entering data values.Decimal point key. Enters a decimal point.Enters minus symbol or sig n chang e key.C hang es positive numbers to neg ative and neg ative numbers to positive.Pressing this key switches to scientific notation data entry .O per ationD isplay2149688RandomGenerates random numbers.Random numbers are three-decimal-place values between0.000 and 0.999. Using this function enables the user to obtain unbiased sampling data derived from random values g enerated by the calculator.<Example>A PPL IC AT IO N S:Building sample sets for statistics or research.0. ***(A random number has been generated.)[R andom D ice]To simulate a die-rolling, a random integ er between 1 and 6can be g enerated by pressing[R andom C oin]0 (heads) or 1 (tails) can be randomly g enerated by pressingTo g[R andom Integer]An integ er between 0 and 99 can be g enerated randomly by pressing To g enerate the next random integ9Function to round calculation results.Even after setting the number of decimal places on the display, the calculator per-forms calculations using a larg er number of decimal places than that which appears on the display . By using this function, internal calculations will be performed using only the displa yed value.A P P L IC AT IO N S:Frequently used in scientific and technical fields, as well as business,when performing chained calculations.<Example>0.65.4599Modify(internally, 0.6)(internally, 0.5555...)Basic ArithmeticKeys, ParenthesesUsed to specify calculations in which certain operations have precedence.You can make addition and subtraction operations have precedence over multiplication and division by enclosing them in parentheses.The four basic operators. Each is used in the same way as a standard calculator:+ (addition), – (subtraction), x (multiplication), and ÷ (division).Finds the result in the same way as a standar d calculator.For calculating percentages. Four methods of calculating percentages are presented as follows.1) $1137.52) $125 reduced by 20% (1)003) 118.754) W hen $1, Xequals (2500)1251012520125151255Percent<Example>C alculates the square root of the value on the display .Calculates the inverse of the value on the display.Squares the value on the display.C ubes the value on the display.C alculates the cube root of the value on the display.C alculates the x th root of y.24416O per ation D isplayCalculates exponential values.2222Inverse, Square, Cube, xth Power of y,Square Root,Cube Root, xth Root of y10 to the Power of x,Common Logarithm <Example>C alculates the value of 10 raised to the x th power.C alculates log arithm, the exponent of the power to which 10 must be raised to equal the g iven value.10003O per ationD isplaye to the Power of x,Natural LogarithmC alculates powers based on the constant e (2.718281828).<Example>510O per ationD isplay C omputes the value of the natural log arithm, the exponent of the power to which e must be raised to equal the g iven value.FactorialsThe product of a g iven positive integ er n multiplied by all the lesser positive integers from 1 to n-1 is indicated by n! and called the factorial of n.APPL IC ATIO N S:Used in statistics and mathematics. In statistics, this function is used in calculations involving combinations and permutations.<Example>c.fn! = 1 x 2 x 3 x …xnO per ation D isplay76464A PPL IC AT IO N S:Used in statistics (probability calculations) and in simulation hypoth-eses in fields such as medicine, pharmaceutics, and physics. Also,can be used to determine the chances of winning in lotteries.Permutations, Combinations<Example>T his function finds the number of different possible orderings in selecting r objects from a set of n objects. For example, there are six differentways of ordering the letters ABC in groups of three letters—ABC , AC B,BAC , BC A, C AB, and C BA.The calculation equation is3P 3 = 3 x 2 x 1= 6 (ways).T his function finds the number of ways of selecting r objects from a set of n objects. For example, from the three letters ABC , there are three ways we can extract groups of two different letters—AB, AC , and C B.T he calculation equation is 3C 2.O per ation D isplayTime CalculationC onvert 24° 28’ 35” (24 degrees, 28 minutes, 35 sec-onds) to decimal notation. T hen convert 24.476° to sexagesimal notation.C onver ts a sexagesimal value displayed in degrees, minutes, seconds to decimal notation. Also, conver ts a decimal value to sexagesimalnotataion (degrees, minutes, seconds).Inputs values in sexagesimal notation (degrees, minutes, seconds).