Dynamical properties of low dimensional CuGeO3 and NaV2O5 systems

Product Data Sheet Mobilgear SHC 46MHigh Technology, Ultra High Viscosity,Synthetic Open Gear Oil Product DescriptionMobilgear SHC 46M is a speciality ultra high viscosity lubricant primarily intended for use in heavily loaded, low and medium speed gears where boundary lubrication conditions prevail.Mobilgear SHC 46M is formulated from wax-free synthetic base stocks which have exceptional oxidation and thermal properties and remarkable excellent low-temperature fluidity. The combination of a naturally highviscosity index and a unique additive system enables the product to provide outstanding performance under severe high and low temperature operating conditions. The base stocks have inherently low traction properties which result in low fluid friction in the load zone of non-conforming surfaces such as gears. Reduced fluidfriction produces lower oil operating temperatures and improved gear efficiency.Mobilgear SHC 46M exceeds Falk minimum viscosity requirements for intermittent lubrication of gears.Mobilgear SHC 46M does not contain any solvent.BenefitsMobilgear SHC 46M offers the following benefits:•Extended gear life resulting from outstanding load-carrying, anti-wear and tackiness properties derived from proprietary thick film lubrication•Reduction in pump replacement costs due to their pumpability at ambient temperature•Improved safety gear efficiency and lower operating temperatures arising from low traction properties •Reduced lubricant consumption and disposal costs•Absence of hard-packed deposits and the elimination of the need for gear cleaning with subsequent reduction in downtime and maintenance costs•Improved safety through absence of solvent•Easier distribution/meteringApplicationMobilgear SHC 46M is primarily recommended for open gears where asphaltic, diluent-containing lubricantshave historically been used. Mobilgear SHC 46M is also recommended for the replacement of semi-fluid greases.Mobilgear SHC 46M can be applied with conventional single line or dual line spray systems without costly circulation system modifications. It is pumpable at temperatures as low as +6°C but, at lower temperatures, the product may have to be heated to achieve the proper spray distribution pattern.Health and SafetyBased on available toxicological information, it has been determined that this product poses no significant health risk when used and handled properly.Details on handling, as well as health and safety information, can be found in the Material Safety Data Bulletin which can be obtained through Mobil Oil Company Ltd., by telephoning 01372 22 2000.Typical physical characteristics are given in the table. These are intended as a guide to industry and are not necessarily manufacturing or marketing specifications.Typical CharacteristicsMobilgear SHC 46MTest MethodISO VG-46000Viscosity, cSt at 40ºC ASTM D4*******Viscosity, cSt at 100ºC ASTM D4451250Viscosity Index ASTM D2270210Specific Gravity ASTM D12980.924Flash Point, ºC ASTM D92228Pour Point, ºC ASTM D97+6Rust Protection, Distilled Water ASTM D665PassRust Protection, Synthetic Sea Water ASTM D665PassCopper Corrosion ASTM D1301A/1BTimken OK Load, kg ASTM D250926.7FZG Gear Test, Fail Stage DIN 5135412+4-Ball Wear, Scar mm ASTM D22660.9754-Ball Weld Load, kg ASTM D2596315Colour ASTM D1500 1.5Due to continual product research and development, the information contained herein is subject to change without notice.Mobil Oil Company LimitedActing as Agent for Mobil Lubricants UK LimitedExxonMobil House, Ermyn WayLeatherhead, Surrey, KT22 8UXTelephone: 01372 22 2000。

暗物质英语定义Dark matter, also known as invisible matter, is a mysterious substance that makes up a significant portion of the universe. Although it cannot be directly observed, its existence is inferred from its gravitational effects on visible matter. In this article, we will explore the definition of dark matter, its properties, and its implications for our understanding of the cosmos.Dark matter is believed to account for approximately 85% of the total matter in the universe. Its presence is necessary to explain the observed rotational velocities of galaxies and the gravitational lensing effects observed in clusters of galaxies. Unlike ordinary matter, dark matter does not interact with electromagnetic radiation, making it invisible to telescopes and other instruments that rely on light detection.One of the key properties of dark matter is its non-baryonic nature. Baryonic matter, which includes protons and neutrons, makes up only a small fraction of the total matter in the universe. Dark matter, on the other hand, consists of particles that do not interact via the strong nuclear force, which binds protons and neutrons together. Instead, dark matter particles are thought to interact primarily through gravity and weak nuclear forces.The exact nature of dark matter remains unknown, but several theoretical candidates have been proposed. One possibility is that dark matter consists of weakly interacting massive particles (WIMPs). These hypothetical particles would have masses larger than those of ordinary matter particles and would interact only weakly with other particles. Another candidate is the axion, a hypothetical particle that could explain the absence of certain symmetry violations in the strong nuclear force.The search for dark matter is a major focus of modern astrophysics and particle physics. Scientists employ a variety of experimental techniques to detect or indirectly infer the presence of dark matter. These include direct detection experiments, which aim to detect the rare interactions between dark matter particles and ordinary matter, and indirect detection experiments, which look for the products of dark matter annihilation or decay.Understanding the nature of dark matter is crucial for our understanding of the universe's evolution and structure formation. The presence of dark matter has profound implications for the Big Bang theory and the formation of galaxies and galaxy clusters. It is believed that dark matter played a crucial role in the formation of the large-scale structure of the universe, acting as a gravitational seed for the formation of galaxies and galaxy clusters.In addition to its gravitational effects, dark matter also influences the distribution of ordinary matter. The presence of dark matter affects the growth of structures in the universe, leading to the formation of cosmic web-like structures composed of filaments and voids. These structures can be observed through large-scale surveys of galaxies and the cosmic microwave background radiation.In conclusion, dark matter is a mysterious substance that constitutes a significant portion of the universe. Although invisible and non-baryonic, its presence is inferred from its gravitational effects on visible matter. The search for dark matter is an active area of research, with scientists employing various experimental techniques to shed light on its nature. Understanding dark matter is crucial for our understanding of the universe's evolution and structure formation.。

IA ... DIA ... SDescriptionThe rack and pinion pneumatic actuator IA Motion combines innovative design features with the latest technology, materials and protection coatings available, resulting in one of the highest grade pneumatic actuators on the market.Product features• FunctionIA...D double acting IA...S single acting• Nominal torque15 ÷ 10007 Nm(double acting at 6 bar air supply)• Supply pressure 3 ÷ 8 bar (IA1000D 3 ÷ 7 bar) • Supply fluids Filtered air or neutral gas • Working temperature -40°C ÷ 80°C• ConnectionMounting face for valves according to EN ISO 5211,for solenoid valves and accessories to VDI/VDE 3845 (NAMUR)• LubricationFactory lubricated for the life of actuator under normal working conditions• ATEXActuator IP68, standard in compliance with ATEX 94/9/ECDesign properties• Compact design with identical body and end caps for double acting and spring return types, allowing field conversion by adding or removing spring cartridges.• Body made of extruded aluminium with internal and external ALODUR ® corrosion protection, with honed cylinder surface for a higher cycle life and lower coefficient of friction.• Symmetric rack and pinion design for high-cycle life and fast operation. Reverse rotation can be accomplished by inverting the pistons.• Two independent external travel stop adjustments, enabling an easy and precise adjustment of -5°÷15° / 75°÷95°, in order to get a precise valve positioning.• One-piece blow-out proof, electroless nickel-plated drive shaft with bearing guided one-piece pinion for improved safety and max. cycle life.• Fully machined piston teeth for accurate low backlash rack and pinion engagement and maximum efficiency.• Pistons standard anodised for higher life.• Multifunction position indicator, adaptable to all kinds of limit and proximity switches.• Preloaded spring cartridges with coated springs for simple versatile range and corrosion resistance. Spring return actuator can be disassembled without danger on field.• High quality bearings and seals for low friction, high cycle life and a wide operating temperature range.• End caps, anodized and Polyester ® coated (RAL 5021).• All used screws in stainless steel for life time corrosion resistance.