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# 3d Coordinate Geometry Problems Pdf
When a line segment in space is oriented so. Find the slope of a line, which passes through point А(5, -3) and meets y axis at 7. Rene Descartes (1596 – 1650), the mathematician & philosopher who also said “I think therefore I am. It describes a two-dimensional plane in terms of two perpendicular axes: x and y. Amplitude: I Period: 5 47. And then do the same but following a horizontal line to find the y-coordinate. In the right-handed system, one of the axes ($$x$$-axis) is directed to the right, the other $$y$$-axis is directed vertically upwards. The full list of Algorithm Titles is shown below, and active links indicate the algorithms that have been posted and are now accessible. Any point can be located within one of the four quadrants in the coordinate plane using a specific ordered pair of numbers, called its _____. The standards call for learning mathematical content in the context of real-world situations, using mathematics to solve problems, and developing “habits of mind” that foster mastery of mathematics content as well as mathematical understanding. Algebra Worksheets & Printable. The first challenge asks you to use just compass bearings to move to Elsa, the second is a little trickier and asks for coordinates. It is a writing paper that has fine lines arranged in a regular grid pattern which serves as a guide for drawing. pdf Vector Geometry of grids and 2D shapes. They make excellent classroom activities. You can select different variables to customize these Area and Perimeter Worksheets for your needs. Consider a straight line L in the above Cartesian coordinate plane formed by x axis and y axis. Weekly Weather Track. you will get a brand new printable PDF worksheet on Full Year 10th Grade Review. Vectors in 2D and 3D [all mathematics is done by specifying position of the spacecraft and the moon relative to some coordinate system, say centered at the earth; that's how orbits are calculated at JPL and NASA]. 13 Extensions: Maximizing, adjustments for 3+ variables. Next week, Tues 9/4, quiz chapter 1. 3 Difference estimates and Harnack inequality 125 6. Coordinate_Transformations. tfluxphi, ht. MAKE_2D function to convert a three-dimensional geometry to a two-dimensional geometry; and you can use the resulting geometry to perform a transformation into a geometry with the desired number of dimensions. These three. 1 Introduction 166 7. Read the problem at least three times before trying to solve it. GaDOE Math GPS via TI-83 Part 5 If you experience technical difficulties with any webcast, please call the Help Desk at (800) 869-1011 for assistance. We present a new unified algorithm for optimizing geometric energies and computing positively oriented simplicial mappings. There is no cost or registration required to practice your math on the AAAMath. (2) We propose a novel Omni-scale Graph Network to learn the feature from both human appearance and 3D geometry structure. doing some math without knowing what the symbols on the page mean. Based on the estimated camera positions and pictures themselves a 3D polygon mesh, representing the object surface, is build by PhotoScan. Panning and zooming a map involves changing the location and size of the viewport, and navigation is typically tied into keyboard and mouse events. This mean value is taken over the Klein polyhedra of integer 3D lattices with determinants in , where is an increasing parameter. This website is created solely for Jee aspirants to download pdf, eBooks, study materials for free. Interpret box-and-whisker plots M. What formula will give you the correct mid-point of the line segment?. Use plots to visualize data. Looking for an online maths program that’s curriculum aligned, tracks student progress and frees you up to teach? Help students excel, book demo on 1300833194. This activity has students use graphing, surface area, volume, unit conversion, Pythagorean Theorem, and more!. 8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Game The game allows users to figure out and to practice using the coordinate plane for giving the "address" or exact location of particular points. By using this website, you agree to our Cookie Policy. The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. 3D geometry is introduced with rectangular prisms. from x to u • Example: Substitute. to a 3D object coordinate frame and to reconstruct unknown 3D objects through triangulation. Cartesian coordinates allow one to specify the location of a point in the plane, or in three-dimensional space. Common Core and Mathematics: Grades K–5 > Module 6 > Reading: Geometry and Measurement and Data _____ Assessment of Geometry The Geometry standards cover a wide range of topics, from shapes to coordinate planes to transformations to lines and angles. This then was the problem—to give an introductory course in modern algebra and geomety—and I have proceeded on the assumption that neither is complete without the other, that they are truly two sides of the same coin. You can construct a three-dimensional coordinate systemby passing a z-axis perpendicular to both the x- and y-axes at the origin. Good luck. pdf Factors, Multiples, Primes, Prime Factors, LCM and HCF. This was Part One of the Math Land Amusement Park Task and set up the situation and the graph for Math Land Amusement Park. used in a 3D editor as a reference. The standard form for the equation is given by:. The constructions that can be used to solve this problem are based—among other things—on the property that a parallel projection between two coplanar lines preserves distances. While this document can be read serially, most technology stacks will not encounter all of the problems described herein. Primary SOL G. 1, 5, 7, 9 They are a special set of problems that may look the same at first glance, but which require different mathematical ideas. The geometry center has information in several places on this problem, the best being an article describing a way of filling space by unit circles (discontinuously). Geometry Vocabulary with definitions. Teaching geometry well involves knowing how to recognise interesting geometrical problems and theorems, appreciating the history and cultural context of geometry, and understanding the many and varied uses to which geometry is put. y z x u=(ux,uy,uz) v=(vx,vy,vz) w=(wx,wy,wz) (x0,y0,z0) • Solution: M=RT where T is a. Notice that the y-coordinate for both points did not change, but the value of the x-coordinate changed from 5 to -5. These worksheets are printable PDF exercises of the highest quality. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value. CBSE NCERT Solutions for Class 9 Maths Chapter 3: Descriptions. There is an x, y,andz coordinate. Weekly Weather Track. Each coordinate can be any real number. Nothing more. 3D geometry is introduced with rectangular prisms. Grade 5 geometry worksheets. Help typing in your math problems. A significant percentage of the questions on Level 1 is devoted to plane Euclidean geometry and measurement, which is not tested directly on Level 2. For this purpose to acquire mathematical. Coordinate geometry is the combination of geometry and algebra to solve the problems. If A ( x 1, y 1) and B( x 2, y 2,), then. TEACHER NOTE : (Optional Anticipatory Set) Before class, review and print copy of Activity 1 for each student: Coordinate Place Four and Independent Worksheet 1. (2) We propose a novel Omni-scale Graph Network to learn the feature from both human appearance and 3D geometry structure. As in the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, which has coordinates (0;0;0). Teaching geometry well involves knowing how to recognise interesting geometrical problems and theorems, appreciating the history and cultural context of geometry, and understanding the many and varied uses to which geometry is put. In the menu bar, go to Concept > 3D Curve. Geometry and Measurement. / Procedia Social and Behavioral Sciences 8 (2010) 686–693 third phase, they underwent teaching and learning phase and were given assessment questions to. Grade 5 geometry worksheets. As a function, we can consider the perimeter or area of a figure or, for example, the volume of a body. The next stage is building geometry. Are there any global min/max? Solution: Partial derivatives f x = 6x2 6xy 24x;f y = 3x2 6y: To find the critical points, we solve f x = 0 =)x2 xy 4x= 0 =)x(x y 4) = 0 =)x= 0 or x y 4 = 0 f y = 0 =)x2. Geometry Worksheets Area and Perimeter Worksheets. Even though the ultimate goal of elegance is a complete coordinate free. So: 5 is the x-coordinate, and 8 is the y-coordinate. Whether you are teaching kindergartens how to count, youngsters how to multiply, teens how to factor polynomials, or adults how to understand Ohm’s law, you will find what you need at The Math Worksheet Site. Geometry and Spatial Sense, Grades 4 to 6 is a practical guide that teachers will find useful in helping students to achieve the curriculum expectations outlined for Grades 4 to 6 in the Geometry and Spatial Sense strand of The Ontario Curriculum, Grades 1–8: Mathematics, 2005. This method of solving geo problems (often called coordinate bashing) can. Enter 2 sets of coordinates in the 3 dimensional Cartesian coordinate system, (X 1, Y 1, Z 1) and (X 2, Y 2, Z 2), to get the distance formula calculation for the 2 points and calculate distance between the 2 points. It is clear that the curvature is large at the top of the loops and smaller lower down. Changing Coordinate Systems • Problem: Given the XYZ orthogonal coordinate system, find a transformation, M, that maps XYZ to an arbitrary orthogonal system UVW. Since the late 1940s and early 1950s, differential geometry and the theory of manifolds has developed with breathtaking speed. 9 Introduction to 3D: functions in 2 variables, 3D graphing 2. The standards for kindergarten through grade 8 prepare students for higher mathematics. Distance of a point from the plane We now consider this problem. Such a vector is called the position vector of the point P and its. CHAPTER 16 Coordinate Geometry 107 POSTTEST 113 ANSWERS 121 reviewed in the first eight chapters to solve these problems. called solid analytic geometry. 6 Capacity in two dimensions 144 6. 2) Perpendicular Lines in a Coordinate Plane: In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is -1. Our grade 4 geometry worksheets cover topics such as classifying angles, triangles and quadrilaterals, areas and perimeters and coordinate grids. The Equation of a Line Two points, point and slope, slope and y-int: Graphing and Plotting. Some rules found in spherical geometry include: There are no parallel lines. Coordinate Geometry for Transformations – Free Worksheet! Transformations – Free Worksheet! (As promised on p. Volume of cubes and rectangular prisms: word problems (6-FF. The v -coordinates are v=0 which corresponds to the eyelevel and horizon line. We say that an equation makes a true statement if the. In this video, the instructor shows how to find an unknown coordinate given the other coordinate of that point and the equation that passes thought the point. Math A/B (1998-2010) REGENTS RESOURCES. Graph paper is completely essential for a range of subject matter. –All nonzero scalar multiples of (x,y,1) form an equivalence class of points that project to the same 2D Cartesian point (x,y). Worksheets > Math > Grade 5 > Geometry. While this document can be read serially, most technology stacks will not encounter all of the problems described herein. The point A(–6, 4) and the point B(8, –3) lie on the line L. Plenty of online activities and lessons that explore the world of Math! emathematics. College Algebra, Geometry, and Trigonometry Placement Tests College Algebra Placement Test Items in the College Algebra Test focus on algebra knowledge and skills in a variety of content. −Affine transformations in OpenGL. Vectors in two dimensions 2 2. First, we will create the geometry of the airfoil. Geometry helps you to bring together both sides of your brain. 15 – Shapes on the Coordinate. INTRODUCTION The general aim of mathematics is stated as making an individual acquire the mathematical knowledge needed in daily basis , teaching how to solve problems , making him/her have a method of solving problems and acquiring reasoning methods (Altun,2008). It helps in visualizing the problem in order to get a better understanding of the theoretical concepts. …The coordinate plane is the x-axis…which is horizontal…and the y-axis which is vertical. Problem 37. Author: Fletcher Dunn Publisher: CRC Press ISBN: 1568817231 Size: 58. It says that the area of the square whose side is the hypothenuse of the triangle is equal to the sum of the areas of the squares whose sides are the two legs of the triangle. So, if we consider a nodal coordinate matrix nodesthe y-coordinate of the nthnode is. Coordinate Geometry Expansions & Factorisation - pdf Financial Arithmetic - pdf Fractions - Addition and simplication Linear Equations - pdf Number system exercises - worksheet Probability - pdf Sets & Venn Diagrams. To solve real-life problems, such as finding how many times to air a radio commercial in Ex. Set students up for success in Geometry and beyond! Explore the entire Geometry curriculum: angles, geometric constructions, and more. 1 Introduction 166 7. Students solve real world problems through the application of algebraic and geometric concepts. They're all free to watch! ↓. If appropriate, draw a sketch or diagram of the problem to be solved. Weekly Weather Track. The node coordinates are stored in the nodal coordinate matrix. Free geometry worksheets with visual aides, model problems, exploratory activities, practice problems, and an online component. View StewartCalcET8_12_01. Lesson 14. Sometimes words can be ambiguous. 2 Working Copy: February 7, 2017 The square on the grid has an area of 16 square units. A second fifth grader says. Introduction to 3D Coordinate Geometry Ex 28. During the second phase, both group were introduced the basic concept of the Coordinate Geometry and mathematical problem solving session. Trigonometry. In this post, I’d like to shed some light on computational geometry, starting with a brief overview of the subject before moving into some practical advice based on my own experiences (skip ahead if you have a good handle on the subject). MATH TOOLBOX. This is just a grid that you'll be drawing all sorts of cool math things on -- like lines. Pythagoras tells us that c = √(x 2 + y 2) Now we make another triangle with its base along the "√(x 2 + y 2)" side of the previous triangle, and going up to the far. CHAPTER 16 Coordinate Geometry 107 POSTTEST 113 ANSWERS 121 reviewed in the first eight chapters to solve these problems. The spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates,$(\\rho,\\phi,\\theta)$, where$\\rho$ represents the radial distance of a point from a fixed origin,$\\phi$ represents the zenith angle from the positive z-axis and$\\theta$ represents the azimuth angle from the positive x-axis. Arihant books maths and coordinate geometry (get full book for free)) links and pdf( y t study. Geometry, Set C3: Coordinate Systems, pdf. the fixed coordinate system, respectively. 2) Perpendicular Lines in a Coordinate Plane: In a coordinate plane, two non-vertical lines are perpendicular if and only if the product of their slopes is -1. three-dimensional GOAL 1 Graph linear equations in three variables and evaluate linear functions of two variables. The Pythagorean theorem was reportedly formulated by the Greek mathematician and philosopher Pythagoras of Samos in the 6th century BC. - [Narrator] So coordinate geometry,…if we think about it,…is really more algebra than geometry,…which is why I'm showing it in the algebra chapter. problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, and/or equations. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value. Reflections, Translations, and Rotations. Throughout high school there is a focus on analyzing properties of two- and three-dimensional shapes, reasoning about geometric relationships, and using the coordinate system. Tenth Grade (Grade 10) Math Worksheets, Tests, and Activities. 2 Some estimates 167 7. These problems incorporate some decimals into the calculations. 5 CEUs; Geometry 101 Beginner to Intermediate Level $110. The point C lies on l1 and has x-coordinate equal to p. The coordinate plane is also known as the x-y plane and the Cartesian plane, so named after its discoverer, Mr. You can choose to include answers and step-by-step solutions. Computational geometry algorithms for software programming including C++ code, basic lmath, a book store, and related web site links. Learn the coordinate geometry formulas with the help of an example. Contents 1. C E vADlylk 3r YibgAh Atrs h Grve UsHeQrbv 8eId F. The work of [Coros et al. The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant*. n professorofmathematicsinthesheffieldscientificschool yaleuniversity and akthubsullivangale,ph. Word problem #3:. This activity has students use graphing, surface area, volume, unit conversion, Pythagorean Theorem, and more!. Browse to and select the geometry file you downloaded earlier. Here is a range of free geometry worksheets for 3rd graders. com website. Also browse for more study materials on Mathematics here. Thanks to the author’s fun and engaging style, you’ll enjoy thinking about math like a programmer. It has now been four decades since David Mumford wrote that algebraic ge-ometry “seems to have acquired the reputation of being esoteric, exclusive, and. Similar and Congruent Shapes. We will now use. basic geometry worksheets. Our grade 4 geometry worksheets cover topics such as classifying angles, triangles and quadrilaterals, areas and perimeters and coordinate grids. Roughly translating in Greek as "Earth Measurement", it is concerned with the properties of space and figures. While it is not the intent of this lesson to graph planes in space, it is important to understand how the graphs of the three-variable equations planes behave in space. Here is a list of topics: 1. For this purpose to acquire mathematical. In the menu bar, go to Concept > 3D Curve. The three mutually perpendicular lines in a space which divides the space into eight parts and if these perpendicular lines are the coordinate axes, then it is said to be a coordinate system. My problem is finding what A is. This worksheet is a supplementary seventh grade resource to help teachers, parents and children at home and in school. Math 8 – Socrative Space Race “Team Task” We ended this part of the unit with some "App Smashing". Considerations: Geometry Strategies for Middle School T/TAC W&M 2004 2 Geometry Strategies for Middle School This Considerations Packet describes strategies middle school mathematics teachers can incorporate into their teaching of geometry. Rotation in 3D That works in 2D, while in 3D we need to take in to account the third axis. To read more, Buy study materials of 3D Geometry comprising study notes, revision notes, video lectures, previous year solved questions etc. For JEE, three-dimensional geometry plays a major role as a lot of questions are included in the exam. Extended: 26D Vector Addition and 26E Scalar Multiplication (10-11) File Size: 4237 kb: File Type: pdf. In particular it is central to the mathematics students meet at school. Make math easy with our math problem solver tool and calculator. Assorted Math Skill Games for Children – Kindergarten to 7th Grade. 3D graphing is an important tool used by structural engineers to describe locations in space to fellow engineers. If you have adopted the CPM curriculum and do not have a teacher edition, please contact our Business Office at (209) 745-2055 for information to obtain a copy. The u3d file can then be embedded into a pdf with pdflatex and the movie15 package. Geometry in Later Schooling. GaDOE Math GPS via TI-83 Part 5 If you experience technical difficulties with any webcast, please call the Help Desk at (800) 869-1011 for assistance. Rotation in 3D That works in 2D, while in 3D we need to take in to account the third axis. The command ndgrid will produce a coordinate consistent matrix in the sense that the mapping is (i,j) to (x i;y j) and thus will be called coordinate consistent indexing. Lesson 14. Check out our ever growing collection of free math worksheets Free Elementary Math Worksheets Free Geometry Worksheets pdf Download Geometry Worksheets with bar graphs line graphs pie charts and other geometric shapes and their areas and volumes perimeter symmetry find the circumference geometry words problems includes problems of 2D and 3D. Click here to learn more. , jxOjand mutually orthogonal Also, the length of A is. −OpenGL matrix operations and arbitrary geometric transformations. Rene Descartes (1596 – 1650), the mathematician & philosopher who also said “I think therefore I am. You can think of reflections as a flip over a designated line of reflection. It can also be denoted S B!Ato emphasize that Soperates on B-coordinates to produce A-coordinates. It is a writing paper that has fine lines arranged in a regular grid pattern which serves as a guide for drawing. Free and no registration required. Spherical geometry considers spherical trigonometry which deals with relationships between trigonometric functions to calculate the sides and angles of spherical polygons. When geometry is at lower levels like MathCounts/AMC/Early to Mid AIME I usually don't have a problem. 1 Introduction 166 7. Summary of Coordinate Geometry Formulas; Summary of Coordinate Geometry Formulas. The third row and third column look like part of the. local min/local max/saddle point. pdf), Text File (. The IDTF file is converted to u3d with an external binary file. The work of [Coros et al. Then {v1,v2,v3} will be a basis for R3. A cylinder is best suited for cylindrical coordinates since its. Chapter 28 Introduction to 3D Coordinate Geometry Ex 28. Co-ordinate Geometry: Practice Questions Solve the given practice questions based on co-ordinate geometry. Similarly, each point in three dimensions may be labeled by three coordinates (a,b,c). Once you have selected the desired geometry file, click to create the curve. A vector is essentially a line segment in a specific position, with both length and direction, designated by an arrow on its end. Rene Descartes (1596 – 1650), the mathematician & philosopher who also said “I think therefore I am. You can create a front view by defining. Graph paper is completely essential for a range of subject matter. Author: Fletcher Dunn Publisher: CRC Press ISBN: 1568817231 Size: 58. This video tutorial provides a basic introduction into coordinate geometry. Now that you've solved the parallax problem, use the same skills you used there to proposal that shows that you can win the race. 7 CEUs; Math All-In-One (Arithmetic, Algebra, and Geometry Review)$120. As a function, we can consider the perimeter or area of a figure or, for example, the volume of a body. Here is a list of topics: 1. (d) Show that p satisfies p2 – 4p – 16 = 0. • Major features of the part should be used to establish the basic coordinate system, but are not necessary defined as datum. They may check their thinking. Review a book such as SAT Math Essentialsor Acing the SAT 2006by LearningExpress to be sure you’ve got all the skills you need to achieve the best possible math score on the SAT. Bellwork Students will turn in their homework. 9 Eigenvalue of a set 157 7 Dyadic coupling 166 7. Lesson 14. The next stage is building geometry. The Cartesian coordinates (also called rectangular coordinates) of a point are a pair of numbers (in two-dimensions) or a triplet of numbers (in three-dimensions) that specified signed distances from the coordinate axis. The geographic coordinate system. Free Geometry calculator - Calculate properties of planes, coordinates and 3d shapes step-by-step This website uses cookies to ensure you get the best experience. Geometry in Later Schooling. Irodov-Problems_in_General_Physics (1). (A cheat sheet on 3D Geometry is also available on this website. The y-coordinates already match. EDIT: I am adding few pdf links for time being. 7 CEUs; Math All-In-One (Arithmetic, Algebra, and Geometry Review) $120. In seeking to coordinate Euclidean, projective, and non-Euclidean geometry. JEE in 40Days PDF-Master JEE Main Physics, Chemistry, Mathematics in Just 40 Days; CENGAGE. A is the u-coordinate of the y-axis vanishing point V1 and B is the u-coordinate of the x-axis vanishing point V2. Each book in this series provides explanations of the various topics in the course and a substantial number of problems for the student to try. There is an x-coordinate that can be any real number, and there is a y-coordinate that can be any real number. 3D geometry design software is an interactive geometry software that can be used by school kids, teachers and schools to make math calculations easier. II I About This Test. R3 is the space of 3 dimensions. I will add more, do follow me for updates. Keep reading to learn about some of these math problem solving strategies. When we express a vector in a coordinate system, we identify a vector with a list of numbers, called coordinates or components, that specify the geometry of the vector in terms of. pdf Straight Lines. Plenty of online activities and lessons that explore the world of Math! emathematics. Geometry and coordinate geometry. This website is created solely for Jee aspirants to download pdf, eBooks, study materials for free. Geometry in Later Schooling. Algebra I - Math in the Real World - developed by Kelly Muzzy and Lori Schramm. Even though the ultimate goal of elegance is a complete coordinate free. Sudents learn about the three-dimensional Cartesian coordinate system. Many ideas of dimension can be tested with finite geometry. You must specify a coordinate system for the current drawing to take advantage of coordinate transformation capabilities. In the menu bar, go to Concept > 3D Curve. Math Test Series About Us Established in 2009, Our flagship 'IIT JEE Online Preparation Course' has been growing rapidly and thousands of students across India take its advantage every year. applied to fabrication and 3D printing [Bickel et al. Young Princess Elisabeth had successfully attacked a problem in elementary geometry using coordinates. To our knowledge, this work is among the early attempts to address this problem. to a 3D object coordinate frame and to reconstruct unknown 3D objects through triangulation. 3 Difference estimates and Harnack inequality 125 6. pdf Vector Geometry of grids and 2D shapes. Lesson 12. Analytic geometry, also called coordinate geometry, mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry. Weekly Weather Track. Many ideas of dimension can be tested with finite geometry. This free online math web site will help you learn mathematics in a easier way. Plane and Solid Geometry. togrammetry, ComputerVision, Robotics and related fields. geometry, we confined to the Cartesian methods only. This video tutorial provides a basic introduction into coordinate geometry. The choice of the axioms and the investigation of their relations to one another is a problem which, since the time of Euclid, has been discussed in numerous. Some navigation problems ask us to find the groundspeed of an aircraft using the combined forces of the wind and the aircraft. Using these sheets will help your child to: recognize and identify a range of 2d and 3d shapes; recognise and identify right angles and lines of symmetry; recognise and identify parallel lines; identify the faces, edges and vertices of 3d shapes;. We present a new unified algorithm for optimizing geometric energies and computing positively oriented simplicial mappings. v1 and v2 span the plane x +2z = 0. NOTE The signs of the coordinates of a point determine the quadrant in which the point lies. Coordinate System in 3D Geometry. The coordinate plane is also known as the x-y plane and the Cartesian plane, so named after its discoverer, Mr. If we start at the origin of such a system, we can give. three-dimensional GOAL 1 Graph linear equations in three variables and evaluate linear functions of two variables. Also browse for more study materials on Mathematics here. At the moment, the introductory portion of such a development of geometry can be found, in greater detail than is given in this article, in Chapters 4{7 of H. In fact, this approach to school geometry has been taught to teachers in. Euclidean Geometry by Rich Cochrane and Andrew McGettigan. Algebra, Expressions and Equations, Geometry, and Statistics and Probability. This type of design task and way of using a coordinate system increase the students’ abilities to problem solve. (a) Find an equation for L in the form ax + by + c = 0, where a, b and c are integers. difficult problems in stochastic geometry. Let's say we want the distance from the bottom-most left front corner to the top-most right back corner of this cuboid: First let's just do the triangle on the bottom. I will add more, do follow me for updates. To assign a coordinate system to the current drawing Do one of the following: On the status bar, click the down arrow next to Coordinate System and click Library. For example, the yaw matrix essentially performs a 2D rotation with respect to the x-y plane, while leaving the z- coordinate unchanged. MATH 294 FALL 1987 FINAL # 2 294FA87FQ2. Example Question #1 : Coordinate Geometry There is a line segment between the two points (5, 10) and (3, 6). Related SOL G. Here’s a fun hands-on activity cum worksheet that introduces kids to the concept of graphing. The x-axis indicates the horizontal direction while the y-axis indicates the vertical direction of the plane. Chapter 28 Introduction to 3D Coordinate Geometry Ex 28. The standards call for learning mathematical content in the context of real-world situations, using mathematics to solve problems, and developing “habits of mind” that foster mastery of mathematics content as well as mathematical understanding. 7 CEUs; Math All-In-One (Arithmetic, Algebra, and Geometry Review)$120. Check out our collection for free graphing worksheets for kindergarten age children. Browse to and select the geometry file you downloaded earlier. Lesson 12. A significant percentage of the questions on Level 1 is devoted to plane Euclidean geometry and measurement, which is not tested directly on Level 2. Coordinate Geometry - In Cartesian coordinate system,each point has an x-coordinate representing its horizontal position and a y-coordinate representing its vertical position. The exposition serves a narrow set of goals (see §0. Grade level: 3. Set students up for success in Geometry and beyond! Explore the entire Geometry curriculum: angles, geometric constructions, and more. MAKE_3D function to convert a two-dimensional geometry to a three-dimensional geometry, or the SDO_CS. h represents our y-coordinate of 18 Write as Equation: 81a + 9b + c = 18 Calculate Letters i,j,k,l from Point 3 (3, 8): j represents our x-coordinate of 3 i is our x-coordinate squared → 3 2 = 9 k is always equal to 1 l represents our y-coordinate of 8 Write as Equation: 9a + 3b + c = 8 The following practice problem has been generated for you:. ppt from MATH GEOMETRY at Redwood High, Larkspur. Lines and Slopes. Solve multiple-step problems that use ratios, proportions, and percentages. There are many 3D geometry software downloads available on the internet and it can be used as per the requirement. We are going to do all of our work in a three-dimensional coordinate system. distance d, from A to B = midpoint, M, of AB =. When geometry is at lower levels like MathCounts/AMC/Early to Mid AIME I usually don't have a problem. The only assumption is that the reader has a working knowledge of linear algebra. \All Geometry is Algebra" Many geometry problems can be solved using a purely algebraic approach - by placing the geometric diagram on a coordinate plane, assigning each point an x/y coordinate, writing out the equations of lines and circles, and solving these equations. As in the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, which has coordinates (0;0;0). Find a square with vertices on the coordinate grid whose area is n square units, where n is a whole number between 1 and 10, or show that there is. Please feel free to contact me with any suggestions, corrections or comments. The real numerical problems occur near the origin, near the eye. Coordinate-based methods are preferable when we either know a priori that the geometry is in a known, simple position and orientation in the coordinate system, or we pre-process it so that it is. It is imperative to know exactly what the problem is asking. (Sidney Luxton), 1860-1939. Teaching geometry well involves knowing how to recognise interesting geometrical problems and theorems, appreciating the history and cultural context of geometry, and understanding the many and varied uses to which geometry is put. The answers and explanations are given for the practice questions. A good knowledge of vectors will help to solve problems related to 3D Geometry. Therefore (x,y) = (4, -2). Links and pdf(( like subscribe comment down your email your email address. Export triangulated mesh into a pdf with a 3D object. Chapter 1: Coordinates, points and lines. These fundamental principles are called the axioms of geometry. The command ndgrid will produce a coordinate consistent matrix in the sense that the mapping is (i,j) to (x i;y j) and thus will be called coordinate consistent indexing. This then was the problem—to give an introductory course in modern algebra and geomety—and I have proceeded on the assumption that neither is complete without the other, that they are truly two sides of the same coin. Problem 37. Geometry and Spatial Sense, Grades 4 to 6 is a practical guide that teachers will find useful in helping students to achieve the curriculum expectations outlined for Grades 4 to 6 in the Geometry and Spatial Sense strand of The Ontario Curriculum, Grades 1–8: Mathematics, 2005. Click on the links below to view sample pages. ” Of course, it consists of two perpendicular numbers lines, the x- and y-axis, which define a grid that. X Y Coordinate Graph Paper Printable. With accessible examples, scenarios, and exercises perfect for the working developer, you’ll start by exploring functions and geometry in 2D and 3D. There are some facts that we can rely upon. You can choose to include answers and step-by-step solutions. This then was the problem—to give an introductory course in modern algebra and geomety—and I have proceeded on the assumption that neither is complete without the other, that they are truly two sides of the same coin. (2) We propose a novel Omni-scale Graph Network to learn the feature from both human appearance and 3D geometry structure. pdf Find Percentage Decrease. Teacher will ask for the length of the following segments. problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, and/or equations. Please note: Although we have taken care to create all files so that they are as accurate as possible, some files may not print accurately. Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. Review a book such as SAT Math Essentialsor Acing the SAT 2006by LearningExpress to be sure you’ve got all the skills you need to achieve the best possible math score on the SAT. Cartesian coordinates allow one to specify the location of a point in the plane, or in three-dimensional space. Math Test Series About Us Established in 2009, Our flagship 'IIT JEE Online Preparation Course' has been growing rapidly and thousands of students across India take its advantage every year. I will add more, do follow me for updates. includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step This website uses cookies to ensure you get the best experience. Math solution free, houghton mifflin algebra and trigonometry book 2 step by step, ti84 trig, free ratio printable worksheets, rewrite as a system for x(t) and v(t) differential equations. Extend the set {v1,v2} to a basis for R3. Solving Systems Of Equations Word Problem. Converting into this will change the X and Y units to degrees, but leaves the X units as meters. A graphing tool to plot and interactively explore 2D math functions in the coordinate plane. The Equation of a Line Two points, point and slope, slope and y-int: Graphing and Plotting. Areas of Trapezoids Worksheet 3 RTF. Virtual Manipulatives - Glencoe. When thinking of 3-D, our initial thoughts ranged around planes and intersecting lines, three dimensional coordinate geometry and all sorts of 'hard and scary' maths. 40 CHAPTER 1. 1 Vector addition and multiplication by a scalar We begin with vectors in 2D and 3D Euclidean spaces, E2 and E3 say. Area calculator See Polygon area calculator for a pre-programmed calculator that does the arithmetic for you. The point A(–6, 4) and the point B(8, –3) lie on the line L. ), continue to serve important roles, but a digital 3D reference atlas is preferred for informatics-based workflows, data visualization, and integration across brains, labs, and data types. 13 Extensions: Maximizing, adjustments for 3+ variables. In finite geometry. 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A complete list of Math workbooks is available below. Algebra Worksheets & Printable. Contents 1. 1 Q2(iv) Introduction to 3D Coordinate Geometry Ex 28. Primary Resources - free worksheets, lesson plans and teaching ideas for primary and elementary teachers. coordinate plane, linear equations in three variables require three-dimensional space to be graphed. 13 – Interpreting Points on a Coordinate Plane. They make excellent classroom activities. MAKE_3D function to convert a two-dimensional geometry to a three-dimensional geometry, or the SDO_CS. 2 Page 4 of 82 August 28, 2018. DeTurck Math 241 002 2012C: Solving the heat equation 4/21. This website is created solely for Jee aspirants to download pdf, eBooks, study materials for free. v1 and v2 span the plane x +2z = 0. So: 5 is the x-coordinate, and 8 is the y-coordinate. Panning and zooming a map involves changing the location and size of the viewport, and navigation is typically tied into keyboard and mouse events. Introduction to 3D Coordinate Geometry Ex 28. In mathematics, analytic geometry (also called coordinate geometry) describes every point in three-dimensional space by an ordered triplet of real numbers known as the coordinates of the point. As abstract thinking progresses, geometry becomes much more about analysis and reasoning. An important student resource for any high school math student is a Schaum’s Outline. Coordinate Geometry. (2) We propose a novel Omni-scale Graph Network to learn the feature from both human appearance and 3D geometry structure. pdf Indices - Subtraction Rule. Its major improvements over the state-of-the-art are: adaptive partition of vertices into coordinate blocks with the blended local-global strategy, introduction of new distortion energies for repairing inverted and degenerated simplices, modification of standard rotation. congruent segments. This method of solving geo problems (often called coordinate bashing) can. The matrix of the resulting. 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Geometry, Surfaces, Curves, Polyhedra Written by Paul Bourke. We have a complete K-12 math curriculum library. This enables geometric problems to be solved algebraically and provides geometric insights into algebra. The standards for kindergarten through grade 8 prepare students for higher mathematics. Sudents learn about the three-dimensional Cartesian coordinate system. Description Of : Arihant Coordinate Geometry Session 1 Feb 10, 2020 - By Patricia Cornwell ^ Free eBook Arihant Coordinate Geometry Session 1 ^ coordinate geometry booster book for iit jee coordinate geometry booster for iit jee main and advanced has been conceptualised and produced for aspirants of various engineering entrance examinations it. 13 – Interpreting Points on a Coordinate Plane. The left-brain is the more logical, technical field, whereas the right-brain is the part that visualizes and where the artist gets their creative inspiration from. Gazette 80, November 1996. Plots & Geometry - powered by WebMath. Free Geometry calculator - Calculate properties of planes, coordinates and 3d shapes step-by-step This website uses cookies to ensure you get the best experience. A plane is a flat, two-dimensional surface that extends infinitely far. We present a new unified algorithm for optimizing geometric energies and computing positively oriented simplicial mappings. At least 20% of CAT questions each year are from Geometry alone. Thanks to the author’s fun and engaging style, you’ll enjoy thinking about math like a programmer. State an equation of this function. The Cartesian coordinate system should be familiar to you from earlier math and physics courses The vector A is readily written in terms of the cartesian unit vectors xO, yO, and zO A DxOA xCyOA yCzOA z In linear algebra xO, yO, and zOare known as basis vectors, each having unit length, i. Areas of Trapezoids Worksheet 3 RTF. A good knowledge of vectors will help to solve problems related to 3D Geometry. This worksheet is a supplementary seventh grade resource to help teachers, parents and children at home and in school. The angle between a position vector and an axis 6 5. this enabled the author to squeeze about 2000 problems on plane geometry in the book of volume of ca 600 pages thus embracing practically all the known problems and theorems of elementary geometry. Bibliography: 27 titles. For our first set of problems, we will look at some basic ideas of how to transform English sentence descriptions of math into an equation and viceversa. Coordinate system 3D essentially is all about representations of shapes in a 3D space, with a coordinate system used to calculate their position. Reading and plotting points on a coordinate grid is also covered. The chapter provides a compact, gentle introduction to the fundamental geometric relations that underly image-based 3D measurement. Letter G Worksheets For Toddlers. We summarize the chapter:. 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College Algebra, Geometry, and Trigonometry Placement Tests College Algebra Placement Test Items in the College Algebra Test focus on algebra knowledge and skills in a variety of content. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. As in the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, which has coordinates (0;0;0). 8 Beurling estimate 154 6. 3d coordinate geometry book pdf Crusade for justice the autobiography of ida b wells pdf, geometry, we confined to the Cartesian methods only. ratios on a coordinate plane. The coordinate plane is also known as the x-y plane and the Cartesian plane, so named after its discoverer, Mr. Try it free!. Good luck. 12 Vectors and the Geometry of Space 12. Volume of cubes and rectangular prisms: word problems (6-FF. Geometry, Set C3: Coordinate Systems, pdf. Download CBSE Class 9 Maths Chapter 3 Solutions As PDF. Geometry and coordinate geometry. Area calculator See Polygon area calculator for a pre-programmed calculator that does the arithmetic for you. Multiple Choice Math Problems with Solutions. There is an x-coordinate that can be any real number, and there is a y-coordinate that can be any real number. Many of the problems are worked out in the book, so the. pdf - Free download Ebook, Handbook, Textbook, User Guide PDF files on the internet quickly and easily. 3D PDF Pro is PROSTEP's powerful client solution for animating the 3D models embedded in 3D PDF documents. Geometry Worksheets. –(x,y,w) coordinates form a 3D projective space. Answers to practice 01. The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant*. If the problem persists, please contact the site's administrator. If AB + BC = AC, then B is between A and C. Skills in Mathematics for JEE Main and Advanced for Vectors and 3D Geometry has been revised carefully to help aspirants learn to tackle the mathematical problems with the help of session-wise. One is free to make trouble for oneself and use an inconvenient coordinate system. Related SOL G. Here are collected all the Euclidean Geometry problems (with or without aops links) from the problem corner (only those without any constest's source) and the geometry articles from the online magazine ''Mathematical Excalibur''. Grade 5 geometry worksheets. The relationship between Cartesian coordinates and Euclidean geometry is well known. problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, and/or equations. Some rules found in spherical geometry include: There are no parallel lines. Click Map Setup tabCoordinate System panelAssign. • We plot points as we did in two dimensions, but we try to use slanted lines to depict depth. EDIT: I am adding few pdf links for time being. coordinate system. To find out the coordinates of a point in the coordinate system you do the opposite.
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2020-12-04 04:43:19
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https://discuss.pytorch.org/t/batchnorm-fine-tuning/62130
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# Batchnorm fine tuning
Hi,
tl;dr : Does batchnorm use “learned” mean/variance to normalize data when in eval() mode? If so how do I get to mean/variance of batchnorm ( Not the affine transformation after normalization ).
Longer version:
I have a trained network. I have some new unlabeled data with the same categories as before but a slightly different domain. For simplicity let’s say I trained a “dog type” classifier on images taken during the day and now I want to fine-tune on new images that were taken at night.
I want to update the normalization factors in group norm to reflect the new data’s statistics. To the best of my understanding group norm during inference = 1) normalization with learned mean/std + 2) a learned affine transformed.
I only see the parameters of the affine transform. Is there a way to get to the mean/std and change it.
I tried to bypass this by training the network with a constant loss of zero ( since the mean/var are not dependent on the loss ). This did not do anything…
Thanks,
Dan
according to this there should be some parameters such as running_mean , running_std.
How do I get to them and can I change them- preferably by running a bunch of unlabeled input examples so that the running mean is slowly adjusted toward a new mean.
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2022-05-25 02:08:01
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https://answers.opencv.org/question/51740/write-data-to-text-file/
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# Write data to text file
Hi, I am trying to read a ".csv" file that contains double valued data like this:
1 0.0922279 0.0977822 0.0567845 0.0308806 0.0491464 0.0436114
1 0.277745 0.130502 0.118348 0.0365346 0.0669739 0.00924114
1 0.137181 0.140128 0.116455 0.122226 0.151289 0.129282
1 0.182825 0.0810142 0.0811148 0.0588809 0.147732 0.0410181
The above is just an example. The real ".csv" file is 269 row by 3780 column (269 x 3780).
I try to read the csv file, put the values in a Mat and then write that Mat to another file called "data.txt".
However, when I open the data.txt nothing is written except a value of 1. This is obviously wrong as, the data to the file would have contained the same values of the original "csv" file. This is the code:
#include <opencv2/core/core.hpp>
#include <opencv2/highgui/highgui.hpp>
#include <opencv2/imgproc/imgproc.hpp>
#include <opencv2/ml/ml.hpp> // for CvMLData
#include <iostream>
#include <fstream>
using namespace std;
using namespace cv;
int main()
{
ofstream data("data.txt"); //
CvMLData mlData;
mlData.read_csv("HOG_data.csv"); // File containing 269 row by 3780 column
if(mlData.get_values() == nullptr)
{
cout << "Unable to open file !" << endl;
return 1;
}
//const Mat m = mlData.get_values(); <---Also tried this;
const CvMat *m = mlData.get_values();
Mat dataset;
dataset = m;
for(int i = 0; i < dataset.rows; i++)
{
for(int j =0; j < dataset.cols; j++)
{
data << dataset.at<float>(i,j) << " ";
}
data << endl;
}
waitKey();
return 0;
}
Any suggestions or something i've probably overlooked ?
edit retag close merge delete
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2020-02-17 09:32:27
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https://www.groundai.com/project/spectral-gaps-for-the-linear-surface-wave-model-in-periodic-channels/
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Spectral gaps for the linear surface wave model in periodic channels
# Spectral gaps for the linear surface wave model in periodic channels
F.L. Bakharev , K. Ruotsalainen, J. Taskinen
###### Abstract
We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width . Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as : the endpoints are determined within corrections of order . The width of the first bands is shown to be . Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.
Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia111The first named author was supported by the St. Petersburg State University grant 6.38.64.2012 as well as by the Chebyshev Laboratory - RF Government grant 11.G34.31.0026 and by JSC “Gazprom Neft”. The first and third named authors were also supported by the Academy of Finland project ”Functional analysis and applications”.
University of Oulu, Department of Electrical and Information Engineering, Mathematics Division, P.O. Box 4500, FI-90401 Oulu, Finland
University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, FI-00014 Helsinki, Finland.
## 1 Introduction
### 1.1 Overview of the results
Research on wave propagation phenomena in periodic media has been very active during many decades. The topics and applications include for example photonic crystals, meta-materials, Bragg gratings of surface plasmon polariton waveguides, energy harvesting in piezoelectric materials as well as surface wave propagation in periodic channels, which is the subject of this paper. A standard mathematical approach consists of linearisation and posing a spectral problem for an elliptic, hopefully self-adjoint, equation or system.
Early on it was noticed that waves propagating in periodic media have spectra with allowed bands separated by forbidden frequency gaps. This phenomenon was first discussed by Lord Rayleigh [22]. It has also attracted some interest in coastal engineering because it provides a possible means of protection against wave damages [11, 14], for example by varying the bottom topography by periodic arrangements of sandbars. The existence of forbidden frequencies is conventionally related to Bragg reflection of water waves by periodic structures. Here, Bragg reflection is an enhanced reflection which occurs when the wavelength of an incident surface wave is approximately twice the wavelength of the periodic structure. This mechanism works, if the waves are relatively long so that the depth changes can effect them [14].
A similar phenomenon may also happen, when waves are propagating along a channel with periodically varying width. In [9], and later [13], the authors studied a channel, the wall of which had a periodic stepped structure. Using resonant interaction theory they were able to verify that significant wave reflection could occur. These results are based on the assumption of small wall irregularities.
Gaps in the continuous spectrum for equations or systems in unbounded waveguides have been studied in many papers, and we refer to [7] for an introduction to the topic. In [19] the authors studied the linear elasticity system and proved the existence of arbitrarily (though still finitely) many gaps, the number of them depending on a small geometric parameter; the approach is similar to Section 3.1, below, and the result is analogous to Corollary 3.2. In the setting of the linear water-wave problem, spectral gaps have been studied in [8], [12], [17] and [4], though the point of view is different from the present work.
In this paper we consider surface wave propagation using the linear water wave equation with spectral Steklov boundary condition on the free water surface, see the equations (1.8)–(1.10), which are called the original problem here. The water-filled domain forms an unbounded periodic channel consisting of infinitely many identical bounded containers connected by apertures of width , see Figure 1.1. The first results, Theorem 3.1 and Corollary 3.2 show that the essential spectrum of the original problem (which is expected to be non-empty due to the unboundedness of the domain) has gaps, and the number of them can be made arbitrarily large depending on the parameter . An explanation of this phenomenon can be outlined rather simply using the Floquet-Bloch theory, though a lot of technicalities will eventually be involved. Namely, if , the domain becomes a disjoint union of infinitely many bounded containers, and the water-wave problem reduces to a problem on a bounded domain (we call it the limit problem), hence it has a discrete spectrum consisting of an increasing sequence of eigenvalues . On the other hand, for , one can use the Gelfand transform to render the original problem into another bounded domain problem depending on the additional parameter . For each fixed this problem again has a sequence of eigenvalues . Moreover, by results of [15], [18], Theorem 3.4.6, and [16], Theorem 2.1, the essential spectrum of the problem (1.8)–(1.10) equals
σ=∞⋃k=1Υϵk , Υϵk={Λϵk(η):η∈[0,2π)}, (1.1)
where the sets are subintervals of the positive real axis, or bands of the spectrum. (For the use of this so called Bloch spectrum in other problems, see for example [FG], or [2].) In general, those bands may overlap making connected, but in Theorem 3.1 we obtain asymptotic estimates for the lower and upper endpoints of : we show that for all and and for some constants . In view of (1.1) this implies the existence of a spectral gap between and for small and such that . However, since the estimates depend also on , we can only open a gap for finitely many , though the number of gaps tends to infinity as .
The asymptotic position (as ) of the gaps is determined more accurately in Theorems 3.5 and 3.6: those main results state that
Υϵk=(Λ0k+Akϵ+O(ϵ3/2),Λ0k+Bkϵ+O(ϵ3/2))
where the numbers depend linearly on the three dimensional capacity of the set . This result also ensures that in case the bands do not degenerate into single points, which means that the spectrum of the original problem indeed has a genuine band-gap structure. Facts concerning the numbers , are discussed after Theorem 3.6.
As for the structure of this paper, we recall in Section 1.2 the exact formulation of the linear water-wave problem, its variational formulation as well as the parameter dependent problem arising from the Gelfand transform, and the limit problem. Section 2 contains the formal asymptotic analysis which relates the spectral properties of the original problem with the limit problem and which is rigorously justified in Secion 3. The main results, Theorems 3.1, 3.5 and 3.6 as well as Corollary 3.2 are also given in Section 3. The proofs are based on the max-min principle and construction of suitable test functions adjusted to the geometric characteristics of the domains under study.
Acknowledgement. The authors want to thank Prof. Sergey A. Nazarov for many discussions on the topic of this work.
### 1.2 Formulation of the problem, operator theoretic tools
Let us proceed with the exact formulation of the problem. We consider an infinite periodic channel (see (1.7)), consisting of water containers connected by small apertures of diameter . The coordinates of the points in the channel are denoted by , and stands for the projection of to the plane . We choose the coordinate system in such a way that the axis of the channel is in -direction and the free surface is in the plane .
###### Definition 1.1.
We describe the geometric assumptions on the periodicity cell in detail, as well as some related technical tools including the cut-off funcions. Let us denote by a domain with a Lipschitz boundary and compact closure such that its intersections with - and -planes are simply connected planar domains with positive area and contain the points and with , respectively; these points are fixed throughout the paper. Then the periodicity cell and its translates are defined by setting (see Figure 1.1)
ϖ={x∈ϖ∙:x3<0,x1∈(0,1)},ϖj={x:(x1−j,x2,x3)∈ϖ}, j∈Z. (1.2)
Furthermore, we assume that the set is a bounded planar domain containing the origin and that the boundary is at least -smooth. We assume that is so small that the set is contained in and . We define the apertures between the container walls as the sets
θϵj={x=(j,x′):ϵ−1(x′−(P2,P3))∈θ}, j∈Z. (1.3)
It is plain that for for all , by the choice of . We shall need at several places a cut-off function
χθ∈C∞0(R3), (1.4)
which is equal to one in a neighbourhood of the set and vanishes outside another compact neighbourhood of . More precisely, we require that
(supp(χθ)+(0,P2,P3))∩{x1=0}⊂∂ϖ, (supp(χθ)+(0,P2,P3)) ∩{x1>0}⊂ϖ (1.5)
(this is possible by the specifications made on ) and vanishes, if or . We also assume that , when . Furthermore, denoting , it follows from the above specifications that , if ; in particular vanishes on the free water surface . Finally, we shall need the scaled cut-off functions
Xϵj=χθ(ϵ−1(x−Pj)). (1.6)
It is plain that also vanishes on for and that for , .
###### Definition 1.2.
The periodic water channel is defined by
Πϵ=⋃j∈Z(ϖj∪θϵj), (1.7)
and it will be the main object of our investigation. The free surface of the channel is denoted by , and the wall and bottom part of the boundary is . The boundary of the isolated container , the periodicity cell, consists of the free surface and the wall and bottom with two apertures and .
###### Remark 1.3.
We shall use the following general notation. Given a domain , the symbol stands for the natural scalar product in , and , , for the standard Sobolev space of order on . The norm of a function belonging to a Banach function space is denoted by . For and , (respectively, ) stands for the Euclidean ball (resp. ball surface) with centre and radius . By (respectively, , , etc.) we mean positive constans (resp. constants depending on a parameter ) which do not depend on functions or variables appearing in the inequalities, but which may still vary from place to place. The gradient and Laplace operators and act in variable , unless otherwise indicated.
In the framework of the linear water-wave theory we consider the spectral Steklov problem in the channel ,
−Δuϵ(x)=0 for all x∈Πϵ, (1.8) ∂nuϵ(x)=0 for a.e. x∈Σϵ, (1.9) ∂zuϵ(x)=λϵuϵ(x) for a.e. x∈Γϵ. (1.10)
Here is the velocity potential, is a spectral parameter related to the frequency of harmonic oscillations and the acceleration of gravity . By the geometric assumptions made above, the outward normal derivative is defined almost everywhere on . It coincides with on the free surface .
The rest of this section is devoted to presenting the operator theoretic tools which will be needed later to prove our results: Gelfand transform, variational formulation of the boundary value problems, and max-min-formulas for eigenvalues. The spectral problem (1.8)–(1.10) can be transformed into a family of spectral problems in the periodicity cell using the Gelfand transform. We briefly recall its definition:
v(y,z)↦ V(y,z,η)=1√2π∑j∈Zexp(−iη(z+j))v(y,z+j), (1.11)
where on the left while and on the right. As is well known, the Gelfand transform establishes an isometric isomorphism between the Lebesgue spaces,
L2(Πϵ)≃L2(0,2π;L2(ϖ)),
where is the Lebesgue space of functions with values in the Banach space endowed with the norm
∥V;L2(0,2π;B)∥=(∫2π0∥V(η);B∥2dη)1/2.
The Gelfand transform is also an isomorphism from the Sobolev space onto for . The space consists of Sobolev functions which satisfy the quasi-periodicity conditions
u(0,x′)=e−iηu(1,x′), (0,x′)∈θϵ0, (1.12) ∂x1u(0,x′)=e−iη∂x1u(1,x′), (0,x′)∈θϵ0, (1.13)
whereas is the Sobolev space with the condition (1.12) only.
Applying the Gelfand transform to the differential equation (1.8) and to the boundary conditions (1.9)–(1.10), we obtain a family of model problems in the periodicity cell parametrized by the dual variable ,
−ΔUϵ(x;η)=0, x∈ϖ, (1.14) ∂nUϵ(x;η)=0, x∈σϵ, (1.15) ∂zUϵ(x;η)=Λϵ(η)Uϵ(x;η), x∈γ, (1.16) Uϵ(0,x′;η)=e−iηUϵ(1,x′;η), x∈θϵ0, (1.17) ∂x1Uϵ(0,x′;η)=e−iη∂x1Uϵ(1,x′;η), x∈θϵ0. (1.18)
Here, is a new notation for the spectral parameter . More details on the use of the Gelfand-transform can be found e.g. in [19], Section 2.
The apertures disappear at so in that case the also quasi-periodicity conditions cease to exist. Hence, we can consider the problem (1.14)–(1.18) as a singular perturbation of the limit spectral problem
−ΔU0(x)=0, x∈ϖ, (1.19) ∂nU0(x)=0, x∈σ, (1.20) ∂zU0(x)=Λ0U0(x), x∈γ (1.21)
with as a spectral parameter.
Our approach to the spectral properties of model and limit problems is similar to [20], Sections 1.2, 1.3.. We first write the variational form of the problem (1.14)–(1.18) for the unknown function as
(∇Uϵ,∇V)ϖ=Λϵ(Uϵ,V)γ, V∈H1ϵ,η(ϖ), (1.22)
and the corresponding variational formulation of the limit problem for reads as
(∇U,∇V)ϖ=Λ(U,V)γ, V∈H1(ϖ). (1.23)
We denote by the space endowed with the new scalar product
(u,v)ϵ=(∇u,∇v)ϖ+(u,v)γ, (1.24)
and define a self-adjoint, positive and compact operator using
(Bϵ(η)u,v)ϵ=(u,v)γ. (1.25)
The problem (1.22) is then equivalent to the standard spectral problem
Bϵ(η)u=Mϵu (1.26)
with another spectral parameter
Mϵ=(1+Λϵ)−1. (1.27)
Clearly, the spectrum of consist of 0 and a decreasing sequence of eigenvalues, which moreover can be calculated from the usual min-max formula
Mϵk(η)=minEkmaxv∈Ek(Bϵ(η)v,v)ϵ(v,v)ϵ, (1.28)
where the minimum is taken over all subspaces of co-dimension . Using (1.24) and (1.25), we can write a max-min formula for the eigenvalues of the problem (1.22):
Λϵk(η)=1Mϵk(η)−1=maxEkminv∈Ek(∇v,∇v)ϖ+(v,v)γ(v,v)γ−1 (1.29) = maxEkminv∈Ek∥∇v;L2(ϖ)∥2∥v;L2(γ)∥2
On the other hand, the connection (1.27) and the properties of the sequence mean that the eigenvalues (1.29) form an unbounded sequence
0≤Λϵ1(η)≤Λϵ2(η)≤…≤Λϵk(η)≤…→+∞. (1.30)
The eigenfunctions can be assumed to form an orthonormal basis in the space . The functions are continuous and -periodic (see for example [5], Ch. 9). Hence the sets
Υϵk={Λϵk(η):η∈[0,2π)} (1.31)
are closed connected segments, which may degenerate into single points; their relation to the original problem was already mentioned in (1.1).
The spectral concepts of the limit problem (1.19)–(1.21) can be treated in the same way as in (1.24)–(1.30). Since the quasi-periodicity conditions vanish for , the space is replaced by ; the norm induced by (1.24) is now equivalent to the original Sobolev norm of . We denote by the operator defined as in (1.24)–(1.25). The limit problem has an eigenvalue sequence like (1.30), however, neither the eigenvalues nor the operator depend on (cf. [19], Section 3). The first eigenvalue equals , and the first eigenfunction is the constant function. Analogously to (1.29) we can write
Λ0k=maxFkminv∈Fk∥∇v;L2(ϖ)∥2∥v;L2(γ)∥2, (1.32)
where again is running over all subspaces of codimension . We denote by
(U0k)∞k=1 (1.33)
an -orthonormal sequence of eigenfunctions corresponding to the eigenvalues (1.32).
###### Lemma 1.4.
For all there exists a constant such that
|U0k(x)|≤Ck , |∇U0k(x)|≤Ck (1.34)
for all , (and hence for all , ).
###### Proof.
Let for example (the other case is treated similarly), and define the domains with boundary such that and still so small that
G2∩{x1=0}⊂∂ϖ and ¯¯¯¯¯¯G2∩{x1>0}⊂ϖ. (1.35)
As a consequence, these domains are smooth enough so that we can use the local elliptic estimates [1], Theorem 15.2, to the solutions of the equation (1.19): this yields for every , a constant such that
∥U0k;Hl+1(Gn∩ϖ)∥≤Cl,k(∥U0k;Hl−1(Gn+1∩ϖ)∥+∥U0k;L2(Gn+1∩ϖ)∥)
for . Applying this first with and we get a bound for and then, with and , for . The standard embeddings and imply the result.
## 2 The formal asymptotic procedure
### 2.1 The case of a simple eigenvalue
To describe the asymptotic behaviour (as ) of the eigenvalues of the problem (1.14)-(1.18) we consider first the case is a simple eigenvalue of the problem (1.19)-(1.21) for some fixed . Let us make the following ansatz:
Λϵk(η)=Λ0k+ϵΛ′k(η)+˜Λϵk(η), (2.1)
where is a correction term and a small remainder to be evaluated and estimated. In this section we derive the expression (2.13) for , cf. also (2.18) and (2.19), and the remainder will be treated in Section 3.2
The corresponding asymptotic ansatz for the eigenfunction reads as follows:
Uϵk(x;η) = U0k(x) + χ0(x)wk0(ϵ−1(x−P0))+χ1(x)wk1(ϵ−1(x−P1)) + ϵU′k(x;η)+˜Uϵk(x;η),
where is as in (1.33). The functions and are of boundary layer type, and is given above (1.6).
The boundary layers depend on the “fast” variables (“stretched” coordinates)
ξj=(ξj1,ξj2,ξj3)=ϵ−1(x−Pj),j=0,1.
They are needed to compensate the fact that the leading term in the expansion (2.1) does not satisfy the quasi-periodicity conditions (1.17)–(1.18). By Lemma 1.4 and the mean value theorem, the eigenfunction has the representation
U0k(x)=U0k(Pj)+O(ϵ),x∈θϵj
near the points . We look for and as the solutions of the problems
Δξ0wk0(ξ0)=0, ξ01>0, ∂ξ01wk0(ξ0)=0, ξ0∈{0}×(R2∖¯¯¯θ), wk0(ξ0)=ak0, ξ0∈{0}×θ,
and
Δξ1wk1(ξ1)=0, ξ11<0, ∂ξ11wk1(ξ1)=0, ξ1∈{0}×(R2∖¯¯¯θ), wk1(ξ1)=ak1, ξ1∈{0}×θ
in the half spaces and , respectively; the meaning of the numbers will be explained below. Both of the functions can be extended to even harmonic functions in the exterior of the set :
Δξjwkj(ξj)=0, ξj∈R3∖({0}ׯ¯¯θ), (2.3) wkj(ξj)=akj, ξj∈∂({0}ׯ¯¯θ).
Furthermore, the problem (2.3) admits a solution (see [21])
wkj(ξj)=akjcap3θ|ξj|+˜wkj(ξj), (2.4) ˜wkj(ξj)=O(|ξj|−2) , ∇ξj˜wkj(ξj)=O(|ξj|−3), (2.5)
where is the 3-dimensional capacity of the set and (2.5) concerns large -behaviour. Moreover, the solution has a finite Dirichlet integral:
(2.6)
for some constant .
We aim to choose the coefficients such that satisfies the quasi-periodicity conditions (1.17)–(1.18) . Clearly, for each
Uϵk(P0;η) = e−iηUϵk(P1;η), ∂x1Uϵk(P0;η) = e−iη∂x1Uϵk(P1;η),
which together with the asymptotic expansion (2.1) yield the relations
U0k(P0)+ak0=e−iη(U0k(P1)+ak1) and ak0=−e−iηak1
for the coefficients. Hence,
ak1=−eiηak0,ak0=12(e−iηU0k(P1)−U0k(P0)). (2.7)
Now we can write a model problem for the main asymptotic correction term :
−ΔU′k(x;η)=ΔWk(x) x∈ϖ, (2.8) (∂z−Λ0k)U′k(x;η)=Λ′k(η)U0k(x), x∈γ, (2.9) ∂nU′k(x;η)=0, x∈σ, (2.10)
where we denote
Wk(x)=(1∑j=0χj(x)akjcap3(θ)|x−Pj|),x∈ϖ,. (2.11)
In addition to , the problem (2.8)–(2.10) will also determine the number in a unique way for every and . This will follow by requiring the solvability condition to hold in the Fredholm alternative, see Lemma 2.1 and its proof, below. Indeed, using the Green formula and the normalization in (1.33) we write ( is the surface measure):
Λ′k(η)=Λ′k(η)∥U0k;L2(γ)∥2= (2.12) = ∫γ(∂zU′k(x;η)−Λ0kU′k(x;η))¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)ds(x)= = ∫∂ϖ(¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)∂nU′k(x;η)−U′k(x;η)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∂nU0k(x))ds(x)= = ∫ϖ¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)ΔU′k(x;η)=−∫ϖ¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)ΔWk(x)dx.
Taking into account that the last integral converges absolutely and using the Green formula again yield
Λ′k(η) = limr→01∑j=0∫Sr(Pj)∩ϖ¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)∂n(−akjcap3(θ)|x−Pj|)ds(x) = −2πcap3(θ)(ak0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(P0)+ak1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(P1));
see (2.11) and Remark 1.3 for notation. According to (2.7) we finally obtain
Λ′k(η)=πcap3(θ)|U0k(P0)−e−iηU0k(P1)|2. (2.13)
###### Lemma 2.1.
Choosing as in (2.13), the problem (2.8)–(2.10) has a solution .
###### Proof.
The variational formulation of the problem (2.8)–(2.10) reads as
(∇U′k,∇V)ϖ−Λ0k(U′k,V)γ=(∇Wk,∇V)ϖ−Λ′k(U0k,V)γ. (2.14)
We remark that the function is harmonic in , and since equals constant one in a neighbourhood of , the function vanishes there, hence, and are smooth as well as uniformly bounded everywhere in . Moreover, .
Using the definition of the operator (cf. (1.24), (1.25) and the remarks above (1.32)) we can rewrite (2.14) as follows:
(U′k,V)0−(Λ0k+1)(BU′k,V)0=(Wk,V)0−Λ′k(BU0k,V)0−(BWk,V)0 (2.15)
which means that must be a solution of the equation
(B−M0k)U′k=−M0k(Wk−BWk−Λ′kBU0k). (2.16)
Notice that is the solution of the homogeneous problem (2.16), so, by the Fredholm alternative, (2.16) is solvable, if and only if the right hand side of it is orthogonal to the function . This condition is satisfied by choosing as above, since
(Wk−BWk−Λ′kBU0k,U0k)0=(∇Wk,∇U0k)ϖ−Λ′k∥U0k;L2(γ)∥2=0,
by (2.12) and . This last identity follows from the first Green formula, because the normal derivative of vanishes on due to the properties of the function , see below (1.5).
### 2.2 The case of a multiple eigenvalue
In this section we complete the asymptotic analysis by studying the behaviour of eigenvalues in the case some has multiplicity greater than one: we have
Λ0k−1<Λ0k=…=Λ0k+m−1<Λ0k+m.
The ansatz (2.1) is used again. Furthermore, as in (1.33) we denote by an orthonormal system of eigenfunctions associated with the eigenvalue . Any eigenfunction corresponding to can be presented as a linear combination
U0(x)=m−1∑j=0αjU0k+j(x).
Analogously to (2.1) we introduce the asymptotic ansatz
Uϵ(x;η) = U0(x) + χ0(x)wk0(ϵ−1(x−P0))+χ1(x)wk1(ϵ−1(x−P1)) + ϵU′(x;η)+˜Uϵ(x;η).
Using the same argumentation as in the previous section we construct the boundary layers , , which satisfy the conditions
wkj(ξj)=akjcap3θ|ξj|+O(|ξj|−2);
here the coefficients come from the equations (2.7), where is replaced by . The main asymptotic term is also treated in the same way as in Section 2.1. To use the Fredholm alternative for finding , , we write
Λ′k+j(η)αj=Λ′k+j(η)(U0,U0k+j)γ,
and making use of the Green formula as above we get
Λ′k+j(η)αj=m−1∑l=0βljαj,
where
βlj=πcap3(θ)(U0k+l(P0)−e−iηU0k+l(P1))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(U0k+j(P0)−e−iηU0k+j(P1)).
Hence, is an eigenvalue of the matrix . This matrix has rank one, because it can be represented in the form , where is a vector with components , . This means that
Λ′k(η) = πcap3(θ)m−1∑l=0|U0k+l(P0)−e−iηU0k+l(P1)|2, (2.18) Λ′k+j(η) = 0,1≤j≤m−1. (2.19)
## 3 Existence and position of spectral gaps
### 3.1 Existence of gaps
The first estimate on the eigenvalues of the problem (1.22) can now be stated as follows.
###### Theorem 3.1.
For any there are numbers and such that for every and any dual variable , the eigenvalues of the problem (1.22) and the eigenvalues of the limit problem (1.23) are related as follows:
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2020-10-21 08:17:20
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https://www.shaalaa.com/question-bank-solutions/o-centre-circle-find-length-radius-if-chord-length-24-cm-distance-9-cm-centre-circle-properties-chord-circle_81476
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SSC (English Medium) Class 8Maharashtra State Board
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# O is Centre of the Circle. Find the Length of Radius, If the Chord of Length 24 Cm is at a Distance of 9 Cm from the Centre of the Circle. - SSC (English Medium) Class 8 - Mathematics
ConceptProperties of Chord of a Circle
#### Question
O is centre of the circle. Find the length of radius, if the chord of length 24 cm is at a distance of 9 cm from the centre of the circle.
#### Solution
Join OA.
Let the perpendicular drawn from point O to the chord AB be P.
We know that the perpendicular drawn from the centre of the circle to the chord bisects the chord.
So, AP = "AB"/2 = 24/2 = 12 cm
In Δ OPA,
We apply the Pythagoras theorem,
OP² + AP² = OA²
⇒ 9² + 12² = OA²
⇒ OA² = 81 +144 = 225
⇒ OA = √225 15 cm
Hence, the radius of the circle is 15 cm.
Is there an error in this question or solution?
#### APPEARS IN
Balbharati Solution for Balbharati Class 8 Mathematics (2019 to Current)
Chapter 17: Circle : Chord and Arc
Practice Set 17.1 | Q: 3 | Page no. 116
Solution O is Centre of the Circle. Find the Length of Radius, If the Chord of Length 24 Cm is at a Distance of 9 Cm from the Centre of the Circle. Concept: Properties of Chord of a Circle.
S
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2020-01-26 11:03:17
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https://dergipark.org.tr/en/pub/toleho/writing-rules
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The Title Page
Chapter Title
The chapter title needs be short. It can be two title lines (all in UPPER CASE), each containing a maximum of 26 characters (including blank spaces), with no word hyphenated from the first to the second line.
It is also possible to opt for the title: subtitle format. That is, THE TITLE ALL IN UPPER CASE: The Subtitle in Lower Case. In this instance, the subtitle line can contain 30 characters (including blank spaces).
Author’s Name
Right under the chapter title, the name of the author appears in online, followed by the name of his/her institution and country on the next line.
Same format for additional authors.
Abstract
The abstract should be between 120 and 150 words, including keywords. Please limit keywords to five or not more than seven, and avoid using obvious ones such as “tourism” or “wellness”.
Biosketch
The biosketch should include the name(s), the postal/email address of the first author, and a very brief statement about the research interest(s) of the author(s). Its length, whether for single or for all co-authors, should be between 60 and 75 words.
Acknowledgments
To protect the anonymity of the review process, acknowledgments are included in the title page. If eventually accepted for publication, appropriate format will be suggested at that point.
Note: To insure anonymity, name(s) and biosketch(es) of the author(s) will be deleted by the editors if the article is selected to be sent to a panel of outside referees.
The Manuscript
The article should be made up of six distinct parts: the introduction, literature review, method, findings and discussion, conclusion and recommendation and appendix (optional) followed by references, tables, and figures, as outlined below.
Subsections / Sub-Subsections can be used only for the sections 2, 3, 4 and 5.
Example;
3. Method
3.1. Sampling
3.2. Measure
3.3. Data Analysis
Framework of Paper:
Abstract*
1. Introduction*
2. Literature Review*
3. Method*
4. Findings and Discussion*
5. Conclusion and Recommendation\Implications
6. Appendix (optional)
References*
The Introduction Section
The heading for this section is simply INTRODUCTION (IN UPPER CASE).
The purpose of this section is to set the stage for the main discussion.
It is preferred that this section ends by stating the purpose of the chapter, but without outlining what sequentially will follow.
If the introduction is short, it appears as one undivided piece. A long introduction of more than 1,500 words can be subdivided.
The Main Section
This is the main body of the chapter, headed with a section heading capturing the theme/scope/nature of the chapter, ALL IN UPPER CASE. Often this heading is somewhat similar to the chapter title itself.
Its opening discussion begins immediately after the section heading. This should include a literature review on the topic so that the book becomes a documentation of work-to-date in the topic area. Please use present tense (not past tense) for the literature review.
The study methodology, if applicable, is then introduced. Then the chapter proceeds to discuss the study findings and their theoretical and practical applications. The discussion in this section is Subtitled as Appropriate (again in a Level 2 heading, in italics).
In general, this is how this discussion section is headed/sub headed.
The Conclusion Section
This section, headed simply CONCLUSION, can begin with a restatement of the research problem, followed by a summary of the research conducted and the findings.
It then proceeds to make concluding remarks, offering insightful comments on the research theme, commenting on the contributions that the study makes to the formation of knowledge in this field, even also suggesting research gaps and themes/challenges in years ahead.
To do justice to the chapter, this section should not be limited to one or two paragraphs. Its significance/contribution deserves to be insightfully featured here, including remarks which they had been added to the earlier sections would have been premature.
If the CONCLUSION section is longer than 1,000 words (an average length), one may choose to subdivide it into appropriate Subheadings in Italics.
Tables and Figures
Each table (single space) or figure appears on a separate sheet at the end of the chapter, with all illustrations considered as Figures (not charts, diagrams, or exhibitions).The title for tables should be above whereas titles for figures should appear below the table.
Both tables and figures are identified with Arabic numerals, followed with a very brief one-line descriptive title (about 10 words). Example:
Table 1: Table Title (Times New Roman, Regular, 11pt, Centered)
TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE TABLE
(Reference –If necessary)
Figure 1: Figure Title (Times New Roman, Regular, 11pt, Centered)
(Reference – If necessary)
The data in tables should be presented in columns with non-significant decimal places omitted. All table columns must have extremely brief headings.
Clean and uncrowded tables and figures are sought. Notes and comments, including references, are incorporated in the paper text, where the table or figure is first mentioned. If any remain, they are “telegraphically” footnoted, using alphabetic superscripts (not asterisks). References, if not already in the text, take this format: (Oncel, 2015:34). All such references are also included fully in the Reference list. Tables and figures generated by the author need not be sourced. Proof of permission to reproduce previously published material must be supplied with the paper.
Tables should not be boxed and girded. No vertical bars can be added and the use of horizontal bars should be limited to 3 or 4, to mark the table heading and its end. See recent issues of Annals for examples.
Figures should be in “camera ready” or “ready-to-go” format suitable for reproduction without retouching. No figures (or tables) can be larger than one page, preferably ½ pages or less in size. All lettering, graph lines, and points on graphs should be sufficiently large to permit reproduction.
When essential, it can be also published photographs (preferably black and white), to be submitted electronically at the end of the paper.
Only very few tables and figures (preferably, less than five in total) central to the discussion can be accommodated. The rest, including those with limited value/data, should be deleted and instead their essence incorporated into the body of the text. All tables and figures (including photos) must appear in “portrait”, not “landscape”, format.
In-Text Citation
The format for making references in the text is as follows:
• Single reference: Emir (2013) states that . . . . Or it is emphasized that . . . . (Emir, 2013).
• Multiple references: (Aksöz 2017; Bayraktaroğlu 2016; Özel 2014; Yilmaz, 2013; Yüncü 2013). Please note that authors in this situation appear in alphabetical order (also note the use of punctuation and spacing).
• Using specific points from a paper, including direct quotations or referring to a given part of it: (Asmadili & Yüksek 2017, pp. 16-17).This reference appears at the end of the quotation. Please note that there is no space between the colon and the page numbers.
• Longer quotations (50 words or longer) appear indented on both margins, ending with the reference: . . . (2004, p. 37).
• Multi-author sources, when cited first in the paper, should name all co-authors, for example (Gunay Aktas, Boz, & Ilbas 2015); thereafter, the last name of the first author, followed with et al (Gunay Aktas et al. 2015). Please note that et al is not followed with a period.
• References to personal communication appear parenthetically: . . . (Interview with the minister of tourism in 2006) and are not included in the reference list.
• Works by association, corporation, government policies: First citation: United Nations World Tourism Organization (UNWTO), 2014). For subsequent citation: (UNWTO, 2014). Please avoid introducing acronyms which are used less than about five times in the whole text.
• Unfamiliar terms, particularly those in foreign languages, need to appear in italics, followed with their meaning in parenthesis.
• The whole text must be written in the third person. The only exception is when the usage occurs in direct quotes.
• For the sake of uniformity and consistency, American spelling should be used throughout the paper. Please utilize the Spell Check feature of the computer (click on the American spelling option) to make sure that all deviations are corrected, even in direct quotations (unless the variation makes a difference in the discussion).
• The use of bullets and numbers to list itemized points or statements should be avoided. If it is necessary to delineate certain highlights or points, then this can be worked out in a paragraph format: …. One, tourism…. implemented. Two, a search goal …. is understood. Three, ….
• All amounts, both in the text and in tables/figures, must be given in American dollars; when important, their equivalents may be added in parentheses. If the chapter does not deal with the United States, please use “US$” in first instance, and only “$” subsequently.
• Numbers under 10 are spelled out, but all dollar amounts appear in Arabic numerals.
• Please use % after numbers (ie, 15%, not 15 percent).
• Frequent use of keywords or pet words must be avoided. If the chapter is dealing with “wellness tourism” it should be recognized that the reader knows that the chapter is dealing with this subject. Such uses/repetitions must be carefully avoided.
• Please use “tourist” when referring to the person (and please avoid using “traveler” and “visitor”—unless the article is defining and distinguishing among them) and use “tourism” when discussing the industry/phenomenon. “Travel” and “tourism” are not used synonymously.
• Very long or very short paragraphs should be avoided (average length: 15 lines or 150 words).
References
The heading for this bibliographic list is simply REFERENCES, and is centered. All entries under this heading appear in alphabetic order of authors. Only references cited in the text are listed and all references listed must be cited in the text. Reference lists of all chapters are eventually consolidated by the volume editor into one and placed at the end of the book.
Journal Articles:
Dogru T., Isik, C., & SirakayaTurk E. (2019). The Balance of Trade and Exchange Rates: Theory and Contemporary Evidence From Tourism, Tourism Management, 74 (4), pp. 12-23.
Sezgin, E., & Duz, B. (2018). Testing the proposed “GuidePerf” scale for tourism: performances of tour guides in relation to various tour guiding diplomas. Asia Pacific Journal of Tourism Research, 23 (2), pp. 170-182.
Ozan, A. E. (2015). Perceived Image Of Cittaslow By Tourism Students: The Case of Faculty of Tourism, Anadolu University-Turkey. Annals of Faculty of Economics, 1 (2), pp. 331-339.
Online Journal Articles:
Yuksek, G. (2013). Role of Information Technologies In Travel Business And Case Of Global Distribution System: AMADEUS, AJIT‐e: OnlineAcademic Journal ofInformation Technology, 4(12), pp. 17-28, Retrieved from //…..
Conference Prooceedings:
Yilmaz, A., & Yetgin, D. (2017). Assessment on Thermal Tourism Potential in Eskisehir through the Tour Guides’ Perspective. 5th International Research Forum on Guided Tours, (5th IRFGT), University of Roskilde, Denmark, pp.70-84.
Book:
Kozak, N. (2014). Academic Journal Guides of Turkey, 1st Edition, Ankara: Detay Publishing
Article or Chapter in Edited Book:
Kaya-Sayarı, B., & Yolal, M. (2019). The Postmodern Turn in Tourism Ethnography: Writing against Culture. In H. Andrews, T. Jimura, & L. Dixon (Eds), Tourism Ethnographies, Ethics, Methods, Application and Reflexivity (pp. 157-173). New York, NY: Routledge.
More than one Contribution by the Same Author:
Coşkun, I.O., & Ozer, M. (2014). Reexamination of the Tourism Led Growth Hypothesis under Growth and Tourism Uncertainties in Turkey, European Journal of Business and Social Sciences, 3(8), pp. 256-272.
Coşkun, I.O., & Ozer, M. (2011). MGARCH Modeling of Inbound Tourism Demand Volatility in Turkey. Management of International Business and Economic Systems (MIBES) Transactions International Journal, 5(1), pp. 24-40.
If an author has two or more publications in the same year, they are distinguished by placing a, b, etc. after the year. For example, 1998a or 1998b, and they are referred to accordingly in the text.
Thesis/Dissertation:
Toker, A. (2011). The Role of Tourist Guides at Sustainability of Cultural Tourism: Ankara Sample, Unpublished Master’s Thesis, Anadolu University, Eskisehir, Turkey.
Bayraktaroğlu, E. (2019). Establishing Theoretical Background of Destination Value, Unpublished Doctoral Dissertation, Anadolu University, Eskişehir, Turkey.
Same as journal articles (with article title, volume number, etc., as above).
Internet:
Name of the Site, Date, Title of the Article/Publication Sourced .
If the date the site was visited is important: 2004 Title of the Article/Publication Sourced < //www…..> (18 November 2005).
Personal Communications/Interviews:
NB In all above instances, the author’s name lines up with the left margin, the publication date
Making Submissions via DergiPark
The article—prepared according to above specifications (covering text, references, tables, and figures)—should be sent to Journal of Tourism, Leisure and Hospitality (TOLEHO) via DergiPark. Please follow this link to reach the submission page.
Please, use the links below to access the visual descriptions of the application steps;
Abstracting & Indexing
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Making Submissions via DergiPark
The article—prepared according to author guidelines (covering text, references, tables, and figures)—should be sent to Journal of TOLEHO via DergiPark.
Please, use the links below to access the visual descriptions of the submission steps;
Full Open Access Strategy
Journal of TOLEHO is fully sponsored by Anadolu University Faculty of Tourism. Therefore there aren't any article submission, processing or publication charges
There are also no charges for rejected articles, no proofreading charges, and no surcharges based on the length of an article, figures or supplementary data etc. All items (editorials, corrections, addendums, retractions, comments, etc.) are published free of charge.
Journal of TOLEHO is an open access journal which means that all content is freely available without charge to the users or institutions. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the author. This is in accordance with the BOAI definition of open access.
Therefore, all articles published will be immediately and permanently free to read and download. All items has their own unique URL and PDF file.
All items published by the Journal of Tourism, Leisure and Hospitality are licensed under a Creative Commons Attribution 4.0 International License.
The licence permits others to use, reproduce, disseminate or display the article in any way, including for commercial purposes, so long as they credit the author for the original creation.
Authors retain copyright and grant the journal exclusive right of first publication with the work simultaneously licensed under a Creative Commons Attribution 4.0 International License.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
However, Anadolu University Press can also demand to make an additional license agreement with the corresponding author of the study after first publication, in order to publish the manuscript in full text on various other platforms (web page, databases, indexes, promotion circles and etc.).
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2023-01-29 18:09:11
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https://math.stackexchange.com/questions/476338/help-with-formulating-a-linear-programming-problem
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# Help with formulating a linear programming problem
I have the following linear programming problem I would like to be verified:
I have sketched the problem in the following picture:
Here is my attempted solution:
I figured that I have ten variables (corresponding to the colored lines, between each building): $x_1, x_2, ..., x_{10}$ which are the amounts (in tons) of raw materials from Source 1 --> Plant A, Source 1 --> Plant B, Source 2 --> Plant A, Source 2 --> Plant B, Plant A --> Market 1, Plant A --> Market 2, ............, Plant B --> Market 3.
The formulation I get is the following:
$$\text{minimize}\;\;\; 1x_1 + 1.5x_2 + 2x_3 + 1.5x_4 + 4x_5 + 2x_6 + 1x_7 + 3x_8 + 4x_9 + 2x_{10}$$ $$\text{subject to}\;\;\;x_1 + x_2 = 10$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x_3 + x_4 = 15$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x_5 + x_8 = 8$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x_6 + x_9 = 14$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x_7 + x_{10} = 3$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x_1 \ge 0, x_2 \ge 0, \cdots, x_{10} \ge 0.$$
I would appreciate if someone could verify the correctness of my answer :) Thank you for any help!
• It looks correct and well presented to me. You might consider to reduce the number of variables. Example: $x_8 = 8 - x_5$. You put the source and market constraints as equalities. In real situations, they might be inequalities. – Axel Kemper Aug 26 '13 at 7:25
I think the term $1x_9$ is wrong in the minimisation, it should be $4x_9$. Rest is correct. Proceed further.
Your answer looks correct although I would advise you for the sake of clarity to introduce new variables separately for markets, plants and sources (so for instance $x_i$ for the sources $y_i$ for the plants etc.). Also, instead of writing equality signs in some of the constraints, it might be neater to write inequality signs,(for instance, the 3 market centers require... actually means at least) although it doesn't really matter for the outcome and your answer would also be considered correct. Good job :-)
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2019-09-18 16:23:41
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https://xadupre.github.io/machinelearningext/mlnetdocs/idataviewtypesystem.html
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# IDataView Type System¶
## Overview¶
The IDataView system consists of a set of interfaces and classes that provide efficient, compositional transformation of and cursoring through schematized data, as required by many machine-learning and data analysis applications. It is designed to gracefully and efficiently handle both extremely high dimensional data and very large data sets. It does not directly address distributed data, but is suitable for single node processing of data partitions belonging to larger distributed data sets.
