text: segment defined by two points in the world frame. The Toolbox supports named joint configurations and these are shown in the table If we display the value of puma we see that the $$\alpha_j$$ values are now displayed in red to indicate that they are symbolic constants. This is a modified version of a paper submitted to ICRA2020. Line 3 defines the unique lengths of the Puma robot, and line 4 defines the kinematic chain in At line 11 we compute the inverse kinematics of each pose in the trajectory pose in 3D space. For a user transitioning from MATLAB the most significant difference is using a single call to the ikine_LM method. major refactorings led to the current state of three toolboxes: Robotics Toolbox In this Python3 for Robotics course, you will master essential Python3 knowledge in order to get started with ROS smoothly. Robotics-toolbox-python - The Robotics Toolbox for Python #opensource. Build on the Spatial Math Toolbox for Python [SMTB-P] which provides objects to A Puma robot can also be specified in ETS format [Corke07] as a sequence of simple rigid-body transformations – pure translation started in 2008 but ultimately failed for lack of ongoing resources to complete Modular approach to interfacing to different graphics libraries, simulators and physical robots. constructors, methods which wrap base functions, and type checking. simplicity and expressiveness of the OpenAI Gym API which was designed as a Compute inverse dynamics via recursive Newton-Euler formulation. fate. and components can be extracted such as the rotation submatrix (line 11) and translation (line 15). The coding in this course is done in Python 3 using ROS Noetic. This, the tenth release of the Toolbox, represents over twenty years of development and a substantial level of maturity. Issues can be reported via GitHub issues or patches submitted as equivalent class is DHRobot is similarly limited, but a new class ERobot Jacobians can also be computed for symbolic joint variables as for forward kinematics above. $$\SO{3}$$ or an arbitrary matrix. This applies to the visualizers Vpython and Swift, as well as pybullet and ROS. rigid-body motions as SE(2) and SE(3) matrices or twists in parent link. For 25 years the Robotics Toolbox for MATLAB ® has been used for teaching and research worldwide. Google Groups was established in 2008 and as of 2020 has over 1400 members. The Toolbox provides capability to import URDF-xacro files without ROS. This paper describes its successor – the Robotics Toolbox for Python. Attempts to compose with a non SE3 instance would result in a TypeError. Previous message: [robotics-worldwide] 2009 ICRA Workshop: Brain Machine Interfaces for Neuroprostheses and Robot Control Next message: [robotics-worldwide] Postdoctoral opportunity in the area of medical robotics - Imperial College, London, UK Support Python notebooks which allows publication of static notebooks (for example via GitHub) and interactive online notebooks (MyBinder.org). The Toolbox adopts a “when needed” approach to many dependencies and will only attempt However, subsequent expression manipulation is slow. 0. means that for the user, working with symbols is as easy as working with A solution is not possible if the specified transform describes equations, class inheritance diagrams and figures – all of which is converted Use of UniCode characters to make console output easier to read. row is a joint coordinate. PrismaticMDH. The first We therefore need tools to represent quantities such as rigid-body other robotics packages. sin(phi)*sin(psi)*sin(theta) + cos(phi)*cos(psi). Introduction. There are also functions to quaternions, vectors, twists and argument handling. method, rne_spatial(), and SMTB-P provides a set of classes for spatial RevoluteDH objects for a revolute joint described using standard DH example, $$\SE{3} * \SE{3} \rightarrow \SE{3}$$ but $$\SE{3} + \SE{3} \rightarrow \mathbb{R}^{4 \times 4}$$, and equality testing for a It provides many of the essential tools necessary for Support branched, but not closed-loop or parallel, robots, Utilize WebGL and Javascript graphics technologies. elementary transform sequence into a series of links and joints – link frames are declared 0. represent rotations as SO(2) and SE(3) matrices as well as unit-quaternions; This toolbox brings robotics-specific functionality to Python, and leverages Python's advantages of portability, ubiquity and support, and the capability of the open-source ecosystem for linear algebra (numpy, scipy), graphics (matplotlib, three.js, WebGL), interactive development (jupyter, jupyterlab, mybinder.org), and documentation (sphinx). approach is cumbersome and reduces code clarity, while the second cannot ensure RTB-M could only animate a robot in a figure, and there was limited but Know more here.. 2| Pyro. return with an error. The code is documented with ReStructured Text format The toolbox provides such definitions wrapped as class definitions, for example: We can now easily perform standard kinematic operations. Kluwer Academic, 1987. Australian Robot Association, pages 319–330, Melbourne, July 1995. transformations (matrices $$\in \SE{n}$$ or twists $$\in \se{n}$$), Provided models, such as for Panda or Puma, are again encapsulated as classes: and kinematic operations are performed using methods with the same name Featherstone, Robot Dynamics Algorithms. The underlying datatypes in all cases are 1D and 2D NumPy and a minimalist block-diagram simulation tool [bdsim]. pull requests. is the Yoshikawa measure computed for just the task space translational degrees 3, pp. So far only the following functions and their unit tests have been ported: rot2 rotx, roty, rotz trot2 trotx, troty, trotz se2, se3 r2t t2r rpy2r, rpy2t (was just rpy2tr in the original) tr2rpy rpy2jac Licence ----- … For example: There is strong similarity to the equivalent MATLAB case apart from the use of numbers. Python version supports a simple, but universal API to a robot inspired by the It can produce vivid 3D effects using anaglyphs viewed with colored glasses. Options such as vpython can be used to specify additional dependencies to be installed. Some provided robot models have an analytical solution coded, for example: where we have specified a left-handed, elbow up and wrist no-flip configuration. provides abstraction classes SE3, Twist3, SO3, UnitQuaternion, Every robot link can have a collision shape in addition to the shape Other classes available are PrismaticDH, RevoluteMDH and This arrays. iDynTree is a library of robot elements calculations for control, estimation, and recreation. devices. rotations (matrices $$\in \SO{n}$$ or $$\so{n}$$, Euler or roll-pitch-yaw It was designed to provide a simple, clear, and concise interface to quickly simulate and evaluate common robot concepts, such as kinematics, dynamics, trajectory generations, and calibration. At line 12 we convert this into an array, with one row per time step, and each The x-coordinate of the end-effector is algorithm implemented in Python or C. The same C code as used by RTB-M is called directly from Python, and does not use NumPy. Animations can be recorded as MP4 files or animated GIF files which are useful for inclusion in GitHub markdown documents. There are several areas of innovation compared to the MATLAB version of the Toolbox. At a singular configuration. I have searched on the internet, but couldn't find anything. robot is described by a set of ELink objects, each of which points to its given by line 7. We use classes and data encapsulation to address all these issues. other link in order to determine the pose of that link’s coordinate frame. Swift is a web-based visualizer using three.js to provide high-quality 3D animations. where puma.qr is an example of a named joint configuration. National Conf. [-sin(theta), sin(phi)*cos(theta), cos(phi)*cos(theta)]], 0.15005*sin(q_0) - 0.0203*sin(q_1)*sin(q_2)*cos(q_0) - 0.4318*sin(q_1)*cos(q_0)*cos(q_2) - 0.4318*sin(q_2)*cos(q_0)*cos(q_1) + 0.0203*cos(q_0)*cos(q_1)*cos(q_2) + 0.4318*cos(q_0)*cos(q_1), Yoshikawa manipulability is 0.07861716534599998, Asada manipulability is 0.004374613728166503, Yoshikawa manipulability is 0.11118146146764128, $$\dot{m} = \mat{J}_m(\vec{q}) \dvec{q}$$, [-0. PyRobot - Python for Robotics Written by Nikos Vaggalis Thursday, 11 July 2019 We've grown accustomed to hearing about Python being used for pretty much everything, albeit mostly for data science. and we can use the mouse to rotate and zoom the plot. version of RNE rne_python has been implemented as well. Robotics System Toolbox provides a library of robotics algorithms and tools to design, simulate, and test robotics application. [Corke96]. By default the Toolbox behaves like the MATLAB version with a plot method: which will plot the robot at the specified joint configurmation, or animate it if q is an $$m \times 6$$ matrix, using This Create Own Python Robotics Simulator using pygames course is a small initiative for stepping towards AI and , use of most widely popular computer language PYTHON for everybody who are involve in robotics in our country and are just wandering with embedded system and mechanics . Each element is an array with one row per time step, and each row is a joint coordinate vector. It includes commercially available industrial robot models to model your robot applications and reference examples of common industrial robot applications. velocity, acceleration, momentum, force and inertia. For a manipulator with $$n < 6$$ DOF SMTB-P In line 7 we pass this to the constructor for an ERobot which partitions the These benefits come at a price in terms of execution time due to the overhead of For example, the previous example could be written Test coverage is uploaded to \times 1 \times 1\) box centered at $$(1,0,0)$$ can be tested against all, or In Proc. Welcome the QUT Centre for Robotics Open Source website. Composition with a single-valued (scalar) pose instance broadcasts the scalar In this article, I am trying to write down the important 10 Python libraries for Robotics. Pybotics is an open-source Python toolbox for robot kinematics and calibration. The pose T can also be displayed as a 3D coordinate frame: Rotation can also be represented by a unit quaternion. With familiar, simple yet powerful functions; the clarity of Python syntax; but without the complexity of ROS; users from beginner to advanced will find this a powerful open-source toolset for ongoing robotics education and research. used for rendering. URDF models. Levenberg-Marquadt minimization, and additional status results are also ROBOPY is a python implimentation of The Robotics Toolbox (RTB) (The brainchild of Peter Corke) for Matlab. defaults to zero, and affects both the search time and the solution found, since in general terms of elementary transforms. For models defined using standard or modified DH notation we use a classical version of the recursive Newton-Euler unit-quaternion has to respect the double mapping. a manipulator may have several multiple joint configurations which result in the same end-effector pose. which again demonstrates several alternative constructors. For 25 years the Robotics Toolbox for MATLAB® has been used for teaching and research worldwide. the default PyPlot backend which draws a “noodle robot” using the PyPlot backend. sin(phi)*sin(psi) + sin(theta)*cos(phi)*cos(psi)]. that symbolic operations work as expected. We could plot the joint coordinates as a function of time using the convenience Features: Easy to read for understanding each algorithm’s basic idea. $$3 \times 3$$ array could be an element of $$\SE{2}$$ or If the answer is yes, then learning Python is mandatory for you. We can also specify any using matplotlib. The Python Toolbox supports several approaches to computing dynamics. To specify a kinematic model using DH notation, we create a new subclass of DHRobot and pass the superclass constructor Robotics Stack Exchange is a question and answer site for professional robotic engineers, hobbyists, researchers and students. RTB-M and the equivalent functionality exists in RTB-P which makes use of the could add an extra dimension to the matrices representing rigid-body The course is designed for beginners with little, to no experience using ROS. Spatial Maths Toolbox for Python (SMTB-P) [SMTB-P]. version control tools including rcs, cvs, svn and git. Peter Corke peter.corke.ieee at gmail.com Tue Apr 14 04:42:55 PDT 2009. for MATLAB and Machine Vision Toolbox for MATLAB (1999) both of which are built and integer exponentiation (repeated composition) using the ** operator. I'm a Mechatronics student at Southern Polytechnic State University.This an animation with Matlab Robotics Toolbox for our Robotics class. of freedom. modified), elementary transform sequences [Corke07] [Haviland20], and URDF-style which makes many models from the ROS universe available. rne_python() rne_python (Q, QD = None, QDD = None, gravity = None, fext = None, debug = False, basewrench = False) [source] ¶. Learn about robotics using MATLAB ® from Professor Peter Corke, a recognized leader in the field. elements. IEEE Robotics and Automation Magazine, 3(1):24–32, Sept. 1996. which uses an RK45 numerical integration from the SciPy package to solve for the joint trajectory q given the conventions. servers, Google code and currently GitHub and maintained under a succession of Corke. Hear Corke talk about Robotics Toolbox and Robotics System Toolbox™, and learn about how Robotics System Toolbox can be used. For transformations or unit-quaternions, or place them in a list. For ERobot instances we can also compute the manipulability whole robots, discrete links, and objects in the world. For branched robots, with multiple end-effectors, the name of the frame of interest must be provided. Discussion Forums Register with Us. The final approach to manipulator modeling is to an import a URDF file. robotic manipulator modelling, simulation and control which is essential for robotics education and research. Still frames and animations can be recorded. RTB-P, and its dependencies, can be installed simply by: which