• Full compliance to the latest specifications: EN ISO 5211, VDI/VDE 3845, NAMUR and ATEX (Directive 94/9/CE).• Every single actuator is tested and provided with a unique serial number for traceability.❷❹❷❹❷❹❷❹IA...D ✈IA...S ✈IA...DIA...SIA050-100IA200-1000FunctionMaterialsYour benefits• High quality actuator designed for high-cycle life.• Multiple mounting circles and shafts to fit most quarter turn valves.• Easy conversion from double to single acting and vice versa.• Lower inventory with greater flexibility.• Position indicator with graduated ring indicating acurate angle.• Two external travel stop adjustmentsfor easy valvepositioning -5°÷15° / 75°÷95°.• Extensive size range to fit the requested torque at lowest costs.• Full compliance to latest worldwide standards.IA...D double acting actuatorAir supplied to port ❷ moves pistons toward endposition.(→ 90° counterclockwise rotation)IA...S single acting actuatorAir supplied to port ❷ moves pistons toward endposition, compressing springs (→ 90° counterclockwise rotation)Air supplied to port ❹ moves pistons toward center position.(→ 90° clockwise rotation)Air failure allows springs to move pistons toward center position(→ 90° clockwise rotation)IA200D .F05 - F0714❶❷❸❹Type codeAvailable options:• 5 different external coatings.• Stainless steel AISI 303, 430 or 316 drive shaft.• High and low temperature versions.• 0 ÷ 90° adjustable travel stop.• Cost efficient lock out capability.• Other drive shaft connections.• Rotation 120° and 180° and intermediate such as 135°.• 3 position actuators.• Stainless steel actuators.Please contact our technical department for more information about these options.0 ÷ 90° adjustable travel stopMd0º90ºMd0º90ºIA...D IA...STorques [Nm]Opening angle Opening angleTorques [Nm]1) IA045 S33, IA050 - IA550. S12 = Standard version of InterApp. Other number of springs on requestAAAAACCCCBBBB B ØZ1ØZ1ØZ1ØZ1ØZ1PPPP P CIA ... D IA ... St O ”t C ”[kg]t O ”t C ”[kg]0,150,200,750,200,250,90,20,251,150,250,31,260,250,31,70,30,351,90,30,353,00,40,53,40,40,54,20,50,64,81) BSP / ISO 228 / DIN 259V(l) Volume in litre, V O = OPEN, V C = CLOSE To calculate the air consumption, multiply the volume in litre by the supply pressure.t O / t C t O = opening time / t C = closing time, in secondsThe above mentioned operating times are obtained under the following conditions:- Air supply pressure min. 5,5 bar (80 psi) - at room temperature - medium clean air - actuator stroke 90° - actuator without resistance load Caution: obviously, during operation, if one or more of the above listed criteria differ, the operating time will be different.The technical data are noncommittal and do not assure you of any properties. Please refer to our general sales conditions. Modifications without notice. AccessoriesOur wide range of accessories includes all kinds of position indicators, solenoid valves, positioners, Bus systems, manual emergency overdrives, etc. Please refer to the corresponding documentation or download it from our website.Limit switchesSolenoid valveProximity switchAS-InterfacePositionerActuator size, solenoid valve and air supply pipe accordingtable below.© 2020 InterApp AG, all rights reserved。

Analyzing the Properties of Pigmentsand DyesPigments and dyes are two types of colorants used in various industries such as art, fashion, cosmetics, and printing. Both pigments and dyes have unique properties that make them suitable for different applications. In this article, we will analyze the properties of pigments and dyes.What are Pigments?Pigments are colorants that do not dissolve in the medium in which they are dispersed. Pigments are insoluble but dispersible in water, oil, or another medium. Pigments are used in a wide range of applications such as paints, inks, plastics, ceramics, and textiles. Pigments come in a variety of forms such as powders, pastes, or granules.The Properties of PigmentsPigments have several unique properties that make them ideal for various applications. Here are some of the properties of pigments:1. LightfastnessOne of the essential properties of pigments is lightfastness. The lightfastness of a pigment refers to its ability to retain its color when exposed to light. Pigments with a high level of lightfastness are resistant to fading, whereas pigments with low lightfastness will fade quickly.2. OpacityOpacity is the ability of pigments to block light. Pigments with high opacity can cover a surface entirely, whereas pigments with low opacity will allow some of the underlying surface color to show through.3. Chemical StabilityPigments must be chemically stable when exposed to various chemicals, whether they come into contact with solvents, acids, or bases. Any chemical reaction with the medium can cause a change in color or degrade the quality of the pigment.4. Particle SizeThe particle size of a pigment determines its dispersibility and the resulting color intensity. Smaller particles make pigments more translucent, whereas larger particles make pigments more opaque.5. Color StrengthThe color strength of a pigment is the intensity of its color when used at maximum concentration. Pigments with high color strength require less material to produce vivid, vibrant colors.What are Dyes?Dyes are colorants that dissolve in the medium in which they are used. Dyes are soluble in water, oil, or another medium. Dyes are used in a wide range of applications such as textiles, paper, leather, and food. Dyes come in various forms such as liquids or powders.The Properties of DyesDyes have several unique properties that make them ideal for various applications. Here are some of the properties of dyes:1. SolubilitySolubility is the ability of dyes to dissolve in liquid or other mediums. It is the essential property of dyes, which allows it to penetrate deep into the fiber and produce a vibrant color.2. WashfastnessWashfastness is the ability of dyes to resist fading when exposed to water. Dyes with high washfastness will retain their color even after repeated exposure to water and detergents.3. LightfastnessLightfastness is the ability of dyes to resist fading when exposed to light. Dyes with high lightfastness will retain their color even when exposed to sunlight or artificial light sources.4. AffinityThe affinity of dyes is the ability to attach themselves to the surface of the material they are applied to. Dyes with high affinity are more likely to produce uniform and vibrant colors.5. Color RangeOne of the essential properties of dyes is the ability to produce a wide range of colors. Dyes can create bright and vivid colors in various shades, hues, and tones.In ConclusionPigments and dyes are two unique types of colorants used for various applications. Pigments are insoluble but dispersible in water, oil, or another medium, while dyes are soluble. Pigments have properties such as lightfastness, opacity, and chemical stability, while dyes have properties such as solubility, washfastness, and affinity. Understanding the properties and differences between pigments and dyes can help you choose the best one for your specific needs.。

SOIL MECHANICSLECTURE NOTESLECTURE # 1SOIL AND SOIL ENGINEERING* The term Soil has various meanings, depending upon the general field in which it is being considered.*To a Pedologist ... Soil is the substance existing on the earth's surface, which grows and develops plant life.*To a Geologist ..... Soil is the material in the relative thin surface zone within which roots occur, and all the rest of the crust is grouped under the term ROCK irrespective of its hardness.*To an Engineer .... Soil is the un-aggregated or un-cemented deposits of mineral and/or organic particles or fragments covering large portion of the earth's crust.* Soil Mechanics is one of the youngest disciplines of Civil Engineering involving the study of soil, its behavior and application as an engineering material.*According to Terzaghi (1948): "Soil Mechanics is the application of laws of mechanics and hydraulics to engineering problems dealing with sediments and other unconsolidated accumulations of solid particles produced by the mechanical and chemical disintegration of rocks regardless of whether or not they contain an admixture of organic constituent."* Geotechnical Engineering ..... Is a broader term for Soil Mechanics.* Geotechnical Engineering contains:- Soil Mechanics (Soil Properties and Behavior)- Soil Dynamics (Dynamic Properties of Soils, Earthquake Engineering, Machine Foundation)- Foundation Engineering (Deep & Shallow Foundation)- Pavement Engineering (Flexible & Rigid Pavement)- Rock Mechanics (Rock Stability and Tunneling)- Geosynthetics (Soil Improvement)Soil Formation* Soil material is the product of rock* The geological process that produce soil isWEATHERING (Chemical and Physical).* Variation in Particle size and shape depends on:- Weathering Process- Transportation Process* Variation in Soil Structure Depends on:- Soil Minerals- Deposition Process* Transportation and DepositionFour forces are usually cause the transportation and deposition of soils1- Water ----- Alluvial Soil 1- Fluvial2- Estuarine3- Lacustrine4- Coastal5- Marine2- Ice ---------- Glacial Soils 1- Hard Pan2- Terminal Moraine3- Esker4- Kettles3- Wind -------- Aeolin Soils 1- Sand Dunes2- Loess4- Gravity ----- Colluvial Soil 1- TalusWhat type of soils are usually produced by the different weathering & transportation process?