<Example>A PPLIC AT IO N S:Used in calculations of angles and angular velocity in physics, andlatitude and long itude in g eography.242835O per ationD isplay Repeat last key operation to return to the previous display.C onvert to decimal notationFractional CalculationsAdd 3 and , and convert to decimal notation.<Example>Inputs fractions and converts mutually between fractions and decimals.C onverts between mixed numbers and improper fractions.31257C onvert to an improper fractionPress once to return to the previous displayC onvert to decimal notationPress once to return to the previous displayA PPL IC AT IO N S:T here is a wide variety of applications for this function becausefractions are such a basic part of mathematics. T his function is usefulfor calculations involving electrical circuit resistance.O per ation D isplay1257Stores displayed values in memories A~F, X , Y, M.Recalls values stored in A~F, X , Y, M.Adds the displayed value to the value in the independent memory M.Memory Calculations<Example 1>252773O per ation D isplayTemporary memories~(Enter 0 for M)<Example 2>Subtracts the displayed value from the value in the independent memory M.Independent memory O per ation D ispla yC alculates $/¥ at the desig nated exchang e rate.110265102750$1 = ¥110¥26,510 = $?$2,750 = ¥?Solve for x first and then solve for y using x.Last Answer Memory<Example>y =4 ÷ xandx =2 +3O per ationDisplay234Automatically recalls the last answer calculated by pressingThe ang le from a point 15 meters from a building to the hig hest floor of the building is 45°. How tall is the building ?Trigonometric Functions[DEG mode]V APPL IC AT IO NS:Trig onometric functions are useful in mathematics and various eng ineering calculations. They are often used in astronomical observations, civil eng i-neering and in calculations involving electrical circuits, as well as in calcula-tions for physics such as parabolic motion and wave motion.C alculates the sine of an angle.C alculates the cosine of an angle.C alculates the tangent of an angle.<Example>451515O per ation D isplaysin θ =ba tan θ =bccos θ =ca acbθTrig onometric functions determine the ratio of three sides of a rig ht triang le. The combinations of the three sides are sin, cos, and tan. Their relations are:Arc trig onometric functions, the inverse of trig onomet-ric functions, are used to determine an angle from ratios of a rig ht triang le. The combinations of the three sides are sin -1, cos -1, and tan -1. Their relations are;Arc Trigonometric Functions[DEG mode](arc sine) Determines an angle based on the ratio b/a of two sides of a right triangle.(arc cosine) Determines an angle based on the ratio c/a for two sides of a right triangle.(arc tangent) Determines an angle based on the ratio a/b for two sides of a right triangle.<Example>At what ang to climb 80 meters in 100 meters?80100O per ation D isplayθ = sin -1b aθ = cos -1ca θ = tan -1bccabθHyperbolic FunctionsThe hyperbolic function is defined by using natural exponents in trig o-nometric functions.A PPL IC AT IO N S:Hyperbolic and arc hyperbolic functions are very useful in electrical engineering and physics.Arc hyperbolic functions are defined by using natural logarithms in trig ono-metric functions.Coordinate ConversionyxyxC onverts rectang ular coordinates to polar coordinates (x,y r, θ)C onverts polar coordinates to rectang ular coordinates (r, θ x, y )Splits data used for dual-variable data input.Displays r, θ and x, y. (Cxy or r θ)←←←←<Example>Determine the polar coordinates (r, θ) when the rectang u-lar coordinates of Point P are (x = 7, y = 3).[D E G m ode]A PPL IC AT IO N S:Coordinate conversion is often used in mathematics and eng ineering , espe-cially for impedance calculations in electronics and electrical eng ineering .737.623.2O per ationD isplay←←Binary, Pental, Octal,Decimal, and HexadecimalOperations (N-Base)This calculator can perform conversions between numbers expressed in binary, pental, octal, decimal, and hexadecimal systems. It can also perform the four basic arithmetic operations, calculations with parentheses and memory calculations using binary, pental, octal, decimal, and hexadecimal numbers. In addition, the calculator can carry out the log ical operations AN D, O R, N O T, N EG, X O R, and X N O R on binary, pental, octal, and hexadecimal numbers.C onverts to the binary system. "b" appears.C onverts to the pental system. "P" appears.C onverts to the octal system. "o" appears.C onverts to the hexadecimal system. "H" appears.Converts to the decimal system. "b", "P", "o", and "H" disappear from the display.C onversion is performed on the displayed value when these keys are pressed.