While IDataView is one interface in this system, colloquially, the term IDataView is frequently used to refer to the entire system. In this document, the specific interface is written using fixed pitch font as IDataView.
IDataView is the data pipeline machinery for ML.NET. The ML.NET codebase has an extensive library of IDataView related components (loaders, transforms, savers, trainers, predictors, etc.). More are being worked on.
The name IDataView was inspired from the database world, where the term table typically indicates a mutable body of data, while a view is the result of a query on one or more tables or views, and is generally immutable. Note that both tables and views are schematized, being organized into typed columns and rows conforming to the column types. Views differ from tables in several ways:
• Views are immutable; tables are mutable.
• Views are composable – new views can be formed by applying transformations (queries) to other views. Forming a new table from an existing table involves copying data, making them decoupled—the new table is not linked to the original table in any way.
• Views are virtual; tables are fully realized/persisted.
Note that immutability and compositionality are critical enablers of technologies that require reasoning over transformation, like query optimization and remoting. Immutability is also key for concurrency and thread safety.
This document includes a very brief introduction to some of the basic concepts of IDataView, but then focuses primarily on the IDataView type system.
Why does IDataView need a special type system? The .NET type system is not well suited to machine-learning and data analysis needs. For example, while one could argue that typeof(double[]) indicates a vector of double values, it explicitly does not include the dimensionality of the vector/array. Similarly, there is no good way to indicate a subset of an integer type, for example integers from 1 to 100, as a .NET type. In short, there is no reasonable way to encode complete range and dimensionality information in a System.Type.
In addition, a well-defined type system, including complete specification of standard data types and conversions, enables separately authored components to seamlessly work together without surprises.
### Basic Concepts¶
IDataView, in the narrow sense, is an interface implemented by many components. At a high level, it is analogous to the .Net interface IEnumerable<T>, with some very significant differences.
While IEnumerable<T> is a sequence of objects of type T, IDataView is a sequence of rows. An IDataView object has an associated ISchema object that defines the IDataView’s columns, including their names, types, indices, and associated metadata. Each row of the IDataView has a value for each column defined by the schema.
Just as IEnumerable<T> has an associated enumerator interface, namely IEnumerator<T>, IDataView has an associated cursor interface, namely IRowCursor. In the enumerable world, an enumerator object implements a Current property that returns the current value of the iteration as an object of type T. In the IDataView world, an IRowCursor object encapsulates the current row of the iteration. There is no separate object that represents the current row. Instead, the cursor implements methods that provide the values of the current row, when requested. Additionally, the methods that serve up values do not require memory allocation on each invocation, but use sharable buffers. This scheme significantly reduces the memory allocations needed to cursor through data.
Both IDataView and IEnumerable<T> present a read-only view on data, in the sense that a sequence presented by each is not directly mutable. “Modifications” to the sequence are accomplished by additional operators or transforms applied to the sequence, so do not modify any underlying data. For example, to normalize a numeric column in an IDataView object, a normalization transform is applied to the sequence to form a new IDataView object representing the composition. In the new view, the normalized values are contained in a new column. Often, the new column has the same name as the original source column and “replaces” the source column in the new view. Columns that are not involved in the transformation are simply “passed through” from the source IDataView to the new one.
Detailed specifications of the IDataView, ISchema, and IRowCursor interfaces are in other documents.
### Column Types¶
Each column in an IDataView has an associated column type. The collection of column types is open, in the sense that new code can introduce new column types without requiring modification of all IDataView related components. While introducing new types is possible, we expect it will also be relatively rare.
All column type implementations derive from the abstract class ColumnType. Primitive column types are those whose implementation derives from the abstract class PrimitiveType, which derives from ColumnType.
### Representation Type¶
A column type has an associated .Net type, known as its representation type or raw type.
Note that a column type often contains much more information than the associated .Net representation type. Moreover, many distinct column types can use the same representation type. Consequently, code should not assume that a particular .Net type implies a particular column type.
### Standard Column Types¶
There is a set of predefined standard column types, divided into standard primitive types and vector types. Note that there can be types that are neither primitive nor vector types. These types are not standard types and may require extra care when handling them. For example, a PictureType value might require disposing when it is no longer needed.
Standard primitive types include the text type, the boolean type, numeric types, and key types. Numeric types are further split into floating-point types, signed integer types, and unsigned integer types.
A vector type has an associated item type that must be a primitive type, but need not be a standard primitive type. Note that vector types are not primitive types, so vectors of vectors are not supported. Note also that vectors are homogeneous—all elements are of the same type. In addition to its item type, a vector type contains dimensionality information. At the basic level, this dimensionality information indicates the length of the vector type. A length of zero means that the vector type is variable length, that is, different values may have different lengths. Additional detail of vector types is in a subsequent section. Vector types are instances of the sealed class VectorType, which derives from ColumnType.
This document uses convenient shorthand for standard types:
• TX: text
• BL: boolean
• R4, R8: single and double precision floating-point
• I1, I2, I4, I8: signed integer types with the indicated number of bytes
• U1, U2, U4, U8: unsigned integer types with the indicated number of bytes
• UG: unsigned type with 16-bytes, typically used as a unique ID
• TS: timespan, a period of time
• DT: datetime, a date and time but no timezone
• DZ: datetime zone, a date and time with a timezone
• U4[100-199]: A key type based on U4 representing legal values from 100 to 199, inclusive
• V<R4,3,2>: A vector type with item type R4 and dimensionality information [3,2]
See the sections on the specific types for more detail.
The IDataView system includes many standard conversions between standard primitive types. A later section contains a full specification of these conversions.
### Default Value¶
Each column type has an associated default value corresponding to the default value of its representation type, as defined by the .Net (C# and CLR) specifications.
The standard conversions map source default values to destination default values. For example, the standard conversion from TX to R8 maps the empty text value to the value zero. Note that the empty text value is distinct from the missing text value, as discussed next.
### Missing Value¶
Most of the standard primitive types support the notion of a missing value. In particular, the text type, floating-point types, signed integer types, and key types all have an internal representation of missing. We follow R’s lead and denote such values as NA.
Unlike R, the standard primitive types do not distinguish between missing and invalid. For example, in floating-point arithmetic, computing zero divided by zero, or infinity minus infinity, produces an invalid value known as a NaN (for Not-a-Number). R uses a specific NaN value to represent its NA value, with all other NaN values indicating invalid. The IDataView standard floating-point types do not distinguish between the various NaN values, treating them all as missing/invalid.
A standard conversion from a source type with NA to a destination type with NA maps NA to NA. A standard conversion from a source type with NA to a destination type without NA maps NA to the default value of the destination type. For example, converting a text NA value to R4 produces a NaN, but converting a text NA to U4 results in zero. Note that this specification does not address diagnostic user messages, so, in certain environments, the latter situation may generate a warning to the user.
Note that a vector type does not support a representation of missing, but may contain NA values of its item type. Generally, there is no standard mechanism faster than O(N) for determining whether a vector with N items contains any missing values.
For further details on missing value representations, see the sections detailing the particular standard primitive types.
### Vector Representations¶
Values of a vector type may be represented either sparsely or densely. A vector type does not mandate denseness or sparsity, nor does it imply that one is favored over the other. A sparse representation is semantically equivalent to a dense representation having the suppressed entries filled in with the default value of the item type. Note that the values of the suppressed entries are emphatically not the missing/NA value of the item type, unless the missing and default values are identical, as they are for key types.
A column in an ISchema can have additional column-wide information, known as metadata. For each string value, known as a metadata kind, a column may have a value associated with that metadata kind. The value also has an associated type, which is a compatible column type.
For example:
• A column may indicate that it is normalized, by providing a BL valued piece of metadata named IsNormalized.
• A column whose type is V<R4,17>, meaning a vector of length 17 whose items are single-precision floating-point values, might have SlotNames metadata of type V<TX,17>, meaning a vector of length 17 whose items are text.
• A column produced by a scorer may have several pieces of associated metadata, indicating the “scoring column group id” that it belongs to, what kind of scorer produced the column (for example, binary classification), and the precise semantics of the column (for example, predicted label, raw score, probability).
The ISchema interface, including the metadata API, is fully specified in another document.
## Text Type¶
The text type, denoted by the shorthand TX, represents text values. The TextType class derives from PrimitiveType and has a single instance, exposed as TextType.Instance. The representation type of TX is an immutable struct known as DvText. A DvText value represents a sequence of characters whose length is contained in its Length field. The missing/NA value has a Length of -1, while all other values have a non-negative Length. The default value has a Length of zero and represents an empty sequence of characters.
In text processing transformations, it is very common to split text into pieces. A key advantage of using DvText instead of System.String for text values is that these splits require no memory allocation—the derived DvText references the same underlying System.String as the original DvText does. Another reason that System.String is not ideal for text is that we want the default value to be empty and not NA. For System.String, the default value is null, which would be a more natural representation for NA than for empty text. By using a custom struct wrapper around a portion (or span) of a System.String, we address both the memory efficiency and default value problems.
## Boolean Type¶
The standard boolean type, denoted by the shorthand BL, represents true/false values. The BooleanType class derives from PrimitiveType and has a single instance, exposed as BooleanType.Instance. The representation type of BL is the DvBool enumeration type, logically stored as sbyte:
DvBool | sbyte Value ——–:|:————- NA | -128 False | 0 True | 1
The default value of BL is DvBool.False and the NA value of BL is DvBool.NA. Note that the underlying type of the DvBool enum is signed byte and the default and NA values of BL align with the default and NA values of I1.
There is a standard conversion from TX to BL. There are standard conversions from BL to all signed integer and floating point numeric types, with DvBool.False mapping to zero, DvBool.True mapping to one, and DvBool.NA mapping to NA.
## Number Types¶
The standard number types are all instances of the sealed class NumberType, which is derived from PrimitiveType. There are two standard floating-point types, four standard signed integer types, and four standard unsigned integer types. Each of these is represented by a single instance of NumberType and there are static properties of NumberType to access each instance. For example, to test whether a variable type represents I4, use the C# code type == NumberType.I4.
Floating-point arithmetic has a well-deserved reputation for being troublesome. This is primarily because it is imprecise, in the sense that the result of most operations must be rounded to the nearest representable value. This rounding means, among other side effects, that floating-point addition and multiplication are not associate, nor satisfy the distributive property.
However, in many ways, floating-point arithmetic is the best-suited system for arithmetic computation. For example, the IEEE 754 specification mandates precise graceful overflow behavior—as results grow, they lose resolution in the least significant digits, and eventually overflow to a special infinite value. In contrast, when integer arithmetic overflows, the result is a non- sense value. Trapping and handling integer overflow is expensive, both in runtime and development costs.
The IDataView system supports integer numeric types mostly for data interchange convenience, but we strongly discourage performing arithmetic on those values without first converting to floating-point.
### Floating-point Types¶
The floating-point types, R4 and R8, have representation types System.Single and System.Double. Their default values are zero. Any NaN is considered an NA value, with the specific Single.NaN and Double.NaN values being the canonical NA values.
There are standard conversions from each floating-point type to the other floating-point type. There are also standard conversions from text to each floating-point type and from each integer type to each floating-point type.
### Signed Integer Types¶
The signed integer types, I1, I2, I4, and I8, have representation types Sytem.SByte, System.Int16, System.Int32, and System.Int64. The default value of each of these is zero. Each of these has a non-zero value that is its own additive inverse, namely (-2)^^{8n-1}, where n is the number of bytes in the representation type. This is the minimum value of each of these types. We follow R’s lead and use these values as the NA values.
There are standard conversions from each signed integer type to every other signed integer type. There are also standard conversions from text to each signed integer type and from each signed integer type to each floating-point type.
Note that we have not defined standard conversions from floating-point types to signed integer types.
### Unsigned Integer Types¶
The unsigned integer types, U1, U2, U4, and U8, have representation types Sytem.Byte, System.UInt16, System.UInt32, and System.UInt64, respectively. The default value of each of these is zero. These types do not have an NA value.
There are standard conversions from each unsigned integer type to every other unsigned integer type. There are also standard conversions from text to each unsigned integer type and from each unsigned integer type to each floating- point type.
Note that we have not defined standard conversions from floating-point types to unsigned integer types, or between signed integer types and unsigned integer types.
## Key Types¶
Key types are used for data that is represented numerically, but where the order and/or magnitude of the values is not semantically meaningful. For example, hash values, social security numbers, and the index of a term in a dictionary are all best modeled with a key type.
The representation type of a key type, also called its underlying type, must be one of the standard four .Net unsigned integer types. The NA and default values of a key type are the same value, namely the representational value zero.
Key types are instances of the sealed class KeyType, which derives from PrimitiveType.
In addition to its underlying type, a key type specifies:
• A count value, between 0 and int.MaxValue, inclusive
• A “minimum” value, between 0 and ulong.MaxValue, inclusive
• A Boolean value indicating whether the values of the key type are contiguous
Regardless of the minimum and count values, the representational value zero always means NA and the representational value one is always the first valid value of the key type.
Notes:
• The Count property returns the count of the key type. This is of type int, but is required to be non-negative. When Count is zero, the key type has no known or useful maximum value. Otherwise, the legal representation values are from one up to and including Count. The Count is required to be representable in the underlying type, so, for example, the Count value of a key type based on System.Byte must not exceed 255. As an example of the usefulness of the Count property, consider the KeyToVector transform implemented as part of ML.NET. It maps from a key type value to an indicator vector. The length of the vector is the Count of the key type, which is required to be positive. For a key value of k, with 1 ≤ k ≤ Count, the resulting vector has a value of one in the (k-1)th slot, and zero in all other slots. An NA value (with representation zero) is mapped to the all- zero vector of length Count.
• For a key type with positive Count, a representation value should be between 0 and Count, inclusive, with 0 meaning NA. When processing values from an untrusted source, it is best to guard against values bigger than Count and treat such values as equivalent to NA.
• The Min property returns the minimum semantic value of the key type. This is used exclusively for transforming from a representation value, where the valid values start at one, to user facing values, which might start at any non-negative value. The most common values for Min are zero and one.
• The boolean Contiguous property indicates whether values of the key type are generally contiguous in the sense that a complete sampling of representation values of the key type would cover most, if not all, values from one up to their max. A true value indicates that using an array to implement a map from the key type values is a reasonable choice. When false, it is likely more prudent to use a hash table.
• A key type can be non-Contiguous only if Count is zero. The converse however is not true. A key type that is contiguous but has Count equal to zero is one where there is a reasonably small maximum, but that maximum is unknown. In this case, an array might be a good choice for a map from the key type.
• The shorthand for a key type with representation type U1, and semantic values from 1000 to 1099, inclusive, is U1[1000-1099]. Note that the Min value of this key type is outside the range of the underlying type, System.Byte, but the Count value is only 100, which is representable in a System.Byte. Recall that the representation values always start at 1 and extend up to Count, in this case 100.
• For a key type with representation type System.UInt32 and semantic values starting at 1000, with no known maximum, the shorthand is U4[1000-*].
There are standard conversions from text to each key type. This conversion parses the text as a standard non-negative integer value and honors the Min and Count values of the key type. If a parsed numeric value falls outside the range indicated by Min and Count, or if the text is not parsable as a non-negative integer, the result is NA.
There are standard conversions from one key type to another, provided:
• The source and destination key types have the same Min and Count values.
• Either the number of bytes in the destination’s underlying type is greater than the number of bytes in the source’s underlying type, or the Count value is positive. In the latter case, the Count is necessarily less than 2k, where k is the number of bits in the destination type’s underlying type. For example, U1[1-*] can be converted to U2[1-*], but U2[1-*] cannot be converted to U1[1-*]. Also, U1[1-100] and U2[1-100] can be converted in both directions.
## Vector Types¶
### Introduction¶
Vector types are one of the key innovations of the IDataView system and are critical for high dimensional machine-learning applications.
For example, when processing text, it is common to hash all or parts of the text and encode the resulting hash values, first as a key type, then as indicator or bag vectors using the KeyToVector transform. Using a k-bit hash produces a key type with Count equal to 2^^k, and vectors of the same length. It is common to use 20 or more hash bits, producing vectors of length a million or more. The vectors are typically very sparse. In systems that do not support vector-valued columns, each of these million or more values is placed in a separate (sparse) column, leading to a massive explosion of the column space. Most tabular systems are not designed to scale to millions of columns, and the user experience also suffers when displaying such data. Moreover, since the vectors are very sparse, placing each value in its own column means that, when a row is being processed, each of those sparse columns must be queried or scanned for its current value. Effectively the sparse matrix of values has been needlessly transposed. This is very inefficient when there are just a few (often one) non-zero entries among the column values. Vector types solve these issues.
A vector type is an instance of the sealed VectorType class, which derives from ColumnType. The vector type contains its ItemType, which must be a PrimitiveType, and its dimensionality information. The dimensionality information consists of one or more non-negative integer values. The VectorSize is the product of the dimensions. A dimension value of zero means that the true value of that dimension can vary from value to value.
For example, tokenizing a text by splitting it into multiple terms generates a vector of text of varying/unknown length. The result type shorthand is V<TX,*>. Hashing this using 6 bits then produces the vector type V<U4[0-63],*>. Applying the KeyToVector transform then produces the vector type V<R4,*,64>. Each of these vector types has a VectorSize of zero, indicating that the total number of slots varies, but the latter still has potentially useful dimensionality information: the vector slots are partitioned into an unknown number of runs of consecutive slots each of length 64.
As another example, consider an image data set. The data starts with a TX column containing URLs for images. Applying an ImageLoader transform generates a column of a custom (non-standard) type, Picture<*,*,4>, where the asterisks indicate that the picture dimensions are unknown. The last dimension of 4 indicates that there are four channels in each pixel: the three color components, plus the alpha channel. Applying an ImageResizer transform scales and crops the images to a specified size, for example, 100x100, producing a type of Picture<100,100,4>. Finally, applying a ImagePixelExtractor transform (and specifying that the alpha channel should be dropped), produces the vector type V<R4,3,100,100>. In this example, the ImagePixelExtractor re-organized the color information into separate planes, and divided each pixel value by 256 to get pixel values between zero and one.
### Equivalence¶
Note that two vector types are equivalent when they have equivalent item types and have identical dimensionality information. To test for compatibility, instead of equivalence, in the sense that the total VectorSize should be the same, use the SameSizeAndItem method instead of the Equals method (see the ColumnType code below).
### Representation Type¶
The representation type of a vector type is the struct VBuffer<T>, where T is the representation type of the item type. For example, the representation type of V<R8,10> is VBuffer<double>. When the vector type’s VectorSize is positive, each value of the type will have length equal to the VectorSize.
The struct VBuffer<T>, sketched below, provides both dense and sparse representations and encourages cooperative buffer sharing. A complete discussion of VBuffer<T> and associated coding idioms is in another document.
Notes:
• VBuffer<T> contains four public readonly fields: Length, Count, Values, and Indices.
• Length is the logical length of the vector, and must be non-negative.
• Count is the number of items explicitly represented in the vector. Count is non-negative and less than or equal to Length.
• When Count is equal to Length, the vector is dense. Otherwise, the vector is sparse.
• The Values array contains the explicitly represented item values. The length of the Values array is at least Count, but not necessarily equal to Count. Only the first Count items in Values are part of the vector; any remaining items are garbage and should be ignored. Note that when Count is zero, Values may be null.
• The Indices array is only relevant when the vector is sparse. In the sparse case, Indices is parallel to Values, only the first Count items are meaningful, the indices must be non-negative and less than Length, and the indices must be strictly increasing. Note that when Count is zero, Indices may be null. In the dense case, Indices is not meaningful and may or may not be null.
• It is very common for the arrays in a VBuffer<T> to be larger than needed for their current value. A special case of this is when a dense VBuffer<T> has a non-null Indices array. The extra items in the arrays are not meaningful and should be ignored. Allowing these buffers to be larger than currently needed reduces the need to reallocate buffers for different values. For example, when cursoring through a vector valued column with VectorSize of 100, client code could pre-allocate values and indices arrays and seed a VBuffer<T> with those arrays. When fetching values, the client code passes the VBuffer<T> by reference. The called code can re-use those arrays, filling them with the current values.
• Generally, vectors should use a sparse representation only when the number of non-default items is at most half the value of Length. However, this guideline is not a mandate.
See the full IDataView technical specification for additional details on VBuffer<T>, including complete discussion of programming idioms, and information on helper classes for building and manipulating vectors.
## Standard Conversions¶
The IDataView system includes the definition and implementation of many standard conversions. Standard conversions are required to map source default values to destination default values. When both the source type and destination type have an NA value, the conversion must map NA to NA. When the source type has an NA value, but the destination type does not, the conversion must map NA to the default value of the destination type.
Most standard conversions are implemented by the singleton class Conversions in the namespace Microsoft.MachineLearning.Data.Conversion. The standard conversions are exposed by the ConvertTransform.
### From Text¶
There are standard conversions from TX to the standard primitive types, R4, R8, I1, I2, I4, I8, U1, U2, U4, U8, and BL. For non- empty, non-missing TX values, these conversions use standard parsing of floating-point and integer values. For BL, the mapping is case insensitive, maps text values { true, yes, t, y, 1, +1, + } to DvBool.True, and maps the values { false, no, f, n, 0, -1, - } to DvBool.False.
If parsing fails, the result is the NA value for floating-point, signed integer types, and boolean, and zero for unsigned integer types. Note that overflow of an integer type is considered failure of parsing, so produces an NA (or zero for unsigned). These conversions map missing/NA text to NA, for floating-point and signed integer types, and to zero for unsigned integer types.
These conversions are required to map empty text (the default value of TX) to the default value of the destination, which is zero for all numeric types and DvBool.False for BL. This may seem unfortunate at first glance, but leads to some nice invariants. For example, when loading a text file with sparse row specifications, it’s desirable for the result to be the same whether the row is first processed entirely as TX values, then parsed, or processed directly into numeric values, that is, parsing as the row is processed. In the latter case, it is simple to map implicit items (suppressed due to sparsity) to zero. In the former case, these items are first mapped to the empty text value. To get the same result, we need empty text to map to zero.
### Floating Point¶
There are standard conversions from R4 to R8 and from R8 to R4. These are the standard IEEE 754 conversions (using unbiased round-to-nearest in the case of R8 to R4).
### Signed Integer¶
There are standard conversions from each signed integer type to each other signed integer type. These conversions map NA to NA, map any other numeric value that fits in the destination type to the corresponding value, and maps any numeric value that does not fit in the destination type to NA. For example, when mapping from I1 to I2, the source NA value, namely 0x80, is mapped to the destination NA value, namely 0x8000, and all other numeric values are mapped as expected. When mapping from I2 to I1, any value that is too large in magnitude to fit in I1, such as 312, is mapped to NA, namely 0x80.
### Signed Integer to Floating Point¶
There are standard conversions from each signed integer type to each floating- point type. These conversions map NA to NA, and map all other values according to the IEEE 754 specification using unbiased round-to-nearest.
### Unsigned Integer¶
There are standard conversions from each unsigned integer type to each other unsigned integer type. These conversions map any numeric value that fits in the destination type to the corresponding value, and maps any numeric value that does not fit in the destination type to zero. For example, when mapping from U2 to U1, any value that is too large in magnitude to fit in U1, such as 312, is mapped to zero.
### Unsigned Integer to Floating Point¶
There are standard conversions from each unsigned integer type to each floating-point type. These conversions map all values according to the IEEE 754 specification using unbiased round-to-nearest.
### Key Types¶
There are standard conversions from one key type to another, provided:
• The source and destination key types have the same Min and Count values.
• Either the number of bytes in the destination’s underlying type is greater than the number of bytes in the source’s underlying type, or the Count value is positive. In the latter case, the Count is necessarily less than 2^^k, where k is the number of bits in the destination type’s underlying type. For example, U1[1-*] can be converted to U2[1-*], but U2[1-*] cannot be converted to U1[1-*]. Also, U1[1-100] and U2[1-100] can be converted in both directions.
The conversion maps source representation values to the corresponding destination representation values. There are no special cases, because of the requirements above.
### Boolean to Numeric¶
There are standard conversions from BL to each of the signed integer and floating point numeric. These map DvBool.True to one, DvBool.False to zero, and DvBool.NA to the numeric type’s NA value.
## Type Classes¶
This chapter contains information on the C# classes used to represent column types. Since the IDataView type system is extensible this list describes only the core data types.
### ColumnType Abstract Class¶
The IDataView system includes the abstract class ColumnType. This is the base class for all column types. ColumnType has several convenience properties that simplify testing for common patterns. For example, the IsVector property indicates whether the ColumnType is an instance of VectorType.
In the following notes, the symbol type is a variable of type ColumnType.
• The type.RawType property indicates the representation type of the column type. Its use should generally be restricted to constructing generic type and method instantiations. In particular, testing whether type.RawType == typeof(int) is not sufficient to test for the standard U4 type. The proper test is type == NumberType.I4, since there is a single universal instance of the I4 type.
• Certain .Net types have a corresponding DataKind enum value. The value of the type.RawKind property is consistent with type.RawType. For .Net types that do not have a corresponding DataKind value, the type.RawKind property returns zero. The type.RawKind property is particularly useful when switching over raw type possibilities, but only after testing for the broader kind of the type (key type, numeric type, etc.).
• The type.IsVector property is equivalent to type is VectorType.
• The type.IsNumber property is equivalent to type is NumberType.
• The type.IsText property is equivalent to type is TextType. There is a single instance of the TextType, so this is also equivalent to type == TextType.Instance.
• The type.IsBool property is equivalent to type is BoolType. There is a single instance of the BoolType, so this is also equivalent to type == BoolType.Instance.
• Type type.IsKey property is equivalent to type is KeyType.
• If type is a key type, then type.KeyCount is the same as ((KeyType)type).Count. If type is not a key type, then type.KeyCount is zero. Note that a key type can have a Count value of zero, indicating that the count is unknown, so type.KeyCount being zero does not imply that type is not a key type. In summary, type.KeyCount is equivalent to: type is KeyType ? ((KeyType)type).Count : 0.
• The type.ItemType property is the item type of the vector type, if type is a vector type, and is the same as type otherwise. For example, to test for a type that is either TX or a vector of TX, one can use type.ItemType.IsText.
• The type.IsKnownSizeVector property is equivalent to type.VectorSize > 0.
• The type.VectorSize property is zero if either type is not a vector type or if type is a vector type of unknown/variable length. Otherwise, it is the length of vectors belonging to the type.
• The type.ValueCount property is one if type is not a vector type and the same as type.VectorSize if type is a vector type.
• The Equals method returns whether the types are semantically equivalent. Note that for vector types, this requires the dimensionality information to be identical.
• The SameSizeAndItemType method is the same as Equals for non-vector types. For vector types, it returns true iff the two types have the same item type and have the same VectorSize values. For example, for the two vector types V<R4,3,2> and V<R4,6>, Equals returns false but SameSizeAndItemType returns true.
### PrimitiveType Abstract Class¶
The PrimitiveType abstract class derives from ColumnType and is the base class of all primitive type implementations.
### TextType Sealed Class¶
The TextType sealed class derives from PrimitiveType and is a singleton- class for the standard text type. The instance is exposed by the static TextType.Instance property.
### BooleanType Sealed Class¶
The BooleanType sealed class derives from PrimitiveType and is a singleton-class for the standard boolean type. The instance is exposed by the static BooleanType.Instance property.
### NumberType Sealed Class¶
The NumberType sealed class derives from PrimitiveType and exposes single instances of each of the standard numeric types, R4, R8, I1, I2, I4, I8, U1, U2, U4, U8, and UG.
### DateTimeType Sealed Class¶
The DateTimeType sealed class derives from PrimitiveType and is a singleton-class for the standard datetime type. The instance is exposed by the static DateTimeType.Instance property.
### DateTimeZoneType Sealed Class¶
The DateTimeZoneType sealed class derives from PrimitiveType and is a singleton-class for the standard datetime timezone type. The instance is exposed by the static DateTimeType.Instance property.
### TimeSpanType Sealed Class¶
The TimeSpanType sealed class derives from PrimitiveType and is a singleton-class for the standard datetime timezone type. The instance is exposed by the static TimeSpanType.Instance property.
### KeyType Sealed Class¶
The KeyType sealed class derives from PrimitiveType and instances represent key types.
Notes:
• Two key types are considered equal iff their kind, min, count, and contiguous values are the same.
• The static IsValidDataKind method returns true iff kind is U1, U2, U4, or U8. These are the only valid underlying data kinds for key types.
• The inherited KeyCount property returns the same value as the Count property.
### VectorType Sealed Class¶
The VectorType sealed class derives from ColumnType and instances represent vector types. The item type is specified as the first parameter to each constructor and the dimension information is inferred from the additional parameters.
• The DimCount property indicates the number of dimensions and the GetDim method returns a particular dimension value. All dimension values are non- negative integers. A dimension value of zero indicates unknown (or variable) in that dimension.
• The VectorSize property returns the product of the dimensions.
• The IsSubtypeOf(VectorType other) method returns true if this is a subtype of other, in the sense that they have the same item type, and either have the same VectorSize or other.VectorSize is zero.
• The inherited Equals method returns true if the two types have the same item type and the same dimension information.
• The inherited SameSizeAndItemType(ColumnType other) method returns true if other is a vector type with the same item type and the same VectorSize value.
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2019-10-15 09:40:37
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https://codereview.meta.stackexchange.com/questions/907/is-just-reviewing-the-code-good-enough-of-an-answer
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Is "just reviewing the code" good enough of an answer?
Looking back at some of my last few answers (here and especially here), I started wondering if these answers were good enough.
Allow me to expand: in these answers I went through the code almost line by line, and noted everything that jumped at me; by the time I was done, I already had a quite lengthy answer and decided to stop there.
In both cases, re-reading myself I find my answers look like code bashing sessions, shredding the OP's code to pieces. That's certainly symptomatic of my "read-the-code-and-note-everything-I-don't-like" approach, and could make me sound like a harsh dude that comes along and bitches about everything I see in someone else's code.
In this case, I hinted between the lines that a more OOP approach would be much better, and ended up intentionally leaving out an actual alternative approach, 1 because my answer was getting way too long, and 2 because I figured I'd leave room for other answers. I was thanked, upvoted and my answer was even accepted, and when I told the OP that the acceptance might be premature my answer kept its green tick.
Then another [excellent if not epic] answer came, which basically picked up where I left off and, without the slightest hint at the OP's code, gave the alternative approach I had in mind.
I don't mind that my answer was un-accepted and that the other one got the green tick - it's the OP's call and it's all fair game, and that other answer is really absolutely great.
I made the effort of downloading the OP's code, building it, running it, reviewing it, writing a code review to the best of my abilities, ...and yet the OP deems the other answer as most useful, the one that I could have written by merely reading the OP's post - again, not to take anything away from that great answer (which even quotes mine).
So the question is, exactly what is expected from the "best answer" on this site? An actual code review? Or an alternative approach? The best of both worlds, even if it means an answer so long that only the OP will read entirely (if we're lucky)?
• That anonymous upvote is just agreeing with me being a harsh dude that comes along and bitches about everything I see in someone else's code, isn't it? :) Oct 26, 2013 at 0:27
• Present. :-) That upvote is mainly for the constructiveness of the question... although I also do this with all of my answers. Anyway, I may post an answer here later.
– Jamal Mod
Oct 26, 2013 at 0:30
• 4. It sounds like you find it easier to go through all the details first, and then when you're done you realize that there are high-level problems too. There is nothing stopping you from doing this in that order and then putting your analysis of the high-level problems and overall advice at the top of your answer, as if you had thought of it first. 5. Both details and high-level stuff are important. Whether you look at details first or high-level stuff first is not. Oct 27, 2013 at 21:31
• Not sure that this is what you're asking at all, but I think it's worthwhile to mix a little sugar with the vinegar and tell people what they are doing well also... even if it's just "this seems roughly the right approach" or "nice project" or "this compiles without warnings - good job!". It's easy (and can be helpful) to nit-pick on code style, but if an answer comes across as highly negative or "critical" then the poster might feel downhearted... and that is a worse outcome than settling for less than perfect code. Feb 9, 2015 at 4:58
A good answer is whatever the OP can learn the most from. Depending on the skill level of the OP, either a line-by-line critique or a high-level review could be more appropriate.
Line-by-line critiques
A line-by-line critique is often needed for beginners whose code is too confused or buggy. A pitfall, though, is that often beginners produce code where every line needs work. With a review consisting of a dozen items, you would be lucky if the OP could keep just three points in mind when writing his/her next program.
You do have a bit of a responsibility, as a teacher, to make your answer digestible. The more disastrous the original code is, the more challenging and more important it is to keep the review simple. To that end, you could
• Prioritize
With Question 31502, I would consider
If Expr = FalseIf Not Expr
and
i = i + 1i += 1
to be nitpicks, since their badness only affects the line itself. When you have 11 bullet points, less is more.
Here's an example of an answer with prioritization. Even though I've raised about a dozen issues, I consider only four to be serious.
• Group issues into themes, and summarize
Here's an example of one of my reviews that I think was successful. Even though I found lots of lines of bad code, you can get the gist by reading the headings. You can also skip straight to the revised code to see how it should be done, because reading good code is also an effective way to learn. Finally, the summary gives a root-cause assessment of why the original code was so convoluted to begin with.
Some common categories might be: bugs, design issues, flow-of-control issues, naming suggestions, and nitpicks.
• Highlight key words, name your ideas
If your suggestions have names, they are more likely to be memorable. Soundbites and slogans work!
That said, here's one of my line-by-line reviews that didn't follow those simplification principles. I can't honestly judge my own work — do you think it's effective?
High-level reviews
If the code could benefit from a major overhaul, then a line-by-line critique, no matter how thorough, is just missing the forest for the trees. In fact, it gives the OP a false impression that the code is salvageable. So, with the OOP dice issue, it pays to think before you write, and sometimes you need to discard part of your review once you realize that the code needs a major overhaul.
Summary
Writing clear and concise reviews is as hard as writing good code. There are reviews that get the job done (a "core dump" of your brain), and then there are reviews that express key ideas eloquently. How meta!
• Wow. Just.. Wow. If only I could give you a meta-bounty for that one! [...] then a line-by-line critique, no matter how thorough, is just missing the forest for the trees this is it. Oct 27, 2013 at 2:42
• I gotta give credit for this answer, too. I suppose I should keep answering more questions to really understand this.
– Jamal Mod
Oct 27, 2013 at 3:09
• @200_success I've just put your teachings into application, check this out - I actually started going line-by-line, and then scrapped the whole thing! And I don't regret anything! Oct 30, 2013 at 0:33
• I think it's a good idea to do a high-level review in addition to the line-by-line review. Line-by-line does tend to miss the forest. It might be harder to review at a high level, because you have to comprehend the whole program and consider whether there are better ways to do it. Feb 9, 2015 at 5:05
We need to make sure that we review the code and not the person. Based on experience reviewing code for my co-workers, code review comments tend to work better when written about the code. E.g., the following comments come across--especially in a textual medium--as constructive.
• This function should be refactored into two smaller functions, each with a single purpose and a descriptive name.
• We're missing a test case here when i is a multiple of PostsPerPage.
• Consider renaming this variable to diagLog dl is a bit terse.
Comments like "You forgot the check for nullptr" are often (subconsciously) reinterpreted as "retailcoder is calling me stupid. I'm not stupid. Now I have to argue to prove that I'm not stupid."
Remember, there's a human being on the other end of a code review. This person has placed themselves in an inherently vulnerable position: they've asked for feedback. Whoah. That's a scary place to be in, especially on the Internet.
I haven't reviewed any code here yet. I'm still getting a feel for what the norms are. As I write this, the norms still being developed by the community. :-)
• Welcome aboard! "That's a scary place to be in, especially on the internet" - this would be true on many forums, fortunately CR (SE in general) isn't one. The community is quite active at moderating the site, ...and we're a friendly bunch in general.. when we've had our coffee :) Nov 15, 2013 at 11:06
• @retailcoder: And unlike SO, we don't hate fun (as much). :-)
– Jamal Mod
Nov 15, 2013 at 20:31
• We're missing… (&we're a friendly bunch…(Mat's Mug)) being co-opted unasked rubs me all the wrong way - my 1st impulse is to ask "Who we?" (I might exempt persons addressing me thus if I see reason to assume "we" spent 99+ hours cooperating - I might give my brother a frown.) Jan 25, 2018 at 9:12
The way I see it, there's no right or wrong answer. An answer is an answer, regardless of how it's approached. Personally, I think having your own "style" helps you personalize your answers a bit, while there's nothing wrong with changing it up for the better.
I'll offer my thoughts on both forms, and you can decide for yourself.
First form:
Depending on how you look at this one, it may look a bit hard to read. I'd say that's because there's no code written for separation, but it doesn't mean you need to force out written code. Overall, this is like, "Here are the main points. Follow (and/or check off each one), and you're good." Although it's not organized (depending on how you've written it), it just gets right to the point. The "harshness" factor shouldn't matter; if it does, then perhaps the OP is not courageous enough to take any needed criticism.
Second form:
This is probably how most code reviews will be approached, at least that's what I've seen. This also looks a bit more "formal" as well. You could also say that it's not "spewing" out the points, as with the first form. This one does take longer to read, but it may very well be easier to read. Despite the essay-like form, it could be a better pleasure to read when one is willing to read through it all.
So, which is better? Well, as I see it, they both have their pros and cons. In my opinion, I think the answerer should first go with what feels more comfortable. If the OP and/or anyone else finds any issues with said format, then it could be changed. The answerer could also adapt to the OP's code depending on the desired review and/or severity of the issues. In other words, not every answer has to follow the same form. If the contents of the answer are spot-on, then chances are that anyone interested will read through the answer regardless of the form.
• What about an answer like the object-oriented dice, which doesn't actually review code but rather teaches a much better approach that mootinizes most of the OP's code? Oct 26, 2013 at 21:01
• Then I think such an answer is necessary. If the entire code needs to be redone, then the bullet points won't be sufficient. But they can still be used to emphasize the important points.
– Jamal Mod
Oct 26, 2013 at 21:04
As an observation on the topic in question, under the section "Actual Code Review" for the question "Win Forms design pattern", you write
You're using a SplitContainer, and your form is resizable. That's good. Now I realize this isn't WPF, but in terms of layout, I'm sure you could do better
My (humble) pointers are as follows:
1 because my answer was getting way too long
Agreed
I find my answers look like code bashing sessions, shredding the OP's code to pieces
Not just the code
Stay on topic. e.g.
Is “just reviewing the code” good enough of an answer?
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2022-05-25 09:17:11
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https://www.physicsforums.com/threads/continuity-on-piecewise-function.511225/
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# Continuity on piecewise function
1. Jul 1, 2011
### Wables
1. The problem statement, all variables and given/known data
[10 Marks] At which points is the following function continuous and at which point is it discontinuous. Explain the types of discontinuity at each point where the function is discontinuous. Then at each point of the discontinuity, if possible, find a value for f(x) that makes it continuous or one sided continuous.
f(x) =
-2x if -1$\leq$x<1
-2/(x-1) if 1<x<2
x-2 if x> 2
2. Relevant equations
Test continuity at point:
f(a) is defined
lim f(x) exists
x->a
lim f(x) = f(a)
x->a
Continuity at Endpoints
lim f(x) = f(a) = left continuous
x->a-
lim f(x) = f(a) = right continuous
x->a+
3. The attempt at a solution
Im thinking what I need to do, is:
Check for continuity at the points -1, 1, 2.
Then I would classify any discontinuities as either removable, jump, or infinite discontinuities.
But that last part, im not sure what its asking?
What I have is this so far:
Discontinuous at x=1 and x=2.
At x=1: Jump discontinuity
At x=2: Jump discontinuity
Im not sure if im supposed to do this or if its right, but I did it anyway:
At x=-1, the function is right continuous on the interval [-1, 1). The function is also continuous on (1, 2) and (2, infinity)
2. Jul 1, 2011
### HallsofIvy
Staff Emeritus
Basically correct but the discontinuity at x= 1 is NOT a "jump" discontinuity:
$$\lim_{x\to 1^+} f(x)= \lim_{x\to 1}\frac{-2}{x- 1}= -\infty$$
3. Jul 1, 2011
### Wables
Oh really? Cool! Thanks! So by stating the intervals of continuity, I satisfied the last part of the problem? Cause I was not sure what it was asking..