- Boulders- Gravel Cohesionless- Sand (Physical)- Silt Cohesive- Clay (Chemical)* These soils can be- Dry- Saturated - Fully- Partially* Also they have different shapes and texturesLECTURE # 2SOIL PROPERTIESPHYSICAL AND INDEX PROPERTIES1- Soil Composition- Solids- Water-Air2- Soil Phases- Dry- Saturated * Fully Saturated* Partially Saturated- Submerged3- Analytical Representation of Soil:For the purpose of defining the physical and index properties of soil it is more convenient to represent the soil skeleton by a block diagram or phase diagram. 4- Weight - Volume Relationships:WeightW t = W w + W sVolumeV t = V v + V s = V a + V w + V s 1- Unit Weight - Density* Also known as- Bulk Density- Soil Density-Unit Weight-Wet DensityRelationships Between Basic Properties:Examples:________________________________________________________________________Index PropertiesRefers to those properties of a soil that indicate the type and conditions of the soil, and provide a relationship to structural properties such as strength, compressibility, per meability, swelling potential, etc.________________________________________________________________________1- PARTICLE SIZE DISTRIBUTION* It is a screening process in which coarse fractions of soil are separated by means of series of sieves.* Particle sizes larger than 0.074 mm (U.S. No. 200 sieve) are usually analyzed by means of sieving. Soil materials finer than 0.074 mm (-200 material) are analyzed by means of sedimentation of soil particles by gravity (hydrometer analysis).1-1 MECHANICAL METHODU.S. Standard Sieve:Sieve No. 4 10 20 40 60 100 140 200 -200Opening in mm 4.76 2.00 0.84 0.42 0.25 0.149 0.105 0.074 -Cumulative Curve:* A linear scale is not convenient to use to size all the soil particles (opening from 200 mm to 0.002 mm).* Logarithmic Scale is usually used to draw the relationship between the % Passing and the Particle size.Example:Parameters Obtained From Grain Size Distribution Curve:1- Uniformity Coefficient C u (measure of the particle size range)Cu is also called Hazen CoefficientCu = D60/D10C u < 5 ----- Very UniformC u = 5 ----- Medium UniformC u > 5 ----- Nonuniform2- Coefficient of Gradation or Coefficient of Curvature C g(measure of the shape of the particle size curve)C g = (D30)2/ D60 x D10C g from 1 to 3 ------- well graded3- Coefficient of Permeabilityk = C k (D10)2 m/secConsistency Limits or Atterberg Limits:- State of Consistency of cohesive soil1- Determination of Liquid Limit:2- Determination of Plastic Limit:3- Determination of Plasticity IndexP.I. = L.L. - P.L. 4- Determination of Shrinkage Limit5- Liquidity Index:6- Activity:SOIL CLASSIFICATION SYSTEMS* Why do we need to classify soils ?To describe various soil types encountered in the nature in a systematic way and gathering soils that have distinct physical properties in groups and units.* General Requirements of a soil Classification System:1- Based on a scientific method2- Simple3- Permit classification by visual and manual tests.4- Describe certain engineering properties5- Should be accepted to all engineers* Various Soil Classification Systems:1- Geologic Soil Classification System2- Agronomic Soil Classification System3- Textural Soil Classification System (USDA)4-American Association of State Highway Transportation Officials System (AASHTO) 5- Unified Soil Classification System (USCS)6- American Society for Testing and Materials System (ASTM)7- Federal Aviation Agency System (FAA)8- Others1- Unified Soil Classification (USC) System:The main Groups:G = GravelS = Sand.........................M = SiltC = Clay........................O = Organic........................* For Cohesionless Soil (Gravel and Sand), the soil can be Poorly Graded or Well GradedPoorly Graded = PWell Graded = W* For Cohesive Soil (Silt & Clay), the soil can be Low Plastic or High Plastic Low Plastic = LHigh Plastic = HTherefore, we can have several combinations of soils such as:GW = Well Graded GravelGP = Poorly Graded GravelGM = Silty GravelGC = Clayey GravelPassing Sieve # 4SW = Well Graded SandSP = Poorly Graded SandSM = Silty SandSC = Clayey SandPassing Sieve # 200ML = Low Plastic SiltCL = Low Plastic ClayMH = High Plastic SiltCH = High Plastic ClayTo conclud if the soil is low plastic or high plastic use Gassagrande's Chart________________________________________________________________________ 2- American Association of State Highway Transportation Officials System (AASHTO):- Soils are classified into 7 major groups A-1 to A-7Granular A-1 {A-1-a - A-1-b}(Gravel & Sand) A-2 {A-2-4 - A-2-5 - A-2-6 - A-2-6}A-3More than 35% pass # 200A-4Fine A-5(Silt & Clay) A-6A-7Group Index:_________________________________________________ ___3- Textural Soil Classification System (USDA)* USDA considers only:SandSiltClayNo. Gravel in the System* If you encounter gravel in the soil ------- Subtract the % of gravel from the 100%.* 12 Subgroups in the systemExample: ********MOISTURE DENSITY RELATIONSHIPS(SOIL COMPACTION)INTRODUCTION:* In the construction of highway embankments, earth dams, and many other engineering projects, loose soils must be compacted to increase their unit weight.* Compaction improves characteristics of soils:1- Increases Strength2- Decreases permeability3- Reduces settlement of foundation4- Increases slope stability of embankments* Soil Compaction can be achieved either by static or dynamic loading:1- Smooth-wheel rollers2- Sheepfoot rollers3- Rubber-tired rollers4- Vibratory Rollers5- Vibroflotation_____________________________________________________________________________________________General Principles:* The degree of compaction of soil is measured by its unit weight, , and optimum moisture content, w c.* The process of soil compaction is simply expelling the air from the voids.or reducing air voids* Reducing the water from the voids means consolidation.Mechanism of Soil Compaction:* By reducing the air voids, more soil can be added to the block. When moisture is added to the block (water content, w c, is increasing) the soil particles will slip more oneach other causing more reduction in the total volume, which will result in adding moresoil and, hence, the dry density will increase, accordingly.* Increasing W c will increaseUp to a certain limit (Optimum moister Content, OMC)After this limitIncreasing W c will decreaseDensity-Moisture RelationshipKnowing the wet unit weight and the moisture content, the dry unit weight can be determined from:The theoretical maximum dry unit weight assuming zero air voids is:I- Laboratory Compaction:* Two Tests are usually performed in the laboratory to determine the maximum dry unit weight and the OMC.1- Standard Proctor Test2- Modified Proctor TestIn both tests the compaction energy is:1- Standard Proctor TestFactors Affecting Compaction:1- Effect of Soil Type2- Effect of Energy on Compaction3- Effect of Compaction on Soil Structure4- Effect of Compaction on Cohesive Soil PropertiesII- Field CompactionFlow of Water in SoilsPermeability and Seepage* Soil is a three phase medium -------- solids, water, and air* Water in soils occur in various conditions* Water can flow through the voids in a soil from a point of high energy to a point of low energy.* Why studying flow of water in porous media ???????1- To estimate the quantity of underground seepage2- To determine the quantity of water that can be discharged form a soil3- To determine the pore water pressure/effective geostatic stresses, and to analyze earth structures subjected to water flow.4- To determine the volume change in soil layers (soil consolidation) and settlement of foundation.* Flow of Water in Soils depends on:1- Porosity of the soil2- Type of the soil - particle size- particle shape- degree of packing3- Viscosity of the fluid - Temperature- Chemical Components4- Total head (difference in energy) - Pressure head- Velocity head- Elevation headThe degree of compressibility of a soil is expressed by the coefficient of permeability of the soil "k."k cm/sec, ft/sec, m/sec, ........Hydraulic GradientBernouli's Equation:For soilsFlow of Water in Soils1- Hydraulic Head in SoilTotal Head = Pressure head + Elevation Headh t = h p + h e- Elevation head at a point = Extent of that point from the datum- Pressure head at a point = Height of which the water rises in the piezometer above the point.- Pore Water pressure at a point = P.W.P. = g water . h p*How to measure the Pressure Head or the Piezometric Head???????Tips1- Assume that you do not have seepage in the system (Before Seepage)2- Assume that you have piezometer at the point under consideration3- Get the measurement of the piezometric head (Water column in the Piezometer before seepage) = h p(Before Seepage)4- Now consider the problem during seepage5- Measure the amount of the head loss in the piezometer (Dh) or the drop in the piezometric head.6- The piezometric head during seepage = h p(during seepage) = h p(Before Seepage) - DhGEOSTATIC STRESSES&STRESS DISTRIBUTIONStresses at a point in a soil mass are divided into two main types:I- Geostatic Stresses ------ Due to the self weight of the soil mass.II- Excess Stresses ------ From structuresI. Geostatic stressesI.A. Vertical StressVertical geostatic stresses increase with depth, There are three 3 types of geostatic stresses1-a Total Stress, s total1-b. Effective Stress, s eff, or s'1-c Pore Water Pressure, uTotal Stress = Effective stress + Pore Water Pressures total = s eff + uGeostatic Stress with SeepageWhen the Seepage Force = H g sub -- Effective Stress s eff = 0 This case is referred asBoiling or Quick ConditonI.B. Horizontal Stress or Lateral Stresss h = k o s'vk o = Lateral Earth Pressure Coefficients h is always associated with the vertical effective stress, s'v.never use total vertical stress to determine s h.II. Stress Distribution in Soil Mass:When applying a load on a half space medium the excess stresses in the soil will decrease with depth.Like in the geostatic stresses, there are vertical and lateral excess stresses.1. For Point LoadThe excess vertical stress is according to Boussinesq (1883):- I p = Influence factor for the point load- Knowing r/z ----- I1 can be obtained from tablesAccording to Westergaard (1938)where h = s (1-2m / 2-2m) m = Poisson's Ratio2. For Line LoadUsing q/unit length on the surface of a semi infinite soil mass, the vertical stress is:3. For a Strip Load (Finite Width and Infinite Length): The excess vertical stress due to load/unit area, q, is:Where I l = Influence factor for a line load3. For a Circular Loaded Area:The excess vertical stress due to q is:。