includes basic visualization using matplotlib. less than a second. The imperative for a Python version has long existed and the first port was Last updated on 05-Jan-2021. retrospect) early design decisions hinder development. This paper presents examples of the use of Python in robotics and mechatronics education, including computer vision (OpenCV), ... the Matlab Robotics Toolbox of Peter Cork e [19], but. $$\SE{3} + \SE{3} \rightarrow \mathbb{R}^{4 \times 4}$$, Any of the SMTB-P pose classes can contain a list of values, Overloaded operators support broadcasting. Pybotics: Python Toolbox for Robotics Nicholas A. Nadeau1 1 Department of Automated Manufacturing Engineering, École de Technologie Supérieure, Montréal, QC H3C 1K3, Canada DOI: 10.21105/joss.01738 Software • Review • Repository • Archive Submitted: 13 September 2019 Published: 30 September 2019 License Authors of papers retain a NumPy array can contain symbolic The design goals of this version can be summarised as new functionality: A superset of the MATLAB Toolbox functionality. IKsolution(q=array([ 2.1012, 2.9416, 2.9355, 1.6526, -0.953 , -2.809 ]), success=True, reason=''), tz(0.672) * Rz(q0) * ty(0.2337) * Ry(q1) * tz(0.4318) * tx(0.0203) * ty(-0.0837) * Ry(q2) * tz(0.4318) * Rz(q3) * Ry(q4) * Rz(q5), ┌───┬────────┬────────┬───────┬────────────────────────────────────────────────┐, │id │ link │ parent │ joint │ ETS │, ├───┼────────┼────────┼───────┼────────────────────────────────────────────────┤, │ 0 │ link0 │ - │ │ tz(0.672) * Rz(q0) │, │ 1 │ link1 │ link0 │ None │ ty(0.2337) * Ry(q1) │, │ 2 │ link2 │ link1 │ None │ tz(0.4318) * tx(0.0203) * ty(-0.0837) * Ry(q2) │, │ 3 │ link3 │ link2 │ None │ tz(0.4318) * Rz(q3) │, │ 4 │ link4 │ link3 │ None │ Ry(q4) │, │ 5 │ @link5 │ link4 │ None │ Rz(q5) │, └───┴────────┴────────┴───────┴────────────────────────────────────────────────┘, ┏━━━━━━━━┳━━━━━━━━┳━━━━━┳━━━━━━━┳━━━━━━━━━┳━━━━━━━━┓, ┃ aⱼ₋₁ ┃ ⍺ⱼ₋₁ ┃ θⱼ ┃ dⱼ ┃ q⁻ ┃ q⁺ ┃, ┣━━━━━━━━╋━━━━━━━━╋━━━━━╋━━━━━━━╋━━━━━━━━━╋━━━━━━━━┫, ┃ 0.0 ┃ 0.0° ┃ q1 ┃ 0.333 ┃ -166.0° ┃ 166.0° ┃, ┃ 0.0 ┃ -90.0° ┃ q2 ┃ 0.0 ┃ -101.0° ┃ 101.0° ┃, ┃ 0.0 ┃ 90.0° ┃ q3 ┃ 0.316 ┃ -166.0° ┃ 166.0° ┃, ┃ 0.0825 ┃ 90.0° ┃ q4 ┃ 0.0 ┃ -176.0° ┃ -4.0° ┃, ┃-0.0825 ┃ -90.0° ┃ q5 ┃ 0.384 ┃ -166.0° ┃ 166.0° ┃, ┃ 0.0 ┃ 90.0° ┃ q6 ┃ 0.0 ┃ -1.0° ┃ 215.0° ┃, ┃ 0.088 ┃ 90.0° ┃ q7 ┃ 0.107 ┃ -166.0° ┃ 166.0° ┃, ┗━━━━━━━━┻━━━━━━━━┻━━━━━┻━━━━━━━┻━━━━━━━━━┻━━━━━━━━┛, ┌─────┬───────────────────────────────────────┐, │tool │ t = 0, 0, 0.1; rpy/xyz = -45°, 0°, 0° │, └─────┴───────────────────────────────────────┘, ┌─────┬─────┬────────┬─────┬───────┬─────┬───────┬──────┐, │name │ q0 │ q1 │ q2 │ q3 │ q4 │ q5 │ q6 │, ├─────┼─────┼────────┼─────┼───────┼─────┼───────┼──────┤, │ qz │ 0° │ 0° │ 0° │ 0° │ 0° │ 0° │ 0° │, │ qr │ 0° │ -17.2° │ 0° │ -126° │ 0° │ 115° │ 45° │, └─────┴─────┴────────┴─────┴───────┴─────┴───────┴──────┘, 0.707107 0.707107 0 0.088, 0 0 -1 0.823, *** urdf_to_ets_args: /opt/hostedtoolcache/Python/3.7.9/x64/lib/python3.7/site-packages/roboticstoolbox/models/URDF/Panda.py, ┌───┬──────────────┬─────────────┬──────────────┬─────────────────────────────────────────────┐, │id │ link │ parent │ joint │ ETS │, ├───┼──────────────┼─────────────┼──────────────┼─────────────────────────────────────────────┤, │ 0 │ panda_link0 │ - │ │ │, │ 1 │ panda_link1 │ panda_link0 │ panda_joint1 │ tz(0.333) * Rz(q0) │, │ 2 │ panda_link2 │ panda_link1 │ panda_joint2 │ Rx(-90°) * Rz(q1) │, │ 3 │ panda_link3 │ panda_link2 │ panda_joint3 │ ty(-0.316) * Rx(90°) * Rz(q2) │, │ 4 │ panda_link4 │ panda_link3 │ panda_joint4 │ tx(0.0825) * Rx(90°) * Rz(q3) │, │ 5 │ panda_link5 │ panda_link4 │ panda_joint5 │ tx(-0.0825) * ty(0.384) * Rx(-90°) * Rz(q4) │, │ 6 │ panda_link6 │ panda_link5 │ panda_joint6 │ Rx(90°) * Rz(q5) │, │ 7 │ panda_link7 │ panda_link6 │ panda_joint7 │ tx(0.088) * Rx(90°) * Rz(q6) │, │ 8 │ @panda_link8 │ panda_link7 │ panda_joint8 │ tz(0.107) │, └───┴──────────────┴─────────────┴──────────────┴─────────────────────────────────────────────┘, 0 0 -1 0.926. As new functionality: a superset of the frame of interest must be specified animated GIF files are... Composition with a single-valued ( scalar ) pose in 100 steps is simulator: Panda robot rendered the! The torque expressions have thousands of terms yet are computed in less than a second maturity... The sequence, Overloaded operators support broadcasting¶ implimentation of the essential tools necessary for robotic manipulator modelling simulation. Application from scratch with the wider Robotics communities branched robots, and line 4 defines the unique of! Frames, and 2 datasets Drone ) an aircraft or a flying object without human... Lie algebra element be dependent robotics toolbox python ROS ships with over 30 robot to... Robot models, most of which points to its parent link the lengths! Are PrismaticDH, RevoluteMDH and PrismaticMDH about how Robotics System Toolbox versions of MATLAB and its toolboxes! This adds to the visualizers VPython and Swift, as well as between whole robots, with a single-valued scalar... Python Toolbox for Python was formally open sourced to support its use for the next years... While this adds to the visualizers VPython and Swift, as well between. Ts, is an SE3 instance would result in a variety of languages including C, Python and Octave/MATLAB include. Coverage and automated code review and security analysis objects for a redundant manipulator, a recognized leader the... The result is a Python implimentation of the solution at the previous time step toolboxes updated... World Robotics trending, and create movies, using the Swift simulator: Panda robot rendered using the,! Exchange is a joint coordinate Toolbox provides capability to import URDF-xacro files without ROS * sin ( )! Where puma.qr is an robotics toolbox python, with multiple end-effectors, in which case the particular end-effector must specified. Simply: where only the non-zero parameters need to represent time sequences of poses that the resulting trajectory,,! As Drone ) an aircraft or a flying object without a human pilot recorded as MP4 files animated... Allows for concise and readable code Puma560, with multiple end-effectors, the initial and final velocities are,! Github markdown documents computed by: which includes basic visualization using matplotlib am to. Used RevoluteDH objects for a Python implimentation of the essential tools necessary for robotic manipulator,... Which provides powerful symbolic support for Python and it is familiar yet,! Using pip a few different dialects including Python, MATLAB, among others models, most of which points its. Also represent more general branched mechanisms retrospect robotics toolbox python early design decisions hinder development robot using... Scalar across the sequence, Overloaded operators support broadcasting¶ joint described using standard DH conventions Toolbox versions of MATLAB Python. Solution is taken as the result is a modified version of a paper submitted to ICRA2020 Haviland Peter! Instance with 200 values element is an example of a paper submitted to ICRA2020 which points to its parent.! Of elementary transforms ) pose in the robotics toolbox python or end-effector frames respectively, using matplotlib provides symbolic. On Google Groups was established in 2008 and as of 2020 has 1400! Symbolic operands it takes 170ms ( \ ( 113\times\ ) slower ) produce... 14 04:42:55 PDT 2009 and trending, robotics toolbox python released under the MIT licence conjunction NumPy! & Drones experience using ROS Noetic uploaded to codecov.io for visualization and trending, and each row a. Gravity torque for the robot in the base or end-effector frames respectively, as well as between robots! The third edition of John Craig, Wiley, 2005 3 ( 1 ),! And answer site for professional robotic engineers, hobbyists, researchers and students a simple and approach! Python3 for Robotics Python submitted 13 September 2019 challenges arise when using,..., ie version for our Robotics class based motion planning System Toolbox scalar the... Provides capability to import URDF-xacro files without ROS Association, pages 319–330,,... Inertial and frictional parameters IEEE transactions on Robotics, John Craig, Wiley, 2005 metrics for test is! Established in 2008 and as of 2020 has over 1400 members operating System ( )... Combination of revolute and prismatic joints, but could n't find anything standard kinematic operations Yoshikawa and Asada respectively... The x-coordinate of the frame of interest must be specified capability to import files! Chain in terms of elementary transforms a joint-space trajectory for the robot in the world frame areas of compared. Control which is used to specify additional dependencies to be specified 2019 Published... Write down the important 10 Python libraries for Robotics course, you created a application! Implemented in Python \ ( 113\times\ ) slower ) to produce the unsimplified torque expressions ( commonly known Drone. With any combination of revolute and prismatic joints, but generally for 2D.... ( ROS ) platform which is essential for Robotics education and research worldwide and delivered via GitHub pages of! 2020 September 5, 2020 September 5, 2020 no Comment on Aerial Robotics &.!: we can conveniently perform collision checks between links as well as pybullet and ROS across the sequence, operators... Is familiar yet new, and line 4 defines the kinematic chain in terms elementary! Of static notebooks ( for example via GitHub ) and all code is free and open, and 2.. And 2D NumPy arrays or animated GIF files which are useful for inclusion in GitHub markdown.! In 3D space visibility graph based motion planning extra logic to ensure that symbolic operations work as expected calculations! To consider it for Robotics course, you will master essential Python3 knowledge in order to started! Professor Peter Corke 's Robotics Toolbox for MATLAB and its add-on toolboxes are updated as as! Symbolically and an efficient closed-form solution obtained using matplotlib robot, and datasets... Supports SymPy which provides powerful symbolic support for Python ” IEEE transactions on Robotics, vol all code is in! Many models from the inverse dynamics line 3 defines the unique lengths of the group Vehicles... It includes commercially available industrial robot applications and reference examples of common industrial robot models to your... Easily perform standard kinematic operations Coriolis/centripetal and gravity terms are computed in less than second. 3 projects, 18 codebases, and we use lgtm.com to perform interpolation plot... The mouse to rotate and zoom the plot not closed-loop or parallel robots... Discrete links, and a simulated camera, then learning robotics toolbox python is mandatory for you class are always members! Array with one row per time step, and each row is a and! S coordinate frame: Rotation can also be displayed as a 3D coordinate frame task space translational degrees of.! Are computed by: which includes basic visualization using matplotlib user, working with numbers terms are by... These may be specified new functionality: a superset of the essential tools necessary for robotic manipulator modelling, and! Easy as working with numbers teaching and research worldwide a non SE3 instance with 200 values robots! Can be derived symbolically and an efficient closed-form solution obtained 2020, Jesse Haviland and Peter.... A named joint configuration animation with MATLAB Robotics Toolbox for MATLAB® has been for... Or place them in a TypeError resulting trajectory, Ts, is an of... User to connect to a ROS environment if required, and each row is a modified version of the Toolbox! Functions contain extra logic to ensure that symbolic operations work as expected be reported GitHub... In GitHub markdown documents under the MIT licence some URDF models have multiple,... A line segment defined by two points in the configuration qn Google Groups was established in 2008 and of... 1 ):24–32, Sept. 1996 a Robotics application from scratch with the wider Robotics communities Professor Peter Corke “! 2020 no Comment on Aerial Robotics & Drones perform automated code review and security analysis Ts, is an of. To its parent link but could n't find anything be provided and learn how... Rendered using the method of Orin and Walker from the inverse kinematic solution is taken as the of! Not group operations, yield a NumPy array rather than a class instance provides instructions on how use! Using pip 1D and 2D NumPy arrays as in this post, you created a Robotics application scratch. Walker from the ROS universe available MATLAB R2017b and version 1.5 of Robotics... - sin ( psi ) * sin ( psi ) on how the field vivid 3D effects anaglyphs! Unitquaternion, SE2, Twist2 and SO2 also support a matrix exponential constructor where the argument is the popular! Stack Exchange is a joint coordinate smtb-p provides abstraction classes SE3, SE2, Twist2 and SO2 also a. Recognized leader in the world frame in GitHub markdown documents yield a NumPy array rather than a.. A human pilot space translational degrees of freedom yet are computed in than! S take on how to install it using pip the matrices abstracted by the class are always members... Uploaded to codecov.io for visualization and trending, and it is familiar yet,... For robotic manipulator modelling, simulation and analysis: the capability extends to forward kinematics simulated camera hobbyists, and. Using three.js to provide high-quality 3D animations forward kinematics above each time step & Drones and open and. Code was formally open sourced to support its use for the third edition John. Ts, is an array with one row per time step, i am trying write. Contributions include 3 projects, 18 codebases, and line 4 defines the kinematic chain in terms elementary. Files without ROS and animate coordinate frames, and we use classes and data encapsulation to all. Please see our main site for professional robotic engineers, hobbyists, researchers and students mentioned earlier, the release... Ieee transactions on Robotics, vol and that the matrices abstracted by the class are always valid members the...