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2017-11-23 11:56:34
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http://sioc-journal.cn/Jwk_hxxb/CN/Y2000/V58/I7/748
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### 非典型双亲性β-二酮稀土配合物LB膜荧光稳定性 的研究
1. 山东大学教育部胶体与界面化学重点实验室
• 发布日期:2000-07-15
### Investigation of fluorescence stability of a series of atypical amphiphilic β-diketone rare earth complex LB films
Zhang Renjie;Yang Kongzhang
• Published:2000-07-15
The fluorescence stability of a series of atypical amphiphilic rare earth complex LB films is studied under the following two conditions: (1) measure the LB films every 40 seconds; (2) measure the LB films once a week. The fluorescence intensity decreases slowly in a linear relationship for the first case, nearly 96% of the original intensity is kept after 30 times of irradiation. The fluorescence intensity decreases in a monoexponential relationship for the second case. The time τ that the fluorescence intensity decayed to 1/e of its original value is about 10 weeks, e.g., the τ value of n[Eu(TTA) ~3Phen] : n(AA)=1 : 1 LB film is 11.4 weeks. When the LB films are excited for the same times (> 1), fluorescence of the LB films for the first case is more intense than that for the second case. The fluorescence stability was studied by the UV-vis spectra and low angle X-ray diffraction. The decreased absorbance of the rare earth complexes in LB films contributed to the decreased fluorescence intensity with time passing by. However, all the LB films can emit detectable fluorescence even after half one year, indicating that the rare earth complexes in LB films have better fluorescence intensity. The reason can be attributed to the high ordered orientation of the rare earth complexes in LB films.
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2022-08-20 05:09:24
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http://tex.stackexchange.com/questions/47981/unexpected-conditional-branch-with-ifdim?answertab=active
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# Unexpected conditional branch with \ifdim
I have a problem with the following code:
\newtoks\sectoks
\sectoks={\noindent}
\newtoks\subsubjectstyle
\subsubjectstyle={\emitsectglue 1\the\sectoks}
\newtoks\postsectoks
\postsectoks={\par\smallskip\noindent\kern-1sp\hskip1sp}
\long\def\subsubject#1\par{%
{\the\subsubjectstyle#1}\the\postsectoks}
\def\emitsectglue#1{%
\ifdim\lastskip=1sp
\nobreak
\else
\vskip0pt plus#1\baselineskip
\penalty-\numexpr#1*100+50\relax
\vskip0pt plus-#1\baselineskip
\vskip#1\baselineskip
\fi}
\long\def\blockquote#1\eol{%
\emitsectglue 1
\begingroup
\raggedright\narrower\noindent #1
\smallskip
\endgroup\noindent}
\tracingall
\subsubject
The following subsubject will choose the first if's true-branch.
\subsubject
But the following blockquote will not. Why is that?
\blockquote
Be conservative in what you send, liberal in what you receive
\eol
\bye
When using the \subsubject command followed by another \subsubject, the macros work as expected and do not add the vertical glue & negative penalty.
However, when using the exact same code with another command, all of a sudden the right \if-branch doesn't get selected. I don't understand why is that.
-
Could you think of a better question title? Problem with X titles tend to be not the best titles ... – doncherry Mar 14 '12 at 9:42
@doncherry: You're right, the title isn't good. But I'm having trouble coming up with something descriptive. Would you have a suggestion? – morbusg Mar 14 '12 at 10:06
Not really, unfortunately. This question generally is too hardcore-TeX-y for me to understand. How about Plain-TeX \if doesn't work as expected? But perhaps @egreg can come up with an even better title? – doncherry Mar 14 '12 at 10:16
That italic backslash looks funny :) – doncherry Mar 14 '12 at 10:17
@doncherry: hmm, \if is a TeX primitive, not plain-tex IIRC, so maybe this title is better? – morbusg Mar 14 '12 at 13:34
show 5 more comments
Add \relax after \hskip1sp: TeX expands tokens when looking for a skip specification and, when \blockquote follows, the specification is not yet complete, so the expansion evaluates \ifdim before the skip is inserted.
If \subsubject follows, there is no problem, because the first token in the replacement text is { which stops expansion.
I would brace the argument to \emitsecglue and get rid of a couple of possible spurious space sources:
\newtoks\sectoks
\sectoks={\noindent}
\newtoks\subsubjectstyle
\subsubjectstyle={\emitsectglue{1}\the\sectoks}
\newtoks\postsectoks
\postsectoks={\par\smallskip\noindent\kern-1sp\hskip1sp\relax}
\def\subsubject#1\par{%
{\the\subsubjectstyle#1}\the\postsectoks}
\def\emitsectglue#1{%
\ifdim\lastskip=1sp
\nobreak
\else
\vskip0pt plus#1\baselineskip
\penalty-\numexpr#1*100+50\relax
\vskip0pt plus-#1\baselineskip
\vskip#1\baselineskip
\fi}
\long\def\blockquote#1\eol{%
\emitsectglue{1}%
\begingroup
\raggedright\narrower\noindent #1%
\smallskip
\endgroup\noindent}
\subsubject
The following subsubject will choose the first if's true-branch.
\subsubject
But the following blockquote will not. Why is that?
\blockquote
Be conservative in what you send, liberal in what you receive
\eol
\bye
-
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2014-03-16 17:25:36
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https://www.semanticscholar.org/paper/Iterations-of-rational-functions%3A-which-hyperbolic-Przytycki/caff0eeaa5443faa4225dd951a9808545c0d020c
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# Iterations of rational functions: which hyperbolic components contain polynomials?
@inproceedings{Przytycki1994IterationsOR,
title={Iterations of rational functions: which hyperbolic components contain polynomials?},
author={Feliks Przytycki},
year={1994}
}
• Feliks Przytycki
• Published 1994
• Mathematics
• Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is… CONTINUE READING
#### Citations
##### Publications citing this paper.
SHOWING 1-7 OF 7 CITATIONS
## On Connectivity of Fatou Components concerning a Family of Rational Maps
• Mathematics
• 2014
## Connectedness of Julia Sets of Rational Functions
• Mathematics
• 2001
## On Rational Maps with Two Critical Points
• John W. Milnor
• Mathematics, Computer Science
• Experimental Mathematics
• 1997
#### References
##### Publications referenced by this paper.
SHOWING 1-4 OF 4 REFERENCES
## On the iteration of a rational function: Computer experiments with Newton's method
• Mathematics
• 1983
## The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift
• Mathematics
• 1990
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2020-05-29 13:36:33
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https://www.physicsforums.com/threads/mean-value-theorem-applications.195263/
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Mean Value Theorem Applications
1. Nov 1, 2007
kingwinner
Q: Prove that if f: R^n -> R is defined by f(x)=arctan(||x||), then |f(x)-f(y)| <= ||x-y|| FOR ALL x,y E R^n.
[<= means less than or equal to]
Theorem: (a corollary to the mean value theorem)
Suppose f is differentiable on an open, convex set S and ||gradient [f(x)]|| <= M for all x E S. Then |f(b) - f(a)| <= M ||b-a|| for all a,b E S.
Now the trouble is that f(x)=arctan(||x||) is not differentiable at x=0 E R^n since the partial derivatives doesn't exist at x=0. Even worse, notice that S = R^n \ {0} is not convex.
Then how I can show that "f is differentiable on an open, convex set S"? (what is S in this case?) I strongly believe that this is an important step because if the conditions in the theorem are not fully satisifed, then there is no guarantee that the conclusion will hold. But this seems to be the only theorem that will help. What should I do?
Next, how can I prove that the conclusion is true FOR ALL x,y E R^n ?
Thanks for explaining!
Last edited: Nov 1, 2007
2. Nov 1, 2007
Dick
If you want to use that theorem, just break the problem into cases. Suppose x and y are points such that 0 is not on the line between x and y. Then you shouldn't have any trouble finding an open convex set containing x and y that doesn't contain 0. Now what happens if 0 is on the line? You could break things up more, but I think it's easier to think about a sequence x_n->x where 0 is not on the line connecting x_n and y.
Last edited: Nov 1, 2007
3. Nov 2, 2007
kingwinner
How can I break the problem into cases such that in each case, f is differentiable on an open, convex set S?
4. Nov 2, 2007
Dick
Do one case at a time. If 0 is not on the line between x and y, then what? If it is, then you are going to have a lot of trouble finding an open convex set on which f is differentiable containing x and y. Obviously. So give up that hope. Please reread my post.
5. Nov 2, 2007
Dick
If 0 is on the line, you could try breaking the problem up by arguing about the difference between x and 0 and y and 0, but that doesn't involve a single convex domain of differentiability. And it's certainly more complicated.
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2017-01-17 04:52:03
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https://www.proofwiki.org/wiki/Category:Commutativity
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# Category:Commutativity
Let $\circ$ be a binary operation.
Two elements $x, y$ are said to commute (with each other) if and only if:
$x \circ y = y \circ x$
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2022-09-24 20:16:02
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http://nodus.ligo.caltech.edu:8080/40m/page147?&attach=0&rsort=Category
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40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
40m Log, Page 147 of 344 Not logged in
ID Date Author Type Category Subject
12112 Sat May 7 09:40:40 2016 ericqUpdateLSCGreen PDH demod lowpass
As I was looking at filter designs, it seemed difficult to get 40dB of supression at 2F with a bandpass without going to a pretty high order, which would mean a fair number of lossy inductors.
I'll keep working on it. Maybe we don't need 40dB...
12113 Sun May 8 08:39:21 2016 ranaUpdateLSCGreen PDH demod lowpass
Indeed. This is why the LSC PDs have a 2f notch in addition to the 1f resonance. In recent versions, we also put a 2f notch in the feedback of the preamp which comes after the diode but before the mixer. The overall 1f to 2f ratio that we get is in the 50-60 dB region. I don't think we have to go that far with this thing; having a double LC already seems like it should be pretty good, or we could have a single LC bandpass with a 2f notch all in one Pomona box.
12114 Tue May 10 03:44:59 2016 ericqUpdateLSCRelocked
ALSX noise is solidly within past acceptable performance levels. The DRFPMI was locked on four out of six attempts.
Some housekeeping was done:
• PMC aligned
• Static alignment voltages of X end PZT mirrors offloaded by turning mount screws
• Rough comissioning of AUX X dither alignment
• Locking scripts reverted to AUX X Innolight voltage/temperature sign convention
The recombination of the QPD signals to common / differential is imperfect, and limited how well we could keep the interferometer aligned, since the QPD at X has changed. This needs some daytime work.
Some sensing matrix measurements were made, to be meditated upon for how to 1F the DRMI.
Other to-dos:
• Bandpass + notch combo for green refl PDs
• SRCL, and to a lesser extant, MICH feedforward subtraction (See DARM vs. other length DOF coherence plot below)
• Fiber couple AUX X light
• Make IFO work good
As an aside, Gautam and I noticed numerous green beams coming from inside the vacuum system onto the PSL table. They exist only when green is locked to the arms. Some of them come out at very non-level angles and shine in many places. This doesn't make me feel very happy; I suppose we've been living with it for some time.
12124 Fri May 20 17:36:06 2016 gautamUpdateLSCNew stands for TransMon/Oplev QPDs
As we realized during the EX table switch, the transmitted beam height from the arm is not exactly 4" relative to the endtable, it is more like 4.75" at the X-end (yet to be investigated at the Y-end). As a result, the present configuration involves the steering optics immediately before the Oplev and TransMon QPDs sending the beam downwards at about 5 degrees. Although this isn't an extremely large angle, we would like to have things more level. For this purpose, Steve has ordered some Aluminium I-beams (1/2 " thick) which we can cut to size as we require. The idea is to have the QPD enclosures mounted on these beams and then clamped to the table. One concern was electrical isolation - but Steve thinks Delrin washers between the QPD enclosure and the mount will suffice. We will move ahead with getting these machined once I investigate the situation at the Y end as well.. The I beams should be here sometime next week...
12138 Fri May 27 02:52:53 2016 ericqUpdateLSCRestoring high BW single arm control
I've been futzing with the common mode servo, trying to engage the AO path with POY for high bandwidth control of a single arm lock. I'm able to pull in the crossover and get a nice loop shape, but keep getting tripped up by the offset glitches from the CM board gain steps, so can't get much more than a 1kHz UGF.
As yutaro measured, these can be especially nasty at the major carrier transitions (i.e. something like 0111->1000). This happens at the +15->+16dB input gain step; the offset step is ~200x larger than the in-loop error signal RMS, so obviously there is no hope of keeping the loop engaged when recieving this kind of kick. Neither of the CM board inputs are immune from this, as I have empirically discovered. I can turn down the initial input gain to try and avoid this step occuring anywhere in the sequence, but then the SNR at high frequencies get terrible and I inject all kinds of crud into the mode cleaner, making the PC drive furious.
I think we're able to escape this when locking the full IFO because the voltages coming out of REFL11 are so much larger than the puny POY signals so the input-referred glitches aren't as bad. I think in the past, we used AS55 with a misaligned ITMX for this kind of single arm thing, which probably gives better SNR, but the whole point of this is to keep the X arm aligned and lock it to the Y-arm stabilized PSL.
12142 Wed Jun 1 09:06:38 2016 SteveUpdateLSCNew stands for TransMon/Oplev QPDs
Machined from I-beam 6061 T6 Aluminum 5" x 0.5 x 3.25
Quote: As we realized during the EX table switch, the transmitted beam height from the arm is not exactly 4" relative to the endtable, it is more like 4.75" at the X-end (yet to be investigated at the Y-end). As a result, the present configuration involves the steering optics immediately before the Oplev and TransMon QPDs sending the beam downwards at about 5 degrees. Although this isn't an extremely large angle, we would like to have things more level. For this purpose, Steve has ordered some Aluminium I-beams (1/2 " thick) which we can cut to size as we require. The idea is to have the QPD enclosures mounted on these beams and then clamped to the table. One concern was electrical isolation - but Steve thinks Delrin washers between the QPD enclosure and the mount will suffice. We will move ahead with getting these machined once I investigate the situation at the Y end as well.. The I beams should be here sometime next week...
Atm2, version 2 "pdstand" will allow you to clamp from any direction ( Koji was right )
12188 Thu Jun 16 11:25:00 2016 JohannesUpdateLSCY-Arm round-trip loss measurement with ALS
Using the ALS green beat and armlength feedback I mapped an IR resonance of the Y-Arm by stepping through a ramp of offset values.
First I optimized the IR alignment with the dither scripts while LSC kept the arm on resonance, and then transitioned the length control to ALS. The beat frequency I obtained between the Y-arm green and the PSL was about 25 MHz. Then I applied a controlled ramp signal (stepping through small offset increments applied to LSC-ALSY_OFFSET, while logging the readback from channels LSC-TRY_OUT16 and ALS-Y_FC_SERVO_INMON with an averaging time of 1s.
The plots show the acquired data with fits to $T(x)=\frac{T_0}{1+\frac{(x-x_0)^2}{\mathrm{HWHM}^2}}+\mathrm{offset}$ and $f(x)=mx+b$, respectively.
The fits, weighted with inverse rms uncertainty of the data points as reported by the cds system, returned HWHM = 0.6663 ± 0.0013 [offset units] and m = -0.007666 ± 0.000023 [MHz/offset unit], which gives a combined FWHM = 10,215 ± 36 Hz. The error is based purely on the fit and does not reflect uncertainties in the calibration of the phase tracker.
This yields a finesse of 388.4 ± 1.4, corresponding to a total loss (including transmissivities) of 16178 ± 58 ppm. These uncertainties include the reported accuracies of FSR and phase tracker calibration from elog 9804 and elog 11761.
The resulting loss is a little lower than that of elog 11712, which was done before the phase tracker re-calibration. Need to check for consistency.
12189 Thu Jun 16 12:06:59 2016 ericqUpdateLSCRF Amp installed at POY11 RF output
I have installed a ZFL-500LN on the RF output of POY11. This should reduce the effect of the CM board voltage offsets by increasing the size of the error signal coming into the board. Checking with an oscilloscope at the LSC rack, the single arm PDH peak to peak voltage was something like 4mV, now it is something like 80mV
The setup is similar to the REFL165 situation, but with the amplifier in proximity with the PD, instead of at the end of a long cable at the LSC rack.
The PD RF output is T'd between an 11MHz minicircuits bandpass filter and a 50 Ohm terminator (which makes sure that signals outside of the filter's passband don't get reflected back into the PD). The output of the filter is connected directly to the input of the ZFL-500LN, which is powered (temporarily) by picking off the +15V from the PD interface cable via Dsub15 breakout. (I say temporarily, as Koji is going to pick out some fancy pi-filter feedthrough which we can use to make a permanent power terminal on the PD housing.)
The max current draw of this amplifier is 60mA. Gazing at the LSC interface (D990543), I think the +15V on the DSUB cable is being passed from the eurocard crate; I don't see any 15V regulator, so maybe this is ok...
The free swinging PDH signal looked clean enough on a scope. Jamie is doing stuff with the framebuilder, so I can't look at spectra right now. However, turning the POY whitening gain down to +18dB from +45dB lets the Y arm lock on POY with all other settings nominal, which is about what we expect from the nominal +23dB gain of the amplifier.
I would see CM board offsets of ~5mV before, which was more a little more than a linewidth before this change. Now it will be 5% of that, and hopefully more manageable.
12205 Tue Jun 21 04:01:09 2016 ericqUpdateLSCY arm @ 30kHz UGF w/POY, AO
With the newly amplified POY signal, locking the mode cleaner to the Y arm at ~30kHz bandwidth was quite straightforward. The offset jumps still happen, and are visible in POY11_I_ERR, but are never big enough to cause much power degradation in TRY (except when turning on CM board boosts, but its still not enough to lose lock). The script which accomplishes this is at scripts/YARM, and is in the svn. The MC2/AO crossover is at about 150Hz with 40deg margin.
For now, I'm using IN1 of the CM board, because I haven't removed the op27s that I put into IN2's gain stages. I believe the slew rate limitations of these prevent them from working completely during the offset jumps. I'll put AD829s back soon.
At first, I had ITMX misalgined to use AS55 as an out of loop sensor, then I aligned and locked the X arm on POX to compare.
Weirdly enough, locking the mode cleaner to the Y arm with 30kHz UGF and two boosts on make no real visible difference in the X arm control signal. This is strange, as the whole point of this affair was to remove the presumably large influence of frequency noise on the X arm signals... Maybe this is injecting too much POY sensor noise?
12210 Wed Jun 22 08:40:42 2016 ranaUpdateLSCY arm @ 30kHz UGF w/POY, AO
Below 100 Hz, I suppose this means that the X arm is now limited by the quadrature sum of the X and Y arm seismic noise.
12535 Thu Oct 6 03:56:43 2016 ericqUpdateLSCRevival Attempt
[ericq, Gautam, Lydia]
We spent some time tonight trying to revive the PRFPMI. (Why PR instead of DR? Not having to deal with SRM alignment and potentially get a better idea of our best-case PRG). After the usual set up and warm up, we found ourselves unable to hold on to the PRMI while the arms flash. In the past, this was generally solved through clever trigger matrix manipulations, but this didn't really work tonight. We will meditate on the solution.
12547 Tue Oct 11 02:48:43 2016 ericqUpdateLSCRevival Attempts
Still no luck relocking, but got a little further. I disabled the output of the problematic PRM OSEM, it seems to work ok. Looking at the sensing of the PRMI with the arms held off, REFL165 has better MICH SNR due to its larger seperation in demod angle. So, I tried the slightly odd arrangement of 33I for PRCL and 165Q for MICH. This can indefinitely hold through the buzzing resonance. However, I haven't been able to find the sweet spot for turning on the CARM_B (CM_SLOW) integrator, which is neccesary for turning up the AO and overall CARM gain. This is a familiar problem, usually solved by looking at the value far from resonance on either side, and taking the midpoint as the filter module offset, but this didn't work tonight. Tried different gains and signs to no avail.
12587 Fri Oct 28 15:46:29 2016 gautamSummaryLSCX/Y green beat mode overlap measurement redone
I've been meaning to do this analysis ever since putting in the new laser at the X-end, and finally got down to getting all the required measurements. Here is a summary of my results, in the style of the preceeding elogs in this thread. I dither aligned the arms and maximized the green transmission DC levels, and also the alignment on the PSL table to maximize the beat note amplitude (both near and far field alignment was done), before taking these measurements. I measured the beat amplitude in a few ways, and have reported all of them below...
XARM YARM
o BBPD DC output (mV), all measured with Fluke DMM
V_DARK: +1.0 +3.0 V_PSL: +8.0 +14.0 V_ARM: +175.0 +11.0
o BBPD DC photocurrent (uA)
I_DC = V_DC / R_DC ... R_DC: DC transimpedance (2kOhm) I_PSL: 3.5 5.5 I_ARM: 87.0 4.0
o Expected beat note amplitude I_beat_full = I1 + I2 + 2 sqrt(e I1 I2) cos(w t) ... e: mode overlap (in power) I_beat_RF = 2 sqrt(e I1 I2) V_RF = 2 R sqrt(e I1 I2) ... R: RF transimpedance (2kOhm) P_RF = V_RF^2/2/50 [Watt] = 10 log10(V_RF^2/2/50*1000) [dBm]
= 10 log10(e I1 I2) + 82.0412 [dBm]
= 10 log10(e) +10 log10(I1 I2) + 82.0412 [dBm]
for e=1, the expected RF power at the PDs [dBm] P_RF: -13.1 -24.5
o Measured beat note power (measured with oscilloscope, 50 ohm input impedance) P_RF: -17.8dBm (81.4mVpp) -29.8dBm (20.5mVpp) (38.3MHz and 34.4MHz) e: 34 30 [%] o Measured beat note power (measured with Agilent RF spectrum analyzer) P_RF: -19.2 -33.5 [dBm] (33.2MHz and 40.9MHz) e: 25 13 [%]
I also measured the various green powers with the Ophir power meter:
o Green light power (uW) [measured just before PD, does not consider reflection off the PD]
P_PSL: 16.3 27.2 P_ARM: 380 19.1
Measured beat note power at the RF analyzer in the control room P_CR: -36 -40.5 [dBm] (at the time of measurement with oscilloscope) Expected -17 - 9 [dBm] (TO BE UPDATED)
Expected Power: (TO BE UPDATED) Pin + External Amp Gain (25dB for X, Y from ZHL-3A-S) - Isolation trans (1dB) + GAV81 amp (10dB) - Coupler (10.5dB)
The expected numbers for the control room analyzer in red have to be updated.
The main difference seems to be that the PSL power on the Y broadband PD has gone down by about 50% from what it used to be. In either measurement, it looks like the mode matching is only 25-30%, which is pretty abysmal. I will investigate the situation further - I have been wanting to fiddle around with the PSL green path in any case so as to facilitate having an IR beat even when the PSL green shutter is closed, I will try and optimize the mode matching as well... I should point out that at this point, the poor mode-matching on the PSL table isn't limiting the ALS noise performance as we are able to lock reliably...
12611 Sat Nov 12 01:09:56 2016 gautamUpdateLSCRecovering DRMI locking
Now that we have all Satellite boxes working again, I've been working on trying to recover the DRMI 1f locking over the last couple of days, in preparation for getting back to DRFPMI locking. Given that the AS light levels have changed, I had to change the whitening gains on the AS55 and AS110 channels to take this into account. I found that I also had to tune a number of demod phases to get the lock going. I had some success with the locks tonight, but noticed that the lock would be lost when the MICH/SRCL boosts were triggered ON - when I turned off the triggering for these, the lock would hold for ~1min, but I couldn't get a loop shape measurement in tonight.
As an aside, we have noticed in the last couple of months glitchy behaviour in the ITMY UL shadow sensor PD output - qualitatively, these were similar to what was seen in the PRM sat. box, and since I was able to get that working again, I did a similar analysis on the ITMY sat. box today with the help of Ben's tester box. However, I found nothing obviously wrong, as I did for the PRM sat. box. Looking back at the trend, the glitchy behaviour seems to have stopped some days ago, the UL channel has been well behaved over the last week. Not sure what has changed, but we should keep an eye on this...
12619 Wed Nov 16 03:10:01 2016 gautamUpdateLSCDRMI locked on 1f and 3f signals
After much trial and error with whitening gains, demod phases and overall loop gains, I was finally able to lock the DRMI on both 1f and 3f signals! I went through things in the following order tonight:
1. Lock the arms, dither align
2. Lock the PRMI on carrier and dither align the PRM to get good alignment
3. Tried to lock the DRMI on 1f signals - this took a while. I realized the reason I had little to no success with this over the last few days was because I did not turn off the automatic unwhitening filter triggering on the demod screens. I had to tweak the SRM alignment while looking at the AS camera, and also adjust the demod phases for AS55 (MICH is on AS55Q) and REFL55 (SRCL is on REFL55I). Once I was able to get locks of a few seconds, I used the UGF servos to set the overall loop gain for MICH, PRCL and SRCL, after which I was able to revert the filter triggering to the usual settings
4. Once I adjusted the overall gains and demod phases, the DRMI locks were very stable - I left a lock alone for ~20mins, and then took loop shape measurements for all 3 loops
5. Then I decided to try transfering to 3f signals - I first averaged the IN1s to the 'B' channels for the 3 vertex DOFs using cds avg while locked on the 1f signals. I then set a ramp time of 5 seconds and turned the gain of the 'A' channels to 0 and 'B' channels to 1. The transition wasn't smooth in that the lock was broken but was reacquired in a couple of seconds.
6. The lock on 3f signals was also pretty stable, the current one has been going for >10 minutes and even when it loses lock, it is able to reacquire in a few seconds
I have noted all the settings I used tonight, I will post them tomorrow. I was planning to try a DRFPMI lock if I was successful with the DRMI earlier tonight, but I'm calling it a night for now. But I think the DRMI locking is now back to a reliable level, and we can push ahead with the full IFO lock...
It remains to update the auto-configure scripts to restore the optimized settings from tonight, I am leaving this to tomorrow as well...
Updated 16 Nov 2016 1130am
Settings used were as follows:
1f/3f DOF Error signal Whitening gain (dB) Demod phase (deg) Loop gain Trigger
DRMI Locking 16 Nov 2016
1f MICH (A) AS55Q 0 -42 -0.026 POP22I=1
1f PRCL (A) REFL11I 18 18 -0.0029 POP22I=1
1f SRCL (A) REFL55I 18 -175 -0.035 POP22I=10
3f MICH (B) REFL165Q 24 -86 -0.026 POP22I=1
3f PRCL (B) REFL33I 30 136 -0.0029 POP22I=1
3f SRCL (B) REFL165I and REFL33I - - -0.035 POP22I=10
12620 Wed Nov 16 08:14:43 2016 SteveUpdateLSCDRMI locked on 1f and 3f signals
Nice job.
Quote: After much trial and error with whitening gains, demod phases and overall loop gains, I was finally able to lock the DRMI on both 1f and 3f signals! I went through things in the following order tonight: Lock the arms, dither align Lock the PRMI on carrier and dither align the PRM to get good alignment Tried to lock the DRMI on 1f signals - this took a while. I realized the reason I had little to no success with this over the last few days was because I did not turn off the automatic unwhitening filter triggering on the demod screens. I had to tweak the SRM alignment while looking at the AS camera, and also adjust the demod phases for AS55 (MICH is on AS55Q) and REFL55 (SRCL is on REFL55I). Once I was able to get locks of a few seconds, I used the UGF servos to set the overall loop gain for MICH, PRCL and SRCL, after which I was able to revert the filter triggering to the usual settings Once I adjusted the overall gains and demod phases, the DRMI locks were very stable - I left a lock alone for ~20mins, and then took loop shape measurements for all 3 loops Then I decided to try transfering to 3f signals - I first averaged the IN1s to the 'B' channels for the 3 vertex DOFs using cds avg while locked on the 1f signals. I then set a ramp time of 5 seconds and turned the gain of the 'A' channels to 0 and 'B' channels to 1. The transition wasn't smooth in that the lock was broken but was reacquired in a couple of seconds. The lock on 3f signals was also pretty stable, the current one has been going for >10 minutes and even when it loses lock, it is able to reacquire in a few seconds I have noted all the settings I used tonight, I will post them tomorrow. I was planning to try a DRFPMI lock if I was successful with the DRMI earlier tonight, but I'm calling it a night for now. But I think the DRMI locking is now back to a reliable level, and we can push ahead with the full IFO lock... It remains to update the auto-configure scripts to restore the optimized settings from tonight, I am leaving this to tomorrow as well...
12630 Mon Nov 21 14:02:32 2016 gautamUpdateLSCDRMI locked on 3f signals, arms held on ALS
Over the weekend, I was successful in locking the DRMI with the arms held on ALS. The locks were fairly robust, lasting order of minutes, and was able to reacquire by itself when it lost the lock in <1min. I had to tweak the demod phases and loop gains further compared to the 1f lock with no arms, but eventually I was able to run a sensing matrix measurement as well. A summary of the steps I had to follow:
• Lock on 1f signals, no arms, and run sensing lines, adjust REFL33 and REFL 165 demod phases to align PRCL, MICH and SRCL as best as possible to REFL33I, REFL165Q and REFL165I respectively
• I also set the offsets to the 'B' inputs at this stage
• Lock arms on ALS, engage DRMI locking on 3f signals (the restore script resets some values like the 'B' channel offsets, so I modified the restore script to set the offsets I most recently measured)
• I was able to achieve short locks on the settings from the locking with no arms - I set the loop gains using the UGF servos and ran some sensing lines to get an idea of what the final demod phases should be
• Adjusted the demod phases, locked the DRMI again (with CARM offset = -4.0), and took another sensing matrix measurement (~2mins). The data was analyzed using the set of scripts EricQ has made for this purpose, here is the result from a lock yesterday evening (the radial axis is meant to be demod board output volts per meter but the calibration I used may be wrong)
I've updated the appropriate fields in the restore script. Now that the DRMI locking is somewhat stable again, I think the next step towards the full lock would be to zero the CARM offset and turning on the AO path.
On the downside, I noticed yesterday that ITMY UL shadow sensor readback was glitching again - for the locking yesterday, I simply held the output of that channel to the input matrix, which worked fine. I had already done some debugging on the Sat. Box with the help of the tester box, but unlike the PRM sat. box, I did not find anything obviously wrong with the ITMY one... I also ran into a CDS issue when I tried to run the script that sets the phase tracker UGF - the script reported that the channels it was supposed to read (the I and Q outputs of the ALS signal, e.g. C1:ALS-BEATX_FINE_I_OUT) did not exist. The same channels worked on dataviever though, so I am not sure what the problem was. Some time later, the script worked fine too. Something to look out for in the future I guess..
12638 Wed Nov 23 16:21:02 2016 gautamUpdateLSCITMY UL glitches are back
Quote: As an aside, we have noticed in the last couple of months glitchy behaviour in the ITMY UL shadow sensor PD output - qualitatively, these were similar to what was seen in the PRM sat. box, and since I was able to get that working again, I did a similar analysis on the ITMY sat. box today with the help of Ben's tester box. However, I found nothing obviously wrong, as I did for the PRM sat. box. Looking back at the trend, the glitchy behaviour seems to have stopped some days ago, the UL channel has been well behaved over the last week. Not sure what has changed, but we should keep an eye on this...
I've noticed that the glitchy behaviour in ITMY UL shadow sensor readback is back - as mentioned above, I looked at the Sat. Box and could not find anything wrong with it, perhaps I'll plug the tester box in over the Thanksgiving weekend and see if the glitches persist...
12644 Tue Nov 29 11:07:37 2016 SteveUpdateLSCITMY UL glitches are back
400 days plot. Satelite amp ITMY has been swapped with ETMY
Unlabeled sat.amps are labeled. This plot only makes sense if you know the Cuh-Razy sat amp locations.
12648 Wed Nov 30 01:47:56 2016 gautamUpdateLSCSuspension woes
Short summary:
• Looks like Satellite boxes are not to blame for glitchy behaviour of shadow sensor PD readouts
• Problem may lie at the PD whitening boards (D000210) or with the Contec binary output cards in c1sus
• Today evening, similar glitchy behaviour was observed in all MC1 PD readout channels, leading to frequent IMC unlocking. Cause unknown, although I did work at 1X5, 1X6 today, and pulled out the PD whitening board for ITMY which sits in the same eurocrate as that for MC1. MC2/MC3 do not show any glitches.
Detailed story below...
Part 1: Satellite box swap
Yesterday, I switched the ITMY and ETMY satellite boxes, to see if the problems we have been seeing with ITMY UL move with the box to ETMY. It did not, while ITMY UL remained glitchy (based on data from approximately 10pm PDT on 28Nov - 10am PDT 29 Nov). Along with the tabletop diagnosis I did with the tester box, I concluded that the satellite box is not to blame.
Part 2: Tracing the signal chain (actually this was part 3 chronologically but this is how it should have been done...)
So if the problem isn't with the OSEMs themselves or the satellite box, what is wrong? I attempted to trace the signal chain from the satellite box into our CDS system as best as I could. The suspension wiring diagram on our wiki page is (I think) a past incarnation. Of course putting together a new diagram was a monumental task I wasn't prepared to undertake tonight, but in the long run this may be helpful. I will put up a diagram of the part I did trace out tomorrow, but the relevant links for this discussion are as follows (? indicates I am unsure):
1. Sat box (?)--> D010069 via 64pin IDE connector --> D000210 via DB15 --> D990147 via 4pin LEMO connectors --> D080281 via DB25 --> ADC0 of c1sus
2. D000210 backplane --> cross-connect (mis)labelled "ITMX white" via IDE connector
3. c1sus CONTEC DO-32L-PE --> D080478 via DB37 --> BO0-1 --> cross-connect labelled "XY220 1Y4-33-16A" via IDE --> (?) cross-connect (mis)labelled "ITMX white" via IDE connector
I have linked to the DCC page for the various parts where available. Unfortunately I can't locate (on new DCC or old or elog or wiki) drawings for D010069 (Satellite Amplifier Adapter Board), D080281 ("anti-aliasing interface)" or D080478 (which is the binary output breakout box). I have emailed Ben Abbott who may have access to some other archive - the diagrams would be useful as it is looking likely that the problem may lie with the binary output.
So presumably the first piece of electronics after the Satellite box is the PD whitening board. After placing tags on the 3 LEMOs and 1 DB15 cable plugged into this board, I pulled out the ITMY board to do some tabletop diagnosis in the afternoon around 2pm 29Nov.
Part 3: PD whitening board debugging
This particular board has been reported as problematic in the recent past. I started by inserting a tester board into the slot occupied by this board - the LEDs on the tester board suggested that power-supply from the backplane connectors were alright, confirmed with a DMM.
Looking at the board itself, C4 and C6 are tantalum capacitors, and I have faced problems with this type of capacitor in the past. In fact, on the corresponding MC3 board (which is the only one visible, I didn't want to pull out boards unnecessarily) have been replaced with electrolytic capacitors, which are presumably more reliable. In any case, these capacitors do not seem to be at any fault, the board receives +/-15 V as advertised.
The whitening switching is handled by the MAX333 - this is what I looked at next. This IC is essentially a quad SPDT switch, and a binary input supplied via the backplane connector serves to route the PD input either through a whitening filter, or bypass it via a unity gain buffer. The logic levels that effect the switching are +15V and 0V (and not the conventional 5V and 0V), but according to the MAX333 datasheet, this is fine. I looked at the supply voltage to all ICs on the board, DC levels seemed fine (as measured with a DMM) and I also looked at it on an oscilloscope, no glitches were seen in ~30sec viewing stretch. I did notice something peculiar in that with no input supplied to the MAX333 IC (i.e. the logic level should be 15V), the NO and NC terminals appear shorted when checked with a DMM. Zach has noticed something similar in the past, but Koji pointed out that the DMM can be fooled into thinking there is a short. Anyway, the real test was to pull the logic input of the MAX333 to 0, and look at the output, this is what I did next.
The schematic says the whitening filter has poles at 30,100Hz and a zero at 3 Hz. So I supplied as "PD input" a 12Hz 1Vpp sinewave - there should be a gain of ~x4 when this signal passes through the path with the whitening filter. I then applied a low frequency (0.1Hz) square wave (0-5V) to the "bypass" input, and looked at the output, and indeed saw the signal amplitude change by ~4x when the input to the switch was pulled low. This behaviour was confirmed on all five channels, there was no problem. I took transfer functions for all 5 channels (both at the "monitor" point on the backplane connector and on the front panel LEMOs), and they came out as expected (plot to be uploaded soon).
Next, I took the board back to the eurocrate. I first put in a tester box into the slot and measured the voltage levels on the backplane pins that are meant to trigger bypassing of the whitening stage, all the pins were at 0V. I am not sure if this is what is expected, I will have to look inside D080478 as there is no drawing for it. Note that these levels are set using a Contec binary output card. Then I attached the PD whitening board to the tester board, and measured the voltages at the "Input" pins of all the 5 SPDT switches used under 2 conditions - with the appropriate bit sent out via the Contec card set to 0 or 1 (using the button on the suspension MEDM screens). I confirmed using the BIO medm screen that the bit is indeed changing on the software side, but until I look at D080478, I am not sure how to verify the right voltage is being sent out, except to check at the pins on the MAX333. For this test, the UL channel was indeed anomalous - while the other 4 channels yielded 0V (whitening ON, bit=1) and 15V (whitening OFF, bit=0), the corresponding values for the UL channel were 12V and 10V.
I didn't really get any further than this tonight. But this still leaves unanswered questions - if the measured values are faithful, then the UL channel always bypasses the whitening stage. Can this explain the glitchy behaviour?
Part 4: MC1 troubles
At approximately 8pm, the IMC started losing lock far too often - see the attached StripTool trace. There was a good ~2hour stretch before that when I realigned the IMC, and it held lock, but something changed abruptly around 8pm. Looking at the IMC mirror OSEM PD signals, all 5 MC1 channels are glitching frequently. Indeed, almost every IMC lockloss in the attached StripTool is because of the MC1 PD readouts glitching, and subsequently, the damping loops applying a macroscopic drive to the optic which the FSS can't keep up with. Why has this surfaced now? The IMC satellite boxes were not touched anytime recently as far as I am aware. The MC1 PD whitening board sits in the same eurocrate I pulled the ITMY board out of, but squishing cables/pushing board in did not do anything to alleviate the situation. Moreover, MC2 and MC3 look fine, even though their PD whitening boards also sit in the same eurocrate. Because I was out of ideas, I (soft) restarted c1sus and all the models (the thinking being if something was wrong with the Contec boards, a restart may fix it), but there was no improvement. The last longish lock stretch was with the MC1 watchdog turned off, but as soon as I turned it back on the IMC lost lock shortly after.
I am leaving the autolocker off for the night, hopefully there is an easy fix for all of this...
12652 Wed Nov 30 17:08:56 2016 gautamUpdateLSCBinary output breakout box removed
[ericq, gautam]
To diagnose the glitches in OSEM readouts, we have removed one of the PCIE BO D37 to IDE50 adaptor boxes from 1X5. All the watchdogs were turned off, and the power to the unit was cut before the cables on the front panel were removed. I am working on the diagnosis, I will update more later in the evening. Note that according to the c1sus model, the box we removed supplies backplane logic inputs that control whitening for ITMX, ITMY, BS and PRM (in case anyone is wondering/needs to restore damping to any of these optics). The whitening settings for the IMC mirrors resides on the other unit in 1X5, and should not be affected.
12653 Thu Dec 1 02:19:13 2016 gautamUpdateLSCBinary output breakout box restored
As we suspected, the binary breakout board (D080478, no drawing available) is simply a bunch of tracks printed on the PCB to route the DB37 connector pins to two IDE50 connectors. There was no visible damage to any of the tracks (some photos uploaded to the 40m picasa). Further, I checked the continuity between pins that should be connected using a DMM.
I got a slightly better understanding of how the binary output signal chain is - the relevant pages are 44 and 48 in the CONTEC manual. The diagram on pg44 maps the pins on the DB37 connector, while the diagram on pg 48 maps how the switching actually occurs. The "load" in our case is the 4.99kohm resistor on the PD whitening board D000210. Following the logic in the diagram on pg48 is easy - setting a "high" bit in the software should pull the load resistor to 0V while setting a "low" bit keeps the load at 15V (so effectively the whole setup of CONTEC card + breakout board + pull-up resistor can be viewed as a simple NOT gate, with the software bit as the input, and the output connected to the "IN" pin of the MAX333).