Professor & Head of DepartmentNT Bishop, MA(Cambridge), PhD(Southampton), FRASSenior LecturersJ Larena, MSc(Paris), PhD(Paris)D Pollney, PhD(Southampton)CC Remsing, MSc(Timisoara), PhD(Rhodes)Vacant LecturersEOD Andriantiana, PhD(Stellenbosch)V Naicker, MSc(KwaZulu-Natal)AL Pinchuck, MSc(Rhodes), PhD(Wits)Lecturer, Academic Development M Lubczonok, Masters(Jagiellonian)Mathematics (MA T) is a six-semester subject and Applied Mathematics (MAP) is a four-semester subject. These subjects may be taken as major subjects for the degrees of BSc, BA, BJourn, BCom, BBusSci, BEcon and BSocSc, and for the diploma HDE(SEC).To major in Mathematics, a candidate is required to obtain credit in the following courses: MAT1C; MAM2; MAT3. See Rule S.23.To major in Applied Mathematics, a candidate is required to obtain credit in the following courses: MAT1C, MAM2; MAP3. See Rule S.23.The attention of students who hope to pursue careers in the field of Bioinformatics is drawn to the recommended curriculum that leads to postgraduate study in this area, in which Mathematics is a recommended co-major with Biochemistry, and for which two years of Computer Science and either Mathematics or Mathematical Statistics are prerequisites. Details of this curriculum can be foundin the entry for the Department of Biochemistry, Microbiology and Biotechnology.See the Departmental Web Page http://www.ru.ac.za/departments/mathematics/ for further details, particularly on the content of courses.First-year level courses in MathematicsMathematics 1 (MAT1C) is given as a year-long semesterized two-credit course. Credit in MAT1C must be obtained by students who wish to major in certain subjects (such as Applied Mathematics, MATHEMATICS (PURE AND APPLIED)Physics and Mathematical Statistics) and by students registered for the BBusSci degree.Introductory Mathematics (MAT1S) is recommended for Pharmacy students and for Science students who do not need MAT1C or MAT1C1.Supplementary examinations may be recommended for any of these courses, provided that a candidate achieves a minimum standard specified by the Department.Mathematics 1L (MA T1L) is a full year course for students who do not qualify for entry into any of the first courses mentioned above. This is particularly suitable for students in the Social Sciences and Biological Sciences who need to become numerate or achieve a level of mathematical literacy. A successful pass in this course will give admission to MA T1C.First yearMAT1CThere are two first-year courses in Mathematics for candidates planning to major in Mathematics or Applied Mathematics. MAT1C1 is held in thefirst semester and MAT1C2 in the second semester. Credit may be obtained in each course separately and, in addition, an aggregate mark of at least 50%will be deemed to be equivalent to a two-creditcourse MAT1C, provided that a candidate obtains the required sub-minimum (40%) in each component. Supplementary examinations may be recommended in either course, provided that a candidate achieves a minimum standard specified by the department. Candidates obtaining less than 40% for MAT1C1 are not permitted to continue with MAT1C2.MAT1C1 (First semester course): Basic concepts (number systems, functions), calculus (limits,continuity, differentiation, optimisation, curvesketching, introduction to integration), propositional calculus, mathematical induction, permutations, combinations, binomial theorem, vectors, lines andplanes, matrices and systems of linear equations.MAT1C2 (Second semester course): Calculus (integration, applications of integration, improper integrals), complex numbers, differential equations, partial differentiation, sequences and series.MAT1S (Semester course: Introductory Mathematics) (about 65 lectures)Estimation, ratios, scales (log scales), change of units, measurements; Vectors, systems of equations, matrices, in 2-dimensions; Functions: Review of coordinate geometry, absolute values (including graphs); Inequalities; Power functions, trig functions, exponential functions, the number e (including graphs); Inverse functions: roots, logs, ln (including graphs); Graphs and working with graphs; Interpretation of graphs, modeling; Descriptive statistics (mean, standard deviation, variance) with examples including normally distributed data; Introduction to differentiation and basic derivatives; Differentiation techniques (product, quotient and chain rules); Introduction to integration and basic integrals; Modeling, translation of real-world problems into mathematics.MAT 1L: Mathematics Literacy This course helps students develop appropriatemathematical tools necessary to represent and interpret information quantitatively. It also develops skills and meaningful ways of thinking, reasoning and arguing with quantitative ideas in order to solve problems in any given context.Arithmetic: Units of scientific measurement, scales, dimensions; Error and uncertainty in measure values.Fractions and percentages - usages in basic science and commerce; use of calculators and spreadsheets. Algebra: Polynomial, exponential, logarithmic and trigonometric functions and their graphs; modelling with functions; fitting curves to data; setting up and solving equations. Sequences and series, presentation of statistical data.Differential Calculus: Limits and continuity; Rules of differentiation; Applications of Calculus in curvesketching and optimisation.Second Year Mathematics 2 comprises two semesterized courses,MAM201 and MAM202, each comprising of 65 lectures. Credit may be obtained in each course seperately. An aggregate mark of 50% will grant the two-credit course MAM2, provided a sub-minimumof 40% is achieved in both semesters. Each semester consists of a primary and secondary stream which are run concurrently at 3 and 2 lectures per week, respectively. Additionally, a problem-based course in Mathematical Programming contributes to the class record and runs throughout the academic year.MAM201 (First semester):Advanced Calculus (39 lectures): Partial differentiation: directional derivatives and the gradient vector; maxima and minima of surfaces; Lagrange multipliers. Multiple integrals: surface and volume integrals in general coordinate systems. Vector calculus: vector fields, line integrals, fundamental theorem of line integrals, Green’s theorem, curl and divergence, parametric curves and surfaces.Ordinary Differential Equations (20 lectures): First order ordinary differential equations, linear differential equations of second order, Laplace transforms, systems of equations, series solutions, Green’s functions.Mathematical Programming 1 (6 lectures): Introduction to the MATLAB language, basic syntax, tools, programming principles. Applicationstaken from MAM2 modules. Course runs over twosemesters.MAM202 (Second semester):Linear Algebra (39 lectures): Linear spaces, inner products, norms. Vector spaces, spans, linear independence, basis and dimension. Linear transformations, change of basis, eigenvalues, diagonalization and its applications.Groups and Geometry (20 lectures): Number theory and counting. Groups, permutation groups, homomorphisms, symmetry groups in 2 and 3 dimensions. The Euclidean plane, transformations and isometries. Complex numbers, roots of unity and introduction to the geometry of the complex plane.Mathematical Programming 2 (6 lectures): Problem-based continuation of Semester 1.Third-year level courses inMathematics and Applied Mathematics Mathematics and Applied Mathematics are offered at the third year level. Each consists of four modules as listed below. Code TopicSemester Subject AM3.1 Numerical analysis 1 Applied MathematicsAM3.2 Dynamical systems 2 Applied Mathematics AM3.4 Partial differentialequations 1 Applied Mathematics AM3.5 Advanced differentialequations 2 Applied MathematicsM3.1 Algebra 2 MathematicsM3.2 Complex analysis 1 MathematicsM3.3 Real analysis 1 MathematicsM3.4 Differential geometry 2 Mathematics Students who obtain at least 40% in all of the above modules will be granted credit