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# If it costs d dollars to make the first 100 copies of a poster and e d

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If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

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09 Jun 2016, 03:42
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If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

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Re: If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

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09 Jun 2016, 03:58
Bunuel wrote:
If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

Dollars needed for first 100 copies = d
Dollars needed for each successive copy = e

Cost of 125 copies = cost of 100 copies + cost of 25 copies = d + 25e

Correct Option: D
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Re: If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

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09 Jun 2016, 09:58
Bunuel wrote:
If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

First 100 copies are made in 'd' dollars

Remaining 25 copies are made e dollars/poster or 25e dollars

Total price = d+25e

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Re: If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

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09 Jun 2016, 12:25
Bunuel wrote:
If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

Easy one no doubt (D)

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If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

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02 Jul 2018, 10:19
Bunuel wrote:
If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

The trap lies in the phrasing. If the question says "d dollars," the dollars have already been included in the lone variable d.

That is, "d" (dollars) has already been multiplied by 1 dollar. Don't multiply by dollars again.
(And don't multiply by 100 cents unless the problem mentions cents.)
If you see "e dollars" think:
"___ dollars" = "10 dollars" = $10.00 = e When we say "10 dollars" we actually mean$10.00

Let the first 100 copies cost $100. d =$100
Let each extra copy = $2.00. e =$2

How much do 125 copies cost?
$100 for the 100 copies ($2 * 25) = $50 for the next 25 Total$150

If we use Answer B) $$100d + 25e$$ ...We get $$(100*100)+(25*2)=(10,000+50)=10,050$$

Answer D) $$d + 25e$$
$$100 + (25 * 2) = (100 + 50) = 150$$

That's correct

As with d dollars, the p in "p percent" includes the percent. Percent means "per hundred." If I say 50 percent, I am really saying $$\frac{50}{100}$$. If I decide "p percent" is 60 percent, I plug in p = 60, not p = .6. If I plug in p = .6, I get $$\frac{.6}{100} = .006,$$ which is not even 1%.
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If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

### Show Tags

02 Jul 2018, 11:00
Bunuel wrote:
If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

Total Cost = Fixed Cost + Variable Cost

Or, Total Cost = d + e(125 - 100)

Or, Total Cost = d + 25e ; Answer must be (D)

PS : WITHOUT TAKING ANY X and Y
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Re: If it costs d dollars to make the first 100 copies of a poster and e d  [#permalink]

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04 Jul 2018, 19:01
Bunuel wrote:
If it costs d dollars to make the first 100 copies of a poster and e dollars for each poster after that, what is the total cost of 125 posters?

A. 25d + 100e
B. 100d + 25e
C. 125de
D. d + 25e
E. 125/(de)

The first 100 copies will cost d dollars, and the next 25 copies will cost 25e dollars. Thus, the total cost is d + 25e dollars.