Since I was satisfied with the physical condition of the BO breakout board, I re-installed the box on 1X5. Then, with the help of a breakout board, I diagnosed the situation further - I monitored the voltage to the pins on the backplane connector to the whitening boards while switching the MEDM switches to toggle the whitening state. For all channels except ITMY UL, the behaviour was as expected, in line with the preceeding paragraph - the voltage swings between ~0V and ~15V. As mentioned in my post yesterday, the ITMY UL channel remains dodgy, with voltages of 12.84V (bit=1) and 10.79V (bit=0). So unless I am missing something, this must point to a faulty CONTEC card? We do have spares, do we want to replace this? It also looks like this problem has been present since at least 2011...
In any case, why should this lead to ITMY UL glitching? According to the MAX333 datasheet, the switch wants "low"<0.8V and "high">2.4V - so even if the CONTEC card is malfunctioning and the output is toggling between these two states, the condition should be that the whitening stage is always bypassed for this channel. The bypassed route works just fine, I measured the transfer function and it is unity as expected.
So what could possibly be leading to the glitches? I doubt that replacing the BO card will solve this problem. One possibility that came up in today's meeting is that perhaps the +24V to the Sat. Box. (which is used to derive the OSEM LED drive current) may be glitching - of course we have no monitor for this, but given that all the Sat. Amp. Adaptor boards are on 1X5 near the Acromag, perhaps Lydia and Johannes can recommission the PSL diagnostic Aromag to a power supply monitoring Acromag?
What do these glitches look like anyway? Here is a few second snapshot from one of the many MC1 excursions from yesterday - the original glitch itself is very fast, and then that gives an impulse to the damping loop which eventually damps away.
And here is one from when there was a glitch when the tester box was plugged in to the ITMY signal chain (so we can rule out anything in the vacuum, and also the satellite box itself as the glitches seem to remain even when boxes are shuffled around, and don't migrate with the box). So even though the real glitch happens in the UL channel (note the y axes are very different for the channels), the UR, LR and LL channels also "feel" it. recall that this is with the tester box (so no damping loops involved), and the fact that the side channel is more immune to it than the others is hard to explain. Could this just be electrical cross-coupling?
Still beats me what in the signal chain could cause this problem.
Some good news - Koji was running some tests on the modified WFS demod board and locked the IMC for this. We noticed that MC1 seemed well behaved for extended periods of time unlike last night. I realigned the PMC and IMC, and we have been having lock streches of a few hours as we usually have. I looked at the MC1 OSEM PD readbacks during the couple of lock losses in the last few hours, and didn't notice anything dramatic . So if things remain in this state, at least we can do other stuff with the IFO... I have plugged in the ITMY sat. box again, but have left the watchdog disabled, lets see what the glitching situation is overnight... The original ITMY sat. box has been plugged into the ETMY DAQ signal chain with a tester box. The 3 day trend supports the hypothesis the sat. box is not to blame. So I am plugging the ETMY suspension back in as well...
12654 Thu Dec 1 08:02:57 2016 SteveUpdateLSCglitching ITMY_UL_LL
12657 Fri Dec 2 11:56:42 2016 gautamUpdateLSCMC1 LEMO jiggled
I noticed 2 periods of frequent IMC locklosses on the StripTool trace, and so checked the MC1 PD readout channels to see if there were any coincident glitches. Turns out there wasnt BUT - the LR and UR signals had changed significantly over the last couple of days, which is when I've been working at 1X5. The fast LR readback was actually showing ~0, but the slow monitor channel had been steady so I suspected some cabling shenanigans.
Turns out, the problem was that the LEMO connector on the front of the MC1 whitening board had gotten jiggled ever so slightly - I re-jiggled it till the LR fast channel registered similar number of counts to the other channels. All looks good for now. For good measure, I checked the 3 day trend for the fast PD readback for all 8 SOS optics (40 channels in all, I didn't look at the ETMs as their whitening boards are at the ends), and everything looks okay... This while situation seems very precarious to me, perhaps we should have a more robust signal routing from the OSEMs to the DAQ that is more immune to cable touching etc...
12664 Mon Dec 5 15:05:37 2016 gautamUpdateLSCMC1 glitches are back
For no apparent reason, the MC1 glitches are back. Nothing has been touched near the PD whitening chassis today, and the trend suggests the glitching started about 3 hours ago.. I had disabled the MC1 watchdog for a while to avoid the damping loop kicking the suspension around when these glitches occur, but have re-enabled it now. IMC is holding lock for some minutes... I was hoping to do another round of ringdowns tonight, but if this persists, its going to be difficult...
12674 Thu Dec 8 10:13:43 2016 SteveUpdateLSCglitching ITMY_UL has a history
12860 Wed Mar 1 17:25:28 2017 SteveUpdateLSCMCREFL condition pictures
Gautam and Steve,
Our MCREFL rfpd C30642GH 2x2mm beeing investigated for burned spots.
Atm1, unused - brand new pd
Atm2,3,4 MCREFL in place was not moved
More pictures will be posted on 40m Picassa site later.
12891 Fri Mar 17 14:49:09 2017 gautamUpdateLSCMCREFL condition pictures
I did a quick measurement of the beam size on the MC REFL PD today morning. I disabled the MC autolocker while this measurement was in progress. The measurement set up was as follows:
This way I was able to get right up to the heat sink - so this is approximately 2cm away from the active area of the PD. I could also measure the beam size in both the horizontal and vertical directions.
The measured and fitted data are:
The beam size is ~0.4mm in diameter, while the active area of the photodiode is 2mm in diameter according to the datasheet. So the beam is ~5x smaller than the active area of the PD. I couldn't find anything in the datasheet about what the damage threshold is in terms of incident optical power, but there is ~100mW on th MC REFL PD when the MC is unlocked, which corresponds to a peak intensity of ~1.7 W / mm^2...
Even though no optics were intentionally touched for this measurement, I quickly verified that the spot is centered on the MC REFL PD by looking at the DC output of the PD, and then re-enabled the autolocker.
12952 Thu Apr 27 16:41:13 2017 Eric GustafsonUpdateLSC Status of the 40 m PD Frequency Response Fiber System
There two reports in the DCC describing the state of the system as of October 2014 including: (1) Alex Cole’s “T1300618 Automated photodiode Frequency Response Measurement System” and a Wiki created by Alex Cole where there are some instructions on the Master Script at https://wiki.ligo.caltech.edu/ajw?AlexanderCole
And (2) P140021 “Final Report: Automated Photodiode Frequency Response Measurement System for 40m Lab” by Nichin Sreekantaswamy and also as part of Nichin’s report by there is an archive of data at https://wiki-40m.ligo.caltech.edu/Electronics/PDFR%20system
I made a visual inspection of the system and saw that the following fibers collimators are still mounted in alignment mounts and the fiber is attached and pointed at a photodetector but possibly not aligned.
ASP Table
Photodetector Label Fiber Label
REFL11 REFL55 Fiber on mount
REFL33 REFL33 Fiber on mount
REFL55 REFL11 Fiber on mount
REFL165 No Fiber
AS55 AS55 Fiber on mount
MCREFPD MCREFPD Fiber on mount
No PD Loose unlabeled Fiber No mount
ITMX Optics Table
Photodetector Label Fiber Label
POX11 POX11 on mount
Unlabeled PD POP22/POP110 on mount
NO PD POP55 loose fiber No mount
The RF switch seems to be hooked up and there is a fiber running from the Diode Laser module to the fiber splitter module. So REFL 11 and REFL545 seem to be illuminated by the wrong fiber. I’ll try and run the software on Monday and check to see if I need to move the fibers or just relabel them.
13022 Wed May 31 12:58:30 2017 Eric GustafsonUpdateLSCRunning the 40 m PD Frequency Response Fiber System; Hardware and Software
Overall Design
A schematic of the overall subsystem diagran in attachment.
RF and Optical Connections
Starting at the top left corner is the diode laser module. This laser has an input which allows it to be amplitude modulated. The output of the laser is coupled into an optical fiber which is connectorized with an FC/APC connector and is connected to the input port of a 1 by 16 Optical Fiber Splitter. The Splitter produces 16 optical fiber outputs dividing the input laser power into 16 roughly equal optical optical fiber outputs. These optical fibers are routed to the Photodiode Receivers (PD) which are the devices under test. All of the PDs are illuminated simultaneously with amplitude modulated light. The Optical Fiber outputs each have a collimating fiber telescope which is used to focus the light onto the PDs. Optical Fiber CH1 is routed to a broadband flat response reference photodiode which is used to provide a reference to the HP-4395A Network Analyzer. The other Channel outputs are connected to an RF switch which can be programmed to select one of 16 inputs as the output. The selected outputs can then be sent into channel A of the RF Network Analyzer.
RF Switch
The RF switch consists of two 8 by 1 Multiplexers (National Instruments PXI-254x) slotted into a PXI Chassis (National Instruments PXI-1033). The Multiplexers have 8 RF inputs and one RF output and can be programmed through the PXI Chassis to select one and only one of the 8 inputs to be routed to the RF output.) The first 8 Channels are connected to the first 8 inputs of the first Multiplexer. The first Multiplexer’s output is then connected to the Channel 1 input of the second Multiplexer. The remaining PD outputs are connected to the remaining inputs of the second Multiplexer. The output of the second Multiplexer is connected to the A channel of the RF Network Analyzer. Thus it is possible to select any one of the PD RF outputs for analysis.
Software
Something on this tomorrow.
13248 Thu Aug 24 00:39:47 2017 gautamUpdateLSCDRMI locking attempt
Since the single arm locking and dither alignment seemed to work alright after the CDS overhaul, I decided to try some recycling cavity locking tonight.
• First, I locked single arms, ran dither alignment servos, and centered all test mass Oplevs. Note: the X arm dither alignment doesn't seem to work if we use the High-Gain Thorlabs PD as the Transmission PD. The BS loops just seem to pick up large offsets and the alignment actually degrades over a couple of minutes. This needs to be investigated.
• Next, to get good PRM alignment, I manually moved the EPICS sliders till the REFL spot became roughly centered on the CCD screen.
• Then I tried locking PRMI on carrier using the usual C1IFOConfigure script - the lock was caught within ~30 seconds.
• The PRCL and MICH dither servo scripts also ran fine.
• Centered PRM Oplev.
• Next, I tried enabling the PRC angular feedforward.
• OAF model does not automatically revert to its safe.snap configuration on model reboot, so I first manually did this such that the correct filter banks were enabled.
• I was able to turn on the angular feedforward without disturbing the PRMI carrier lock. The angular motion of the POP spot on the CCD monitor was visibly reduced.
• At this point I decided to try DRMI locking.
• I centered the beam on the AS PDs with the simple Michelson.
• Centered the beam on the REFL PDs with PRM aligned and PRC flashing through resonances.
• Restored SRM alignment by eye with EPICS sliders.
• Cavity alignment seemed alright - so I tried to lock DRMI with the old settings (i.e. from DRMI 1f locking a couple of months ago). But I had no success.
• The behaviour of REFL55 (used for SRCL control) has changed dramatically - the analog whitening gain for this PD used to be +18dB, but at this setting, there are frequent ADC overflows. I had to reduce the whitening gain to +6dB to stop the ADC overflows. I also checked to make sure that the whitening setting was "manual" and not triggered.
Why should this have changed? I was just on the AS table and did re-center the beam onto the REFL 55 RFPD, but I had also done this in April/May when I was last doing DRMI locking. But I can't explain the apparent factor of ~4 increase in light level. I think I have some measurements of the light levels at various PDs from April 2017, I will see how the present levels line up.
Of course dataviever won't cooperate when I am trying to monitor testpoints.
I may be missing something obvious, but I am quitting for tonight, will look into this more tomorrow.
Unrelated to this work: looking at the GTRY spot on the CCD monitor, there seems to be some excess angular motion. Not sure where this is coming from. In the past, this sort of problem has been symptomatic of something going wonky with the Oplev loops. But I took loop measurements for ITMY and ETMY PIT and YAW, they look normal. I will investigate further when I am doing some more ALS work.
13250 Thu Aug 24 18:02:16 2017 GabrieleSummaryLSCFirst cavity length reconstruction with a neural network
## 1) Introduction
In brief, I trained a deep neural network (DNN) to recosntuct the cavity length, using as input only the transmitted power and the reflection PDH signals. The training was performed with simulated data, computed along 0.25s long trajectories sampled at 8kHz, with random ending point in the [-lambda/4, lambda/4] unique region and with random velocity.
The goal of thsi work is to validate the whole approach of length reconstruction witn DNN in the Fabry-Perot case, by comparing the DNN reconstruction with the ALS caivity lenght measurement. The final target is to deploy a system to lock PRMI and DRMI. Actually, the Fabry-Perot cavity problem is harder for a DNN: the cavity linewidth is quite narrow, forcing me to use very high sampling frequency (8kHz) to be able to capture a few samples at each resonance crossing. I'm using a recurrent neural network (RNN), in the input layers of the DNN, and this is traine using truncated backpropagation in time (TBPT): during training each layer of RNN is unrolled into as many copies as there are input time samples (8192 * 0.25 = 2048). So in practice I'm training a DNN with >2000 layers! The limit here is computational, mostly the GPU memory. That's why I'm not able to use longer data stretches.
But in brief, the DNN reconstruction is performing well for the first attempt.
## 2) Training simulation
In the results shown below, I'm using a pre-trained network with parameters that do not match very well the actual data, in particular for the distribution of mirror velocity and the sensing noises. I'm working on improving the training.
I used the following parameters for the Fabry-Perot cavity:
The uncertaint is assumed to be the 90% confidence level of a gaussian distribution. The DNN is trained on 100000 examples, each one a 0.25/8kHz long trajectory with random velocity between 0.1 and 5 um/s, and ending point distributed as follow: 33% uniform on the [-lambda/4, lambda/4] region, plus 33% gaussian distribution peaked at the center with 5 nm width. In addition there are 33% more static examples, distributed near the center.
For each point along the trajectory, the signals TRA, POX11_I and POX11_Q are computed and used as input to the DNN.
## 3) Experimental data
Gautam collected about 10 minutes of data with the free swinging cavity, with ALS locked on the arm. Some more data were collected with the cavity driven, to increase the motion. I used the driven dataset in the analysis below.
### 3.1) ALS calibration
The ALS signal is calibrated in green Hz. After converting it to meters, I checked the calibration by measuring the distance between carrier peaks. It turned out that the ALS signal is undercalibrated by about 26%. After correcting for this, I found that there is a small non-linearity in the ALS response over multiple FSR. So I binned the ALS signal over the entire range and averaged the TRA power in each bin, to get the transmission signals as a function of ALS (in nm) below:
I used a peak detection algorithm to extract the carrier and 11 MHz sideband peaks, and compared them with the nominal positions. The difference between the expected and measured peak positions as a function of the ALS signal is shown below, with a quadratic fit that I used to improve the ALS calibration
The result is
z_initial = 1e9 * L*lamba/c *1.26. * ALS
z_corrected = 2.1e-06 z^2 -1.9e-02 z -6.91e+02
The ALS calibrated z error from the peak position is of the order of 3 nm (one sigma)
## 3.2) Mirror velocity
Using the calibrated ALS signal, I computed the cavity length velocity. The histogram below shows that this is well described by a gaussian with width of about 3 um/s. In my DNN training I used a different velocity distribution, but this shouldn't have a big impact. I'm retraining with a different distirbution.
## 4) DNN results
The plot below shows a stretch of time domain DNN reconstruction, compared with the ALS calibrated signal. The DNN output is limited in the [-lambda/4, lambda/4] region, so the ALS signal is also wrapped in the same region. In general the DNN reconstruction follows reasonably well the real motion, mostly failing when the velocity is small and the cavity is simultanously out of resonance. This is a limitation that i see also in simulation, and it is due to the short training time of 0.25s.
I did not hand-pick a good period, this is representative of the average performance. To get a better understanding of the performance, here's a histogram of the error for 100 seconds of data:
The central peak was fitted with a gaussian, just to give a rough idea of its width, although the tails are much wider. A more interesting plot is the hisrogram below of the reconstructed position as a function of the ALS position, Ideally one would expect a perfect diagonal. The result isn't too far from the expectation:
The largest off diagonal peak is at (-27, 125) and marked with the red cross. Its origin is more clear in the plot below, which shows the mean, RMS and maximum error as a function of the cavity length. The second peak corresponds to where the 55 MHz sideband resonate. In my training model, there were no 55 MHz sidebands nor higher order modes.
## 5) Conclusions and next steps
The DNN reconstruction performance is already quite good, considering that the DNN couldn't be trained optimally because of computation power limitations. This is a validation of the whole idea of training the DNN offline on a simulation and then deploy the system online.
I'm working to improve the results by
• training on a more realistic distribution of velocity
• adding the 55 MHz sidebands
• tune the DNN architecture
However I won't spend too much time on this, since I think the idea has been already validated.
13251 Thu Aug 24 18:51:57 2017 KojiSummaryLSCFirst cavity length reconstruction with a neural network
Phenomenal!
13252 Fri Aug 25 01:20:52 2017 gautamUpdateLSCDRMI locking attempt
I tried some DRMI locking again tonight, but had no success. Here is the story.
• I started out by going to the AS table and measuring the light level on the REFL55 photodiode (with PRM aligned and the PRC flashing, but LSC disabled).
• The Ophir power meter reads 13mW
• The DC output of the photodiode shows ~500mV on an oscilloscope.
• Both of these numbers line up well with measurements I made in April/May.
• Returned to the control room and aligned the IFO for DRMI locking - but LSC servos remained disabled.
• At the nominal REFL55 whitening level of +18dB, the REFL 55 signals saturated the ADC (confirmed by looking at the traces on dataviewer).
• But the signals still looked like PDH error signals.
• Lowering the whitening gain to 6dB makes the PDH error signal horns peak around 20,000 counts.
• Could this be indicative of problems with either the analog whitening gain switching or the LSC Demod Boards? To be investigated.
• Tried enabling LSC servos with same settings with which I had success right up till a couple of months ago, but had no success.
• If it is true that the REFL55 signal is getting amplified because of some gain stage not being switched correctly, I should still have been able to lock the SRC with a lowered loop gain - but even lowering the gain by a factor of 10 had no effect on the locking success rate.
Looks like I will have to embark on the REFL55 LSC electronics investigation. I was able to successfully lock the PRC on carrier and sideband, and the Michelson lock also seems to work fine, all of which seem to point to a hardware problem with the REFL55 signal chain.
I did a quick check by switching the output of the REFL55 demod board to the inputs normally used by AS55 signals on the whitening board. Setting the whitening gain to +18dB for these channels had the same effect - ADC overflow galore. So looks like the whitening board isn't to blame. I will have to check the demod board out.
13256 Sat Aug 26 09:56:34 2017 GabrieleSummaryLSCFirst cavity length reconstruction with a neural network
## Update
I included the 55 MHz sideband and higher order modes in my training examples. To keep things simple, I just assumed there are higher order modes up to n+m=4 in the input beam. The power in each HOM is randomly chosen from a random gaussian distribution with width determined from experimental cavity scans. I used a value of 0.913+-0.01 rad for the Gouy phase (again estimated from cavity scans, but in reasonable agreement with the nominal radius of curvature of ETMX)
Results are improved. The plot belows show the performance of the neural network on 100s of experimental data
For reference, the plots below show the performance of the same network on simulated data (that includes sensing noise but no higher order modes)
13258 Mon Aug 28 08:47:32 2017 JamieSummaryLSCFirst cavity length reconstruction with a neural network
Quote: Phenomenal!
truly.
13274 Wed Aug 30 11:04:08 2017 GabrieleSummaryLSCFirst look at neural network reconstruction of PRMI motion
## Introduction
I trained a deep neural network (DNN) to reconstruct MICH and PRCL degrees of freedom in the PRMI configuration. For details on the DNN architecture please refer to G1701455 or G1701589. Or if you really want all the details you can look at the code. I used the following signals as input to the DNN: POPDC, POP22_Q, ASDC, REFL11_I/Q, REFL55_I/Q, AS55_I/Q.
Gautam took some PRMI data in free swinging and driven configuration:
• 1187819331 + 10mins: Free swinging PRMI (after first locking PRMI on carrier and dither aligning).
• 1187820070 + 5mins: PRM driven at low freq.
• 1187820446 + 5mins: BS driven at low freq.
In contrast to the Fabry-Perot cavity case, we don't have a direct measurement of the real PRCL/MICH degrees of freedom, so it's more difficult to assess if the DNN is working well.
## Results
All MICH and PRCL values are wrapped into the unique region [-lambda/4, lambda/4]^2. It's even a bit more complicated than simpling wrapping. Indeed, MICH is periodic over [-lambda/2, lambda/2]. However, the Michelson interferometer reflectivity (as seen from PRC) in the first half of the segment is the same as in the second half, except for a change in sign. This change of sign in Michelson reflectivity can be compensated by moving PRCL by lambda/4, thus generating a pi phase shift in the PRC round trip propagation that compensate for the MICH sign change. Therefore, the unit cell of unique values for all signals can be taken as [-lambda/4, lambda/4] x [-lambda/4, lambda/4] for MICH x PRCL. But when we hit the border of the MICH region, PRCL is also affected by addtion of lambda/4. Graphically, the square regions A B C below are all equivalent, as well as more that are not highlighted:
This makes it a bit hard to un-wrap the resonstructed signal, especially when you add in the factor that in the reconstruction the wrapping is "soft".
The plot below shows an example of the time domain reconstruction of MICH/PRCL during the free swinging period.
It's hard to tell if the positions look reasonable, with all the wrapping going on.
## Two-dimensional maps of signals
Here's an attempt at validating the DNN reconstruction. Using the reconstructed MICH/PRCL signal, I can create a 2d map of the values of the optical signals. I binned the reconstructed MICH/PRCL in a 51x51 grid, and computed the mean value of all optical signals for each bin. The result is shown in the plot below, directly compared with the expectation from a simulation.
The power signals (POP_DC, AS_DC, PO22_Q) looks reasonably good. REFL11_I/Q also looks good (please note that due to an early mistake in my code, I reversed the convention for I/Q, so PRCL signal is maximized in Q instead than in I). The 55MHz signals look a bit less clear...
## Steps forward
• I'm quite confident in the tuning of demodulation phase and signs for REFL11 and POP22, but less so for REFL55 and not sure at all for AS55. So it would be useful to measure a full sensing matrix of PRCL and MICH against those signals, to compare with my simulation
• I'm working on an idea to fine tune the DNN using the real interferometer data, more to follow when the idea crystallizes in a clear form.
13276 Wed Aug 30 19:49:33 2017 gautamUpdateLSCREFL55 demod board debugging
## Summary:
Today I tried debugging the mysterious increase in REFL55 signal levels in the DRMI configuration. I focused on the demod board, because last week, I had tried routing these signals through different channels on the whitening board, and saw the same effect.
Based on my tests, everything on the Demod board seems to work as expected. I need to think more about what else could be happening here - specifically do a more direct test on the whitening board.
## Details:
• The demod board is a modified D990511 (marked up schematic + high-res photo to follow).
• Initially, I tried probing the LO signal levels at various points with the board in the eurocrate itself, with the help of an extender card.
• But this wasn't very convenient, so I pulled the board out to the office area for more testing.
• The 55MHz LO signal going into the board is ~0dBm (measured with Agilent network analyzer)
• I used the active probe to check the LO levels at various points along the signal chain, which mostly consists of attenuators, ERA-5SM amplifiers, and some splitters/phase rotators.
• Everything seemed consistent with the expected levels based on "typical" numbers for gains and insertion losses cited in the datasheets for these devices.
• I couldn't directly measure the level at the LO input to the mixer, but measuring the input to the ERA-5SM immediately before the mixer, barring problems with this amplifier, the LO input of the mixer is being driven at >17dBm which is what it wants.
• Next, I decided to check the gain, gain imbalance and orthogonality of the demodulation.
• For this purpose, I restored the board to the Eurocrate, reconnected the LO input to the board, and used a second Marconi at a slightly offset frequency to drive the PD input at ~0dBm.
• Attachment #1 - The measured outputs look pretty balanced and orthogonal. The gain is consistent with an earlier measurement I made some months ago, when things were "normal". More bullets added after Rana's questions:
• 300 MHz bandwidth oscilloscope used to acquire the data
• I and Q outputs were from the daughter board
Quote: I did a quick check by switching the output of the REFL55 demod board to the inputs normally used by AS55 signals on the whitening board. Setting the whitening gain to +18dB for these channels had the same effect - ADC overflow galore. So looks like the whitening board isn't to blame. I will have to check the demod board out.
All connections have been restored untill further debugging later in the evening.
13280 Thu Aug 31 00:52:52 2017 gautamUpdateLSCREFL55 whitening board debugging
[rana,gautam]
We did an ingenious checkup of the whitening board tonight.
• The board is D990694
• We made use of a tip-tilt DAC channel for this test (specifically TT1 UL, which is channel 1 on the AI board). We disconnected the cable going from the AI board to the TT coil driver board.
• as opposed to using a function generator to drive the whitening filter, this approach allows us to not have to worry the changing offsets as we switch the whitening gain.
• By using the CDS system to generate the signal and also demodulate it, we also don't have to worry about the drive and demod frequencies falling out of sync with each other.
• The test was done by injecting a low frequency (75.13 Hz, amplitude=0.1) excitation to this DAC channel, and using the LSC sensing matrix infrastructure to demodulate REFL55 I and Q at this frequency. Demod phases in these servos were adjusted such that the Q phase demodulated signal was minimized.
• An excitation was injected using awggui into TT1 UL exc channel.
• We then stepped the whitening gains for REFL55_I and REFL55_Q in 3dB steps, waiting 5 seconds for each step. Syntax is z step -s 5 C1:LSC-REFL55_I_WhiteGain +1.0,15 C1:LSC-REFL55_Q_WhiteGain +1.0,15
• Attachment #1 suggests that the whitening filter board is working as expected (each step is indeed 3dB and all steps are equal to the eye).
• Data + script used to generate this plot is in Attachment #2.
I've restored all connections at that we messed with at the LSC rack to their original positions.
The TT alignment seems to be drifting around more than usual after we disconnected one of the channels - when I came in today afternoon, the spot on the AS camera had drifted by ~1 spot diameter so I had to manually re-align TT1.
Quote: Based on my tests, everything on the Demod board seems to work as expected. I need to think more about what else could be happening here - specifically do a more direct test on the whitening board.
13281 Thu Aug 31 03:31:15 2017 gautamUpdateLSCDRMI re-locked!
After our Demod/Whitening electronics investigations suggested nothing obviously wrong, I decided to give DRMI locking another go tonight.
Surprisingly, there was no evidence of REFL55 behaving weirdly tonight, and I was able to easily lock the DRMI on 1f error signals using the recipe I've been using in the last few months.
Not sure what to make of all this .
I got in a ~15 minute lock, but I wasn't prepared to do any sort of characterization/ sensing / attempt to turn on coil-dewhitening, and I'm too tired to try again tonight. I was however able to whiten the error signals, as I have been able to do in the past. There is a ~45Hz bump in MICH that I haven't seen in the past.
I'll try and do some characterization tomorrow eve, but it's encouraging to at least get back to the pre-FB-failure state of locking.
13289 Mon Sep 4 16:30:06 2017 gautamUpdateLSCOplev loop tweaking
Now that the DRMI locking seems to be repeatable again, I want to see if I can improve the measured MICH noise. Recall that the two dominant sources of noise were
1. BS Oplev loop A2L - this was the main noise between 30-60Hz.
2. DAC noise - this dominated between ~60-300Hz, since we were operating with the de-whitening filters off.
In preparation for some locking attempts today evening, I did the following:
1. Added steeper elliptic roll-off filters for the ITMX and ITMY Oplevs. This is necessary to allow the de-whitening filters to be turned on without railing the DAC.
2. Modified the BS Oplev loop to also have steeper high-frequency (>30Hz) roll off. The roll-off between 15-30Hz is slightly less steep as a result of this change.
3. Measured all Oplev loop TFs - UGFs are between 4 Hz and 5 Hz, phase margin is ~30degrees. I did not do any systematic optimization of this for today.
4. Went into the Foton filter banks for all the coil output filters, and modified the "Output" settings to be on "Input crossing", with a "Tolerance" of 10 and a "Timeout" of 3 seconds. These settings are to facilitate smooth transition between the two signal paths (without and with coil-dewhitening). The parameters chosen were arbitrary and not optimized in any systematic manner.
5. After making the above changes, I tried engaging the de-whitening filters on ITMX, ITMY and BS with the arms locked. In the past, I was unable to do this because of a number of issues - Oplev loop shapes and Foton settings among them. But today, the switching was smooth, the single arm locks weren't disturbed when I engaged the coil de-whitening.
Hopefully, I can successfully engage a similar transition tonight with the DRMI locked. The main difference compared to this daytime test is going to be that the MICH control signal is also going to be routed to the BS.
Tasks for tonight, if all goes well:
1. Lock DRMI.
2. Use UGF servos to set the overall loop gains for DRMI DoFs.
3. Reduce PRCL->MICH and SRCL->MICH coupling.
4. Measure loop shapes of all DRMI DoFs.
5. Make sensing matrix measurement.
Unrelated to this work: the PMC was locked near the upper rail of the PZT, so I re-locked it closer to the middle of the range.
Quote: Surprisingly, there was no evidence of REFL55 behaving weirdly tonight, and I was able to easily lock the DRMI on 1f error signals using the recipe I've been using in the last few months.
13290 Mon Sep 4 18:18:29 2017 ranaUpdateLSCdewhite switching: FOTON settings
not immediately necessary, since you have already got it sort of working, but one of these days we should optimize this for real. In the past, we used to do this by putting a o'scope on the coil Vmon during the switching to catch the transient w/ triggering. We download the data/picture via ethernet. Run for loop on tolerance to see what's what.
Went into the Foton filter banks for all the coil output filters, and modified the "Output" settings to be on "Input crossing", with a "Tolerance" of 10 and a "Timeout" of 3 seconds. These settings are to facilitate smooth transition between the two signal paths (without and with coil-dewhitening). The parameters chosen were arbitrary and not optimized in any systematic manner.
13291 Tue Sep 5 02:07:49 2017 gautamUpdateLSCLow Noise DRMI attempt
## Summary:
Tonight, I was able to lock the DRMI, turn on the whitening filters for the sensing PDs, and also turn on the coil de-whitening filters for ITMX, ITMY and BS. However, I didn't see the expected improvement in the MICH spectrum between ~50-300 Hz . Sad.
## Details:
I basically went through the list of tasks I made in the previous elog. Some notes:
• The UGF servos suggested that I had to lower the SRCL gain. I lowered it from -0.055 to -0.025. OLTF measurement using In1/In2 method suggested UGF ~120Hz. I don't know why this should be. Plot to be uploaded later.
• Since we aren't actuating on the ITMs, I was able to leave their coils de-whitened all the time.
• For the BS, it was trickier - I had to play around a little with the "Tolerance" setting in Foton while looking at transients (using DTT, not a scope for now) while switching the filters.
• This transition isn't so robust yet - but eventually I found a setting that worked, and I was able to successfully turn on the de-whitening thrice tonight (but also failed about the same number of times). [GV Oct 6 2017: Remember that the PD whitening has to be turned on for this transition to be successful - otherwise the RMS from the high frequencies saturate the DAC.]
• The locks were pretty stable. One was ~10mins, one was ~15mins, and I broke both deliberately because I was out of ideas as to why the part of the MICH error signal spectrum that I thought was due to DAC noise didn't improve.
• I've made a bunch of shell scripts to help with the various switchings - but now that I think of it, I should make these python scripts.
Attachment #1: Comparison of MICH_ERR with and without the BS de-whitening. Note that the two ITMs have their coils de-whitened in both sets of traces.
Attachment #2: Spectra of MICH output and one of the BS coil outputs in both states. The DAC RMS increases by ~30x when the de-whitening is engaged, but is still well within limits.
So it looks like the switching of paths is happening correctly. The "CDS BIO STATUS" MEDM screen also shows the appropriate bits toggling when I turn the de-whitening on/off. There is no broadband coherence with MCF between 50-300 Hz so it seems unlikely that this could be frequency noise.
Clearly I am missing something. But anyways I have a good amount of data, may be useful to put together the post CDS/electronics modification DRMI noise budget. More analysis to follow.
13294 Tue Sep 5 16:37:47 2017 GabrieleSummaryLSCImproved PRMI deep learning reconstruction
This is an update on the results already presented earlier (refer to elog 13274 for more introductory details). I improved significantly the results with the following tricks:
• I retuned the demodulation phase of AS55, this time ensuring that the (alleged) MICH motion is visible mostly in Q when crossing a carrier resonance. Further fine tunings of phases will be possible once we have a measurement of the length optical matrix
• I fine tuned the netwrok by training it again using the real data. The ides is the following. I started with the network trained on the simulated data, and froze the parameters of the input recurrent layers. I fed the real signal to the network, computed the reconstructed PRCL/MICH, and fed them to my PRMI model to compute simulated signals. I allowed some of the parameters of the models to vary (expecially demodulation phases). I then trained again the network by trying to match the model predicted signals with the real input signals. I allowed only the parameters of the fully connected layers to vary (mostly for technical reasons, I'm working on re-training also the recurrent layers)
An example of time domain reconstruction is visible below. It already looks better than the old results:
As before, to better evaluate the performance I plotted averaged values of the real signals as a function of the reconstructed MICH and PRCL positions. The results are compared with simulation below. They match quite well (left real data, right simualtion expectation)
One thing to better understand is that MICH seems to be somewhat compressed: most of the output values are between -100 and +100 nm, instead of the expected -lambda/4, lambda/4. The reason is still unclear to me. It might be a bug that I haven't been able to track down yet.
13300 Wed Sep 6 23:06:30 2017 gautamUpdateLSCCoil de-whitening switching investigation
## Rana suggested checking if the coil de-whitening switching is actually happening in the analog path. I repeated the test detailed here. Attachments #1 and #2 suggest that all the coils for the BS and ITMs are indeed switching.
### Details:
• The motivation behind this test was the following - the analog path switching is done by applying some logic voltage to a switch, but if this voltage is common among many switches, the hypothesis was that perhaps individual switches were not getting the required voltage to engage the switching.
• This time FM9 (simulated de-whitening) and FM10 (inverse de-whitening) in the coil output filter modules turned off, so as to maintain a flat TF in the digital domain, but engage the de-whitened analog path (turning off FM9 is supposed to do this).
• There is poor coherence in the measurement above 40Hz so the data there should be neglected. It is hard to get a good measurement at higher frequencies because of the pendulum TF + heavy low pass filtering from the analog de-whitening path.
• But between 10-40Hz, we already see the analog de-whitening TF in the measurement.
• For comparison, I have plotted the measured pendulum TFs for one of the coils from an earlier test (all the coils were roughly at the same level).
So it would seem that there is some other noise which has a 1/f^2 shape and is at the same level we expected the DAC noise to be at. Rana suggested checking coherence with MC transmission to see if this could be laser intensity noise.
I also want to re-do the actuator calibrations for the vertex optics again before re-posting the revised noise budget.
13304 Fri Sep 8 12:08:32 2017 GabrieleSummaryLSCGood reconstruction of PRMI degrees of freedom with deep learning
## Introduction
This is an update of my previous reports on applications of deep learning to the reconstruction of PRMI degrees of freedom (MICH/PRCL) from real free swinging data. The results shown here are improved with respect to elog 13274 and 13294. The training is performed in two steps, the first one using simulated data, and the second one fine tuning the parameters on real data.
## First step: training with simulation
This step is exactly the same already described in the previous entries and in my talks at the CSWG and LVC. For details on the DNN architecture please refer to G1701455 or G1701589. Or if you really want all the details you can look at the code. I used the following signals as input to the DNN: POPDC, POP22_Q, ASDC, REFL11_I/Q, REFL55_I/Q, AS55_I/Q. The network is trained using linear trajectories in the PRCL/MICH space, and signals obtained from a model that simulates the PRMI behavior in the plane wave approximation. A total of 150000 trajectories are used. The model includes uncertainties in all the optical parameters of the 40m PRMI configuration, so that the optical signals for each trajectory are actually computed using random optical parameteres, drwn from gaussian distributions with proper mean and width. Also, white random gaussian sensing noise is added to all signals with levels comparable to the measured sensing noise.
The typical performance on real data of a network pre-trained in this way was already described in elog 13274, and although being reasoble, it was not too good.
## Second step: training with real data
Real free swinging data is used in this step. I fine tuned the demodulation phases of the real signals. Please note that due to an old mistake, my convention for phases is 90 degrees off, so for example REFL11 is tuned such that PRCL is maximized in Q instead of I. Regardless of this convention confusion, here's how I tuned the phases:
• REFL11: PRCL is all in Q when crossing the carrier resonance
• REFL55: PRCL is all in Q when crossing the carrier resonance
• AS55: MICH is all in Q when crossing the PRCL carrier resonance
• POP22: signal peaking in Q when crossing carrier or sideband resonances. Carrier resonance crossing gives positive sign
Then I built the following training architecture. The neural network takes the real signals and produces estimates of PRCL and MICH for each time sample. Those estimates are used as the input for the PRMI model, to produce the corresponding simulated optical signals. My cost function is the squared difference of the simulated versus real signals. The training data is generated from the real signals, by selection 100000 random 0.25s long chunks: the history of real signal over the whole 0.25s is used as input, and only the last sample is used for the cost function computation. The weights and biases of the neural network, as well as the model parameters are allowed to change during the learning process. The model parameters are regularized to suppress large deviations from the nominal values.
One side note here. At first sight it might seems weird that I'm actually fedding as input the last sample and at the same time using it as the reference for the loss function. However, you have to remember that there is no "direct" path from input to output: instead all goes through the estimated MICH/PRCL degrees of freedom, and the optical model. So this actually forces the network to tune the reconstruction to the model. This approach is very similar to the auto-encoder architectures used in unsupervised feature learning in image recognition.
## Results
After trainng the network with the two previous steps, I can produce time domain plots like the one below, which show MICH and PRCL signals behaving reasonably well:
To get a feeling of how good the reconstruction is, I produced the 2d maps shown below. I divided the MICH/PRCL plane in 51x51 bins, and averaged the real optical signals with binning determined by the reconstructed MICH and PRCL degrees of freedom. For comparison the expected simulation results are shown. I would say that reconstructed and simulated results match quite well. It looks like MICH reconstruction is still a bit "compressed", but this should not be a big issue, since it should still work for lock acquisition.
## Next steps
There a few things that can be done to futher tune the network. Those are mostly details, and I don't expect significant improvements. However, I think the results are good enough to move on to the next step, which is the on-line implementation of the neural network in the real time system.
13313 Fri Sep 15 16:00:33 2017 gautamUpdateLSCSensing measurement
I've been working on analyzing the data from the DRMI locks last week.
Here are the results of the sensing measurement.
Details:
1. The sensing measurement is done by using the existing sensing matrix infrastructure to drive the actuators for the various DoFs at specific frequencies (notches at these frequencies are turned on in the control loops during the measurement).
2. All the analysis is done offline - I just note down the times at which the sensing lines are turned on and then download the data later. The amplitudes of the oscillators are chosen by looking at the LSC PD error signal spectra "live" in DTT, and by increasing the amplitude until the peak height is ~10x above the nominal level around that frequency. This analysis was done on ~600seconds of data.
3. The actual sensing elements in the various PDs are calculated as follows:
• Calculate the Fourier coefficients at the excitation frequency using the definition of the complex DFT in both the LSC PD signal and the actuator signal (both are in counts). Windowing is "Tukey", and FFT length used is 1 second.
• Take their ratio
• Convert to suitable units (in this case V/m) knowing (i) The actuator discriminant in cts/m and (ii) the cts/V ADC calibration factor. Any whitening gain on the PD is taken into account as well.
• If required, we can convert this to W/m as well, knowing (i) the PD responsivity and (ii) the demodulation chain gain.
• Most of this stuff has been scripted by EricQ and is maintained in the pynoisesub git repo.
The plotting utility is a work in progress - I've basically adapted EricQs scripts and added a few features like plotting the uncertainties in magnitude and phase of the calculated sensing elements. Possible further stuff to implement:
• Only plot those elements which have good coherence in the measurement data. At present, the scripts check the coherence and prompt the user if there is poor coherence in a particular channel, but no vetos are done.
• The uncertainty calculation is done rather naively now - it is just the standard deviation in the fourier coefficient determined from various bins. I am told that Bendat and Piersol has the required math. It would be good to also incorporate the uncertainties in the actuator calibration. These are calculated using the python uncertainties package for now.