for both MAT3 and MAP3, provided that the average of the Applied Mathematics modules is at least 50% AND the average of the Mathematics modules is at least 50%. Students who obtain at least 40% for any FOUR of the above modules and with an average mark over the four modules of at least 50%, will be granted credit in either MAT3 or MAP3. If three or four of the modules are from Applied Mathematics then the credit will be in MAP3, otherwise it will be in MAT3.Module credits may be carried forward from year to year.Changes to the modules offered may be made from time-to-time depending on the interests of the academic staff.Credit for MAM 2 is required before admission to the third year courses.M3.1 (about 39 lectures) AlgebraAlgebra is one of the main areas of mathematics with a rich history. Algebraic structures pervade all modern mathematics. This course introduces students to the algebraic structure of groups, rings and fields. Algebra is a required course for any further study in mathematics.Syllabus: Sets, equivalence relations, groups, rings, fields, integral domains, homorphisms, isomorphisms, and their elementary properties.M3.2 (about 39 lectures) Complex Analysis Building on the first year introduction to complex numbers, this course provides a rigorous introduction to the theory of functions of a complex variable. It introduces and examines complex-valued functions of a complex variable, such as notions of elementary functions, their limits, derivatives and integrals. Syllabus: Revision of complex numbers, Cauchy- Riemann equations, analytic and harmonic functions, elementary functions and their properties, branches of logarithmic functions, complex differentiation, integration in the complex plane, Cauchy’s Theorem and integral formula, Taylor and Laurent series, Residue theory and applications. 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It is a subject which allows students to see mathematics for what it is - a unified whole mixing together geometry, calculus, linear algebra, differential equations, complex variables, calculus of variations and topology.Syllabus: Curves (in the plane and in the space), curvature, global properties of curves, surfaces, the first fundamental form, isometries, the second fundamental form, the normal and principal curvatures, the Gaussian and mean curvatures, the Gauss map, geodesics.AM3.1 (about 39 lectures) Numerical Analysis Many mathematical problems cannot be solved exactly and require numerical techniques. These techniques usually consist of an algorithm which performs a numerical calculation iteratively until certain tolerances are met. These algorithms can be expressed as a program which is executed by a computer. The collection of such techniques is called “numerical analysis”.Syllabus: Systems of non-linear equations, polynomial interpolation, cubic splines, numerical linear algebra, numerical computation of eigenvalues, numerical differentiation and integration, numerical solution of ordinary and partial differential equations, finite differences,, approximation theory, discrete Fourier transform.AM3.2 (about 39 lectures) Dynamical Systems This module is about the dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields of applied mathematics (like control theory and the Lagrangian and Hamiltonian formalisms of classical mechanics). The emphasis is on the mathematical aspects of various constructions and structures rather than on the specific physical/mechanical models. Syllabus Linear systems; Linear control systems; Nonlinear systems (local theory); Nonlinear control systems; Nonlinear systems (global theory); Applications : elements of optimal control and/or geometric mechanics.AM3.4 (about 39 lectures) Partial Differential EquationsThis course deals with the basic theory of partial differential equations (elliptic, parabolic and hyperbolic) and dynamical systems. It presents both the qualitative properties of solutions of partial differential equations and methods of solution. Syllabus: First-order partial equations, classification of second-order equations, derivation of the classical equations of mathematical physics (wave equation, Laplace equation, and heat equation), method of characteristics, construction and behaviour of solutions, maximum principles, energy integrals. Fourier and Laplace transforms, introduction to dynamical systems.AM3.5 (about 39 lectures)Advanced differential equationsThis course is an introduction to the study of nonlinearity and chaos. Many natural phenomena can be modeled as nonlinear ordinary differential equations, the majority of which are impossible to solve analytically. Examples of nonlinear behaviour are drawn from across the sciences including physics, biology and engineering.Syllabus:Integrability theory and qualitative techniques for deducing underlying behaviour such as phase plane analysis, linearisations and pertubations. The study of flows, bifurcations, the Poincare-Bendixson theorem, and the Lorenz equations.Mathematics and Applied Mathematics Honours Each of the two courses consists of either eight topics and one project or six topics and two projects.A Mathematics Honours course usually requires the candidate to have majored in Mathematics, whilst Applied Mathematics Honours usually requires the candidate to have majored in Applied Mathematics. The topics are selected from the following general areas covering a wide spectrum of contemporary Mathematics and Applied Mathematics: Algebra; Combinatorics; Complex Analysis; Cosmology; Functional Analysis; General Relativity; Geometric Control Theory; Geometry; Logic and Set Theory; Measure Theory; Number Theory; Numerical Modelling; Topology.Two or three topics from those offered at the third-year level in either Mathematics or Applied Mathematics may also be taken in the case of a student who has not done such topics before. With the approval of the Heads of Department concerned, the course may also contain topics from Education, and from those offered by other departments in the Science Faculty such as Physics, Computer Science, and Statistics. On the other hand, the topics above may also be considered by such Departments as possible components of their postgraduate courses.Master’s and Doctoral degrees in Mathematics or Applied MathematicsSuitably qualified students are encouraged to proceed to these degrees under the direction of the staff of the Department. Requirements for these degrees are given in the General Rules.A Master’s degree in either Mathematics or Applied Mathematics may be taken by thesis only, or by a combination of course work and a thesis. Normally four examination papers and/or essays are required apart from the thesis. The whole course of study must be approved by the Head of Department.。

Dynamic Motion Planningin Low Obstacle Density Environments Robert-Paul Berretty Mark Overmars A.Frank van der Stappen Department of Computer Science,Utrecht University,P.O.Box80.089,3508TB Utrecht,The Netherlands.AbstractA fundamental task for an autonomous robot is to plan its own motions.Ex-act approaches to the solution of this motion planning problem suffer from highworst-case running times.The weak and realistic low obstacle density(L.O.D.)assumption results in linear complexity in the number of obstacles of the freespace[11].In this paper we address the dynamic version of the motion planningproblem in which a robot moves among moving polygonal obstacles.The ob-stacles are assumed to move along constant complexity polylines,and to respectthe low density property at any given time.We will show that in this situation acell decomposition of the free space of size can be computedin time.The dynamic motion planning problem is then solvedin time.We also show that these results are close to optimal. Keywords:Motion planning,low obstacle density,moving obstacles,cell decompo-sition.1IntroductionRobot motion planning concerns the problem offinding a collision-free path for a robot in a workspace with a set of obstacles from an initial placement to a final placement.The parameters required to specify a placement of the robot are referred to as the degrees of freedom of the robot.The motion planning problem is often studied as a problem in the configuration space,which is the set of parametric representations of the placements of the robot.The free space FP is the sub-space of of placements for which the robot does not intersect any obstacle in.A feasible motion for the robot corresponds to a curve from to in FP(or its closure).Motion planning is a difficult problem.In general,many instances of the robot mo-tion planning problem are P-SPACE-complete,even if the obstacles are stationary[5]. For a constant-complexity robot moving amidst stationary obstacles polynomial time algorithms have been shown to exist.The running time is exponential in the number of degrees of freedom of the robot[7].For an-DOF robot,the complexity of the free space,can be as high as and the motion planning problem will,therefore,in general have a worst case running time close to.Research is partially supported by the Dutch Organization for Scientific Research(N.W.O.).We address the motion planning problem for a robot operating in an environment with moving obstacles.This problem is also referred to as the dynamic motion plan-ning problem.In general,when the obstacles in the workspace are allowed to move, the motion planning problem becomes even more complicated.For example,Reif and Sharir[6]showed that,when obstacles in a3-dimensional workspace are allowed to ro-tate,the motion planning problem is PSPACE-hard if the velocity modulus is bounded, and NP-hard otherwise.