<Example 1>O per ation D isplayHEX(1AC) ©BIN ©PEN ©OCT ©DEC1011 AND 101 = (BIN) ©<Example 2>1AC1011101O per ation D isplayHere is a table of examination results. Input this datafor analysis.<Example 1>Enters data for statistical calculations.C lears data input.Splits data used for dual-variable data input.(Used for dual-variable statistical calculations.)302..O per ationD isplayN o.1234567 8Score30405060708090100N o. of pupils2457121082D ata table 1Select sing le-variable statistics modeThe statistics function is excellent for analyzing qualities of an event. Thoug h primarily used for eng ineering and mathematics, the function is also applied to nearly all other fields including economics and medicine.Statistics FunctionDAT A I N P U T A N D C O R R E C T I O NC alculates the averag e value of the data (sample data x).C alculates the standard deviation for the data (sample data x).C alculates the standard deviation of a data population (sample data x).Displays the number of input data (sample data x).C alculates the sum of the data (sample data x).Calculates the sum of the data (sample data x) raised to the second power.Let’s check the results based on the previous data.69 (averag e value)17.75686128 (standard deviation)17.57839583 (standard deviation of the population)50 (total count of data)3450 (total)N OT E:1.Sample data refers to data selected randomly from the population.2.Standard deviation of samples is determined by the sample data shift from an averag e value.3.Standard deviation for the population is standard deviation when the sample data is deemed a population (full data).“A N S ” K E Y S F O R 1-V A R I A B L E S T AT I S T I C SDA T A C O R R E C T I O N<Example 2>3040502O per ationD isplaySelect single-variable statistics modeC orrection after pressingC orrection prior to pressing(oldest first) order. To reverse the display order to descending Each item is displayed with 'Xn=', 'Yn=', or 'N n=' (n is the sequential number of the data set).Using • Wor appears, more data items can be browsed by pressing • To delete a data set, display an item of the data set to delete, then D ata table 2A PPL IC A T IO N S:Sing le-variable statistical calculations are used in a broad rang e of fields,including eng ineering , business, and economics. They are most often applied to analysis in atmospheric observations and physics experiments, as well as for quality control in factories.45603O per ationDisplayThe table below summarizes the dates in April when cherryblossoms bloom, and the averag e temperature for March inthat same area. Determine basic statistical quantities fordata X and data Y based on the data table.<Example 3>6213D ata table 3Select dual-variable statistics mode and linear reg ression calculation in sub-mode.Year 19831984198519861987198819891990Average temperature 6.27.0 6.88.77.9 6.5 6.18.2Date blossoms bloom 139********7xyDateTemperature 615827......O per ationD isplay 17.175(Averag e for data x)0.973579551(Standard deviation for data x)0.91070028(Standard deviation of the population for data x)9.875(Averag e for data y)3.440826313(Standard deviation for data y)3.218598297(Standard deviation of the population for data y)8(Total count of data)57.4(Sum of data x)418.48(Sum of data x raised to the second power)544.1(Sum of the product of data x and data y)79(Sum of data y)863(Sum of data y raised to the second power)Let’s check the results based on the previous data.C alculates the sum of the product for sample data x and sample data y.C alculates the sum of the data (sample data y).C alculates the sum of the data (sample data y) raised to the second power.C alculates the averag e value of the data (sample data y).In addition to the 1-variable statistic keys, the following keys have been added for calcu-lating2-variable statistics.Calculates the standard deviation of a data population (sample data y).Calculates the standard deviation for the data (sample data y).N OT E:The codes for basic statistical quantities of sample data x and their meaning s are the same as those for sing le-variable statistical calculations.“A N S ” K E Y S F O R 2-V A R I A B L E S T AT I S T I C S。

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I/O Feed.Lin
9
“Epipole-based Visual Servoing for nonholonomic robots”
Epipole kinematics
(planar motion)
Actual robot position Desired robot position
Remark that we are only interested to the epipole x-coordinate.