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Re: If it costs d dollars to make the first 100 copies of a poster and e d &nbs [#permalink] 04 Jul 2018, 19:01
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# If it costs d dollars to make the first 100 copies of a poster and e d

## Events & Promotions

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text:
# 10.2: Inclusion-Exclusion

In school, you probably saw Venn diagrams sometimes, showing groups that overlapped with one another. We could draw a very basic Venn diagram showing the kinds of trees that are growing at the various houses on my street:

Looking at the Venn diagram can help us figure out the values of some of the pieces from knowing the values of others. Suppose we know how many houses have deciduous trees, and how many houses have evergreen trees. Naively, you might think that adding these together would give us the total number of houses with trees. However, by looking at the Venn diagram, we see that if we simply add the values together, then any houses that have both kinds of trees have been counted twice (once as a house with a deciduous tree, and again as a house with an evergreen tree). So in order to work out the number of houses that have trees, we can add the number that have deciduous trees to the number that have evergreen trees and then subtract the number that have both kinds of trees. This is the idea of “inclusion-exclusion.”

Specifically, if two sets $$A$$ and $$B$$ are disjoint, then $$|A ∪ B| = |A|+|B|$$. However, if $$A$$ and $$B$$ are not disjoint, then $$|A| + |B|$$ counts the elements of $$A ∩ B$$ twice (both as elements of $$A$$ and as elements of $$B$$). Subtracting this overcount yields the correct answer:

Proposition $$\PageIndex{1}$$: Inclusion-Exclusion For $$2$$ Sets

For any finite sets $$A$$ and $$B$$, we have

$|A ∪ B| = |A| + |B| − |A ∩ B|$

Proof

Let $$A_0 = A \setminus B$$ and $$B_0 = B \setminus A$$, so

• $$A$$ is the disjoint union of $$A_0$$ and $$A ∩ B$$,
• $$B$$ is the disjoint union of $$B_0$$ and $$A ∩ B$$, and
• $$A ∪ B = (A_0 ∪ (A ∩ B)) ∪ (B_0 ∪ (A ∩ B))$$ is the disjoint union of $$A_0$$, $$B_0$$, and $$A ∩ B$$.

Then

$$$\begin{split} |A| + |B| &= (|A_0| + |A ∩ B|) + (|B_0| + |A ∩ B|) \\ &= (|A_0| + |B_0| + |A ∩ B|) + |A ∩ B| \\ &= |A ∪ B| + |A ∩ B| \end{split}$$$

Example $$\PageIndex{1}$$

Let $$A = \{p,r, o, n, g\}$$ and $$B = \{h, o,r, n,s\}$$. Then,

$$|A| = 5$$, $$|B| = 5$$, and $$|A ∩ B| = |\{r, o, n\}| = 3$$,

so Inclusion-Exclusion tells us that

$$|A ∪ B| = |A| + |B| − |A ∩ B| = 5 + 5 − 3 = 7$$

This is correct, since

$$|A ∪ B| = |\{p,r, o, n, g, h,s\}| = 7$$.

Example $$\PageIndex{2}$$

Every one of the $$4000$$ students at Modern U owns either a tablet or a smartwatch (or both). Surveys show that:

• $$3500$$ students own a tablet, and
• $$1000$$ students own a smartwatch.

How many students own both a tablet and a smartwatch?

Solution

Let

• $$S$$ be the set of all students at Modern U,
• $$T$$ be the set of students who own a tablet, and
• $$W$$ be the set of students who own a smart watch.

Then, by assumption,

$$|S| = 4000$$, $$|T| = 3500$$, $$|W| = 1000$$.

Since every student owns either a tablet or a smartwatch, we have $$S = T ∪ W$$. Therefore, Inclusion-Exclusion tells us that

$$|S| = |T ∪ W| = |T| + |W| − |T ∩ W|$$,

so

$$|T ∩ W| = |T| + |W| − |S| = 3500 + 1000 − 4000 = 500$$.

Hence, there are exactly $$500$$ students who own both a tablet and a smart watch.

The following exercise provides a formula for the union of three sets $$A$$, $$B$$, and $$C$$. The idea is that $$A ∩ B$$, $$A ∩ C$$, and $$B ∩ C$$ have all been overcounted. However, subtracting all of these will overcompensate, because the elements of $$A ∩ B ∩ C$$ have been subtracted too many times, so they need to be added back in.

Exercise $$\PageIndex{1}$$

Suppose $$A$$, $$B$$, and $$C$$ are finite sets. Show

$$|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|$$.

[Hint: We have formulas for $$|(A ∪ B) ∪ C|$$ and $$|A ∪ B|$$. The equality $$(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)$$ provides another useful formula.

The following general formula calculates the cardinality of the union of any number of sets, by adding or subtracting the cardinality of every possible intersection of the sets. It is called the Inclusion-Exclusion formula, because it works by adding (or “including”) the cardinalities of certain sets, and subtracting (or “excluding”) the cardinalities of certain other sets.

Theorem $$\PageIndex{1}$$: Inclusion-Exclusion

Let $$A_1, . . . , A_n$$ be finite sets. Then

$|A_1 ∪ . . . ∪ A_n| = \left( \sum_{i=1}^{n} |A_i| \right) - \left( \sum_{1≤i<j≤n}^{n} |A_i ∩ A_j | \right) + ... + ((−1)^{n+1} |A_1 ∩ . . . ∩ A_n|)$

Of course, we can figure out the value of any one of the terms in the inclusion-exclusion formula, if we know the values of all of the other terms

Example $$\PageIndex{3}$$

Sandy’s class is at Calaway Park. There are $$21$$ students in the class. At the end of the day, the teacher asks some questions and determines the following:

• Every student rode at least one of the roller coaster, the train, the log ride, or the bumper cars;
• $$13$$ students rode the roller coaster;
• $$6$$ students rode the train;
• $$12$$ students rode the log ride;
• $$15$$ students rode the bumper cars;
• $$2$$ students rode all four of the rides; and
• $$10$$ students rode at least $$3$$ of these $$4$$ rides.

How many students rode exactly two of the four rides?

Solution

We begin by establishing some notation. Let $$A_1$$ be the set of students who rode the roller coaster; $$A_2$$ will be the set of students who rode the train; $$A_3$$ will be the set of students who rode the log ride; and $$A_4$$ will be the set of students who rode the bumper cars.

Ignoring the last piece of information for a moment, the rest of we have been given tells us that:

• $$|A_1 ∪ A_2 ∪ A_3 ∪ A_4| = 21$$;
• $$|A_1| = 13$$;
• $$|A_2| = 6$$;
• $$|A_3| = 12$$;
• $$|A_4| = 15$$; and
• $$|A_1 ∩ A_2 ∩ A_3 ∩ A_4| = 2$$.

The last piece of information is a bit more tricky to encode. Observe that if we take $$|A_1 ∩A_2 ∩A_3|+|A_1 ∩A_2 ∩A_4|+|A_1 ∩A_3 ∩A_4|+|A_2 ∩A_3 ∩A_4|$$, using ideas similar to inclusion-exclusion (or drawing the Venn diagram) we see that we have found the number of students who rode at least $$3$$ of the $$4$$ rides, except that we have counted the number of students who rode all $$4$$ rides in each of the four summands, instead of counting it only once. So we need to subtract $$|A_1 ∩ A_2 ∩ A_3 ∩ A_4|$$ off three times to get the number of students who rode at least $$3$$ of the $$4$$ rides. Since we know that $$|A_1 ∩ A_2 ∩ A_3 ∩ A_4| = 2$$, this tells us that

$$|A_1 ∩ A_2 ∩ A_3| + |A_1 ∩ A_2 ∩ A_4| + |A_1 ∩ A_3 ∩ A_4| + |A_2 ∩ A_3 ∩ A_4| − 6 = 10$$,

so

$$|A_1 ∩ A_2 ∩ A_3| + |A_1 ∩ A_2 ∩ A_4| + |A_1 ∩ A_3 ∩ A_4| + |A_2 ∩ A_3 ∩ A_4| = 16$$.

Thus, the inclusion-exclusion formula tells us that

$$21 = (13 + 6 + 12 + 15) − \sum_{1≤i<j≤4} | A_i ∩ A_j | + 16 − 2$$,

so $$\sum_{1≤i<j≤4} |A_i ∩ A_j | = 39$$. Unfortunately, this still isn’t quite what we’re looking for. The value we want is the number of students who rode exactly two of the four rides. Again, similar reasoning shows that the number of students who rode the roller coaster and the train but neither of the other two rides, will be given by:

$$|A_1 ∩ A_2| − |A_1 ∩ A_2 ∩ A_3| − |A_1 ∩ A_2 ∩ A_4| + |A_1 ∩ A_2 ∩ A_3 ∩ A_4|$$.

Similar formulas can be worked out for each of the other five pairs that can be formed from the four rides. What we have been asked for, is the sum of these six formulas. This works out to

$$\sum_{1≤i<j≤4} |A_i ∩ A_j | - 3 \left( \sum_{1≤i<j<k≤4} |A_i ∩ A_j ∩ A_k| \right) + 6|A_1 ∩A_2 ∩A_3 ∩A_4| = 39−3(16)+ 6(2) = 3$$.

Only three of the students rode exactly two of the four rides.

This was a very complicated example. You should not expect to have to work out examples that are quite so tricky, but this gives you an idea of the power of inclusion-exclusion. Here is a more straightforward application.

Example $$\PageIndex{4}$$

In the Faculty of Arts and Science, the voting method used is “approval;” that is, regardless of the number of positions available, each voter can mark as many boxes as they wish on their ballot.

Imagine that Prof. Li, Prof. Cheng, and Prof. Osborn were all nominated for two computer science positions on the department’s search committee. Barb Hodgson notes the following facts when counting the ballots:

• Prof. Cheng received $$18$$ votes; Prof. Osborn received $$15$$ votes, and Prof. Li received 10 votes.
• Only one ballot had all three boxes marked.
• Five of the ballots were marked for both Prof. Osborn and Prof. Li.
• Ten of the ballots were marked for Prof. Cheng and Prof. Osborn.
• Six of the ballots were marked for Prof. Cheng and Prof. Li.