(A similar result was obtained by Sutner and Maass[8].) Canny and Reif[3]showed that dynamic motion planning for a point in the plane,with a bounded velocity modulus and an arbitrary number of convex polygonal obstacles, is NP-hard,even when the obstacles are convex and translate at constant linear veloc-ities.They also showed that the2-dimensional dynamic motion planning problem for a translating robot with bounded velocity modulus,among polygonal obstacles that translate atfixed linear velocity,can be solved using an algorithm that is polygo-nal in the total number of vertices of and,if the number of obstacles is bounded. However,their algorithm takes exponential time in the number of moving obstacles.Van der Stappen et al.[11](see also[10])showed that modelling robots in re-alistic workspaces has a profound influence on the complexity of solving the static motion planning problem,mainly independent of the number of degrees of freedom of the robot.They gave a description of environments with a so-called low obsta-cle density which leads to a surprising gain in efficiency for several instances of the motion planning problem.An environment has the low obstacle density property if any region in the workspace intersects a constant number of obstacles that are larger than the size of the region.(See below for a more precise definition.)Under the low obstacle density assumption,the exact motion planning problem for an-DOF robot was efficiently solved,using the cell decomposition approach.The low obstacle density of the workspace implies a linear combinatorial complexity of the free space, even for-DOF robots.For a robot moving amidst stationary obstacles the cell decomposition of the free space has size and is computable in time. Vleugels[12]extended these results to multiple robots simultaneously operating in the same workspace.De Berg et al.[2]gave an overview of several realistic input models and gave experimental results on scenes based on real input data,which showed that the‘hidden’constant in the low obstacle density assumption was indeed low.We demonstrate that the low obstacle density property can also be used to ef-ficiently plan a motion for a robot with degrees of freedom moving in a2-dimensional workspace with non-stationary obstacles.The obstacles are allowed to translate in the workspace along polyline trajectories,with afixed speed per segment. The motion planning problem is then solved in time,using a cell decomposition of size.Note that these bounds do not depend on(assuming is constant).We also show that this result is close to optimal,by giving an example where the robot has to perform simple motions to get from its start to its goal position.In this paper we willfirst present an overview of the method used in the paper of Van der Stappen et al.[11].The computation of the cell decomposition for the dynamic low obstacle density motion planning problem is treated in Sections3and4; the algorithm to compute a feasible path through the cell decomposition is presented Section5.Section6concludes the paper.2Low Obstacle DensityIn this section we recall some of the definitions and results from the paper by Van der Stappen et al.[11]on motion planning in low density environments.The authors focus in particular on the large class of motion planning problems with configuration spaces of the form,where is the-dimensional workspace and is some -dimensional rest space.Let us use the reach of a robot as a measure for its maximum size;the reach of is defined as the maximum radius that the minimal enclosing hypersphere of the robot,centered at its reference point,can ever have(in any placement of).The reach of the robot is assumed to be comparable to the size of the smallest obstacle.The robot has constant complexity and moves in a workspace with constant-complexity obstacles.The workspace satisfies the static low obstacle density property which is defined as follows.Property2.1Let I R be a space with a set of non intersecting obstacles.Then I R is said to be a static low(obstacle)density space if for any region I R with minimal enclosing hyper-sphere radius,the number of obstacles with minimal enclosing hyper-sphere radius at least intersecting is bounded by a constant. Van der Stappen et al.[11]showed that,under the circumstances outlined above,the complexity of the free space is linear in the number of obstacles.The configuration space contains hyper-surfaces of the form,consisting of placements of the robot in which a robot feature is in contact with an obstacle feature.We shall denote the fact that is a feature of some object or object set by.The arrangement of all(constant-complexity)constraint hyper-surfacesdivides the higher-dimensional configuration space into free cells and forbidden cells.Van der Stappen et al.[11]considered so-called cylindri-fiable configuration spaces which have the property that the subspace —referred to as the base space—can be partitioned into constant complexity regions satisfyingA partition that satisfies this constraint is called a cylindrical partition.In words,the lifting of the region into the configuration space is intersected by a constant number of constraint hyper-surfaces.These hyper-surfaces subdivide the cylinder into constant-complexity free and forbidden cells.The cylindrical partition of therefore almost immediately gives us a cell decomposition of the free portion FP of. Theorem2.2states that the transformation of a cylindrical partition of the base space into a cell decomposition of the free space can be accomplished in time proportional to the size of the cylindrical partition.Theorem2.2[11]Let be the set of regions of a cylindrical partition of a base space and let be the set of region adjacencies.Let the regions of be of constant com-plexity.Then the cell decomposition of the free space calculated by lifting the regions of the base partition into the configuration space consists of constant complexity subcells.Furthermore,the complexity of the decomposition and the time to compute it is.Note that the size of the cylindrical partition determines the size of the cell decompo-sition.The low obstacle density motion planning problem outlined above was shown to yield a cylindrifiable configuration space,in which the workspace is a valid base space.Small and efficiently computable cylindrical partitions of have led to opti-mal cell decompositions and thus efficient solutions to the motion planning problem (see[11]for details).In this paper,we show that the configuration space of the dynamic version of the low obstacle density motion planning problem is cylindrifiable as well.Wefind a cylindrical partition of an appropriate base space that leads to an almost optimal size cell decomposition.3A Dynamic Base Space3.1Problem StatementWe now focus on the dynamic robot motion planning problem,subjected to low ob-stacle density.We show that the framework outlined in Section2can be used to plan a motion for a robot with degrees of freedom,moving in a2-dimensional workspace with non-stationary obstacles.The obstacles translate in the workspace, and can only change speed or direction a constant number of times.We will use a cell-decomposition based on a cylindrical partition,similar to Section2.Since dynamic motion planning is tedious to deal with,we split the problem into sub-problems.We first formally define the problem and state some useful properties of the base space for the dynamic motion planning problem.In Section4,we construct a cylindrical decomposition,and in Section5,we compute the actual path for the robot.The dynamic low obstacle density motion planning problem is defined as follows.The workspace of the robot is the2-dimensional Euclidean space I R and contains a collection of obstacles,each moving along a polyline at constant speed per line segment.The robot has constant complexity and its reach is bounded by, where is a constant and is a lower bound on the minimal enclosing hyper-sphere radii of all obstacles.Each obstacle is polygonal and has constant complexity.Any constraint hyper-surface in the configuration space corresponding to the set of robot placements in which a certain robot feature is in contact with a certain obstacle,is algebraic of bounded degree.The robot is placed at the initial placement at time and has to be at the goal placement at time.At any time between and,the workspace with obstacles satisfies the low obstacle density property.A standard approach when dealing with moving obstacles is to augment the sta-tionary configuration space with an extra time dimension.In this manner,we obtainthe configuration-time space.When planning the motion of our robot through the configuration-time space,we have to make sure that the path is time-monotone—the robot is not allowed to move back in time.Thefirst objective in solving the dynamic low obstacle density motion planning problem is to obtain a cylindrical partition that consists of constant complexity regions.An appropriate choice for a base space is the