• Print a summary of the parameters used in the calculation, as well as sensing elements + uncertainty in cts/m, V/m and W/m, on a separate page.
• Some aesthetics can be improved - I've had some trouble getting the tick intervals to cooperate so I left it as is for the moment.
Also, the value I've used for the BS actuator calibration is not a measured one - rather, I estimated what it will be by scaling the old value by the same ratio which the ITMs have changed by post de-whitening board mods. The ITM actuator coefficients were recently measured here. I will re-do the BS calibrations over the weekend.
Noise budgeting to follow - it looks like I didn't set the AS55 demod phase to the previously determined optimal value of -82degrees, I had left it at -42 degrees. To be fixed for the next round of locking.
13314 Fri Sep 15 17:08:58 2017 gautamUpdateLSCCoil de-whitening switching investigation
I downloaded a segment of data from the time when the DRMI was locked with the BS and ITM coil driver de-whitening switched on, and looked at coherence between MC transmission and the MICH error signal. Attachment #1 doesn't show any broadband high coherence between 60-300Hz, so it cannot explain the noise in the full range between 60-300Hz.
The DQ channel for the MC transmission is recorded at 1024 kHz, so to calculate the coherence, I had to decimate the 16K MICH data.
Since we have the AOM installed, I suppose we can actually measure the intensity noise coupling to MICH by driving a line in the AOM.
I also checked for coherence in the 60-300Hz band between MICH/PRCL and MICH/SRCL, and didn't see any appreciable coherence. Need to think about this more.
Quote: Rana suggested checking coherence with MC transmission to see if this could be laser intensity noise.
13315 Sat Sep 16 10:56:19 2017 ranaUpdateLSCCoil de-whitening switching investigation
The absence of evidence is not evidence of absence.
13328 Fri Sep 22 18:12:27 2017 gautamUpdateLSCDAC noise measurement (again)
I've been working on setting up some scripts for measuring the DAC noise.
In all the DRMI noise budgets I've posted, the coil-driver noise contribution has been based on this measurement, which could be improved in a couple of ways:
• The measurement was made at the output of the AI board - we can make the measurement at the output of the coil driver board, which will be a closer reflection of the actual current noise at the OSEM coils.
• The measurement was made by driving the DAC with shaped random noise - but we can record the signal to the coils during a lock and make the noise measurement by using awg to drive the coil with this signal, with elliptic bandstops at appropriate frequencies to reveal the electronics noise.
1. The IN1 signals to the coils aren't DQ-ed, but ideally this is the signal we want to inject into the coil_EXC channel for this measurement - so I re-locked the DRMI a couple of nights ago and downloaded the coil IN1 channel data for ~5mins for the ITMs and BS.
2. AWGGUI supposedly has a feature that allows you to drive an EXC channel with an arbitrary signal - but I couldn't figure out how to get this working. I did find some examples of this kind of application using the Python awg packages, so I cobbled together some scripts that allows me to drive some channels and place elliptic bandstop filters as I required.
3. I wasted quite a bit of time trying to implement these signals in Python using available scipy functions, on account of me being a DSP n00b . When trying to design discrete-time filters, of course numerical precision errors become important. Initially I was trying to do everything in the "Transfer function (numerator-denominator)" basis, but as Rana pointed out, the way to go is using SOSs. Fortunately, this is a simple additional argument to the relevant python functions, after which elliptic bandstop filter design was trivial.
4. The actual test was done as follows:
• Save EPICS PIT/YAW offsets, set them to 0, disable Oplev servos, and then shut down optic watchdog once the optic is somewhat damped. This is to avoid the optics getting a large kick when disconnecting the DB15 connector from the coil driver board output.
• Disconnect above-mentioned DB15 connector from the appropriate coil driver board output.
• Turn off inputs to coils in filter module EPICs screens. Since the full signal (local damping servo output + Oplev servo output + LSC servo output) to the coil during a DRMI lock will be injected as an excitation, we don't need any other input.
• Use scripts (which I will upload to a git repo soon) to set up the appropriate excitation.
• To measure the spectrum, I used a DB15 breakout board with test-points soldered on and some mini-grabber-to-BNC adaptors, in order to interface the SR785 to the coil driver output. We can use the two input channels of the SR785 to simultaneously measure two coil driver board output channels to save some time.
• Take a measurement of the SR785 noise (at the appropriate "Input Range" setting) with inputs terminated to estimate the analyzer noise floor.
• Just for kicks, I made the measurement with the de-whitening both OFF/ON.
I only managed to get in measurements for the BS and ITMX today. ITMY to be measured later, and data/analysis to follow.
The ITMX and BS alignments have been restored after this work in case anyone else wants to work with the IFO.
Some slow machine reboots were required today - c1susaux was down, and later, the MC autolocker got stuck because of c1iool0 being unresponsive. I thought we had removed all dependency of the autolocker on c1iool0 when we moved the "IFO-STATE" EPICS variable to the c1ioo model, but clearly there is still some dependancy. To be investigated.
ELOG V3.1.3-
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2022-12-08 06:47:57
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http://codeforces.com/problemset/problem/60/E
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E. Mushroom Gnomes
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
Once upon a time in the thicket of the mushroom forest lived mushroom gnomes. They were famous among their neighbors for their magic mushrooms. Their magic nature made it possible that between every two neighboring mushrooms every minute grew another mushroom with the weight equal to the sum of weights of two neighboring ones.
The mushroom gnomes loved it when everything was in order, that's why they always planted the mushrooms in one line in the order of their weights' increasing. Well... The gnomes planted the mushrooms and went to eat. After x minutes they returned and saw that new mushrooms had grown up, so that the increasing order had been violated. The gnomes replanted all the mushrooms in the correct order, that is, they sorted the mushrooms in the order of the weights' increasing. And went to eat again (those gnomes were quite big eaters). What total weights modulo p will the mushrooms have in another y minutes?
Input
The first line contains four integers n, x, y, p (1 ≤ n ≤ 106, 0 ≤ x, y ≤ 1018, x + y > 0, 2 ≤ p ≤ 109) which represent the number of mushrooms, the number of minutes after the first replanting, the number of minutes after the second replanting and the module. The next line contains n integers ai which represent the mushrooms' weight in the non-decreasing order (0 ≤ ai ≤ 109).
Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cin (also you may use %I64d).
Output
The answer should contain a single number which is the total weights of the mushrooms modulo p in the end after x + y minutes.
Examples
Input
2 1 0 6572765451 2
Output
6
Input
2 1 1 8884502821 2
Output
14
Input
4 5 0 100001 2 3 4
Output
1825
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2020-06-06 08:58:31
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http://math.stackexchange.com/questions/70048/mathematical-reason-for-the-validity-of-the-equation-s-1-x2-s
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# Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$
Given the geometric series:
$1 + x^2 + x^4 + x^6 + x^8 + \cdots$
We can recast it as:
$S = 1 + x^2 \, (1 + x^2 + x^4 + x^6 + x^8 + \cdots)$, where $S = 1 + x^2 + x^4 + x^6 + x^8 + \cdots$.
This recasting is possible only because there is an infinite number of terms in $S$.
Exactly how is this mathematically possible?
(Related, but not identical, question: General question on relation between infinite series and complex numbers).
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Are you asking why $\frac{1}{1-x^2} = 1 + x^2 \frac{1}{1-x^2}$? Note that the identification of the formal series $S$ with the function $\frac{1}{1-x^2}$ is valid for $|x| \lt 1$, otherwise it's just a game you can play with formal power series. – t.b. Oct 5 '11 at 12:31
+1 For realizing that it is essential for there to be an infinite number of terms. Infinite sums usually make sense only under some kind of convergence (in a ring of formal power series the convergence is of a very different kind). But yeah, it is the same phenomenon that is underlying the following. Let $S=0.9999\ldots$. Then $$S=0.9+0.09999\ldots=0.9+0.1\cdot S,$$ Therefore $0.9=S-0.1\cdot S=0.9 \cdot S$, so $S=0.9/0.9=1$. – Jyrki Lahtonen Oct 5 '11 at 12:40
Well, I'm basically trying to understand why we can do this recasting. Normally - (I'm assuming) - we cannot recast a finite series. But the above recasting is allowed since the terms go to infinity. ... What is the exact mathematical (i.e. rigorous) basis which allows this manipulation? – UGPhysics Oct 5 '11 at 12:47
You cannot even define the sum of infinitely many elements without some notion of a convergence. In the ring of formal power series (that many of us want to use here!) the convergence is based on the so called $x$-adic topology. Meaning that an infinite sum $\sum_{i=0}^\infty p_i(x)$ makes sense, if and only if the degrees of the lowest degree terms of the power series $p_i(x)$ tend to infinity. Here $p_i(x)=x^{2i}$ is its lowest degree term, and convergence thus follows from the fact that $2i\to\infty$ as $i\to\infty$. – Jyrki Lahtonen Oct 5 '11 at 13:18
IOW those three dots $\ldots$ hide a lot of reasoning. And 'no', the recasting does not make sense without some kind of a convergence allowing infinite sums to be formed in whatever structure we feel is most appropriate (here the complex numbers or the ring of formal power series). – Jyrki Lahtonen Oct 5 '11 at 13:22
show 8 more comments
There is a finite version of which the expression you have is the limit.
Suppose $S=1+x^2+x^4+x^6+x^8$, then we can put
$S+x^{10}=1+x^2(1+x^2+x^4+x^6+x^8)=1+x^2S$
And obviously this can be taken as far as you like, so you can replace 10 with 10,000 if you choose. If the absolute value of $x$ is less than 1, this extra term approaches zero as the exponent increases.
There is also a theory of formal power series, which does not depend on notions of convergence.
-
If x is a matrix, nilpotent to some even power, can we then say, that this holds also for finite series? (Or, more generally, if we have an algebra with zero-divisors, can we formally relax the condition of infiniteness?) – Gottfried Helms Oct 5 '11 at 12:56
Hi @Mark, thanks for the response. | May I know the formal power series version of your response? (I think that's what I'm searching for.) – UGPhysics Oct 5 '11 at 13:21
lhf has put a link to some notes on formal power series, which covers more than any comment I could make. What I wanted to show was that, while it is not possible to recast a finite truncation of your series in the form you gave, it is possible to get close if you include a kind of error term. Roughly, in formal power series the powers of $x$ act as placeholders for the coefficients. Arithmetic operations work as you would expect, but you have to be careful that the calculation for each coefficient of the result is a finite calculation. – Mark Bennet Oct 5 '11 at 13:55
The $n$th partial sum of your series is
\begin{align*} S_n &= 1+x^2+x^4+\cdots +x^{2n}= 1+x^2(1+x^2+x^4+\cdots +x^{2n-2})\\ &= 1+x^2S_{n-1} \end{align*}
Assuming your series converges you get that $$\lim_{n\to\infty}S_n=\lim_{n\to\infty}S_{n-1}=S.$$
Thus $S=1+x^2S$.
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$S = 1 + x^2 \, S$ is true even in the ring of formal power series. No convergence is needed here.
@UGPhysics, if you want to talk about numerical series, where $x$ is a number, then of course you need to worry about convergence. The manipulation you mentioned in the question is really purely formal. That was my point. – lhf Oct 5 '11 at 17:58
So, -to clarify this whole debate-, in the space of real numbers I cannot write an expression like $S$, and claim that I can recast it [as $1 + x^2\,S$] unless the expression is convergent, (or defined within a specified radius of convergence)? – UGPhysics Oct 5 '11 at 18:06
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2013-12-20 12:00:51
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https://erc.wisc.edu/publications/hydrodynamics-and-heat-transfer-of-multiple-droplet-impingement/
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# Hydrodynamics And Heat Transfer Of Multiple Droplet Impingement
Al-Roub, M. A. Hydrodynamics And Heat Transfer Of Multiple Droplet Impingement. University of Wisconsin-Madison, 1996.
The hydrodynamics and heat transfer of single droplets impinging on a heated surface have been used in the previous literature as a basis to assess the full spray impingement processes. In this study the effects of simultaneous impingement of more than one droplet at a time were investigated and compared to single droplet data in order to bridge the gap between the real full spray impingement process and the ideal process of single droplet. For that purpose parcels, groups of droplets with a uniform size, were used as the spray flux.
The effect of the inter-parcel spacing L/D, the parcel’s Weber number, and the surface superheat were tested. The liquid used was water, the inter-parcel spacing was 5-20 L/D, the droplets had a size 200-400 $\mu$m, and reach the wall at We = 70-300, and the surface superheat range was 0-250$\sp\circ$C.
The generated parcels were photographed, during their wall impact, using a high speed camera back lit by a pulsed laser. The heat transfer from the heated surface to the impinging parcels was measured by measuring the surface temperature instantaneously with time, with and without impingement.
In the low superheat regime, a liquid film was shown to deposit on the wall. The film was destabilized the most if impinged by two droplets impinging with We = 200 at (time ratio) TR = 0.5, and when (liquid film thickness) $\delta\sb{f}/d = 1.$ The change of the liquid film thickness produced three different breakup modes.
In the transition and film boiling regimes, the multiple droplet impact ejected more droplets than a single droplet does, however the dispersion of the ejected droplets was smaller than in the single droplet case due to increased coalescence and collision probabilities in the impingement site with the increase of the liquid fraction near the wall. The normal dispersion of the ejected droplets in the transition boiling regime was one order of magnitude higher than that in the film boiling regime. In the film boiling regime tangential dispersion of the droplets is higher than that for the transition boiling regime.
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2023-02-02 08:38:32
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https://proofwiki.org/wiki/Definition:Subspace_Topology
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# Definition:Topological Subspace
## Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a non-empty subset of $S$.
Define:
$\tau_H := \set {U \cap H: U \in \tau} \subseteq \powerset H$
where $\powerset H$ denotes the power set of $H$.
Then the topological space $T_H = \struct {H, \tau_H}$ is called a (topological) subspace of $T$.
The set $\tau_H$ is referred to as the subspace topology on $H$.
## Also known as
The subspace topology $\tau_H$ is also known as the relative topology or the induced topology on $H$.
## Also see
• Results about topological subspaces can be found here.
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2020-02-17 21:55:28
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http://books.duhnnae.com/2017/jul7/150131902351-JLip-versus-Sobolev-Spaces-on-a-Class-of-Self-Similar-Fractal-Foliages.php
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# JLip versus Sobolev Spaces on a Class of Self-Similar Fractal Foliages
1 LJLL - Laboratoire Jacques-Louis Lions 2 IRMAR - Institut de Recherche Mathématique de Rennes
Abstract : For a class of self-similar sets $\Gamma^\infty$ in $\R^2$, supplied with a probability measure $\mu$ called the self-similar measure, we investigate if the $B s^{q,q}\Gamma^\infty$ regularity of a function can be characterized using the coefficients of its expansion in the Haar wavelet basis. Using the the Lipschitz spaces with jumps recently introduced by Jonsson, the question can be rephrased: when does $B s^{q,q}\Gamma^\infty$ coincide with $JLips,q,q;0;\Gamma^\infty$? When $\Gamma^\infty$ is totally disconnected, this question has been positively answered by Jonsson for all $s,q$, $00$, \$1\le p,q
Author: Yves Achdou - Thibaut Deheuvels - Nicoletta Tchou -
Source: https://hal.archives-ouvertes.fr/
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2017-09-24 12:22:28
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http://indexsmart.mirasmart.com/ISMRM2018/PDFfiles/3402.html
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### 3402
Extending the small tip angle approximation to the non-equilibrium initial condition
Bahman Tahayori1,2,3, Zhaolin Chen2, Gary Egan2, and N. Jon Shah 4
1Electrical and Computer Systems Engineering, Monash University, Clayton, Australia, 2Monash Biomedical Imaging, Monash University, Clayton, Australia, 3Medical Physics and Biomedical Engineering Department, Shiraz University of Medical Sciences, Shiraz, Iran (Islamic Republic of), 4Department of Neurology, JARA, RWTH Aachen University, Aachen, Germany
### Synopsis
We have applied Volterra series expansion to the Bloch equation and have calculated the kernels for an arbitrary initial condition. We have shown that small tip angle approximation can be extended to the non-equilibrium initial condition. Simulation results illustrated the validity of the extended small tip angle approximation.
### Introduction
Small Tip Angle (STA) approximation has been widely used in Magnetic Resonance Imaging (MRI) literature to design non-adiabatic slice selective pulses [1-3]. The two key assumptions in STA are thermal equilibrium initial condition and stationary longitudinal magnetisation during the excitation [1-3]. We have extended STA to the non-equilibrium initial condition by expanding the Bloch equation using the Volterra series representation of non-linear dynamical systems. Theoretical as well as simulation results demonstrate that STA approximation can be used to design slice selective pulses even when the bulk magnetisation is not initially at equilibrium.
### Methods
A linear system can be described completely by its impulse response. Volterra series extends the impulse response to nonlinear systems through an infinite sum of higher order convolution integrals using Volterra kernels [4,5]. Bilinear systems have convergent Volterra series expansion and the Volterra kernels can be calculated using matrix exponentials [5]. The Bloch equation, neglecting the relaxation effects, when the excitation is applied in the $x$-direction is written as
$$\dot{ \textbf{M}}= \left[\begin{array}{c}\dot{M}_{x'}\\\dot{M}_{y'}\\\dot{M}_{z'} \end{array}\right] = \left[ \begin{array}{ccc} 0& \Delta \omega & 0 \\ -\Delta \omega & 0 & \omega_{1}(t) \\ 0 & -\omega_{1}(t) & 0 \end{array} \right] \left[\begin{array}{c} {M}_{x'}\\{M}_{y'}\\{M}_{z'} \end{array}\right],$$
where $\dot{ \textbf{M}}$ is the magnetisation vector, $\omega_1$ is the excitation and $\Delta\omega$ represents the off-resonance frequencies that can be generated by gradient fields. Assuming the magnetisation is initially at $\textbf{M}^0 = [M_x^0 \quad M_y^0 \quad M_z^0]^T$, the zeroth and the first kernels for the Bloch equation are
$$h_0 (t) = \left(\begin{array}{c} {M_x^0}\, \cos\!\left(\Delta{}\omega{}\, t\right) +{M_y^0}\, \sin\!\left(\Delta{}\omega{}\, t\right)\\ {M_y^0}\, \cos\!\left(\Delta{}\omega{}\, t\right) - {M_x^0}\, \sin\!\left(\Delta{}\omega{}\, t\right)\\ {M_z^0} \end{array}\right),$$
$$h_1 (t,\tau_1)= \left(\begin{array}{c} {M_z^0}\, \sin\!\left(\Delta{}\omega{}\, \left(t - {\tau{}}_{1} \right)\right)\\ {M_z^0}\, \cos\!\left(\Delta{}\omega{}\, \left(t-{\tau{}}_{1} \right)\right)\\ {M_x^0}\, \sin\!\left(\Delta{}\omega{}\, {\tau{}}_{1}\right) - {M_y^0}\, \cos\!\left(\Delta{}\omega{}\, {\tau{}}_{1}\right) \end{array}\right).$$
### Theoretical Results
For the thermal equilibrium initial condition, the Volterra series expansion clearly shows that $M_{xy} \triangleq M_x+ j M_y$ is the Fourier transform of the excitation $k$-space trajectory proportional to $\omega_1$ [1]. When $M_{xy}^0\neq 0$, the Volterra series expansion shows that $M_z$ is a combination of Fourier sine, Fourier cosine and Fourier transform of the excitation trajectory weighted by $M_x^0$, $M_y^0$ and $M_z^0$, respectively. For special cases when $\textbf{M}^0 = [ 0 \quad M_0 \quad 0]^T$ and $\textbf{M}^0 = [ M_0\quad 0 \quad 0]^T$, where $M_0$ is the thermal equilibrium, the $M_z$ profile at the end of the pulse duration, $t = \tau_p$ , is the Fourier sine and the negative Fourier cosine transform of the excitation pattern, respectively.
### Simulation Results
To illustrate the validity of the theoretical results, we compared the approximation with the numerical solution of the Bloch equation for the following cases
Forward problem: For a rectangular pulse excitation, we have calculated the $M_z$ profile when $\textbf{M}^0 = [ 1 \quad 0 \quad 0]^T$ and $\textbf{M}^0 = [ 0 \quad 1\quad 0]^T$. The resultant $M_z$ profiles for the above initial conditions using the Volterra series kernels are
$$M_z(\Delta\omega, \tau_p) = \mathcal{F}_s\{\omega_1(t)\} = \frac{\omega_1(1-\cos \Delta\omega\tau_p)}{\Delta\omega},$$
and
$$M_z(\Delta\omega, \tau_p) = \mathcal{F}_c\{\omega_1(t)\} = -\frac{\omega_1 \sin \Delta\omega\tau_p}{\Delta\omega},$$
repectively. In the above equations, $\omega_1$ is the pulse amplitude and $\mathcal{F}_s$ and $\mathcal{F}_c$ are Fourier sine and cosine transforms, respectively. The numerical solution of the Bloch equation as well as the presented approximate solutions are shown in Figure 1. These results show that the approximate solution is in agreement with the numerical solution.
Backward problem: Assuming the magnetisation is initially at $\textbf{M}^0 = [ 0 \quad 1\quad 0]^T$, we designed a slice selective pulse to rotate the magnetisation for $\pi/6$ about the $x$-axis. Given that the Fourier cosine transform of a rectangular function is a half a sinc function, the excitation that tips the bulk magnetisation from the $+y$-axis is a truncated half a sinc. The excitation pattern, as well as the resultant magnetisation profile are represented in Figure 2. Similar to the conventional Fourier method, to obtain a refocused slice, the gradient was reversed for half the time of the pulse duration with the same magnitude.
### Conclusions
Here, we provided an alternative approach to address STA approximation using Volterra series expansion. Besides providing an insight into the validity of STA and how it works, we used this approach to extend the STA approximation to the non-equilibrium condition. We specifically looked at the cases where the magnetisation is initially aligned with $x$ or $y$-axes and showed that the resultant profile in the $z$-direction simplifies to Fourier sine and cosine transforms, respectively. The results can be used to design pulses with more efficient slice selectivity, especially for inversion and refocusing pulses.
### Acknowledgements
No acknowledgement found.
### References
1. J. Pauly, D. Nishimura, and A. Macovski, "A k-space analysis of Small-tip-angle excitation," Magnetic Resonance in Medicine, vol. 81, pp. 43-56, 1989.
2. D. Nishimura, Principles of Magnetic Resonance Imaging, Stanford University, 1996.
3. M. Bernstein, K. King, and X. Zhou, Handbook of MRI pulse sequences, Elsevier, 2004.
4. L. Carassale and A. Kareem, "Modeling nonlinear systems by Volterra series," Journal of Engineering Mechanics, vol. 136, 2010.
5. R. Brockett, "The early days of geometric nonlinear control," Automatica, vol. 50, pp. 2203-2224, 2014.
### Figures
Figure 1. $M_z$ profile generated by a rectangular pulse excitation of a 12.8ms duration, (a) when the initial magnetisation was at $[1\quad 0 \quad 0]^T$ and (b) $[0 \quad 1 \quad 0 ]^T$. The STA approximation solution is represented by the solid line and the numerical solution of the Bloch equation at the end of the pulse duration is shown by the dashed line.
Figure 2. (a) The designed pulse to tip the bulk magetisation from $[0 \quad 1 \quad 0 ]^T$ for $\pi/6$ and (b) the resultant slice profile. The gradient was switched on post excitation in the reverse direction to achieve a refocused slice.
Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
3402
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2022-05-22 22:36:43
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https://www.gradesaver.com/textbooks/math/other-math/CLONE-547b8018-14a8-4d02-afd6-6bc35a0864ed/chapter-4-decimals-test-page-325/1
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## Basic College Mathematics (10th Edition)
$18\frac{2}{5}$
18.4 To write it as fraction we need to remove decimal point. here 1 decimal point so it can be done by putting 10 in denominator. i.e. 18.4 = $\frac{184}{10}$ = $\frac{184\div2}{10\div2}$ (dividing by common factor 2) = $\frac{92}{5}$ = $\frac{90+2}{5}$ = $18\frac{2}{5}$
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2021-04-23 18:57:45
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https://www.snapxam.com/calculators/constant-rule-calculator
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# Constant rule Calculator
## Get detailed solutions to your math problems with our Constant rule step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!
Go!
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### Difficult Problems
1
Solved example of constant rule
$\frac{d}{dx}\left(\pi100^2=0\right)$
2
Calculate the power $\pi ^2$
$\frac{d}{dx}\left(9.8696=0\right)$
3
Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable
$\frac{d}{dx}\left(9.8696\right)=\frac{d}{dx}\left(0\right)$
4
The derivative of the constant function ($9.8696$) is equal to zero
$0=\frac{d}{dx}\left(0\right)$
5
The derivative of the constant function ($0$) is equal to zero
$0=0$
6
$0$ equals to $0$
true
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2020-04-09 07:41:14
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http://sioc-journal.cn/Jwk_hxxb/CN/Y1996/V54/I12/1165
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量子化学从头计算法研究C~7~6的分子静电势
1. 贵州大学化学系
• 出版日期:1996-12-15 发布日期:1996-12-15
Ab initio calculation determine the molecular electrostatic potentials of C~7~6 fullerene
WANG YIBO
• Online:1996-12-15 Published:1996-12-15
The geometrical structure of C~7~6 fullerene has been optimized by the semi-empirical molecular orbital MNDO-PM3 method. All-electron ab initio calculation was accomplished at the Hartree-Fock level employing 6-31G basis set to determine the molecular electrostatic potentials (MEP) of the D~2 point group symmetry chiral C~7~6 fullerene. With those of the planiform and radial maps of the MEP are given. The comparisons of the MEP for C~7~6, C~7~0 and C~6~0 fullerenes are carried out.
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2020-08-11 01:11:50
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https://santsys.com/s2blog/mrad-vs-moa/
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# MRAD vs MOA
If you are looking for a new optic for an intermediate or long range rifle, at some point you will be faced with the question, “Should I get an MRAD or MOA optic?”
There is a lot of information out there on the subject, and I am going to do my best to consolidate the parts that really matter (to me) and explain how you will actually use the optic. Hopefully that will help you decide what will work best for you.
First things first, I’m going to make the assumption that you know what the reticle and turrets of a scope are and that you have a basic understanding of how to use a scope. Now that we have that out of the way, some information that is helpful to have beforehand… if you have an optic that is setup with MRAD (Mil) reticle and MRAD (Mil) turrets (Mil/Mil), everything works almost exactly the same as it does for a setup that has an MOA reticle and MOA turrets (MOA/MOA). The only real issues come into play when you have an optic that has MOA turrets and a Mil reticle (MOA/Mil). I’ll dive into all of that more later, but it helps to know that the variations exist beforehand.
So, what is an MRAD and MOA? (slightly technical, but we’ll get into the nuts and bolts soon)
MRAD or Mil or Milliradian or Angular Mil is a standard unit of angular measurement that is a thousandth of a radian. There are $latex 2\pi$ (6.28) radians or $latex 2\pi \times 1000$ milliradians (6,283.185) in a circle.
So what does that mean? 1 Mil = 1 inch @ 1000 inches, 1 Mil = 1 yard @ 1000 yards, 1 Mil = 1 meter @ 1000 meters, and so on. 1 Mil is 1/1000 of any measurement as far as we are concerned.
#### MOA
MOA or Minute of Angle is an angular measurement like a Mil but there are 60 minutes of angle to a degree and 360 degrees to a circle so there are 21,600 MOA to a circle.
What does that mean? 1 MOA = 1.047 inches @ 100 yards, 1 MOA = 2.094 inches @ 200 yards, etc.
The Mil-Dot reticle is a popular military or long range shooting reticle. The Mil-Dots are used for range estimation, windage adjustment, etc. The Mil referenced in both cases is the same. Many older optics offered Mil-Dot reticles with MOA adjustments, a mix of both MRAD and MOA.
Most common scopes offer 1/4 MOA adjustments for MOA based scopes and 0.1 MRAD adjustments for MRAD based scopes. You can also find MOA based scopes with 1/2 MOA or 1/8 MOA turrets, but they are less common.
What all of that basically means is that for an 1/4 MOA turret, every 4 clicks you get 1 MOA movement of the reticle and for an MRAD turret, for every 10 clicks you get 1 MRAD movement of the reticle. To put that into actual numbers, on an MOA based scope, every 1/4 MOA click of adjustment translates to 0.26 inches @ 100 yards and for an MRAD based scope, every 0.1 MRAD click of adjustment translates to 0.36 inches @ 100 yards.
As a good example of how this all comes together, lets say you are shooting at 100 yards with a Mil-Dot style reticle and your bullet impact is 1/2 Mil low. How would you go about making corrections for this? Well, with an MRAD (Mil/Mil) scope where the reticle and turrets use the same measurements you would simply make a 1-to-1 adjustment. You would make a 0.5 (1/2) Mil adjustment of 5 clicks (0.1 Mil turrets = 5 clicks). Pretty simple, right?
If you are using a scope with MOA turrets and an MOA reticle (MOA/MOA) and the shot was 1/2 MOA low, you would do the same thing and make a 2 click adjustment (with 1/4 MOA turrets).
As you can see, when everything matches up, adjustments and corrections are VERY easy!
Now, where it gets more complicated is if you have a scope with MOA turrets and a Mil reticle (MOA/Mil). This calculation is a bit harder, you have to do some math. You would do (Correction in Mils) x 3.438 = (MOA Adjustment), so in the case above you would do 0.5 Mils x 3.438 = 1.72 MOA, so you would make an approximate 7 click adjustment (with 1/4 MOA turrets).
As you can see, the process gets much more complicated when things don’t match up. Bottom line, don’t buy a new optic that doesn’t have a matching turret and reticle, MOA/MOA or Mil/Mil, it doesn’t matter much, as long as they match, things will be easier.
#### Range Estimations
Personally, I think that range estimations with Mil based optics is the way to go. It’s simpler, in many cases and uses much more basic math. The military has used Mil-Dot reticles for ranging basically forever, and is really the most standard. Below I will outline the ranging formulas for Mil (MRAD) optics and MOA optics.
Ranging using a reticle is basically the process of taking a known (or estimated) height of an object, then measuring the height in Mils (or MOA) and plugging it into an equation to get an estimated distance.
$latex \frac{\text{Height of Target (Yards)}}{\text{Size (Mils)}}\times1000 = \text{Distance (Yards)}&s=1$
$latex \frac{\text{Height of Target (Inches)}}{\text{Size (Mils)}}\times27.77 = \text{Distance (Yards)}&s=1$
$latex \frac{\text{Height of Target (Inches)}}{\text{Size (Mils)}}\times25.4 = \text{Distance (Meters)}&s=1$
$latex \frac{\text{Height of Target (Meters)}}{\text{Size (Mils)}}\times1000 = \text{Distance (Meters)}&s=1$
$latex \frac{\text{Height of Target (cm)}}{\text{Size (Mils)}}\times10 = \text{Distance (Meters)}&s=1$
MOA Reticles
$latex \frac{\text{Height of Target (Inches)}}{\text{Size (MOA)}}\times95.5 = \text{Distance (Yards)}&s=1$
$latex \frac{\text{Height of Target (Inches)}}{\text{Size (MOA)}}\times87.3 = \text{Distance (Meters)}&s=1$
$latex \frac{\text{Height of Target (Meters)}}{\text{Size (MOA)}}\times3438 = \text{Distance (Meters)}&s=1$
$latex \frac{\text{Height of Target (cm)}}{\text{Size (MOA)}}\times34.38 = \text{Distance (Meters)}&s=1$
Common size Estimations
Some common estimations of various objects are as follows…
Obeject Size (Inches) Size (Yards) Size (cm)
Standard Door 36 in X 84 in 1 yd X 2.33 yd 91.44 cm X 213.36 cm
License Plate 12 in X 6 in 0.33 yd X 0.17 yd 30.48 cm X 15.24 cm
Wood Pallet 48 in X 48 in 1.33 yd X 1.33 yd 121.92 cm X 121.92 cm
Paper (Letter Size) 8.5 in X 11 in 0.24 yd X 0.31 yd 21.59 cm X 27.94 cm
Concrete Block 16 in X 8 in 0.44 yd X 0.22 yd 40.64 cm X 20.32 cm
White Tailed Deer (Avg Height) 42 in 1.17 yd 106.68 cm
Average Male Height 69 in 1.92 yd 175.26 cm
Average Female Height 64 in 1.78 yd 162.56 cm
Lets use one of these equations to calculate the range of a target…
Say you are shooting a target that is propped up by concrete blocks. You know that the size of a standard block is 16 in X 8 in and that measures 1 Mil in your scope. If we plug that into our equation above, $latex \frac{12\text{in}}{1\text{Mil}}\times25.4 = 304.8\text{m}&s=0$, you get that the target is about 305 m away. Once you know that, you can make the proper adjustments to your optic to get a much closer first round hit.
When doing these calculations, sizing in Yards or Meters is easier for the math, since you are multiplying by 1000 (for Mil calculations). But, sizing objects in yards or meters is often impractical, so it’s generally easier to memorize or write out a “dope” of known objects and their distance calculations beforehand. That way you’re not stuck doing math on the spot.
Note: In many scopes the reticles are only calibrated at a certain magnification. This means that the reticle is etched on the 2nd focal plane and the reticle stays the same throughout any magnification settings. So doing any ranging or adjustment calculations with this type of optic requires you to have the scope on a specific magnification setting. If you can find a First Focal Plane (FFP) optic, where the reticle changes with the magnification and is always true, it is recommended (they usually cost a bit more, and are denoted with “FFP”, in most cases).
If you are looking for more through details on ranging, check out this article on 8541tactical.com.
#### So, what should I buy?
That is generally the million dollar question, right? Obviously everyone’s tastes and budget are a little different, but the one universal point that I think applies to everyone, except those that have more money than they know what to do with, is to start out with something basic and reasonably priced, then as you become a better shooter upgrade as needed. Basically, if you are just getting into shooting, learn to shoot before you spend big money on an optic. No $4,000 optic will make a bad shooter a good shooter. Once you are a good shooter, that$4,000 optic will only make it that much nicer and you will see the benefits of it.
Back to what you should get… I’m a big fan of the Vortex Optics Viper PST series scopes. They work well and are aggressively priced. They also offer nice reticles and have both Mil/Mil and MOA/MOA versions. One of these optics will work very well for the vast majority of shooters, and to be honest, unless you are just looking to spend money, you may never need another scope once you have one. Bottom line, you can get a 6-24x50mm FFP Scope for around $1,000. In my mind, an excellent deal. If you’re just itching to spend some serious money on an optic, you can’t beat a U.S. Optics or a Schmidt and Bender scope. These will all run you$3,000+.
A good place to look around online is swfa.com they usually have decent prices and a really wide selection. So they are generally a good first spot to check out the available options.
#### Conclusion
There are a lot of different scope options out there, the bottom line is find something that will work for you. But the optic wont make any lick of difference if you don’t know how to actually shoot. So don’t spend too much time over thinking your optic. Get something that is in your budget and meets the basic functionality that you need. Then spend any extra money you have on ammunition. Practice makes perfect, and you can’t practice if you spend all of your time researching what scope to get on the internet.
And if you can’t decide on Mil/Mil or MOA/MOA, I’ll make the decision for you… If the scope you want comes in Mil/Mil, get that, if it doesn’t get it in MOA/MOA. Just stay away from Mil/MOA if you can.
True story, I use, and continue to use a \$300.00 fixed 10×42 scope that I purchased 5-6 years ago (that is what is in the pictures used in this post). It’s a SWFA SS MOA/Mil optic and it’s a great scope and has worked really well, especially for your standard 100 yd bench rest shooting. I’m only just now thinking about upgrading to something with variable magnification.
So you don’t have to spend an arm and a leg to get out to the range, and you will be better for it down the road. Only spend what you can afford, and spend the rest on ammo learning to shoot!
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2020-04-09 08:57:23
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https://www.thelittleaussiebakery.com/when-two-plane-mirrors-are-kept-at-90-degree-we-get/
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When two plane mirrors are kept at 90 degree we get?
When two plane mirrors are kept at 90 degree we get?
When you place two plane mirrors at a 90-degree angle, the image of the first mirror is reflected in the second mirror so that the reversed mirror image is reversed again, and you see a true image. (See Glossary, page 73.)
When two mirrors are placed at right angles to each other?
Interestingly, a single mirror produces a single image; another single mirror produces a second image; but when you put the two single mirrors together at right angles, there are three images.
How many images are formed when two plane mirrors are kept at 90 degree to each other draw a ray diagram to support your answer?
Explanation: We get n = 360/90 – 1 = 3. When we simply use n = 360/90 we get an even number 4, but we should get a odd number so we use the other formula. So we get 3 images when two plane mirrors are placed at angle of 90⁰.
At what angle must Two plane mirrors be placed so that incident and resulting reflected rays are always parallel to each other?
The incident ray and reflected ray will always be parallel to each other when the two plane mirrors will be placed at right angle i.e. 90° to each other.
When two plane mirrors are kept at 60 degree we get?
60 degeree=360/60 = 6-1=5 images.
What is the significance of knowing the number of image by two plane mirrors with a given angle?
If two plane mirrors are placed together on one of their edges so as to form a right angle mirror system and then the angle between them is decreased, some interesting observations can be made. One observes that as the angle between the mirrors decreases, the number of images that can be seen increases.
When plane mirror is rotated through an angle?
Thus, for a fixed incident ray, the reflection angle is twice the angle through which the mirror has rotated. When a plane mirror is rotated at an angle $\theta$, the reflected ray rotates through the angle $2\theta$. However, the size of the image remains unchanged only position shifts.
Why are multiple images formed when two mirror are placed at right angle to each other?
Multiple images are formed when two mirrors are placed at right angle to each other because when a ray incident on first mirror from an object it forms an image of the object. So further image of image are also formed. This process continues till no more reflection by either mirror is possible.
How many images will be formed if two plane mirrors are kept at an angle of 20 degrees *?
The number of images formed = (360/40), we get 9 images.
How many images do we see between two plane mirrors?
Infinite images are formed when two plane mirrors are placed parallel to each other, irrespective of the distance.
At what angle must Two plane mirrors?
The angle of inclination between two mirrors should be 90 degrees.
What should be the angle between two plane mirrors so that whatever?
What should be the angle between two plane mirrors, so that whatever be the angle of incident, the incident ray and the reflected ray from the two mirrors be parallel to each other? (A)60∘
What is the angle of reflection of a plane mirror?
Each plane mirror makes an angle of 45 o with the horizontal. Light from the object is turned through 90 0 at each mirror and reaches the eye as shown. The rays from the object are reflected by the top and then reflected again by the bottom into the observer. The image formed is virtual, upright and same size as the object.
What is reflection at plane surface formulae?
Reflect at Plane Surface Formulae includes Laws of reflection, Characteristics of Reflection at Plane Mirror, etc. II law → I, R & N are in a same plane. Light propagates in straight lines in homogeneous media. Rays do not disturb each other upon intersection.
What happens when light is reflected by a plane smooth surface?
When light is reflected by a plane smooth surface, the reflection is regular (specular) and when reflection occurs at a rough surface it is called a diffused reflection. A plane mirror is a flat smooth reflecting surface which forms images by regular reflection.
How to study reflection at plane surface?
Utilize the Cheat Sheet of Reflection at Plane Surface and try to memorize the formulae as you might need them at some point. Try seeking help regarding various concepts of Physics you feel difficult from Physics Formulas. Reflect at Plane Surface Formulae includes Laws of reflection, Characteristics of Reflection at Plane Mirror, etc.
Posted in Life
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2023-02-07 09:03:04
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https://www.edaboard.com/threads/need-help-on-sigma-delta-adc.5736/
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# need help on sigma delta ADC?
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Not open for further replies.
#### 7rots51
Hello
But I have problem with them in noisy environment and in sensors that connected to it with long cables ,Sometimes they hang or lock, if I power off the card then power on it ,it works,I do not know what to do with them,
They are good ADCs ,because they have low pass filter and PGA in them and I can connect the sensors directly to them,but it seems they are not reliable.
bye
#### Bus Master
##### Full Member level 3
Hi there,
I doubt it's a matter of ADC's reliability. These stuff are completely tested upon menufacturing. Are you sure your system is functioning correctly, you do NOT have race conditions, glitches, power supply filteration problems. Are you using a watch dog? if not, enable it (if you have a built in one in your microcontroller) and check if you still have the fault.
Hope this helps.