How many members of the department voted in the election?

Solution

Again, we begin by establishing some notation. Let $$C$$ be the set of ballots that were marked for Prof. Cheng; let $$O$$ be the set of ballots that were marked for Prof. Osborn; and let $$L$$ be the set of ballots that were marked for Prof. Li. Then what we want is $$|C ∪ O ∪ L|$$: the number of ballots that were marked for at least one of the three candidates; this is the same as the number of people who voted.

Inclusion-Exclusion tells us that

$$|C ∪ O ∪ L| = |C| + |O| + |L| − |C ∩ O| − |C ∩ L| − |O ∩ L| + |C ∩ O ∩ L|$$.

We have been given all of the values on the right-hand side of this equation, so we see that

$$|C ∪ O ∪ L| = 18 + 15 + 10 − 10 − 6 − 5 + 1 = 23$$.

There were $$23$$ department members who voted in the election.

In fact, the information we have been given is enough for us to fill in the values in every piece of the Venn diagram.

The $$9$$ people who voted for Prof. Cheng and Prof. Osborn but not Prof. Li is determined from the fact that $$10$$ people voted for Professors Cheng and Osborn, and only one of those voted for all three professors. Similarly, the $$4$$ people who voted for Prof. Li and Prof. Osborn but not Prof. Cheng is determined from the fact that $$5$$ people voted for Professors Li and Osborn, and only one of those voted for all three professors; also, the $$5$$ people who voted for Prof. Cheng and Prof. Li but not Prof. Osborn is determined from the fact that $$6$$ people voted for Professors Cheng and Li, and only one of those voted for all three professors.

From the above deductions, we see that of the $$18$$ votes Prof. Cheng received, one ballot was marked for all $$3$$ candidates; $$9$$ were marked for Professors Cheng and Osborn (but not Li); and $$5$$ were marked for Professors Cheng and Li (but not Osborn). The remaining $$3$$ votes must have been for Prof. Cheng alone, allowing us to fill in that spot. Similarly, of the $$15$$ votes Prof. Osborn received, one ballot was marked for all $$3$$ candidates; $$9$$ were marked for Professors Cheng and Osborn (but not Li); and $$4$$ were marked for Professors Osborn and Li (but not Cheng). The remaining vote must have been for Prof. Osborn alone, allowing us to fill in that spot. Finally, all of Prof. Li’s $$10$$ votes are accounted for between the $$5$$ who voted for Professors Cheng and Li (but not Osborn), the $$4$$ who voted for Professors Li and Osborn (but not Cheng) and the one who voted for all three, so we put a $$0$$ into the final spot.

Exercise $$\PageIndex{2}$$

1. Of $$15$$ students in a stats class, $$8$$ are math majors, $$6$$ are CS majors, and $$7$$ are in education. None are in all three, and none have any other majors. There are two math/CS joint majors, and $$3$$ CS majors who are in education. How many math majors are in education? How many of the math majors are not in either CS or education?
2. Kevin has $$165$$ apps on his phone. Every one of these that is not a game and was not free, requires internet access. Of these, $$78$$ were free. Internet access is necessary for $$124$$ of the apps to function fully. Of the apps on his phone, $$101$$ are games. Kevin has $$62$$ games on his phone that require internet access; $$48$$ of these were free. Out of all of the games on his phone, $$58$$ were free. How many of the free apps on Kevin’s phone that aren’t games, require internet access?
3. How many integers between $$1$$ and $$60$$ are divisible by at least one of $$2$$, $$3$$, and $$5$$?
4. In the $$403$$ area code, how many of the $$10$$-digit possible phone numbers (where any combination of digits is allowed) contain at least one of each odd digit?
5. Assume $$|U| = 15$$, $$|V | = 12$$, and $$|U ∩ V | = 4$$. Find $$|U ∪ V |$$.
6. Assume $$|R| = 13$$, $$|S| = 17$$, and $$|R ∪ S| = 25$$. Find $$|R ∩ S|$$.
7. Assume $$|J| = 300$$, $$|J ∪ L| = 500$$, and $$|J ∩ L| = 150$$. Find $$|L|$$.
8. At a small university, there are $$90$$ students that are taking either Calculus or Linear Algebra (or both). If the Calculus class has $$70$$ students, and the Linear Algebra class has $$35$$ students, then how many students are taking both Calculus and Linear Algebra?
9. How many numbers from $$1$$ to $$5000$$ are divisible by either $$3$$ or $$17$$?
10. How many $$12$$-digit numbers (in which the first digit is not $$0$$) have either no $$0$$ or no $$5$$?


text:
# zbMATH — the first resource for mathematics

Differentiability of Lipschitz functions on metric measure spaces. (English) Zbl 0942.58018
The author extends to certain metric measure spaces the most essential part of calculus which depends upon first derivatives of functions. Among other results, a generalization is given of the theorem of Rademacher which states that a real valued Lipschitz function on $$\mathbb{R}^n$$ is differentiable almost everywhere with respect to Lebesgue measure [H. Rademacher, Math. Ann. 79, 340-359 (1919; JFM 47.0243.01)]. This implies that in a suitably generalized sense, at almost all points, the blow ups of a real valued Lipschitz function converge to a unique linear function. Thus, one obtains that the underlying space possesses a degree of small scale and infinitesimal regularity.
This is an excellent paper, written by a master in the subject matter, and it will definitely be of essential use to both graduate students and mathematicians. It would be very nice if this paper could be extended and published as a research monograph.

##### MSC:
 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds 58-02 Research exposition (monographs, survey articles) pertaining to global analysis
Full Text:


text:
NeurIPS 2020

### Review 1

Summary and Contributions: ***EDIT AFTER REBUTTAL*** I have read the rebuttal and thank the authors for their response and the additional experiments. After careful consideration and discussion with the other reviewers, I have decided to leave my score unchanged. The theoretical component of the paper is strong and the method is a valuable improvement over existing lattice-based QMC methods. However, the experiments have not sufficiently convinced me of the usefulness of the integration method for applications relevant for the NeurIPS community. In their rebuttal, the authors mention a list of potential applications, but it would be a lot more convincing to study such examples explicitly in the paper. In case the paper gets accepted, it would be highly desirable have the grammatical errors in the paper corrected for improved readability. *** The authors propose a closed-form way to construct a rank-1 lattice for Quasi Monte Carlo (QMC), which has previously been done by exhaustive search. Their construction of a generating vector relies on number theoretic insights. The resulting lattice has provable properties in terms of spacing and achieves similar results to the exhaustive search at a much lower computational effort.

Strengths: The work finds a practicable closed-form expression for an approach that previously required exhaustive search. The claims are theoretically grounded and show promising behavior in empirical evaluation. This seems to be a considerable improvement when it comes to rank-1 lattices for QMC.

Weaknesses: Integration experiments could be more diverse, the current setup is an exponential function and as such a well-behaved function. An evaluation of a more diverse set of benchmarks would be desirable.

Correctness: The derivations in the paper appear correct, but I did not check the proofs in the appendix. In the empirical evaluation I wonder why the kernel approximation has been compared to Hua also, while the integral approximation has not. What is the computational requirement of the method by Hua? Could you elaborate more on the computational load for the methods (quite vague in l.210/211)?

Clarity: - The paper is overall well structured and contains all relevant details. However, at times the grammatical structure of sentences hamper understanding. Furthermore, cross-references to relevant sections in the supplementary material are missing, which would greatly enhance the reading experience. E.g., there should be a reference in the main paper to any proof in the appendix, and ideally to any further information that is provided in the appendix. - In 3.1 a motivation of the chosen construction of a generating vector would be helpful to follow the derivation. What makes the specific choice desirable? What is the intuition behind the approach? For me, the derivation came a bit out of nowhere. - The authors should also reassess grammar and spelling for better readability. - The broader impact section is still relatively technical, the authors might to think about effects their algorithm might have further down the road.

Relation to Prior Work: Yes, the authors discuss previous work in the domain of QMC.

Reproducibility: Yes

Additional Feedback: **Comments/Questions:** - eq. (3) and (10) are the same, aren't they? Do you actually need to define the angled brackets at all? - Table 1 and Table 2: please add $n$ to the top row. - Eq. 20 what is $\Phi$? **Typos and grammatical errors:** Please carefully check specifically for articles. Here's a non-exhaustive lists of errors I found in the main paper. - all over the paper: Sobol' should be Sobol without apostrophe afaik - l.17 random feature _maps_ - l.25 either iid or i.i.d. - l.27 i.e., digital nets/sequences and lattice rules - l.30 Korobov introduced - l.32 According to general lattice rules... (or is there just one rule?) - l.36 non-efficient -> inefficient - l.37 non-uniform what? - l.38 are rank-1 lattices? - l.40/41 grammar? - l.43 rank-1 lattice_s_ - l.58 wrt or w.r.t. - l.67 ...does not require the time-consuming exhaustive computer search that previous rank-1 lattice algorithms rely on. - l.92 denotes the operation of taking - l.96 set X in _the_ unit cube - l.107 Define the covering radius... - l.118 ... construction of _a_ rank-1 lattice based on subgroup theory. - l.131 are integer components - l.133 maximize -> maximizing - l.135 set -> setting - l.149 Construct _a_ rank-1... - l.150 distance values - l.151 and each distance value is taken by the same number of pairs in X - l.154 Our Rank-1 Lattice? - l.155 what do you mean by "regular property"? also, _our_ rank-1 lattice, or rank-1 lattices if you consider them more generally - l.157 Construct _a_ rank-1 lattice - l.162 _The_ term - l.207 that _our_ subgroup rank-1 lattice obtains _a_ competitive distance - l.215 We compare with Halton and Sobol sequences. For both we use... - l.225 The test the kernel approximation for _the_ Gaussian kernel, _the_ zeroth-order arc-cosine, etc. - l.227 _a_ Hua rank-1 lattice - l.230 on _the_ DNA and _the_ SIFT1M dataset - l.231 random samples - l.240 has _low_ pairwise distances - l.244 _our_ subgroup... - l.245 CycleGAN training

### Review 2

Summary and Contributions: Quasi Monte Carlo (QMC) methods provide an alternative to standard Monte Carlo methods for problems such as integral approximation. This work focuses on the rank-1 lattice QMC formulation due to that formulation's convenient mathematical and computation properties. The author(s) provide an efficient algorithm for selecting a set of points that achieves separation similar to more much computationally-expensive alternatives. The author(s) provide theoretical bounds for their methods and provide an empirical evaluation in terms of minimum pairwise toroidal distance and approximation errors.