Cartesian product of the2-dimensional workspace and time.This way,the config-uration time space is of the form I R I R I R, where is some-dimensional rest space.3.2Characteristics of the Base SpaceThe base space can be considered as a3-dimensional Euclidean space. In our dynamic motion planning setting,we only consider the work-time space slice I R.Wefirst look at the situation where the obstacles move along a line in the ter,we extend the result to the polyline case.Definition3.1Let and let be a curve in.Then the columnis defined by,where denotes the Minkowski sum operator.The column is the volume swept by in the work-time space as its ref-erence point follows the curve.In our application,the curve describes the translational motion of an obstacle and is therefore time-monotone.A point belongs to if and only if covers the point at time.Definition3.2Let be a square centered at the origin,having side length. Then.The Minkowski sum encloses.No point in has a distance larger than to.We denote the arrangement of the boundaries of the grown obstacle columns by.We will show that this arrangement is of complexity.Let us for a moment consider afixed obstacle at afixed time.We consider the boundary of the grown obstacle.Now,if the reference point of the robot is placed outside,the robot cannot collide with the obstacle.If the reference point of the robot is inside the grown obstacle,there might be configurations in which the the robot intersects the obstacle.Since both the robot and the obstacles have constant complexity,the arrangement of constraint hypersurfaces in at time,when lifted into the configuration space,has constant complexity as well.We exploit this observation to build a partition of the base space.We say that an obstacle is in the proximity of another obstacle if and intersect,hence and intersect.Theorem3.3The complexity of the arrangement of the boundaries of the grown obstacle columns is.Proof:The complexity of the arrangement is determined by the number of ver-tices.A vertex results from an intersection of three columns.A necessary conditionfor three columns to intersect is that the corresponding obstacles are less than apart at some moment in time.We show that the number of such triples is.We charge each such triple to a pair of obstacles.For this we choose the smallest obsta-cle of the three and the one(of the remaining two)that last entered’s proximity. Assume that an obstacle enters the proximity of.(Note that can enter’s proximity at most times because and have constant complexity and both move along line paths.)A third obstacle involved in a triple must al-ready be in the proximity of at the time of arrival of in order to be charged to the pair(,).By Property2.1,there are only larger obstacles in’s proximity at any time,so is chosen from a set of size.As a result,only triples are charged to each of the pairs(,).Each of these triples contribute a constant number of vertices to because the obstacles,, and have constant complexity and move along line paths.Therefore,the complex-ity of is bounded by.It is easy to see that a2-face of a column in thefinal arrangement is divided into a number of parts,of which some are non-convex.The following theorem states that the 2-faces of the arrangement are polygons without holes.This property turns out to be important in the sequel.Theorem3.4The faces of are polygonal and have no holes.Proof:The faces of are formed by the possibly intersecting faces of the columns.Since the columns are polyhedra,the arrangement has polygonal faces.It remains to prove that the faces do not contain holes.A face of the arrangement has a hole iff a column penetrates the interior of this face without intersecting its boundary.We distinguish the bottom and top faces and the side faces of the columns.The bottom and top faces of the columns,i.e.the intersections of the columns with and,are the boundaries of the Minkowski sums of the obstacles at their positions at and and.A grown obstacle cannot be fully contained in another grown obstacle,otherwise the obstacles would also intersect, which is not the case.Therefore,the top and bottom faces of columns are faces without holes.The side faces of the columns are the possibly intersecting walls that connect the top and bottom faces of the columns.Assume,for a contradiction,that(a part of)some side face of has a hole.There must be another columnwhich intersects this face.We call the smallest time coordinate of the hole,and the largest time coordinate.Note that.Without loss of generality, wefix object,such that its speed becomes zero,and adjust the speed of the other objects accordingly.After this transformation,we consider the2-dimensional vertical projection onto the workspace of and(i.e.at and respectively.See Figure1).Note that and are grown using the same square.It is easy to see that,dependent on the location of the obstacle with respect to the projection of,intersects at or which is impossible by assumption.So,the faces of are polygonal and have no holes.If we extend the setting to the case in which obstacles translate along polylines, the complexity of the arrangement does not increase asymptotically—inFigure1:The2-dimensional scene with the dark grey area depicting the intersection of two obstacles at.the proof of Theorem3.3,the chargings to the obstacle pair caused by obstacle are,in the worst case,multiplied by a constant factor.Unfortunately,the2-faces of are no longer polygons without holes.We can resolve this by adding extra faces to the arrangement.For every time at which one of the obstacles changes speed,we add a plane.This way,the area between two successive planes is a work-time space slice where all obstacles move in afixed direction with afixed speed. The arrangements on the newly introduced planes are cross sections of the work-time space.They are arrangements of possibly intersecting grown obstacle boundaries and have linear complexity because the obstacles statisfy the low obstacle density property at any time[11].We compute a triangulation of these2-dimensional arrangements to assure that their faces have no holes.Since we have polyline vertices,the total added complexity is.We will show that every cylinder,defined by a3-cell of the arrange-ment,is intersected by a constant number of constraint hyper-surfaces.We define the coverage of a region.Definition3.5In words,the coverage of a region is the set of obstacles whose columns,which are computed after growing the obstacles,intersect the region.The following result follows from the low density property and the observation that all points in a single 3-cell of the arrangement of column boundaries lie in exactly the same collection of columns.Lemma3.6The regions,defined by the cells of have.Lemma3.7shows that the partition of the base space into regions with is a cylindrical partition.The proof is very similar to the proof of Lemma3.6 of Van der Stappen et al.[11]and has been omitted.Lemma3.7Let be such that.ThenThe only problem is that the complexity of the cells of is not necessarily constant.So,we must refine the partition to create constant complexity subcells.This is discussed in Section4.gadget with fencesFigure 2:The quadratic lowerbound construction.3.3Complexity of the Free SpaceIn the previous subsection we showed that the work-time space of the robot can be partitioned into regions with total combinatorial complexity .Furthermore,by Lemmas 3.6and 3.7,each region,when lifted into the configuration-time space is inter-sected by at most a constant number of constraint hyper-surfaces of bounded algebraic degree.Therefore,a decomposition of the configuration space into free and forbid-den cells of combinatorial complexity exists.Obviously,this bound is an upper bound on the complexity of the free space for our dynamic motion planning setting.Theorem 3.8The complexity of the free space of the dynamic low obstacle densitymotion planning problem is.We will now demonstrate that this bound is worst-case optimal,even in the situation where the robot is only allowed to translate and the obstacles move along lines.To this end,we give a problem instance with obstacles,for which any path for the robot has complexity.Consider the workspace in Figure 2.The grey rectangular robotmust translate from positionto .The gadget in the middle forces the robot to make moves to move from left to right.It can easily be constructed fromstationary obstacles.The big black obstacle at the bottom right moves very slowly to the right.So it takes a long time before the robot can actually get out of the gadget to go to its goal.Now a small obstacle moves from the left to the right,through the gaps in the middle of the gadget.This forces the robot to go to the right as well.Only there can it move slightly further up to let the obstacle pass.But then a new obstacle comes from the right through the gaps,forcing the robot to move to the left of the gadget to let the obstacle pass above it.This is repeated times after which the big obstacle is finally gone and the robot can move to its goal.The robot has to move times through the gadget,each time making moves,leading to a total of moves.As ,the total number of moves is .It is easily verified that at any moment the low obstacle density property is satisfied.Theorem 3.9The complexity of the free space of the dynamic low obstacle