17
Second Step:
feature-based translational strategy
1. Use the actual and desired feature points to compute their centroids, i.e.
2. Translate until the mean distance Da from feature to centroid in actual image, is equal to Dd computed in desired image ,i.e. where
Experimental results
“Epipole-based Visual Servoing for nonholonomic robots”
“Epipole-based Visual Servoing for nonholonomic robots”
13
Zero dynamics
Zero dynamics= “internal dynamics of the system compatible with the output being identically zero”
Epipole-based Visual Servoing for Nonholonomic Mobile Robots via Feedback Linearization
SSIR04- Summer School on Image and Robotics – INRIA Sophia Antipolis
6
“Epipole-based Visual Servoing for nonholonomic robots”
Feedback linearization w.r.t. the epipoles (I)
MAIN IDEA:
Use input/output feedback linearization to control both outputs and (image plane) to track some desired trajectories.
“Epipolar constraint”
“Epipole-based Visual Servoing for nonholonomic robots”
5
Basics on Epipolar Geometry (II)
“Fundamental matrix F ” = keeps most of info on camera displacement
(b) eax(t) = u0
Remark 2. Case (b) can be avoided imposing the trajectory of the actual epipole goes to u0 after that the desired epipole reached u0. Then the controller is stopped.
Control of nonholonomic robot
“Epipole-based Visual Servoing for nonholonomic robots”
7
Feedback linearization w.r.t. the epipoles (II)
Hyp. 1 = Planar motion i.e. (i) (ii) , and are coincident. Only epipoles x-coordinates vary!
Decoupling matrix E is directly obtained from (1) and (2) and depends upon epipoles and also distance d(t) and f.
“Epipole-based Visual Servoing for nonholonomic robots”
Remark 1. It can be shown that the feedback linearization works even if the unknown distance d(t) is substituted with a finite positive constant parameter .
10
General framework
“Epipole-based Visual Servoing for nonholonomic robots”
I/O Feedback Linearizing Controller
Main result: Relative degree r of system is 2 and the I/O Feedback linearizing control
Outline
• • • • • • Introduction Basics on Epipolar Geometry Control of nonholonomic mobile robot along epipoles Visual Servoing strategy Simulation and experimental results Visual Servoing with Panoramic Cameras
11
Proof:
(1) (2)
The relative degree r is the sum of differentiation orders r1 and r2 of both outputs for which inputs appears linearly. Then, from (1) and (2):
law is: with Decoupling Matrix E :
being Let moreover:
and
Being
and
the controller gains and
the two desired output functions.
“Epipole-based Visual Servoing for nonholonomic robots”
Td<Ta
.
“Epipole-based Visual Servoing for nonholonomic robots”
16
First Step (II)
Simulation results
“Epipole-based Visual Servoing for nonholonomic robots”
“Epipole-based Visual Servoing for nonholonomic robots”
2
Basics on pinhole camera model
Consider the point
(world frame). with frame
A pinhole camera is located in
Full-perspective model:
Pixels per unit distance in image
“Epipole-based Visual Servoing for nonholonomic robots”
4
Basics on Epipolar Geometry (I)
• Two distinct views in and . • “baseline” = • “epipole ” intersection of baseline and image plane ;
“Epipole-based Visual Servoing for nonholonomic robots”
18
Simulation results: full servoing
“Epipole-based Visual Servoing for nonholonomic robots”
19
Simulation results:
15
பைடு நூலகம்
Second Step: translational motion from
“Epipole-based Visual Servoing for nonholonomic robots”
First Step (I)
Desired epipole trajectories
Used to apply the previously presented feedback linearizing controller.
“Epipole-based Visual Servoing for nonholonomic robots”
1
Introduction
The problem • Visual servoing • Mobile robots • Nonholonomy The solution ?
• Epipolar Geometry • Feedback linearization • Unknown 3D scene and geometry (only projected scene features)
can be estimated from a set of 8 corresponding points (Luong,Faugeras ’98)