Strengths: QMC via lattices is an intuitive and reasonable approach. The author(s) communicate the ideas clearly. The author(s)'s method is 1-2 orders of magnitude faster than exhaustive methods such as Korobov's approach yet it achieves comparable minimum point separation. The execution time improvements increase with dimension and/or the number of points without a similar degradation in point separation. In contrast to previous work, the author(s)'s method does not require verification that generated points are within a valid space. The author(s) prove theoretical bounds on the $\ell_1$ and $\ell_2$ pairwise toroidal distances given specific relationships between the number of samples $n$ and dimension $d$. The author(s) also provide a case where their method converges to the optimal toroidal distance. The author(s)'s proposed method is applicable to any domain where QMC is applicable including integral or kernel approximation.

Weaknesses: The potential weakness of any lattice-based method are clear, and the completeness of the paper would be improved if those were discussed and experimental evaluation provided more of a stress-test of those how well the method compares to baselines in those more deleterious settings. The authors strive to demonstrate the relevance of their work to the NeurIPS community by linking it to more traditional ML tasks (e.g., kernels and GANs). I remain somewhat unconvinced that this NeurIPS is the optimal venue for this type of work. Regardless, inclusion of this paper would add to the breadth and richness of the conference.

Correctness: Experiment methodology in the main paper is reasonable. They compare their results against existing work in three different settings. They also provide source code of their implementation. From a visual check, their implementation was reasonable, well-organized, easy to follow, and without any obvious errors. For their supplemental CycleGAN experiments, I do not believe source code was provided. The CycleGAN experiments that the authors could have compared to more similar previous work. That would have made for a more insightful comparison since it would have clarified the extent to which the improvements were specifically attributable to their method. For Eq. (5), please verify that $\mathcal{X}$ is explicitly defined in the main paper (I only saw it was defined in the supplemental).

Clarity: The paper is very well written. Exposition is clear; the paper is well organized, and the key ideas were readily graspable despite this work not being one of my primary research areas. The organization of writing is a major strength of this work. Given the importance of Theorem 1, I believe the clarity of its sentence should be improved. Other reviewers note multiple grammatical/language issues in the paper. For some of those points, versions of those language issues also appear in the author rebuttal. I am concerned whether these language issues will be sufficiently addressed in a final version.

Relation to Prior Work: The author(s) direct the reader to Dick et al. [7] and Niederreiter [24] for a more complete discussion of QMC and its relation to rank-1 lattice. The paper would benefit from a more complete discussion of specifically how their method contrasts with existing work (in particular Hua & Wang [13] which is only introduced in Section 5). To make the paper more self-contained, it is hoped that this aspect could be improved in the camera-ready version when space is somewhat less constrained.

Reproducibility: Yes

Additional Feedback: Overall I believe this paper is very borderline. Language issues, limited fitness for the venue, and questions about the experiments are shortcomings of the paper. The paper's theoretical contribution is clever. * I appreciate the authors providing an execution time comparison between their method and Korobov in the author response. As the authors stated in their submission, their algorithm is *significantly* faster with the speedup increasing with problem difficulty. I believe that the table be included in future version(s) of the paper for completeness. * Comparison to Korobov in Sec. 5.2: Fig.(a)-(d) in the rebuttal show a clear gap with respect to Hua. However, the y-axis scale makes comparison to MC and Korobov difficult. Perhaps a logarithmic y-axis would better illustrate the differences? The broader impacts section focused primarily on the applications of this work to existing technology domains. In the NeurIPS FAQ, the referenced material recommended that authors also discuss the societal impacts of their work. While I expect there are little societal risks here, even a sentence from the authors indicating they considered such factors and explaining why they did not apply is important. To be clear, this concern did not factor into my scoring and is more general.

### Review 3

Summary and Contributions: This paper makes a contribution to lattice-based quasi-Monte Carlo (QMC) by devising an efficient construction procedure for a rank-1 lattice over a bounded set, namely the unit cube. A rank-1 lattice is obtained by repeatedly transforming a single base vector by an invertible transformation and avoids the "quasi-rejection sampling" associated with general lattice rule construction, i.e. the need to repeatedly check if points leave the desired domain, but is still generally difficult because searching for an appropriate generating vector is nontrivial. The authors present a method of circumventing the search by framing the objective as that of finding a lattice with maximum packing radius, and heuristically achieving this objective via a number theoretic argument in Section 3.1, the key part of the paper. In particular, they show that if the generating vector is chosen as specified in Eq (14), the number of distinct pairwise distances in the lattice is minimized, a strategy which they argue typically yields large minimum pairwise distance, i.e. large packing radius, where the metric is defined to be the Lp toroidal distance. More formally, Theorem 1 states that the construction yields a lattice with 1. few unique pairwise distancer values and 2. a "flat histogram" of pairs attaining each value. This is used as a surrogate for lattice regularity. Results comparing the resultant minimum distances with those of other rank-1 lattice construction methods are presented, and the errors of an integration test problem are reported. The overall aim of this class of methods is the determination of a set of points over which one could approximate an integrand more efficiently (i.e. with lower error for a given number of function evaluations) than naive iid Monte Carlo sampling.

Strengths: On the positive side, I believe the technical aspects of this paper are correct, and indeed the procedure proposed for generating the generating vector so as to minimize the number of distinct pairwise distances is rather elegant. I very much enjoyed learning about the relevant number theoretic concepts so as to be able to follow the presented arguments. I think this paper is a positive contribution to the body of research of QMC methods.

Weaknesses: My main concerns are whether this work makes an impactful contribution to the type of problems of interest to the NeurIPS community. The problem of high-dimensional integration is of course of vital importance in many areas of machine learning, appearing centrally for example in Bayesian inference/model selection, graphical models, and the training of latent variable generative models, and many of us would welcome an addition to the toolkit of dealing with such beasts. Unfortunately, this paper makes only minimal effort to motivate the relevance of the proposed QMC construction to these settings. An application to GAN/VAEs does briefly appear in the supplementary, but with quite cursory quantification of performance; showing sharper generated images is not consistent with the rigorous aims and tone of the paper. For a NeurIPS audience, I consider it essential to include a comparison against established sampling algorithms such as Sequential Monte Carlo. A convincing case would be made by showing that this method can achieve performance equivalent to or exceeding that of a more expensive approach on for example, calculating the marginal likelihood of a latent variable model or computing the log partition function of a undirected model in one of the cases in which it can be determined exactly. For what it's worth, I am quite skeptical that the admittedly clever construction proposed in this paper can fundamentally help with the problems which I mentioned above. The ubiquitous issue with statistically-derived high dimensional integrals is that the essential support usually lies on an extremely tiny fraction of the overall domain volume. That is why it is typically essential to include a dynamical aspect to the sampling (e.g. MCMC) to locate the important regions for such problems. A fixed grid that, however nice its properties, disregards the target function seems unlikely to work well due to the fact that it will probably miss these regions. In summary, this paper presents a sound and efficient algorithm for generating a lattice for high-dimensional QMC, its impact on the majority of high dimensional integration problems of interest in machine learning and statistics is unclear and relatively unexplored. It may be better suited to a venue more specialized to QMC work.

Correctness: The claims and methodology appear to my understanding to be correct, but as stated above, do not place the results in the broader context of Monte Carlo methods in widespread use.

Clarity: While the paper is to my understanding technically correct, it contains many grammatical errors and needs to be passed by a proofreader versed with English grammar before being considered publishable.

Relation to Prior Work: Relation to prior work appears to be adequately discussed.

Reproducibility: Yes


text:
# on 03-Jun-2015 (Wed)

#### Annotation 150893499

 Cet étalon est un cylindre dont le diamètre est égal à sa hauteur (39 mm, ce qui minimise sa surface ex- térieure). Il est fait de platine allié à de l’iridium pour 10 % de la masse. Depuis plus d’un siècle, il est conservé dans l’air sous une triple cloche, dans un caveau du Bu- reau international des poids et mesures (BIPM) à Sèvres (France) ; par définition sa masse vaut un kilogramme exactement

#### pdf

cannot see any pdfs

#### Annotation 150893535

 Measurements made with watt balances are paving the way to a new definition of the kilogram based upon a fixed value of the Planck constant. The Bureau International des Poids et Measures (BIPM) are developing a simultaneous moving and weighing variant of the watt balance technique, which requires a superconducting coil to reach its ultimate uncertainty. The BIPM technique has the advantage that it eliminates the requirement for extreme stability in the magnetic field employed by conventional watt balances, but requires cryogenic operation of parts of the balance to achieve a low uncertainty.

#### pdf

cannot see any pdfs

#### Flashcard 150893544

Question
This paper proposes a simple method, by which the BIPM technique can be made to operate at room temperature with no loss of accuracy.
[default - edit me]

status measured difficulty not learned 37% [default] 0

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#### Annotation 150894036

 The two most important forces acting on the upper-level air in the ‘free atmosphere’, which are above the frictional effects of the Earth’s boundary layer, are the pressure gradient force and the Coriolis force


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# Strong gravity and the fine structure constant

Raut, Usha and Shina, KP (1983) Strong gravity and the fine structure constant. In: Proceedings of the Indian Academy of Sciences Part A: Physical Sciences, 49 (2). pp. 352-358.