density motion planning problem for a translating robot is .Actually,the example shows a much stronger result.Not only does it give a bound on the complexity of the free space,but also on the complexity of a single cell in the free space and on the complexity of any dynamic motion planning algorithm. Theorem3.10The complexity of any algorithm for the dynamic low obstacle density motion planning problem(even for a translating robot)is lower bounded by.4Decomposing the Base SpaceWe still need to decompose the arrangement of columns into constant complexity subcells.To this end,we construct a vertical decomposition of the arrangement.Since the vertical decomposition refines the cells of the arrangement, the subcells of thefinal decomposition still have constant-size coverage.The approach we use[1]requires that the columns in the work-time space,as described in Section 3.2,are in general position.This can be achieved by an appropriate perturbation of the vertices of the columns.Before we can calculate a vertical decomposition we have to triangulate the2-faces of the columns.Triangulation does not increase the asymptotic complexity of the arrangement.After triangulation,the2-faces of the arrangement might coincide,though.It is easily verified that the vertical decomposition algorithm still works with these introduced degeneracies.To bound the space we add two hori-zontal planes at time and(the start and goal time)and only consider the area in between.To bound the space in the-and-direction we also add a triangular prism far around the relevant region of the work-time space.4.1The Vertical DecompositionLet be a set of possibly intersecting triangles in3space.The vertical decomposition of the arrangement decomposes each cell of into subcells,and is defined as follows(see[1]):from every point on an edge of—this can be a part of a triangle edge or of the intersection of two triangles—we extend a vertical ray in positive and negative-direction to thefirst triangle above and thefirst triangle below this point.This way we create a vertical wall for every edge,which we call a primary wall.We obtain a multi-prismatic decomposition of into subcells, the multi-prisms,with a unique polygonal bottom and top face;the vertical projections of both faces are exactly the same.However,the number of vertical walls of a cylinder need not be constant and the cylinder may not be simply connected.We triangulate the bottom face as in the planar case.The added segments are extended upward vertically until they meet the top face.The walls thus erected are the secondary walls.Each subcell of the vertical decomposition is now a box with a triangular base and top, connected by vertical walls.(Note that,for navigation purposes,our notion of vertical decomposition is slightly different from other notions of vertical decomposition that construct secondary walls using a planar vertical decomposition of the projections of the top and bottom faces.)Theorem4.1The vertical decomposition of the arrangement in the work-time space consists of constant complexity subcells,and can be com-puted in time.Proof:Tagansky[9]proved that the vertical decomposition of the entire arrange-ment of a set of triangles in I R consists of subcells where is the complexity of the arrangement.Application of this result to the arrangement of grown obstacle column boundaries,which satisfies,yields the complexity bound.We can compute the vertical decomposition using an algorithm by De Berg et al.[1].This algorithm runs in time,where is the combi-natorial complexity of the vertical decomposition.As,the bound follows.To faciliate navigation,we want each subcell to have a constant number of neighbors. The common boundary of a subcell and one of its neighbors can be a secondary wall,a primary wall,or a2-face of the arrangement.It is easy to see that the number of neighbors sharing a primary or a secondary wall with is bounded by a constant.Let us now consider the maximum number of neighbors,sharing a part of a triangle of with.Unfortunately the arrangements of walls ending on the top and bottom side of the triangle can be very different,and can in general be as complex as the complexity of the full decomposition which is only upper-bounded by.Simply connecting the subcells at the top of the triangle to the subcells at the bottom of the triangle could result in a number of neighbors that is hard to bound by anything better than for each subcell.However,as we will show,we can connect the subcells at the top and bottom of a face by a symbolic, infinitely thin tetrahedralization.This tetrahedralization will increase the combina-torial complexity of the vertical decomposition by a factor of at most,but assures that the number of neighbors per subcell is bounded by a constant.Since this method is quite complicated,we dedicate the following subsection to it.This will lead to the following result:Theorem4.2There exists a cylindrical decomposition of the base space for the dy-namic low obstacle density motion planning problem consisting ofconstant complexity subcells and a constant number of neighbors per subcell.This decomposition can be computed in time.4.2Tetrahedralizing between PolygonsTo reduce the number of neighbors of the subcells we will extend the vertical decom-position with a symbolic connecting structure,that increases the total combinatorial complexity of the vertical decomposition by a factor of.As a result,the number of neighbors per subcell of the cell decomposition with the connecting struc-ture will be bounded by a constant.For each face of the arrangement, this structure connects the subcells at the top side with the subcells at the bottom side. The structure we use is a symbolic,infinitely thin tetrahedralization.To simplify the discussion,we assume that the face for which we construct the connecting structure is horizontal.(This is not a constraint,but just a matter of definition.)Throughout this section the vertical direction is parallel to the normal of the face.Both the top and the bottom side of the face contain a triangulated2-dimensional arrangement,say and,created by the intersecting faces and the walls that end on it.Such triangulations with extra vertices in their interior are referred to as Steinertriangulations;the extra vertices are called Steiner points.The arrangements and are normally different;they do not share Steiner points.We separate the top and bottom of every face in the arrangement.Imagine that the top of the face is at height and the bottom at height.We tetrahedralize the space between the top and bottom arrangement,by adding a number of Steiner points between the top and bottom face. (Remember that this is only done in a symbolic way.In reality,the top and bottom face lie in the same plane.The vertical distance is only used to define the adjacencies of the added(flat)subcells.)We distinguish between the convex and the non-convex faces.Note that non-convex faces indeed exist,since a column can cut out a part of another column.The-orem3.4gives us that the cut out parts are never strictly included in the open interior of a2-face of a column.Wefirst show how to tetrahedralize the space between two different Steiner triangulations and of the same convex simple polygon.Our tetrahedralization has two layers joined at height by a Steiner triangulation of.This triangulation has one Steiner point:is triangulated using a star of edges from to all vertices of.Both and are different Steiner triangulations of the same polygon,therefore the vertical projections of the boundaries of and are equivalent.We tetrahedralize between and by adding a face from every edge of to.The result is a tetrahedralized pyramid where each tetrahedron corresponds to a triangle of.To triangulate the complement of this pyramid in the layer between and,we connect the boundaries of and by vertical faces between the boundary edges.For every face introduced by connecting the boundaries,we add a Steiner point in the middle of.We connect to all vertices on and connect each resulting triangle to (see Figure3).These triangles complete the tetrahedralization of the space between and.The tetrahedralization between and is constructed in the same way. It is easy to see that the number of tetrahedra created is linear in the complexity of the triangulations and.Unfortunately,faces need not be convex.So we must also show how to tetrahe-dralize the space between two different Steiner triangulations of the same non-convex simple polygon.(As indicated above we know that the polygon has no holes.This is crucial here.)We again add a Steiner triangulation of between and.In the non-convex case we have to use a more sophisticated Steiner triangulation.For this we use a triangulation by Hershberger and Suri[4]that was originally designed for ray shooting in simple polygons.This triangulation has three important properties.Let be the number of edges of:1.It introduces Steiner points with each Steiner point directly connected tothe boundary of by at least one triangulation edge;2.Every line segment that lies inside intersects at most triangles of;3.The triangulation can be computed in time.We can derive the following lemma from the properties of.Lemma4.3Let be a polygon with vertices and without holes,Let be the triangulation of as described in[4].Let be a triangle inside,and let be the。