Yours,
#### 7rots51
hello
I have WDT in circuit and the supply is provided by a DC/DC converter that its output is a regulated 5VDC,I used 100n and 10 uf TAN capacitors for power pins of ADC and the supply filtering is good.
But in noisy field I think it is difficult to use sigma delta ,and the ADC lost its sequence.
Only with power off and then on after several seconds everything becomes ok.
Regards
#### zoovy
##### Member level 1
7rots51 said:
hello
I have WDT in circuit and the supply is provided by a DC/DC converter that its output is a regulated 5VDC,I used 100n and 10 uf TAN capacitors for power pins of ADC and the supply filtering is good.
But in noisy field I think it is difficult to use sigma delta ,and the ADC lost its sequence.
Only with power off and then on after several seconds everything becomes ok.
Regards
7rots51,
Can you give me alittle more explanation in defining the problem?
I want you to break down the described problem.
Here are some questions. I am sure you are knowledgable. But sometimes seemingly silly questions might get you thinking something you didn't look for.
1) Is the latchup really the ADC? How did you determine that?
2) When it was latched up, could you read/write the status/command etc. registers and get some meaningful information?
3) Can you spare one of the ADC inputs to some reference voltage so that you can test your theory that the ADC is latched up?(switch to that port do conversion and check result type of verification)
4) Which side has long wires (ADC to uP or ADC to the sensor?)
5) If the long wires are on the ADC<->sensor side, do you have protection diodes? (Preferrably clipping the voltage below the supply voltages and/or within the range of the input.)
6) Is there any signal that is floating on the ADC?
7) Last, let us know about the results!!!
Zoovy
#### 7rots51
Hi zoovy
1: before each conversion for example for AD7714 ,I read filter high register if I readback the programmed value I know that the communication between ADC and Micro is OK,else the ADC does not respond or respond with error,then micro waits and WDT reset the micro and also the ADC ,at begining of micro program ADC is initialized and self calibrated and then set to get samples
2: yes ,because the complex chips on the board is a micro and the ADC other components are some TTL logic and DC/DC and optocouplers and RS485 line driver.
when WDT resets the micro the micro again performs the initial of program until reach to test routine of ADC ,and because ADC does not respond it wait here and WDT timeout will occur again and again.
3. when it hangs the sequence of communication will be lost and the test routine says there is an error in ADC.
If I turn the power off then I turn on the power of the card everything becomes ok.But WDT reset on ADC and Micro has no effect
4: sensor to ADC is long
5: I used four diode for clamping (two at each leg)and two resistor for current limiting on input of ADC and I used three capacitor for input filter.
the input for example is from a THC sensor and I used two 100n capacitor for common mode noise rejection and one 1uf cap for differential noise rejection.
6: one leg of input sensor is biased to middle of ADC suply and the common mode voltage is in the range.
7.all you said I had done but the problem exist!!
I do not know why some cards does not work properly.
BTW: some cards work ok in safe area but some may crash in noisy area after some days!
the situation is industerial(the place that I used these cards)
bye
#### mystery
##### Full Member level 2
What is the difference between sigma delta adc's and slope adc's ?
#### techie
I beleive you are talking about AD7730. I have used this with exactly the same problem as you have described. In fact, one of my products was scrapped due to this problem. The problem was that whenever there was a some switching noise in the system from relays, the AD7730 just hangs up, gets reset or loses values of it configuration registers. The only way to recover was either a system reset or somehow determining this status and reinitializing the AD7730. I did try both options but this meant loss of precious real time. As I was using the chip in a batching controller, this was entirely unacceptable. No amount of filtering, shielding, etc etc could get me out of the trouble. So I had to convert the design on some other chip after all the time-to-market loss and customer annoyance.
#### 7rots51
hi techie
I have the problems that you have but my system does not work unless I switch the power off and on,this is funny,how can I solve this problem?
I want to use other ADCs for future use ,But I do not know TI or intersil or maxim sigma delta ADCs have these problems or not??
Please if some one worked with them respond.
The main advantage of sigma delta is its PGA and low pass filter and burnout detection,these features enable us to connect sensors directly to ADC without large amount of signal conditioning circuits and amplifiers,But these faults are dangerous for industerial projects ,because they must work 24 hour/day,7day/week.
please give comments on sigma delta problems and how to solvw the problems! :?:
#### zoovy
##### Member level 1
more questions
7rots51
Here are some more questions
8) The device seems to be quite complicated. To eliminate any software problems can you provide reset from the micro as a software control? That way we can eliminate software mistakes if any. So you initialization routine will reset the IC, initialize then run it.
9) Have you checked your power supply pins? More than anything most of these sensitive devices fail due to power supply problems, ripples, spikes etc. I'd suggest 0.1 ceramic caps directly on the pins of the IC. One per power supply pin.
11)Looks like the part has some test pins which tell you it is ready or not. If you can break your reboot of your micro and hand verify it (when it locks)you might get some additional info.
12) The device seems a full cmos. Is your processor CMOS I/O?
13) I forgot to ask. How embedded is this processor? Does it have a serial port for monitoring/controlling it? Can you jump to a monitor program to hand-manage the device?
Don't forget. Assumption is the most expensive quick decision. I may sound paranoid but somebody said "Expect the unexpected".
Let me tell you a quick story that happened today. A new batch of boards came from our outsourced build-house. This is a quite complicated board with ATM, E1, cpu, SDRAM, AAL processors, Ethernet etc. Everything works on these boards except the E1/T1. Once in a while the system crashes too.
Problem:
The system reference is a 50MHz clock. This happens to be one of the Just-In-Time programmable oscillators.
In case you don't know... The supplier has a special gadget that programs the frequency whatever you like when you order them. Internally it has a PLL and it generates any frequency you like.
Turns out these oscillators are not so stable for some reason. The internal pll is not programmed to correct values. When anything disturbs the device (you blow on in so it cools down or warms up) it oscillates wildly for a while then stabilizes. During this time the instantaneous frequency moves wildly between 30 MHz to 70 MHz. Well the processor is rated for 50 MHz. Guess what happens when the instantaneous frequency goes to 70 at temperature extremes.
Zoovy
#### 7rots51
hi zoovy
I have done everything that you said ,my problem is not n my laboratory ,but my problem is in field ,some cards may have this problem after several dayswhen they work,because my application is industerial and noisy environment(I used sheilded with earth for sensor cables)some thing are unpredictable!
I think my mistake is in using sigma delta ,I think sigma delta is good for sound and audio system design ,they are not reliable for harsh and industerial systems ,I think I must ignore some feature of sigma delta and use other type of ADCs that are 16 bit or more and use some PGIA or IA in front of them for sygnal conditioning and some filtering in analog section and some in software.
I think the architecture of sigma delta is not good for operating in harsh and complicated environment,they are not reliable at all.
Regards
#### zoovy
##### Member level 1
one more quote
7rots51 said:
hi zoovy
I have done everything that you said ,my problem is not n my laboratory ,but my problem is in field ,some cards may have this problem after several dayswhen they work,because my application is industerial and noisy environment(I used sheilded with earth for sensor cables)some thing are unpredictable!
Regards
I am running into the danger of pestering you with my suggestions but I will risk it one more time. Because I hate to give up!
The Analog Devices makes protection devices for their ADCs. Do you think it would be useful if we were to use one of them to find out wheter the problem is happening because of the long wires to the sensor or power-supply design problems? They are usually expensive ($50 > XX >$100).
Below is the link to the devices.
Zoovy
#### zoovy
##### Member level 1
is there a layout problem?
7rots51,
I just remembered a problem in one of our designs. The layout had a problem which the input wire passed very close to the reference input/output of the ADC. The High impedance input got the glitches and created chaos.
Zoovy
#### 7rots51
hi zoovy
5B series of analog device is too expensive for multi channel design.
I want to examine the TI ADC both sigma delta and SAR (sampling ADC) ,if the result was good ,I will use them in my future design,it seems that the price of TI is lower than ADI ADCs.
I need to work on pcb layout design(for high resolution systems) and on ADC selection.
bye
#### techie
try the crystal semi cs5520 series. i have seen these also in some designs though never used myslelf
#### Wowee
##### Junior Member level 2
Hi,7rots51
I'v designed a weigh scale based on AD7714 .
When I apply a known voltage to the AIN1-AIN2 of AD7714(be set to
bipolar,gain128) ,there were always some wrong codes among other
approximately right results.
When I changed the input voltage,also got right results with some wrong
codes.But the wrong codes didn't change ,fixed at
259XXXH,58DXXXH. I was confused with this.Could you encounter that in
Regards
Wowee
#### 7rots51
hi wowee
For weigh scale application I recomment you AD7730.
I did not counter the problem hat you have ,(my application is unipolar)
Totally,I fear to use analog devices sigma delta for my future products ,I think It is better to test TI sigma delta ADCs (their price is also lower than AD ICs),for application that we can use sampling ADC(SAR)it is better to use SAR.
for very fast sampling flash ADC is best.
bye
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2022-07-01 14:01:25
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https://math.stackexchange.com/questions/1135753/help-with-a-cumulative-distribution-function-question
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# Help with a cumulative distribution function question.
This is a question I want to solve:
The random variable $X$ has cdf: F_X(x) = \begin{cases} \begin{align} &0 &&x <0\\ &0.5 + c\sin^2\left(\frac{\pi x}{2}\right) &0 \leq\; &x \leq 1\\ &1 &&x>1 \end{align} \end{cases}
(a) What values can $c$ assume?
(b) Plot the cdf.
(c) Find $P[X > 0]$.
I assume that $x$ is between $0$ and $1$ so $$0.5 + c \sin^2(\pi\times x/2) = 0$$
then $x=1$ and $$c \sin^2(\pi\times 1/2) = -0.5$$
since $c = -0.5$
Is that correct and I need help with the nother parts please.
You have no reason to assume that $0.5 + c \sin^2(\pi\times x/2) = 0$. In fact the definition of $F$ tells you that $$F_X(0) = 0.5 + c \sin^2(\pi\cdot 0/2) = 0.5 \neq 0,$$ so your assumption is false for at least one $x$ in the interval $0 \leq x \leq 1$.
There is also no reason to assume that $c = -0.5$. In fact it cannot be $-0.5.$
I would suggest looking carefully at what happens around $x=1.$ It is possible for a cdf to be discontinuous, but only some kinds of discontinuity are possible.
Hint: A CDF is monotonic increasing and right continuous. However, $\sin^2(x\pi/2)\in [0,1]$
The nature of a cumulative distribution function is that it must be càdlàg, monotonically non-decreasing, and take values between 0 and 1 (inclusive). It does not have to be either continuous or discrete, but may be both.
• Part 1: Being that the distribution may not take values greater than one, and must be non-decreasing, what can the maximum of $c$ be if we know it is equal to $0.5 + c\sin^2\left(\frac{\pi x}{2}\right)$? What is $\sin\left(\frac{\pi}{2}\right)$? Similarly, what is the minimum it can be, remembering that it must be non-decreasing?
• Part 2: Being that $c$ is a constant, there are really only two kinds of shapes it can have, so once you figure out part 1, this isn't so difficult
• Part 3: Make sure you recognize those inequalities which are strict, and those that aren't, and what the difference must mean.
I think the CDF given in the question is wrong as in your problem it is neither a continuous CDF nor a discrete one as LHL (left hand limit) approaching 0 and RHL (right hand limit) approaching 1 do not converge to the same value. Which in this case is not true for F(x) at $0\leq x \leq 1$.
• The distribution is neither continuous nor discrete. It is possible for such a distribution to have a cdf that is not continuous. – David K Feb 6 '15 at 5:03
• Consider that there may be mass points for some values, and continuous probability for others. – Avraham Feb 6 '15 at 5:14
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2019-12-16 13:12:31
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http://www.ijac.net/cn/article/doi/10.1007/s11633-021-1289-9
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STRNet: Triple-stream Spatiotemporal Relation Network for Action Recognition
Citation: Z. W. Xu, X. J. Wu, J. Kittler. STRNet: Triple-stream spatiotemporal relation network for action recognition. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1289-9 doi: 10.1007/s11633-021-1289-9
Citation: Citation: Z. W. Xu, X. J. Wu, J. Kittler. STRNet: Triple-stream spatiotemporal relation network for action recognition. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1289-9
## STRNet: Triple-stream Spatiotemporal Relation Network for Action Recognition
###### Author Bio: Zhi-Wei Xu received the B. Eng. degree in computer science and technology from Harbin Institute of Technology, China in 2017. He is a postgraduate student at School of Artificial Intelligence and Computer Science, Jiangnan University, China. His research interests include computer vision, video understanding and action recognition. E-mail: zhiwei_xu@stu.jiangnan.edu.cn ORCID iD: 0000-0003-1472-431X Xiao-Jun Wu received the B. Sc. degree in mathematics from Nanjing Normal University, China in 1991. He received the M. Sc. and the Ph. D. degrees in pattern recognition and intelligent systems from Nanjing University of Science and Technology, China in 1996 and 2002, respectively. He is currently a professor in artificial intelligent and pattern recognition at the Jiangnan University, China. His research interests include pattern recognition, computer vision, fuzzy systems, neural networks and intelligent systems. E-mail: wu_xiaojun@jiangnan.edu.cn (Corresponding author) ORCID iD: 0000-0002-0310-5778 Josef Kittler received the B. A. degree in electrical science tripos, Ph. D. degree in pattern recognition, and D. Sc. degree from University of Cambridge, UK in 1971, 1974, and 1991, respectively. He is a Distinguished Professor of machine intelligence at Centre for Vision, Speech and Signal Processing, University of Surrey, UK. He conducts research in biometrics, video and image database retrieval, medical image analysis, and cognitive vision. He published the textbook Pattern Recognition: A Statistical Approach and over 700 scientific papers. His publications have been cited more than 66000 times (Google Scholar). He is series editor of Springer Lecture Notes on Computer Science. He currently serves on the Editorial Boards of Pattern Recognition Letters, Pattern Recognition and Artificial Intelligence, and Pattern Analysis and Applications. He also served as a member of the Editorial Board of IEEE Transactions on Pattern Analysis and Machine Intelligence during 1982−1985. He served on the Governing Board of the International Association for Pattern Recognition (IAPR) as one of the two British representatives during the period 1982-2005, and President of the IAPR during 1994−1996. His research interests include robotics, feedback control systems, and control theory. E-mail: j.kittler@surrey.ac.uk ORCID iD: 0000-0002-8110-9205
• Figure 1. Architecture overview of STRNet. Our STRNet consists of three individual branches that focus on learning appearance, motion and temporal relation information, respectively. For comprehensively representing the information of the whole video, we apply two-stage fusion and separable (2+1)D convolution to reinforce the feature learning. Finally, we apply a decision level weight assignment to adjust the classification performance.
Figure 2. Feature visualization of STRNet. The first column is the input frames. The second column is the feature maps of Stem. The third column is the fusion feature maps of stage 3. The last column is the output of spatiotemporal with relation feature maps of stage 5. We rescale the feature maps into original size for good comparison.
Figure 3. The schema of building relation unit, where X denotes the original inputs of the sequential feature maps, and $\tilde{ X}$ denotes the calculated relation maps. The function Fsm(*) is to calculate the similarity measurement. And g denotes the similarity weight vector and Y denotes the final relation response maps.
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##### 出版历程
• 收稿日期: 2020-10-30
• 录用日期: 2021-02-05
• 网络出版日期: 2021-03-23
## STRNet: Triple-stream Spatiotemporal Relation Network for Action Recognition
### English Abstract
Citation: Z. W. Xu, X. J. Wu, J. Kittler. STRNet: Triple-stream spatiotemporal relation network for action recognition. International Journal of Automation and Computing. http://doi.org/10.1007/s11633-021-1289-9 doi: 10.1007/s11633-021-1289-9
Citation: Citation: Z. W. Xu, X. J. Wu, J. Kittler. STRNet: Triple-stream spatiotemporal relation network for action recognition. International Journal of Automation and Computing . http://doi.org/10.1007/s11633-021-1289-9
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2021-04-17 02:50:04
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.259677529335022, "perplexity": 8052.570980741239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038098638.52/warc/CC-MAIN-20210417011815-20210417041815-00483.warc.gz"}
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http://mathoverflow.net/questions/84878/spec-mathbbz-in-absolute-geometry
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# Spec$\mathbb{Z}$ in absolute geometry
What are the obstacles that prevent from defining Spec$\mathbb{Z}$ in absolute geometry? By absolute geometry I mean the geometry over the field with one element F1.
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What do you mean by absolute geometry? – Martin Brandenburg Jan 4 '12 at 13:38
I am no expert, but the question seems rather broad. Is there a more focused version that you could ask, e.g. "why does this idea not work?" – Yemon Choi Jan 4 '12 at 14:58
The problem, as I understand it, is not in defining spec Z as an object over $F_1$. There are actually quite a few definitions of $F_1$ and the first property that any putative definition must have is that it must map to Z. The difficulty is in finding a setup where spec Z has various desired properties, such as admitting a nice compactification that behaves like a curve over $F_1$ (I think this is called the Deninger program). – Jeffrey Giansiracusa Jan 4 '12 at 15:06
The problem is probably not to define Spec Z in absolute geometry but to define the correct variant of absolute geometry where this question and some others will have an obvious and expected answer. – Zoran Skoda Jan 4 '12 at 15:14
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2015-04-01 08:10:51
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https://raisingthebar.nl/2016/11/12/properties-of-standard-deviation/
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12
Nov 16
## Properties of standard deviation
Properties of standard deviation are divided in two parts. The definitions and consequences are given here. Both variance and standard deviation are used to measure variability of values of a random variable around its mean. Then why use both of them? The why will be explained in another post.
### Properties of standard deviation: definitions and consequences
Definition. For a random variable $X$, the quantity $\sigma (X) = \sqrt {Var(X)}$ is called its standard deviation.
#### Digression about square roots and absolute values
In general, there are two square roots of a positive number, one positive and the other negative. The positive one is called an arithmetic square root. The arithmetic root is applied here to $Var(X) \ge 0$ (see properties of variance), so standard deviation is always nonnegative.
Definition. An absolute value of a real number $a$ is defined by
(1) $|a| =a$ if $a$ is nonnegative and $|a| =-a$ if $a$ is negative.
This two-part definition is a stumbling block for many students, so making them plug in a few numbers is a must. It is introduced to measure the distance from point $a$ to the origin. For example, $dist(3,0) = |3| = 3$ and $dist(-3,0) = |-3| = 3$. More generally, for any points $a,b$ on the real line the distance between them is given by $dist(a,b) = |a - b|$.
By squaring both sides in Eq. (1) we obtain $|a|^2={a^2}$. Application of the arithmetic square root gives
(2) $|a|=\sqrt {a^2}.$
This is the equation we need right now.
### Back to standard deviation
Property 1. Standard deviation is homogeneous of degree 1. Indeed, using homogeneity of variance and equation (2), we have
$\sigma (aX) =\sqrt{Var(aX)}=\sqrt{{a^2}Var(X)}=|a|\sigma(X).$
Unlike homogeneity of expected values, here we have an absolute value of the scaling coefficient $a$.
Property 2. Cauchy-Schwarz inequality. (Part 1) For any random variables $X,Y$ one has
(3) $|Cov(X,Y)|\le\sigma(X)\sigma(Y)$.
(Part 2) If the inequality sign in (3) turns into equality, $|Cov(X,Y)|=\sigma (X)\sigma (Y)$, then $Y$ is a linear function of $X$: $Y = aX + b$, with some constants $a,b$.
Proof. (Part 1) If at least one of the variables is constant, both sides of the inequality are $0$ and there is nothing to prove. To exclude the trivial case, let $X,Y$ be non-constant and, therefore, $Var(X),\ Var(Y)$ are positive. Consider a real-valued function of a real number $t$ defined by $f(t) = Var(tX + Y)$. Here we have variance of a linear combination
$f(t)=t^2Var(X)+2tCov(X,Y)+Var(Y)$.
We see that $f(t)$ is a parabola with branches looking upward (because the senior coefficient $Var(X)$ is positive). By nonnegativity of variance, $f(t)\ge 0$ and the parabola lies above the horizontal axis in the $(f,t)$ plane. Hence, the quadratic equation $f(t) = 0$ may have at most one real root. This means that the discriminant of the equation is non-positive:
$D=Cov(X,Y)^2-Var(X)Var(Y)\le 0.$
Applying square roots to both sides of $Cov(X,Y)^2\le Var(X)Var(Y)$ we finish the proof of the first part.
(Part 2) In case of the equality sign the discriminant is $0$. Therefore the parabola touches the horizontal axis where $f(t)=Var(tX + Y)=0$. But we know that this implies $tX + Y = constant$ which is just another way of writing $Y = aX + b$.
Comment. (3) explains one of the main properties of the correlation:
$-1\le\rho(X,Y)=\frac{Cov(X,Y)}{\sigma(X)\sigma(Y)}\le 1$.
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2021-07-28 17:21:21
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https://developer-archive.leapmotion.com/documentation/v2/objc/devguide/Leap_Coordinate_Mapping.html?proglang=objc
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# Coordinate Systems¶
A fundamental task when using the Leap Motion controller in an application is mapping the coordinate values received from the controller to the appropriate application-defined coordinate system.
## Leap Motion Coordinates¶
The Leap Motion Controller provides coordinates in units of real world millimeters within the Leap Motion frame of reference. That is, if a finger tip’s position is given as (x, y, z) = [100, 100, -100], those numbers are millimeters – or, x = +10cm, y = 10cm, z = -10cm.
The Leap Controller hardware itself is the center of this frame of reference. The origin is located at the top, center of the hardware. That is if you touch the middle of the Leap Motion controller (and were able to get data) the coordinates of your finger tip would be [0, 0, 0].
In its normal position, that is on a desk with the user on one side and the computer monitor on the other, the user is “in front” (+z) of the controller and the monitor screen is “behind”(-z) the controller. If the user enables automatic orientation, the Leap Motion software adjusts the coordinate system if the controller is reversed (the green LED is facing away from the user). However, if the user places their controller in a different position (upside down or sideways), the Leap Motion software cannot detect or adjust for this.
Design your application to encourage the user to not get too close to the Leap Motion Controller with visual cues; hovering your hands right over the controller increases the chance that fingers and hands will block each other from view.
## Mapping Coordinates to your Application¶
To use information from the Leap Motion device in your application, you have to interpret that data so it makes sense in your application. For example, to map Leap Motion coordinates to application coordinates, you have to decide which axes to use, how much of the Leap Motion field of view to take advantage of, and whether to use an absolute mapping or a relative mapping.
For 3D applications, it usually makes sense to use all three axes of the Leap Motion device. For 2D applications, you can often discard one axis, typically the z-axis, and map input from the two remaining axes to the 2D application coordinates.
Whether you use two or three axes, you have to decide how much of the Leap Motion range and field of view to use in your application. The field of view is an inverted pyramid. The available range on the x and z axes is much smaller close to the device than it is near the top of the range. If you use too wide a range, or allow too low a height, then the user won’t be able to reach the bottom corners of your application. Consider the following example, which maps the Leap Motion coordinates to the rectangle below (this example requires a plugged-in Leap Motion controller to view):
Notice that when you try to reach the bottom corners, your finger passes out of the device’s field of view and you cannot move the cursor there.
You also have to decide how to scale the Leap Motion coordinates to suit your application (i.e. how many pixels per millimeter in a 2D application). The greater the scale factor, the more affect a small physical movement will have. This can make it easier for the user to move a pointer, for example, from one side of the application to another, but also makes precise positioning more difficult. You will need to find the best balance between speed and precision for your application.
Finally, the difference between coordinate systems may require flipping an axis or three. For example, many 2D window drawing APIs put the origin at the top, left corner of the window, with y values increasing downward. Leap Motion y values increase upwards, so you have to essentially flip the y axis when making the conversion. As another example, the Unity 3D game development system uses a left-handed coordinate system, wheras the Leap Motion software uses a right-handed coordinate system.
Mapping coordinates is much like converting temperatures from celsius to fahrenheit if you imagine each axis as a scale with corresponding start (freezing) and an end (boiling) points:
You can use this formula to make the conversion:
$x_{app} = (x_{leap} - Leap_{start})\frac{Leap_{range}}{App_{range}} + App_{start}$
where:
$Leap_{range} = Leap_{end} - Leap_{start}$$App_{range} = App_{end} - App_{start}$
By changing the start and end points you can change the mapping of coordinates to make movements cover a larger or smaller area in your application. The following example uses this formula on the x and y axes to map an area of the Leap Motion field of view to a rectangular display area. You can play with the controls to set the ranges of the Leap Motion Scale.
Leap X = ...
App X = ...
Leap Y = ...
App Y = ...
If you decrease the range values, a smaller range of physical motion is mapped to the width of the display area, making it easier to move from one side to the other.
The X-center value offsets the center of the X scale to the left or right. Because each hand naturally rests to the right or left of the Leap Motion controller, it can be easier for the user to interact with a full-screen application if you offset the coordinate mapping based on which hand is in use.
The Y-floor slider sets the value on the Leap Motion y axis that maps to the bottom of the display area. A larger value means the user has to hold their hand higher above the controller, which also means that they are more likely to be able to reach the bottom corners.
Enabling the clamp option prevents the cursor from leaving the display area – which can provide a valuable cue to the user as to where they can interact.
A more general method of converting coordinates is to go from the Leap Motion coordinates to a normalized scale running between 0 and 1.0 and then from this normalized scale to the final, application scale. In fact, the LeapInteractionBox class in the Leap Motion API will normalize coordinates for you.
## The Interaction Box¶
The LeapInteractionBox defines a rectilinear area within the Leap Motion field of view.
As long as the user’s hand or finger stays within this box, it is guaranteed to remain in the Leap Motion field of view. You can use this guarantee in your application by mapping the interaction area of your application to the area defined by the InteractionBox instead of mapping to the entire Leap Motion field of view. The LeapInteractionBox class provides the normalizePoint:clamp: method to help map Leap Motion coordinates to your application coordinates (by first normalizing them to the range [0..1]).
The size of the LeapInteractionBox is determined by the Leap Motion field of view and the user’s interaction height setting (in the Leap Motion control panel). The controller software adjusts the size of the box based on the height to keep the bottom corners within the filed of view. If you set the interaction height higher, then the box becomes larger. Users can set the interaction height based on the height they prefer to hold their hands when using the controller. Some users like to hold their hands much higher above the device than others. By using the InteractionBox to map the Leap Motion coordinate system to your application coordinate system, you can accommodate both types of user. A user can also set the interaction height to adjust automatically. If the user moves his or her hands below the current floor of the interaction box, the controller software lowers the height (until it reaches the minimum height); likewise if the user moves above the interaction box, the controller raises the box.
Because the interaction box can change size and position, every LeapFrame object contains an instance of the LeapInteractionBox class. The following example illustrates the behavior of the interaction box. Change your interaction height settings using the Leap Motion control panel to see how it affects the box itself.
Box width = ...
Box height = ...
Box depth = ...
Box center = ...
Box origin = ...
You can use the clamp control in this example to see the affect of clamping coordinates when you use the normalizePoint:clamp: function. If you do not clamp, you can get values which are less than zero or greater than one.
Because the interaction box can change over time, be aware that the normalized coordinates of a point measured in one frame may not match the normalized coordinates of the same real world point, when normalized using the interaction box provided by another frame. Thus straight lines or perfect circles “drawn in the air” by the user may not be straight or smooth if they are normalized across frames. For most applications, this isn’t a significant problem, but you should test your application with the Automatic Interaction Height setting enabled if you aren’t sure.
To guarantee that a set of points tracked across frames is normalized consistently, you can save a single InteractionBox object – ideally the one with the largest height, width, and depth – and use that to normalize all the points.
## Mapping Coordinates with the Interaction Box¶
To use the interaction box, first normalize the point to be mapped and then convert the normalized coordinates to your application coordinate system. Normalizing converts points within the interaction area to the range [0..1] and moves the origin to the bottom, left, back corner. You can then multiply these normalized coordinates by the maximum range for each axis of your application coordinate system, shifting and inverting the axis as necessary.
### Mapping to 2D¶
For example, most 2D drawing coordinate systems put the origin at the top, left corner of the window – and, naturally, don’t use a z-axis. To map Leap Motion coordinates to such a 2D system, you could use the following code:
int appWidth = 800;
int appHeight = 600;
LeapFinger *pointable = frame.pointables.frontmost;
LeapVector *leapPoint = pointable.stabilizedTipPosition;
LeapInteractionBox *iBox = frame.interactionBox;
LeapVector *normalizedPoint = [iBox normalizePoint:leapPoint clamp:YES];
int appX = normalizedPoint.x * appWidth;
int appY = normalizedPoint.y * appHeight;
// If graphics origin is at the top of the window invert y axis:
//int appY = (1 - normalizedPoint.y) * appHeight;
If you are using pointable or palm positions in a 2D application, consider using the LeapPointable.stabilizedTipPosition or LeapHand.stabilizedPalmPosition. These attributes have extra stabilization and filtering to help the user interact with a 2D interface. For 3D applications, the unstabilized LeapPointable.tipPosition and LeapHand.palmPosition are usually more suitable.
### Mapping to 3D¶
To map coordinates to a 3D coordinate system, you must know the scale factors, whether the origin needs to be translated, and whether the target coordinate system uses the right-hand rule or the left-hand rule.
Units of measurement and scale in 3D graphics tend to be determined arbitrarily, but essentially, you need to decide how a movement in the real world should map to a movement in the virtual world.
The origin of coordinates normalized with an interaction box is the bottom, left, rear corner. You can translate the normalized coordinates to move the origin to a more suitable location.
The Leap Motion coordinate system uses the right-hand rule. If your coordinate system uses the left-hand rule, multiply the z-axis by -1 before normalizing (or swap the x and y axes).
The following example illustrates mapping Leap Motion coordinates to a 3D system with the origin of the interaction area at the bottom center of the world, scales the range of the interaction box to 200 3d world units across:
-(LeapVector *) leapPointToWorld:(LeapVector *)leapPoint withBox:(LeapInteractionBox *) iBox
{
LeapVector *normalized = [iBox normalizePoint:leapPoint clamp:false];
float z = normalized.z;
// if changing from right-hand to left-hand rule, use:
//float z = normalized.z * -1.0;
//recenter origin
float x = normalized.x + 0.5;
z += 0.5;
//scale
x *= 100;
float y = normalized.y * 100;
z *= 100;
return [[LeapVector alloc] initWithX:x y:y z:z];
}
### Mapping Right and Left Hands Differently¶
If your application allows full-screen interactions, it can be a good idea to offset the origin of the normalized coordinates differently for each hand. By moving the left hand’s origin to the right and the right hand’s origin to the left, you essentially center each hand’s natural resting point in your application’s interaction area. This makes it less tiring for the left hand to reach the right corner of the application and vice versa.
Leap X = ...
App X = ...
Leap Y = ...
App Y = ...
To shift the origin for each hand, use LeapHand.isLeft to detect which hand is which, and then translate the normalized coordinates right or left:
-(LeapVector *) normalizePointDifferentially:(LeapVector *) leapPoint
usingBox:(LeapInteractionBox *) iBox
forHand:(BOOL) isLeft
whileClamping:(BOOL) clamp
{
LeapVector *normalized = [iBox normalizePoint:leapPoint clamp: false];
float offset = isLeft ? 0.25 : -0.25;
float x = normalized.x + offset;
float y = normalized.y;
//clamp after offsetting
x = (clamp && x < 0) ? 0 : x;
x = (clamp && x > 1) ? 1 : x;
y = (clamp && y < 0) ? 0 : y;
y = (clamp && y > 1) ? 1 : y;
return [[LeapVector alloc] initWithX:x y:y z:0];
}
Note that you must set the clamp parameter to false to use this technique. (If you want clamping, you can do it manually after offsetting the x-coordinate.)
### Increasing Sensitivity¶
Mapping the full extents of the interaction box to the interaction area of your application is not always the best choice. You may want a smaller movement in the real world to have a larger affect in your application. You can simply scale the Leap Motion coordinates as desired without normalizing (see section above). But if you still want the other benefits of the InteractionBox, you can scale the normalized coordinates by a factor greater than one. A factor of two will make a given motion twice as sensitive; a factor of three, three times more sensitive; and so on. If you make things too sensitive, however, your interaction could become jumpy and difficult to control.
You can do the opposite, of course, by multiplying the normalized coordinates by a value less than one, but it often doesn’t make sense – by definition, the user’s motion won’t cover the entire area of your application. It is easier, instead, to define your application interaction area as a smaller area within the application coordinate system (the math is the same).
The following example, adapts the 2D example above to make the movement 1.5 times more sensitive:
int appWidth = 800;
int appHeight = 600;
LeapFinger *pointable = frame.pointables.frontmost;
LeapVector *leapPoint = pointable.stabilizedTipPosition;
LeapInteractionBox *iBox = frame.interactionBox;
LeapVector *normalizedPoint = [iBox normalizePoint:leapPoint clamp:YES];
normalizedPoint = [normalizedPoint times: 1.5]; //scale
normalizedPoint = [normalizedPoint minus:[[LeapVector alloc] initWithX:.25 y:.25 z:.25]]; // re-center
int appX = normalizedPoint.x * appWidth;
int appY = normalizedPoint.y * appHeight;
You will notice that Setting the clamp parameter to true also no longer keeps the final coordinates within your defined bounds. The user’s motions can more easily move outside the application’s interaction area since you are effectively mapping a smaller subset of the interaction box to your application. Providing good visual feedback to keep the user oriented is helpful (as always).
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2022-08-19 05:08:18
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https://digital-library.theiet.org/content/books/10.1049/pbpo129e_ch9
|
http://iet.metastore.ingenta.com
1887
## Offshore energy storage
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Renewable Energy from the Oceans: From wave, tidal and gradient systems to offshore wind and solar — Recommend this title to your library
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This chapter focuses on energy storage situated offshore. Large amounts have already been written on energy storage generally and there would be little value in adding to these outputs. However, there are good justifications in concentrating specifically on storing energy offshore. First, the environment is rather special and it provides resources that may be helpful for energy storage. These resources include (a) hydrostatic head between surface and seabed that may sometimes be large, (b) an effectively infinite amount of thermal ballast enabling a stable reference temperature to be maintained and (c) an unlimited supply of saltwater that may be useful for electrolysis to support hydrogen production. Second, energy storage at the site of renewable energy generation potentially makes better use of expensive electricity transmission lines joining the generation to consumption. Finally, there are opportunities for integrating storage with the primary harvesting of energy that can afford substantial effective reductions in cost and increases in effective performance.
Chapter Contents:
• 9.1 Underwater compressed air energy storage
• 9.1.1 How much exergy is stored per unit volume of air containment
• 9.1.2 Corrections for air density and non-ideal gas behaviour
• 9.1.3 Structural capacity and its relevance to energy storage
• 9.1.4 Exergy versus structural capacity for underwater containments
• 9.1.5 The air ducts
• 9.1.6 Using thermal storage in conjunction with air storage
• 9.1.7 An example system design
• 9.1.8 Sites available for UWCAES
• 9.2 Offshore pumped hydro
• 9.2.1 Exergy storage density for UWPH
• 9.2.2 Key distinctions between UWPH and UWCAES
• 9.2.3 The EC2SC ratio for UWPH
• 9.3 Buoyancy energy storage systems
• 9.4 Offshore thermal energy storage systems
• 9.5 Other concepts
• 9.6 Integrating offshore energy storage with generation
• References
Preview this chapter:
Offshore energy storage, Page 1 of 2
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content/books/10.1049/pbpo129e_ch9
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2019-09-21 15:06:21
|
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https://math.stackexchange.com/questions/509923/simplifying-confluient-hypergeometric-functions
|
# Simplifying confluient hypergeometric functions
I need to simplify the confulent hypergeometric function:
$U(x>1,1/2,y>0)$. I don't know if someone knows a simpler form ?
There is no simpler form, the result can only be expressed in terms of parabolic cylinder functions: $$U\left(x,1/2,y\right)=2^x e^{y/2}D_{-2x}\left(\sqrt{2y}\right).$$
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2019-07-20 16:30:16
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https://www.steakunderwater.com/wesuckless/viewtopic.php?p=23524&
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Linear workflow question
Moderator: SecondMan
Posts: 2
Joined: Fri May 10, 2019 1:32 pm
Linear workflow question
Hi everyone,
I have a linear workflow question. I did not find an answer on the web.
I recently jump on fusion try to do all my work in linear space.
So i convert my input file to linear and use the viewer LUT (usualy OCIO) to color correct and transform to sRGB. Just in the viewer.
Great.
But when i use the tracking node, the magnifier (the zoomed pattern) look linear. Extremely dark. It is unusable in dark scene.
So, am i doing something wrong? Or is there something to set in the preference or somewhere else?
Best
Midgardsormr
Fusionator
Posts: 1089
Joined: Wed Nov 26, 2014 8:04 pm
Location: Los Angeles, CA, USA
Been thanked: 73 times
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Re: Linear workflow question
The Tracker node was designed before the industry really understood linear workflows. It therefore assumes a display gamma input. I usually convert to sRGB with a Gamut to do the track, then delete or passthrough the Gamut when I'm done.
You will see similar issues in a couple of other places: The Ranges node and Ranges tab in Color Correct work better in 709 or sRGB, as does LumaKey. For the CC, I usually bracket it between two Gamuts if I need to use the Ranges. For the LumaKey, I seldom use its RGB output anyway, and the Alpha is always linear, so I just leave it in the display gamma most of the time.
SirEdric
Fusionator
Posts: 1858
Joined: Tue Aug 05, 2014 10:04 am
Real name: Eric Westphal
Been thanked: 111 times
Contact:
Re: Linear workflow question
In fact, the Tracker (apart from the amazingly cool interface of Fu2.5) was the main reason I bought my own Fu license back in 1996...
So...bear with some non-totally-linear-adopted flaws in this case.
The Tracker itself was always just awesome...
Posts: 2
Joined: Fri May 10, 2019 1:32 pm
Re: Linear workflow question
Thanks for those quick replies and the further informations.
Brayan, as you explain for the luma key. Can you confirm that the other keyer (Delta,Ultra,...) work without problems or limitation in a linear workspace. Just to clear any doubt in my brain.
Thanks
Midgardsormr
Fusionator
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Joined: Wed Nov 26, 2014 8:04 pm
Location: Los Angeles, CA, USA
Been thanked: 73 times
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Re: Linear workflow question
DeltaKeyer definitely works just fine in linear. Ultra Keyer I'm not sure about; I've never really had satisfactory results with it regardless of color space. I would guess that it was designed for rec 601. Primatte is voodoo; there's no telling what it will do. (I'm sure there are plenty of people who use it well; I'm not one of them.)
SecondMan
Posts: 3466
Joined: Thu Jul 31, 2014 5:31 pm
Been thanked: 92 times
Contact:
Re: Linear workflow question
Midgardsormr wrote:
Fri May 10, 2019 2:17 pm
The Tracker node was designed before the industry really understood linear workflows. It therefore assumes a display gamma input.
Well, not exactly. The Tracker doesn't really assume anything. You feed it whatever and it will work with whatever. The algorithm works with whatever data is coming in, independent of how it is encoded. What you see in the magnifier is the image as it comes in, without any further processing.
What is a little unfortunate is that while an image fed into the viewer can "sit under a LUT", or rather have a LUT applied to it, Fusion's controls can't. It would be better if Fusion would have the ability to apply a truly global LUT that spans the viewers, colour controls, things like the magnifier in the tracker,... Basically anything visual that refers to something in the image should have the same LUT applied as the image itself, if that makes sense.
The User Interface preferences have an option for "Gamma aware color controls" that tries to mitigate the above somewhat, but it can be rather limited in practice. What if for some reason you want to work in log space and have a log->rec709 display LUT? Or more commonly, what about the many totally arbitrary, creative LUTs that a client supplies to be used as a viewing LUT for their project?
Viewer LUTs are rarely simple gamma corrections these days and the "Gamma aware color controls" would be an approximation at best.
Midgardsormr wrote:
Fri May 10, 2019 2:17 pm
The Ranges node and Ranges tab in Color Correct work better in 709 or sRGB, as does LumaKey.
Yes, but that's not because they assume a gamma corrected input. That's just because a gamma corrected input typically maps more evenly into a 0-1 range. To an extent the same applies to a log image. This can be why in some cases a track works better, or a key. But sometimes it won't. The Ranges and any keyers work completely linearly otherwise, and predictably so.