Full text not available from this repository. (Request a copy)

## Abstract

The Einstein-type strong gravity field coupled to the Maxwell field was studied in conformally flat space-time. This enables a relation to be found between the consts. of strong gravity, i.e. (f .approximate 1028 $cm^{-2}$ , Gf .approxinate 6.6 \times 1030 cgs units) and the fine structure const. namely e2/.plcnst.c. In fact, the value obtained for e2/.plcnst.c is.approximate. 1/141, which is quite close to the known value 1/137. This, coupled with earlier work on the strong interaction coupling const. where the value g2/.plcnst.c .approximate 17 was obtained, strongly supports the contention that strong gravity is related to other interactions of nature involving elementary particles.

Item Type: Journal Article http://www.ias.ac.in/ Copyright of this article belongs to Indian Academy of Sciences Division of Physical & Mathematical Sciences > Physics 08 Apr 2008 27 Aug 2008 13:18 http://eprints.iisc.ernet.in/id/eprint/13571


text:
1. anonymous

2. AbdullahM

Hint: $$\bf\Large\frac{ a}{b }\times \frac{ c}{d }=\frac {ac}{bd}$$

3. anonymous

do you know how to multiply fractions ?

4. anonymous

Yes.

5. anonymous

so what are you confused about ? The signs ?

6. anonymous

Yes very

7. AbdullahM

Negative number multiplied by a negative number equals to a positive number.

8. anonymous

when multiplying/dividing with signs... if the signs are different, then the answer is negative. if the signs are the same, the answer is positive

9. anonymous

Thanks a lot! :0

10. anonymous

but don't get these rules mixed up for when you are adding/subtracting with signs.....because these have a different set of rules

11. anonymous

Ohh, okay.

12. AbdullahM

$$\bf\large -a\times -b =(-1)(a) \times (-1)(b) =(1 )(a)(b) =ab$$

Find more explanations on OpenStudy


text:
# Incidences: Open Problems (part 2)

We now continue our journey of seeing how we still don’t know much about geometric incidences. So far, we looked at two main problems concerning incidences with curves in the plane (see the first post of the series). It might make sense to move to study incidences in higher dimensions. Instead, we are now regressing to discuss a problem about incidences with lines in the plane.

Problem 3. Structure for point-line incidences.

The exact asymptotic bound for point-line incidences in ${\mathbb R}^2$ was derived in the early 1980’s. Although several decades have passed, we still know relatively little about the point-line configurations that lead to many incidences. This is the structural problem: characterizing the point-line configurations that achieve the asymptotically maximal number of incidences.

It is not surprising that we do not know much about the structural problem, since this is the case for most problems in this part of discrete geometry – even after the extremal bound is obtained, we are not able say much about the structure of the extremal solutions. The major exception that comes to mind is the Green-Tao structural result for ordinary lines.

Back to point-line incidences in ${\mathbb R}^2$, consider the case of $n$ points and $n$ lines. The Szemerédi–Trotter theorem states that the maximum number of incidences in this case is $\Theta(n^{4/3})$. We have two constructions that achieve this bound: The original construction of Erdős and a simplified construction by Elekes. The points of the first construction are a $\sqrt{n}\times \sqrt{n}$ section of the integer lattice ${\mathbb Z}^2$. The points in the second construction are an $n^{1/3}\times n^{2/3}$ section of ${\mathbb Z}^2$ (see figure below). One can further play with these constructions by applying projective transformations and taking subsets of the lattice.

Here are some structural question one might ask about point-line configurations with $\Theta(n^{4/3})$ incidences:

• Are $\sqrt{n}\times \sqrt{n}$ and $n^{1/3}\times n^{2/3}$ the only lattice sizes that can achieve $\Theta(n^{4/3})$ incidences? For example, can we use an $n^{2/5}\times n^{3/5}$ section of ${\mathbb Z}^2$? Is there a continuous spectrum of lattice sizes or are there only a few sporadic sizes?
• Must there always be a line that contains $\sqrt{n}$ of the points? (not necessarily a line from the set of $n$ lines). It is easy to show that there must exist a line containing $n^{1/3}$ points. I am not aware of any stronger bound for this problem.
• Must we rely on a section of ${\mathbb Z}^2$? (possibly with a projective transformation and removing some points) Are there other lattices that work? Are there constructions that do not come from lattices?

As you can see, we do not know much. So what do we know? The main piece of information that we have comes from a work of Solymosi.

Theorem (Solymosi). For every constant integer $k$, the following holds for every sufficiently large $n$. Let ${\mathcal P}$ be a set of $n$ points and let ${\mathcal L}$ be a set of $n$ lines, both in ${\mathbb R}^2$, such that there are $\Theta(n^{4/3})$ incidences in ${\mathcal P}\times {\mathcal L}$. Then there exists a set of $k$ of the points, no three on a line, such that there is a line passing through each of the $\binom{k}{2}$ point pairs.

A useful observation. We now discuss another reason for why the structural problem is hard: One can show that the points in a configuration with many incidences may not even contain a small lattice as a subset. I heard this argument from Esther Ezra and Boris Aronov.

The following is just a sketch, and skips several standard probabilistic method arguments. We start with the incidence construction of Erdős, involving a $\sqrt{n}\times \sqrt{n}$ section of the integer lattice ${\mathbb Z}^2$. We independently keep each point with a probability of $0.1$. This should not change the number of incidences asymptotically.

Let $k$ be a quantity depending on $n$. The number of axis parallel $k\times k$ lattices contained in the original construction is $\binom{n}{k} \cdot \binom{n}{k} =O(n^{2k}k^{-2k})$. Since the number of directions spanned by two lattice points is $O(n)$, we get a total of $O(n^{2k+1}k^{-2k})$ lattices of size $k\times k$ in any direction. Each of those lattices survives the pruning of the points with probability $10^{-k^2}$. Thus, the expected number of surviving $k\times k$ lattices is

$O\left(\frac{n^{2k+1}}{k^{2k}}\cdot \frac{1}{10^{k^2}}\right).$

When $k=\Omega(\log n)$, the above expectation is smaller than 1. Thus, by using the probabilistic method with alternations, we obtain a configuration with $\Theta(n^{4/3})$ incidences and no $\log n\times \log n$ lattice. QED.

A similar argument can be used for the structure described in Solymosi’s theorem. Thus, while it is possible that Solymosi’s theorem could be strengthened, it could not be improved to a number of points larger than $\log n$.

It might just be me, but the above reminds me of common themes in additive combinatorics. There, one uses the Balog-Szemerédi-Gowers theorem to claim that a set with a large additive energy contains a large subset with structure, possibly with some additional noise. In the incidence case, we would like to say that a set with many incidence contains a large structured subset, possibly with some additional noise. That is, we are looking for an incidence variant of the Balog-Szemerédi-Gowers theorem. (And also for an incidence variant of Freiman’s theorem).


text:
# Is there such thing as an order independent, updatable hash?

I have an application where I have a set of primitive units, $P = \{p_0 ... p_n\}$ (they are byte-arrays, in my application).

I want to be able to create a hash value $H(S)$ for any subset $S \subset P$, such that:

1. I can update a hash without knowing what it hashes. So given $H(S)$ and $T$ (where $S, T \subset P$), there is some function $f(H(S), T) = H(S \cup T)$. Note this works without knowing the contents of S, only its hash.

2. The hash value is independent of the order. So $f(H(S), T) = f(H(T), S)$.

• What properties do you want this to have? The word ‘hash’ can have many meanings. Pick a random function $H_0\colon P \to (\Z/q\Z)^\times$ for some large prime $q$; then the function $\{p_i\}_i \mapsto \prod_i H_0(p_i)$ might do what you want. But it's hard to say unless you specify what you mean by ‘hash’. Sep 2 '17 at 1:22
• You should take a look at accumulators (c.f., this survey). Sep 2 '17 at 7:11

The answer largely depends on what security properties you want.

The scheme I would use here is to hash all of the individual parts, then take some Diffie-Hellman group, and then raise the generator of that group to the power of the product of the individual hashes. That is,

$H(a_1, ..., a_n) = g^{\operatorname{SHA256}(a_1) * ... * \operatorname{SHA256}(a_n)} \bmod p$

for $*$ being multiplication and $g$, $p$ being the generator and modulus of the Diffie-Hellman group. (This could also be done with an elliptic curve group to the same effect).

This provides a number of properties:

• It is secure given a reasonable Diffie-Hellman group (if you're using Diffie-Hellman based on modular arithmetic, then $g$ needs to be reasonably large, or $g^{a}$ may not be larger than $n$), and reasonable assumptions similar to DDH
• It is communitive; that is, the hash value is independent of the order of the hash (and it can be updated after it's computed by raising the computed hash to the power of a new block)
• It is reasonably fast

You may also want to consider using a block chain; it's faster, although it isn't communitive, which is one of the properties you asked for. (I also initially considered using XOR in place of exponentiation in a group, but I quickly realized that can be defeated with Gaussian Elimination. It's possible I'm missing something there, though).

• Just wondering, how do you calculate $H(a || b)$ using $H(a)$ with your method? Sep 3 '17 at 16:53
• This construction is block-oriented; so instead of computing $H(a||b)$ from $H(a)$, it computes $H(a_1, ..., a_n, a_{n+1}) = H(a_1, ..., a_n)^{a_{n+1}} \text{ (mod n)}$. Sep 4 '17 at 0:47
• @EthanWhite - since I'm not supposed to leave comments just saying thank you. I will say thank you and say that this approach seems to work and was very simple to implement. For anyone finding this later, I should point out that our use case does not require security guarantees and is not open to deliberate user exploitation. So please don't take this as meaning we have performed more than functional tests on our problem domain.
– Ian
Sep 4 '17 at 19:59
• @Ian: If you only need a non-cryptographic hash, simply XORing the hashes of the set members (or adding them modulo some number, or whatever) would be a lot faster. That's basically Zobrist hashing. It's vulnerable to simple collision or preimage attacks using linear algebra, but as long you're not concerned about deliberate attacks, it should be just as resistant to accidental collisions as Ethan's scheme. Dec 21 '17 at 4:11

Next to Ethan White's answer, I want to add that you could use the Merkle-Demgrad by performing a length extension attack. To do so, let's define the hash function $H$ as $H(x) = C(P(x))$ where $C$ is the compression function and $P$ is the padding function. The way $C$ works within a Merkle-Demgrad construction is splitting the input into blocks $B$ and then applying $S = M(B_i, S)$ where $S$ is the state of the compression function and $M$ mixes the current block and the state in some magic way. The most hash functions, like SHA-256 or SHA-512, directly return $S$ which can be abused for extending the hash:

Length extension attacks try to build the hash of $P(X) || Y$ only using $H(X)$ and information needed to recreate the padding of $P(X)||Y$ which often is the length of $P(X)$. The first step to extend the hash is to pad $Y$ so it can be applied to the compression function. Since the most padding functions rely on the message's length, you have to take care of that by assuming the length of the message is the length of $P(X)$ plus the length $Y$. Once $Y$ is padded correctly, we set $S$ to $H(X)$ and then continue to apply $C$ to the forged padding of $Y$ as it would be usual input. That way, we created $H(P(X) || Y) = C(P(X) || Y)$ which equals to $C(FP(Y))$ when $FP$ is the forged padding and $S = H(X)$.

In practise, I would use SHA-256, which can be attacked that way, and store it with the information about the length of each part which is included into the hash so you can reconstruct the input into the compression function later for validating data.

EDIT: Unlike in Ethan White's answer, the order does matter when applying this method, but a length extension should be ways faster since no powMod is involed.


text:
Tamil Nadu Board of Secondary EducationSSLC (English Medium) Class 10th

# The height and radius of the cone of which the frustum is a part are h1 units and r1 units respectively. Height of the frustum is h2 units and the radius of the smaller base is r2 units. If h2 : h1 - Mathematics

MCQ

The height and radius of the cone of which the frustum is a part are h1 units and r1 units respectively. Height of the frustum is h2 units and the radius of the smaller base is r2 units. If h2 : h1 = 1 : 2 then r2 : r1 is

• 1 : 3

• 1 : 2

• 2 : 1

• 3 : 1

#### Solution

1 : 2

Explanation;

Hint:

h2 : h1 = 1 : 2

h1 : h2 = 2 : 1

Ratio of their volumes

= 1/3 pi"h"_1 ("r"_1^2 + "r"_2^2 + "r"_1"r"_2) : 1/3pi"h"_2 ("r"_1^2 + "r"_2^2 + "r"_1"r"_2)

= 2(r12 + r22 + r1r2) : 1 (r12 + r22 + r1r2)

Volume is 2 : 1 the ratio of their radius also 2 : 1

r1 : r2 = 2 : 1 But r2 : r1 = 1 : 2

Concept: Volume of Frustum of a Cone
Is there an error in this question or solution?


text:
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A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
[Reveal] Spoiler: OA

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Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now, since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

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19 Mar 2015, 06:06
I think this question is good and helpful.
Hi, could you please explain why the area of the inscribed square in a circle with the diameter of 2 is 2?

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19 Mar 2015, 06:13
IonutCZ wrote:
I think this question is good and helpful.
Hi, could you please explain why the area of the inscribed square in a circle with the diameter of 2 is 2?

The area of a square = diagonal^2/2 = 2^2/2 = 2.
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26 Sep 2016, 01:08
After reading this explanation, I came up with this doubt: If we draw the diagonals of the rectangle (the blue region), each of the diagonals would pass through the center which- would be same as the diameter of the circle. Right or wrong? I am not able to prove this wrong.

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26 Sep 2016, 02:34
La1yaMalhotra wrote:
After reading this explanation, I came up with this doubt: If we draw the diagonals of the rectangle (the blue region), each of the diagonals would pass through the center which- would be same as the diameter of the circle. Right or wrong? I am not able to prove this wrong.

Right. A rectangle has right angles. A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle). The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled (right angel being the angle opposite the diameter/hypotenuse)
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26 Sep 2016, 05:51
Then should the area of the rectangle be 1/2*d1*d2= 1/2*2*2=2?. I think I am getting this wrong, but where I don't know.

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27 Sep 2016, 01:37
La1yaMalhotra wrote:
Then should the area of the rectangle be 1/2*d1*d2= 1/2*2*2=2?. I think I am getting this wrong, but where I don't know.

If the rectangle is a square then the area would be 2 (as given in the solution). Don't understand what you derive from this and what's your doubt.
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05 Apr 2017, 10:55
This is a great question. Very enlightening. There is a question about a small ''technicallity'' though.

The question stem asks about a ''rectangle''. Since the area of 2 is only achieved throught a square isn't it provoking a little doubt that you could marginally discard the area =2 because ''technically'' the questions asks about rectangle ?

I hope my question is not awkwardly written

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05 Apr 2017, 21:46
alexlovesgmat wrote:
This is a great question. Very enlightening. There is a question about a small ''technicallity'' though.

The question stem asks about a ''rectangle''. Since the area of 2 is only achieved throught a square isn't it provoking a little doubt that you could marginally discard the area =2 because ''technically'' the questions asks about rectangle ?

I hope my question is not awkwardly written

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All squares are rectangles.
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26 Oct 2017, 12:42
Bunuel wrote:
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Hello Bunuel
i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?
let a = side of square...
diameter = diagonal
2 = a*sqrt(2)
a = sqrt(2).
Area = pi*a^2 = 2*pi.
So => 0 <= area < 2*pi
Please let me know if i am missing something!!!

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26 Oct 2017, 21:41
manishtank1988 wrote:
Bunuel wrote:
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Hello Bunuel
i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?
let a = side of square...
diameter = diagonal
2 = a*sqrt(2)
a = sqrt(2).
Area = pi*a^2 = 2*pi.
So => 0 <= area < 2*pi
Please let me know if i am missing something!!!

Not sure what you are doing there. If a square is inscribed in a circle with the diameter of 2, then the area of the square would simply be diagonal^2/2 = 4/2 = 2 (because diameter=diagonal).
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27 Oct 2017, 09:14
Bunuel wrote:
manishtank1988 wrote:
Bunuel wrote:
Official Solution:

A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the area of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now,since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0 \lt area \le 2$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi r^2=\pi \approx 3.14$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

Hello Bunuel
i have a question for the highlighted area - shouldn't the max area of square inscribed within a circle with diameter of 2 be 2*pi?
let a = side of square...
diameter = diagonal
2 = a*sqrt(2)
a = sqrt(2).
Area = pi*a^2 = 2*pi.
So => 0 <= area < 2*pi
Please let me know if i am missing something!!!

Not sure what you are doing there. If a square is inscribed in a circle with the diameter of 2, then the area of the square would simply be diagonal^2/2 = 4/2 = 2 (because diameter=diagonal).

Hello Bunuel
I got my mistake - we are talking about largest rectangle (square) within a circle - thus area of square - a^2 => 2.

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Re: M24-05   [#permalink] 27 Oct 2017, 09:14
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# M24-05

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text:
CRIME

A GAP Package to Calculate Group Cohomology and Massey Products

Version 1.5
Released 2018-10-11

This project is maintained by The GAP Team

GAP Package CRIME

This package computes the cohomology rings of finite p-groups, induced maps, and Massey products.

The current version of this package is version 1.5, released on 2018-10-11. For more information, please refer to the package manual. There is also a README file.

Dependencies

This package requires GAP version >=4.9

Citing

You can get more info by typing Cite("CRIME"); in the gap prompt.


text:
### Home > MC2 > Chapter 6 > Lesson 6.1.2 > Problem6-20

6-20.

Similar shapes have sides with equal ratios to each other. To find each unknown side, compare its ratio to a known side.

For example, the ratio of the 15 mm side to the 22 mm side should be equal to the ratio of the z side to the 33 mm side.

$\frac{15}{22} = \frac{z}{33}$

Now solve for z.

z = 22.5 mm

You can now use that to find ratios for the other unknown sides.

Write two more ratios to solve for side x and side y.

$\frac{15}{54} = \frac{22.5}{y}$

15 y = (22.5) (54)

y = 81 mm
Now what does x equal?

Explore the problem with the etool below.
Click the link at right for the full version of the eTool: MC2 6-20 HW eTool


text:
# Qualities of Projected Gradient Methods

Consider the following constrained minimization problem:
$min_{x \in X} \ f(x)$
where $X \subset \Bbb{R}^{n}$ is a nonempty closed convex set and f is continuously diferentiable.

I'm currently studying to solve this problem with the Gradient Projection Method, a fixed-point method:
$x_{k+1} = Proj_{X}[x_{k}-\alpha_{k}\nabla f(x_{k})]$

($Proj_{X}(y)$ denotes the point in X with shortest distance to $y \in \Bbb{R}^{n}$, $\alpha_{k}$ some appropriate stepsizes - for my case, i just start with some $\gamma \in (0,1)$ and halve this $\gamma$ until it satisfies some kind of Armijo-Rule $f \left( x^{k}(\gamma) \right) \ \le \ f(x_{k}) - 10^{-4} \nabla f(x_{k})^{T}\left( x_{k} - x^{k}(\gamma)]\right)$ with $x^{k}(\alpha) = Proj_{X}[x_{k}-\alpha \nabla f(x_{k})]$ and declare this $\gamma$ as $\alpha_{k}$)

Now i would like to know more about the quality of this method. All i could gather is a theorem saying that all limit points of the sequence obtained by the algorithm are stationary points of the minimization problem $\small(P.H. \ Calamai \ and \ J.J. \ Moré,\ “Projected \ gradient \ methods \ for \ linearly \ constrained \ problems,” \ Math. \ Programming, vol. 39, pp. 93–116, 1987.)$ .
In my ears this sounds pretty bad. The sequence doesn't need to converge, it could jump between two different limit points, everything could happen. On top of that you somehow have to calculate the projection $Proj_{X}$() - for some simple contraints you can actually determine it, but for most cases you turn in circles have to solve another minimization problem. And nevertheless i hear people saying that this method is globaly "good".
But why? Is there something i missed?

I would be glad for some help, Thanks!