That's not to say they couldn't be improved to be more intuitive. It would be nice if you could set them to use curves that have an adjustable mid gray point for example. I think @GringoFX made a couple of Macros that did some of that, also for a more intuitive Contrast control. I don't think they made it to Reactor yet but I should dig them up one day...
In short, what image processing is concerned, every single tool in Fusion works without issue in a linear space. Its 3D space and rendering are totally linear as well. Whenever and wherever you're choosing to use non-linear images, make sure you know what you're doing and why, because Fusion doesn't care - it's all numbers.
Midgardsormr
Fusionator
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Location: Los Angeles, CA, USA
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Re: Linear workflow question
Maybe I spoke a little imprecisely, but the Tracker's ability to sense contrast does seem to be better near 0.5 than it does near 0.18. Hence, if an image is brought into display gamma, the Tracker may do a better job. That's my experience, at any rate. It might just be confirmation bias at play, though. I should test my assumption more rigorously. For certain the pattern preview is inconvenient.
danell
Posts: 39
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Been thanked: 1 time
Re: Linear workflow question
They should take a look at the code and bring it into 2019 and make it work best with linear workflows. Having to learn what tool needs display luts and so on brings it down a lot
SecondMan
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Re: Linear workflow question
danell wrote:
Sat May 11, 2019 12:11 pm
They should take a look at the code and bring it into 2019 and make it work best with linear workflows.
That's a bit of a broad statement - I'm not too sure what you mean?
In terms of "the code" - the code looks fine. The math in Fusion is pretty solid, as far as I can tell. Of course I can point out issues with Fusion's UI in terms of working with images underneath a LUT, as mentioned above, but that's I feel a generic UI wish (which I will try to create a Wish List topic for, if time allows and/or nobody else beats me to it).
(Instead of fiddling with what already worked just fine, I wish BMD had concentrated a bit more on things like that for starters.)
Things to think about - what does linear even mean? Scene linear is linear, but so is AcesCG. If you simply mean material that is not gamma corrected, what gamma are you talking about? 2? 2.2? 2.6? What about LUTS that aren't neutral? Creative? Entirely subjective? What about working in log space? So whatever UI changes are implemented, they would have to go beyond "just linear" workflows - they should be adaptable to any workflows, any custom colour spaces.
Likewise and beyond that, it's more a tooling challenge. Which you can already do just fine in Fusion today. I've dug up Gregory Chalenko's Contrast Around Color Macro which illustrates this beautifully:
ContrastAroundColor_v03-2.setting
EDIT: here's a legible UI so you can see what the tool does:
The tool is great because it doesn't assume anything, really (apart from having gamma compensation on by default, which I personally would advise against). With it, you can dial contrast around any pivot so it works for anything.
A lot of what we are talking about is basically figuring out what it is you need for your workflow, and you can adjust Fusion's tools a little accordingly so they can make your life easier. It's often a case of adding a few nodes, a control or two and some expressions. You don't need to be a TD for this, and there are plenty of people around that can help if you get stuck.
The one thing you need to be able to do is understand and identify what it is that you need.
Nothing beats a solid understanding of your workflow. Whether it's colour, or transformations, or basic compositing theory. Everyone being serious about compositing needs to get their heads around images, from resolution over bit depth and colour space to combining all of them together. It's not called compositing for nothing. Log/lin spaces, gamma correction, gamut and a variety of LUTs and grading processes should be, if not second nature, not too scary
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2019-09-15 16:03:43
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https://proofwiki.org/wiki/Numbers_Not_Expressible_as_Sum_of_no_more_than_5_Composite_Numbers
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# Numbers Not Expressible as Sum of no more than 5 Composite Numbers
There are $256$ integers which cannot be expressed as the sum of no more than $5$ composite numbers:
$1, 2, 3, \ldots, 1167$
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2020-05-26 19:50:13
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http://algebra2014.wikidot.com/isomorphism-of-ring
|
Isomorphism Of Ring
Formal Definition
An isomorphism $\phi :R\rightarrow R'$ from a ring $R$ to a ring $R'$ is a homomorphism that is one to one and onto $R'$. The rings $R$ and $R'$ are then isomorphic.
Informal Definition
Replace this text with an informal definition.
Example(s)
Let $gcd(r,s)=1$, then the rings $Z_r \times Z_s$ and $Z_{rs}$ are isomorphic.
Non-example(s)
As abelian groups, $\langle \mathbb{Z}, +\rangle$ and $\langle \mathbb{2Z}, +\rangle$ are isomorphic under the map $\phi :\mathbb{Z}\rightarrow\mathbb{Z}$, with $\phi (x) =2x$ for $x\in \mathbb{Z}$. Here $\phi$ is not a ring isomorphism, for $\phi(xy)=2xy$,while $\phi (x)\phi(y)=2x2y=4xy$.
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2022-05-25 03:36:59
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http://crypto.stackexchange.com/questions?page=38&sort=votes
|
# All Questions
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In this paper the term "simultaneously hardcore" is defined as: "We say that a block of bits of $x$ are simultaneously hard-core for a one-way function $f(x)$, if given $f(x)$ they cannot be ...
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### Salsa20 in Davies-Meyer mode with a fixed key (message): is it one-way?
Say I have a function that calculates a pseudo random permutation and this function is easy to invert. For example $P(i) = AES_k(i)$ where $k$ is a publicly known key. So anyone can compute $P(i)$ ...
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Are there any proofs that cryptographic functions on an elliptic curve are any more difficult than the analogues over modulo arithmetic? While at present, ECC appears to be more difficult, as it is ...
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### ASN.1 OID of bcrypt
What is the ASN.1 OID associated to bcrypt (the key derivation scheme)? For instance, PBKDF2 has 1.2.840.113549.1.5.12 and it is therefore possible to store a ...
130 views
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Would something like the following improve security (against rainbow attacks, not brute force)? Assume that $P$ is a user-chosen password, and the objective is to obtain a hash $H$ for password ...
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### Finite fields in elliptic curve
I have an elliptic curve defined over finite field where $S_1=aP$ . Is it valid to say that $S_1P$ can also be computed. $P$ is the generator of the group. What my real question is that. Should '$a$' ...
167 views
### How to Compute C^2 in AES MixColumns Matrix?
In mix Columns we have: $$C(x) = \{03\}X^3 + \{01\}X^2 + \{01\}X^1 + \{02\}$$ In Viktor Fischer's Paper on MixColumn and ...
94 views
### Does there exist a two-pass AKE protocol that is secure in eCK model and also has PFS?
As we can know that the best two-pass AKE protocols with DH message can achieve is the weak form of perfect forward security (wPFS) which guarantees security against the passive adversary. But ...
123 views
### Are there any good examples of Contemporary Mechanical Cryptography?
Are there secure mechanical cryptosystems in use today? Not necessarily alphabetic either, ie digital but mechanical?
117 views
### How to compute the attacker's probability?
Given a random bit string $R=r_1r_2\cdots r_n$, let us encrypt each bit $c_i=E(r_i,k_i)$ where $k_i$ is taken from the stream of encryption keys. Suppose that an attacker can guess the correct $r_i$ ...
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### OpenSSL AES 256-bit Key Management
I am using C and OpenSSL to encrypt files. After experimenting with the OpenSSL command line utility, it makes you enter a passphrase that can be any length, but uses that to create a 256-bit key. ...
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### Given a private RSA key, how do we get the public key?
Is it possible to pre-choose a private RSA key, then obtain a public key from it?
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2014-07-30 00:59:35
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https://orbi.uliege.be/browse?type=author&value=Mennesson%2C+B
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References of "Mennesson, B" in Complete repository Arts & humanities Archaeology Art & art history Classical & oriental studies History Languages & linguistics Literature Performing arts Philosophy & ethics Religion & theology Multidisciplinary, general & others Business & economic sciences Accounting & auditing Production, distribution & supply chain management Finance General management & organizational theory Human resources management Management information systems Marketing Strategy & innovation Quantitative methods in economics & management General economics & history of economic thought International economics Macroeconomics & monetary economics Microeconomics Economic systems & public economics Social economics Special economic topics (health, labor, transportation…) Multidisciplinary, general & others Engineering, computing & technology Aerospace & aeronautics engineering Architecture Chemical engineering Civil engineering Computer science Electrical & electronics engineering Energy Geological, petroleum & mining engineering Materials science & engineering Mechanical engineering Multidisciplinary, general & others Human health sciences Alternative medicine Anesthesia & intensive care Cardiovascular & respiratory systems Dentistry & oral medicine Dermatology Endocrinology, metabolism & nutrition Forensic medicine Gastroenterology & hepatology General & internal medicine Geriatrics Hematology Immunology & infectious disease Laboratory medicine & medical technology Neurology Oncology Ophthalmology Orthopedics, rehabilitation & sports medicine Otolaryngology Pediatrics Pharmacy, pharmacology & toxicology Psychiatry Public health, health care sciences & services Radiology, nuclear medicine & imaging Reproductive medicine (gynecology, andrology, obstetrics) Rheumatology Surgery Urology & nephrology Multidisciplinary, general & others Law, criminology & political science Civil law Criminal law & procedure Criminology Economic & commercial law European & international law Judicial law Metalaw, Roman law, history of law & comparative law Political science, public administration & international relations Public law Social law Tax law Multidisciplinary, general & others Life sciences Agriculture & agronomy Anatomy (cytology, histology, embryology...) & physiology Animal production & animal husbandry Aquatic sciences & oceanology Biochemistry, biophysics & molecular biology Biotechnology Entomology & pest control Environmental sciences & ecology Food science Genetics & genetic processes Microbiology Phytobiology (plant sciences, forestry, mycology...) Veterinary medicine & animal health Zoology Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences Chemistry Earth sciences & physical geography Mathematics Physics Space science, astronomy & astrophysics Multidisciplinary, general & others Social & behavioral sciences, psychology Animal psychology, ethology & psychobiology Anthropology Communication & mass media Education & instruction Human geography & demography Library & information sciences Neurosciences & behavior Regional & inter-regional studies Social work & social policy Sociology & social sciences Social, industrial & organizational psychology Theoretical & cognitive psychology Treatment & clinical psychology Multidisciplinary, general & others Showing results 1 to 20 of 31 1 2 Space-based infrared interferometry to study exoplanetary atmospheresDefrere, Denis ; Léger, A.; Absil, Olivier et alin Experimental Astronomy: Astrophysical Instrumentation and Methods (in press), 1801The quest for other habitable worlds and the search for life among them are major goals of modern astronomy. One way to make progress towards these goals is to obtain high-quality spectra of a large ... [more ▼]The quest for other habitable worlds and the search for life among them are major goals of modern astronomy. One way to make progress towards these goals is to obtain high-quality spectra of a large number of exoplanets over a broad range of wavelengths. While concepts currently investigated in the United States are focused on visible/NIR wavelengths, where the planets are probed in reflected light, a compelling alternative to characterize planetary atmospheres is the mid-infrared waveband (5-20um). Indeed, mid-infrared observations provide key information on the presence of an atmosphere, the surface conditions (e.g., temperature, pressure, habitability), and the atmospheric composition in important species such as H2O, CO2, O3, CH4, and N2O. This information is essential to investigate the potential habitability of exoplanets and to make progress towards the search for life in the universe. Obtaining high-quality mid-infrared spectra of exoplanets from the ground is however extremely challenging due to the overwhelming brightness and turbulence of Earth's atmosphere. In this paper, we present a concept of space-based mid-infrared interferometer that can tackle this observing challenge and discuss the main technological developments required to launch such a sophisticated instrument. [less ▲]Detailed reference viewed: 32 (0 ULiège) The path towards high-contrast imaging with the VLTI: the Hi-5 projectDefrere, Denis ; Absil, Olivier ; Berger, J.-P. et alin Experimental Astronomy: Astrophysical Instrumentation and Methods (in press), 1801The development of high-contrast capabilities has long been recognized as one of the top priorities for the VLTI. As of today, the VLTI routinely achieves contrasts of a few 10$^{-3}$ in the near-infrared ... [more ▼]The development of high-contrast capabilities has long been recognized as one of the top priorities for the VLTI. As of today, the VLTI routinely achieves contrasts of a few 10$^{-3}$ in the near-infrared with PIONIER (H band) and GRAVITY (K band). Nulling interferometers in the northern hemisphere and non-redundant aperture masking experiments have, however, demonstrated that contrasts of at least a few 10$^{-4}$ are within reach using specific beam combination and data acquisition techniques. In this paper, we explore the possibility to reach similar or higher contrasts on the VLTI. After reviewing the state-of-the-art in high-contrast infrared interferometry, we discuss key features that made the success of other high-contrast interferometric instruments (e.g., integrated optics, nulling, closure phase, and statistical data reduction) and address possible avenues to improve the contrast of the VLTI by at least one order of magnitude. In particular, we discuss the possibility to use integrated optics, proven in the near-infrared, in the thermal near-infrared (L and M bands, 3-5 $\mu$m), a sweet spot to image and characterize young extra-solar planetary systems. Finally, we address the science cases of a high-contrast VLTI imaging instrument and focus particularly on exoplanet science (young exoplanets, planet formation, and exozodiacal disks), stellar physics (fundamental parameters and multiplicity), and extragalactic astrophysics (active galactic nuclei and fundamental constants). Synergies and scientific preparation for other potential future instruments such as the Planet Formation Imager are also briefly discussed. [less ▲]Detailed reference viewed: 8 (0 ULiège) The LBTI Fizeau imager – II. Sensitivity of the PSF and the MTF to adaptive optics errors and to piston errorsPatru, F.; Esposito, S.; Puglisi, A. et alin Monthly Notices of the Royal Astronomical Society (2017), 472We show numerical simulations with monochromatic light in the visible for the LBTI Fizeau imager, including opto-dynamical aberrations due here to adaptive optics (AO) errors and to differential piston ... [more ▼]We show numerical simulations with monochromatic light in the visible for the LBTI Fizeau imager, including opto-dynamical aberrations due here to adaptive optics (AO) errors and to differential piston fluctuations, while other errors have been neglected. The achievable Strehl by the LBTI using two AO is close to the Strehl provided by a single standalone AO system, as long as other differential wavefront errors are mitigated. The LBTI Fizeau imager is primarily limited by the AO performance and by the differential piston/tip–tilt errors. Snapshots retain high-angular resolution and high-contrast imaging information by freezing the fringes against piston errors. Several merit functions have been critically evaluated in order to characterize point spread functions and the modulation transfer functions for high-contrast imaging applications. The LBTI Fizeau mode can provide an image quality suitable for standard science cases (i.e. a Strehl above 70 per cent) by performing both at a time: an AO correction better than ≈λ/18 RMS for both short and long exposures, and a piston correction better than ≈λ/8 RMS for long exposures or simply below the coherence length for short exposures. Such results, which can be applied to any observing wavelength, suggest that AO and piston control at the LBTI would already improve the contrast at near- and mid-infrared wavelengths. Therefore, the LBTI Fizeau imager can be used for high-contrast imaging, providing a high-Strehl regime (by both AO systems), a cophasing mode (by a fringe tracker) and a burst mode (by a fast camera) to record fringed speckles in short exposures. [less ▲]Detailed reference viewed: 18 (1 ULiège) The LBTI Fizeau imager - I. Fundamental gain in high-contrast imagingPatru, F.; Esposito, S.; Puglisi, A. et alin Monthly Notices of the Royal Astronomical Society (2017), 472We show by numerical simulations a fundamental gain in contrast when combining coherently monochromatic light from two adaptive optics (AO) telescopes instead of using a single stand-alone AO telescope ... [more ▼]We show by numerical simulations a fundamental gain in contrast when combining coherently monochromatic light from two adaptive optics (AO) telescopes instead of using a single stand-alone AO telescope, assuming efficient control and acquisition systems at high speed. A contrast gain map is defined as the normalized point spread functions (PSFs) ratio of a single Large Binocular Telescope (LBT) aperture over the dual Large Binocular Telescope Interferometer (LBTI) aperture in Fizeau mode. The global gain averaged across the AO-corrected field of view is improved by a factor of 2 in contrast in long exposures and by a factor of 10 in contrast in short exposures (i.e. in exposures, respectively, longer or shorter than the coherence time). The fringed speckle halo in short exposures contains not only high-angular resolution information, as stated by speckle imaging and speckle interferometry, but also high-contrast imaging information. A high-gain zone is further produced in the valleys of the PSF formed by the dark Airy rings and/or the dark fringes. Earth rotation allows us to exploit various areas in the contrast gain map. A huge-contrast gain in narrow zones can be achieved when both a dark fringe and a dark ring overlap on to an exoplanet. Compared to a single 8-m LBT aperture, the 23-m LBTI Fizeau imager can provide a gain in sensitivity (by a factor of 4), a gain in angular resolution (by a factor of 3) and, as well, a gain in raw contrast (by a factor of 2-1000 varying over the AO-corrected field of view). [less ▲]Detailed reference viewed: 24 (5 ULiège) Overview of LBTI: a multipurpose facility for high spatial resolution observationsHinz, P. M.; Defrere, Denis ; Skemer, A. et alin Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2016, August 01)The Large Binocular Telescope Interferometer (LBTI) is a high spatial resolution instrument developed for coherent imaging and nulling interferometry using the 14.4 m baseline of the 2×8.4 m LBT. The ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a high spatial resolution instrument developed for coherent imaging and nulling interferometry using the 14.4 m baseline of the 2×8.4 m LBT. The unique telescope design, comprising of the dual apertures on a common elevation-azimuth mount, enables a broad use of observing modes. The full system is comprised of dual adaptive optics systems, a near-infrared phasing camera, a 1-5 μm camera (called LMIRCam), and an 8-13 μm camera (called NOMIC). The key program for LBTI is the Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS), a survey using nulling interferometry to constrain the typical brightness from exozodiacal dust around nearby stars. Additional observations focus on the detection and characterization of giant planets in the thermal infrared, high spatial resolution imaging of complex scenes such as Jupiter's moon, Io, planets forming in transition disks, and the structure of active Galactic Nuclei (AGN). Several instrumental upgrades are currently underway to improve and expand the capabilities of LBTI. These include: Improving the performance and limiting magnitude of the parallel adaptive optics systems; quadrupling the field of view of LMIRcam (increasing to 20"x20"); adding an integral field spectrometry mode; and implementing a new algorithm for path length correction that accounts for dispersion due to atmospheric water vapor. We present the current architecture and performance of LBTI, as well as an overview of the upgrades. [less ▲]Detailed reference viewed: 12 (1 ULiège) Enabling the direct detection of earth-sized exoplanets with the LBTI HOSTS project: a progress reportDanchi, W.; Bailey, V.; Bryden, G. et alin Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2016, August 01)NASA has funded a project called the Hunt for Observable Signatures of Terrestrial Systems (HOSTS) to survey nearby solar type stars to determine the amount of warm zodiacal dust in their habitable zones ... [more ▼]NASA has funded a project called the Hunt for Observable Signatures of Terrestrial Systems (HOSTS) to survey nearby solar type stars to determine the amount of warm zodiacal dust in their habitable zones. The goal is not only to determine the luminosity distribution function but also to know which individual stars have the least amount of zodiacal dust. It is important to have this information for future missions that directly image exoplanets as this dust is the main source of astrophysical noise for them. The HOSTS project utilizes the Large Binocular Telescope Interferometer (LBTI), which consists of two 8.4-m apertures separated by a 14.4-m baseline on Mt. Graham, Arizona. The LBTI operates in a nulling mode in the mid-infrared spectral window (8-13 μm), in which light from the two telescopes is coherently combined with a 180 degree phase shift between them, producing a dark fringe at the location of the target star. In doing so the starlight is greatly reduced, increasing the contrast, analogous to a coronagraph operating at shorter wavelengths. The LBTI is a unique instrument, having only three warm reflections before the starlight reaches cold mirrors, giving it the best photometric sensitivity of any interferometer operating in the mid-infrared. It also has a superb Adaptive Optics (AO) system giving it Strehl ratios greater than 98% at 10 μm. In 2014 into early 2015 LBTI was undergoing commissioning. The HOSTS project team passed its Operational Readiness Review (ORR) in April 2015. The team recently published papers on the target sample, modeling of the nulled disk images, and initial results such as the detection of warm dust around η Corvi. Recently a paper was published on the data pipeline and on-sky performance. An additional paper is in preparation on β Leo. We will discuss the scientific and programmatic context for the LBTI project, and we will report recent progress, new results, and plans for the science verification phase that started in February 2016, and for the survey. [less ▲]Detailed reference viewed: 17 (0 ULiège) Nulling Data Reduction and On-sky Performance of the Large Binocular Telescope InterferometerDefrere, Denis ; Hinz, P. M.; Mennesson, B. et alin Astrophysical Journal (2016), 824The Large Binocular Telescope Interferometer (LBTI) is a versatile instrument designed for high angular resolution and high-contrast infrared imaging (1.5-13 μm). In this paper, we focus on the mid ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a versatile instrument designed for high angular resolution and high-contrast infrared imaging (1.5-13 μm). In this paper, we focus on the mid-infrared (8-13 μm) nulling mode and present its theory of operation, data reduction, and on-sky performance as of the end of the commissioning phase in 2015 March. With an interferometric baseline of 14.4 m, the LBTI nuller is specifically tuned to resolve the habitable zone of nearby main-sequence stars, where warm exozodiacal dust emission peaks. Measuring the exozodi luminosity function of nearby main-sequence stars is a key milestone to prepare for future exo-Earth direct imaging instruments. Thanks to recent progress in wavefront control and phase stabilization, as well as in data reduction techniques, the LBTI demonstrated in 2015 February a calibrated null accuracy of 0.05% over a 3 hr long observing sequence on the bright nearby A3V star β Leo. This is equivalent to an exozodiacal disk density of 15-30 zodi for a Sun-like star located at 10 pc, depending on the adopted disk model. This result sets a new record for high-contrast mid-infrared interferometric imaging and opens a new window on the study of planetary systems. [less ▲]Detailed reference viewed: 33 (13 ULiège) Models of the η Corvi Debris Disk from the Keck Interferometer, Spitzer, and HerschelLebreton, J.; Beichman, C.; Bryden, G. et alin Astrophysical Journal (2016), 817Debris disks are signposts of analogs to small-body populations of the solar system, often, however, with much higher masses and dust production rates. The disk associated with the nearby star η Crv is ... [more ▼]Debris disks are signposts of analogs to small-body populations of the solar system, often, however, with much higher masses and dust production rates. The disk associated with the nearby star η Crv is especially striking, as it shows strong mid- and far-infrared excesses despite an age of ∼1.4 Gyr. We undertake constructing a consistent model of the system that can explain a diverse collection of spatial and spectral data. We analyze Keck Interferometer Nuller measurements and revisit Spitzer and additional spectrophotometric data, as well as resolved Herschel images, to determine the dust spatial distribution in the inner exozodi and in the outer belt. We model in detail the two-component disk and the dust properties from the sub-AU scale to the outermost regions by fitting simultaneously all measurements against a large parameter space. The properties of the cold belt are consistent with a collisional cascade in a reservoir of ice-free planetesimals at 133 AU. It shows marginal evidence for asymmetries along the major axis. KIN enables us to establish that the warm dust consists of a ring that peaks between 0.2 and 0.8 AU. To reconcile this location with the ∼400 K dust temperature, very high albedo dust must be invoked, and a distribution of forsterite grains starting from micron sizes satisfies this criterion, while providing an excellent fit to the spectrum. We discuss additional constraints from the LBTI and near-infrared spectra, and we present predictions of what James Webb Space Telescope can unveil about this unusual object and whether it can detect unseen planets. [less ▲]Detailed reference viewed: 15 (3 ULiège) Simultaneous Water Vapor and Dry Air Optical Path Length Measurements and Compensation with the Large Binocular Telescope InterferometerDefrere, Denis ; Hinz, P.; Downey, E. et alin Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (2016)The Large Binocular Telescope Interferometer uses a near-infrared camera to measure the optical path length variations between the two AO-corrected apertures and provide high-angular resolution ... [more ▼]The Large Binocular Telescope Interferometer uses a near-infrared camera to measure the optical path length variations between the two AO-corrected apertures and provide high-angular resolution observations for all its science channels (1.5-13 microns). There is however a wavelength dependent component to the atmospheric turbulence, which can introduce optical path length errors when observing at a wavelength different from that of the fringe sensing camera. Water vapor in particular is highly dispersive and its effect must be taken into account for high-precision infrared interferometric observations as described previously for VLTI/MIDI or the Keck Interferometer Nuller. In this paper, we describe the new sensing approach that has been developed at the LBT to measure and monitor the optical path length fluctuations due to dry air and water vapor separately. After reviewing the current performance of the system for dry air seeing compensation, we present simultaneous H-, K-, and N-band observations that illustrate the feasibility of our feedforward approach to stabilize the path length fluctuations seen by the LBTI nuller. [less ▲]Detailed reference viewed: 23 (7 ULiège) Exoplanet science with the LBTI: instrument status and plansDefrere, Denis ; Hinz, P.; Skemer, A. et alin Shaklan, Stuart (Ed.) Techniques and Instrumentation for Detection of Exoplanets VII (2015, September 16)The Large Binocular Telescope Interferometer (LBTI) is a strategic instrument of the LBT designed for high-sensitivity, high-contrast, and high-resolution infrared (1.5-13 $\mu$m) imaging of nearby ... [more ▼]The Large Binocular Telescope Interferometer (LBTI) is a strategic instrument of the LBT designed for high-sensitivity, high-contrast, and high-resolution infrared (1.5-13 $\mu$m) imaging of nearby planetary systems. To carry out a wide range of high-spatial resolution observations, it can combine the two AO-corrected 8.4-m apertures of the LBT in various ways including direct (non-interferometric) imaging, coronagraphy (APP and AGPM), Fizeau imaging, non-redundant aperture masking, and nulling interferometry. It also has broadband, narrowband, and spectrally dispersed capabilities. In this paper, we review the performance of these modes in terms of exoplanet science capabilities and describe recent instrumental milestones such as first-light Fizeau images (with the angular resolution of an equivalent 22.8-m telescope) and deep interferometric nulling observations. [less ▲]Detailed reference viewed: 30 (10 ULiège) First-light LBT Nulling Interferometric Observations: Warm Exozodiacal Dust Resolved within a Few AU of eta CrvDefrere, Denis ; Hinz, P. M.; Skemer, A. J. et alin Astrophysical Journal (2015), 799We report on the first nulling interferometric observations with the Large Binocular Telescope Interferometer (LBTI), resolving the N' band (9.81-12.41 μm) emission around the nearby main-sequence star η ... [more ▼]We report on the first nulling interferometric observations with the Large Binocular Telescope Interferometer (LBTI), resolving the N' band (9.81-12.41 μm) emission around the nearby main-sequence star η Crv (F2V, 1-2 Gyr). The measured source null depth amounts to 4.40% ± 0.35% over a field-of-view of 140 mas in radius (~2.6 AU for the distance of η Crv) and shows no significant variation over 35° of sky rotation. This relatively low null is unexpected given the total disk to star flux ratio measured by the Spitzer Infrared Spectrograph (IRS; ~23% across the N' band), suggesting that a significant fraction of the dust lies within the central nulled response of the LBTI (79 mas or 1.4 AU). Modeling of the warm disk shows that it cannot resemble a scaled version of the solar zodiacal cloud unless it is almost perpendicular to the outer disk imaged by Herschel. It is more likely that the inner and outer disks are coplanar and the warm dust is located at a distance of 0.5-1.0 AU, significantly closer than previously predicted by models of the IRS spectrum (~3 AU). The predicted disk sizes can be reconciled if the warm disk is not centrosymmetric, or if the dust particles are dominated by very small grains. Both possibilities hint that a recent collision has produced much of the dust. Finally, we discuss the implications for the presence of dust for the distance where the insolation is the same as Earth's (2.3 AU). [less ▲]Detailed reference viewed: 30 (10 ULiège) Constraining the Exozodiacal Luminosity Function of Main-sequence Stars: Complete Results from the Keck Nuller Mid-infrared SurveysMennesson, B.; Millan-Gabet, R.; Serabyn, E. et alin Astrophysical Journal (2014), 797Forty-seven nearby main-sequence stars were surveyed with the Keck Interferometer mid-infrared Nulling instrument (KIN) between 2008 and 2011, searching for faint resolved emission from exozodiacal dust ... [more ▼]Forty-seven nearby main-sequence stars were surveyed with the Keck Interferometer mid-infrared Nulling instrument (KIN) between 2008 and 2011, searching for faint resolved emission from exozodiacal dust. Observations of a subset of the sample have already been reported, focusing essentially on stars with no previously known dust. Here we extend this previous analysis to the whole KIN sample, including 22 more stars with known near- and/or far-infrared excesses. In addition to an analysis similar to that of the first paper of this series, which was restricted to the 8-9 mum spectral region, we present measurements obtained in all 10 spectral channels covering the 8-13 mum instrumental bandwidth. Based on the 8-9 mum data alone, which provide the highest signal-to-noise measurements, only one star shows a large excess imputable to dust emission (eta Crv), while four more show a significant (>3sigma) excess: beta Leo, beta UMa, zeta Lep, and gamma Oph. Overall, excesses detected by KIN are more frequent around A-type stars than later spectral types. A statistical analysis of the measurements further indicates that stars with known far-infrared (lambda >= 70 mum) excesses have higher exozodiacal emission levels than stars with no previous indication of a cold outer disk. This statistical trend is observed regardless of spectral type and points to a dynamical connection between the inner (zodi-like) and outer (Kuiper-Belt-like) dust populations. The measured levels for such stars are clustering close to the KIN detection limit of a few hundred zodis and are indeed consistent with those expected from a populat [less ▲]Detailed reference viewed: 24 (8 ULiège) Fundamental Limitations of High Contrast Imaging Set by Small Sample StatisticsMawet, D.; Milli, J.; Wahhaj, Z. et alin Astrophysical Journal (2014), 792In this paper, we review the impact of small sample statistics on detection thresholds and corresponding confidence levels (CLs) in high-contrast imaging at small angles. When looking close to the star ... [more ▼]In this paper, we review the impact of small sample statistics on detection thresholds and corresponding confidence levels (CLs) in high-contrast imaging at small angles. When looking close to the star, the number of resolution elements decreases rapidly toward small angles. This reduction of the number of degrees of freedom dramatically affects CLs and false alarm probabilities. Naively using the same ideal hypothesis and methods as for larger separations, which are well understood and commonly assume Gaussian noise, can yield up to one order of magnitude error in contrast estimations at fixed CL. The statistical penalty exponentially increases toward very small inner working angles. Even at 5-10 resolution elements from the star, false alarm probabilities can be significantly higher than expected. Here we present a rigorous statistical analysis that ensures robustness of the CL, but also imposes a substantial limitation on corresponding achievable detection limits (thus contrast) at small angles. This unavoidable fundamental statistical effect has a significant impact on current coronagraphic and future high-contrast imagers. Finally, the paper concludes with practical recommendations to account for small number statistics when computing the sensitivity to companions at small angles and when exploiting the results of direct imaging planet surveys. [less ▲]Detailed reference viewed: 36 (9 ULiège) L'-band AGPM vector vortex coronagraph's first light on LBTI/LMIRCamDefrere, Denis ; Absil, Olivier ; Hinz, P. et alin Proceedings of SPIE - The International Society for Optical Engineering (2014, July 21)We present the first observations obtained with the L'-band AGPM vortex coronagraph recently installed on LBTI/LMIRCam. The AGPM (Annular Groove Phase Mask) is a vector vortex coronagraph made from ... [more ▼]We present the first observations obtained with the L'-band AGPM vortex coronagraph recently installed on LBTI/LMIRCam. The AGPM (Annular Groove Phase Mask) is a vector vortex coronagraph made from diamond subwavelength gratings. It is designed to improve the sensitivity and dynamic range of high-resolution imaging at very small inner working angles, down to 0.09 arcseconds in the case of LBTI/LMIRCam in the L' band. During the first hours on sky, we observed the young A5V star HR8799 with the goal to demonstrate the AGPM performance and assess its relevance for the ongoing LBTI planet survey (LEECH). Preliminary analyses of the data reveal the four known planets clearly at high SNR and provide unprecedented sensitivity limits in the inner planetary system (down to the diffraction limit of 0.09 arcseconds). © 2014 SPIE. [less ▲]Detailed reference viewed: 28 (8 ULiège) Co-phasing the Large Binocular Telescope: status and performance of LBTI/PHASECamDefrere, Denis ; Hinz, P.; Downey, E. et alin Optical and Infrared Interferometry IV (2014, July 01)The Large Binocular Telescope Interferometer is a NASA-funded nulling and imaging instrument designed to coherently combine the two 8.4-m primary mirrors of the LBT for high-sensitivity, high-contrast ... [more ▼]The Large Binocular Telescope Interferometer is a NASA-funded nulling and imaging instrument designed to coherently combine the two 8.4-m primary mirrors of the LBT for high-sensitivity, high-contrast, and highresolution infrared imaging (1.5-13 μm). PHASECam is LBTI's near-infrared camera used to measure tip-tilt and phase variations between the two AO-corrected apertures and provide high-angular resolution observations. We report on the status of the system and describe its on-sky performance measured during the first semester of 2014. With a spatial resolution equivalent to that of a 22.8-meter telescope and the light-gathering power of single 11.8-meter mirror, the co-phased LBT can be considered to be a forerunner of the next-generation extremely large telescopes (ELT). [less ▲]Detailed reference viewed: 33 (11 ULiège) The LBTI hunt for observable signatures of terrestrial systems (HOSTS) survey: a key NASA science program on the road to exoplanet imaging missionsDanchi, W.; Bailey, V.; Bryden, G. et alin Optical and Infrared Interferometry IV (2014, July 01)The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) program on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi ... [more ▼]The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) program on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). This warm circumstellar dust, analogous to the interplanetary dust found in the vicinity of the Earth in our own system, is produced in comet breakups and asteroid collisions. Emission and/or scattered light from the exozodi will be the major source of astrophysical noise for a future space telescope aimed at direct imaging and spectroscopy of terrestrial planets (exo- Earths) around nearby stars. About 20% of nearby field stars have cold dust coming from planetesimals at large distances from the stars (Eiroa et al. 2013, A&A, 555, A11; Siercho et al. 2014, ApJ, 785, 33). Much less is known about exozodi; current detection limits for individual stars are at best ~ 500 times our solar system's level (aka. 500 zodi). LBTI-HOSTS will be the first survey capable of measuring exozodi at the 10 zodi level (3σ). Detections of warm dust will also reveal new information about planetary system architectures and evolution. We will describe the motivation for the survey and progress on target selection, not only the actual stars likely to be observed by such a mission but also those whose observation will enable sensible extrapolations for stars that will not be observed with LBTI. We briefly describe the detection of the debris disk around η Crv, which is the first scientific result from the LBTI coming from the commissioning of the instrument in December 2013, shortly after the first time the fringes were stabilized. [less ▲]Detailed reference viewed: 15 (1 ULiège) The Hunt for Observable Signatures of Terrestrial Planetary Systems (HOSTS)Defrere, Denis ; Hinz, P.; Bryden, G. et alConference (2014, March)The presence of large amounts of exozodiacal dust around nearby main sequence stars is considered as a potential threat for the direct imaging of Earth-like exoplanets and, hence, the search for ... [more ▼]The presence of large amounts of exozodiacal dust around nearby main sequence stars is considered as a potential threat for the direct imaging of Earth-like exoplanets and, hence, the search for biosignatures (Roberge et al. 2012). However, it is also considered as a signpost for the presence of terrestrial planets that might be hidden in the dust disk (Stark and Kuchner 2008). Characterizing exozodiacal dust around nearby sequence stars is therefore a crucial step toward one of the main goals of modern astronomy: finding extraterrestrial life. After briefly reviewing the latest results in this field, we present the exozodiacal dust survey on the Large Binocular Telescope Interferometer (LBTI). The survey is called HOSTS and is specifically designed to determine the prevalence and brightness of exozodiacal dust disks with the sensitivity required to prepare for future New Worlds Missions that will image Earth-like exoplanets. To achieve this objective, the LBTI science team has carefully established a balanced list of 50 nearby main-sequence stars that are likely candidates of these missions and/or can be observed with the best instrument performance (see companion abstract by Roberge et al.). Exozodiacal dust disk candidates detected by the Keck Interferometer Nuller will also be observed. The first results of the survey will be presented. To precisely detect exozodiacal dust, the LBTI combines the two 8-m primary mirrors of the LBT using N-band nulling interferometry. Interferometric combination provides the required angular resolution (70-90 mas) to resolve the habitable zone of nearby main sequence stars while nulling is used to subtract the stellar light and reach the required contrast of a few 10-4. A Kband fringe tracker ensures the stability of the null. The current performance of the instrument and the first nulling measurements will be presented. [less ▲]Detailed reference viewed: 16 (4 ULiège) L'-band AGPM vector vortex coronagraph's first light on LBTI/LMIRCAMDefrere, Denis ; Absil, Olivier ; Hinz, P. et alPoster (2014, March)We present the first science observations obtained with the L'-band AGPM coronagraph recently installed on LBTI/LMIRCAM. The AGPM (Annular Groove Phase Mask) is a vector vortex coronagraph made from ... [more ▼]We present the first science observations obtained with the L'-band AGPM coronagraph recently installed on LBTI/LMIRCAM. The AGPM (Annular Groove Phase Mask) is a vector vortex coronagraph made from diamond sub-wavelength gratings tuned to the L'-band. It is designed to improve the sensitivity and dynamic range of high-resolution imaging at very small inner working a [less ▲]Detailed reference viewed: 48 (10 ULiège) Exozodi disk models for the HOSTS survey on the LBTIWyatt, Mark; Kennedy, G.; Skemer, A. et alin American Astronomical Society Meeting Abstracts #223 (2014, January 01)This poster describes a simple model for exozodiacal emission that was developed to interpret observations of the Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) project on the ... [more ▼]This poster describes a simple model for exozodiacal emission that was developed to interpret observations of the Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) project on the Large Binocular Telescope Interferometer (LBTI). HOSTS is a NASA-funded key science project using mid-infrared nulling interferometry at the LBTI to seach for faint exozodiacal dust (exozodi) in the habitable zones of nearby stars. The aim was to make a model that includes the fewest possible assumptions, so that it is easy to characterize how choices of model parameters affect what can be inferred from the observations. However the model is also sufficiently complex that it can be compared in a physically meaningful way with the level of dust in the Solar System, and can also be readily used to assess the impact of a detection (or of a non-detection) on the ability of a mission to detect Earth-like planets. Here we describe the model, and apply it to the sample of stars being searched by HOSTS to determine the zodi level (i.e., the number of Solar System zodiacal clouds) that would be needed for a detection for each star in the survey. Particular emphasis is given to our definition of a zodi, and what that means for stars of different luminosity, and a comparison is given between different zodi definitions justifying our final choice. The achievable exozodi levels range from 1-20 zodi for different stars in the prime sample for a 0.01% null depth, with a median level of 2.5 zodi. [less ▲]Detailed reference viewed: 11 (3 ULiège) Target Selection for the LBTI Hunt for Observable Signatures of Terrestrial Planetary SystemsWeinberger, Alycia J.; Roberge, A.; Kennedy, G. et alin American Astronomical Society Meeting Abstracts #223 (2014, January 01)The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). About 20 ... [more ▼]The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). About 20% of field stars have cold debris disks created by the collisions and evaporation of planetesimals. Much less is known about warm circumstellar dust, such as that found in the vicinity of the Earth in our own system. This dust is generated in asteroidal collisions and cometary breakups, and current detection limits are at best ~500 times our system's level, i.e. 500 zodi. LBTI-HOSTS will be the first survey capable of measuring exozodi at the 10 zodi level (3σ). Exozodi of this brightness would be the major source of astrophysical noise for a future space telescope aimed at direct imaging and spectroscopy of habitable zone terrestrial planets. Detections of warm dust will also reveal new information about planetary system architectures and evolution. We describe the target star selection by the LBTI Science Team to satisfy the goals of the HOSTS survey -- to fully inform target selection for a future exoEarth mission. We are interested in actual stars likely to be observed by a mission and stars whose observation will enable sensible extrapolations to those stars that cannot be observed. We integrated two approaches to generate the HOSTS target list. The mission-driven approach concentrates on F, G, and K-type stars that are the best targets for future direct observations of exoEarths, thereby providing model-independent “ground truth” dust observations. However, not every potential target of a future exoEarth mission can be observed with LBTI. The sensitivity-driven approach selects targets based only on what exozodi sensitivity could be achieved, without consideration of exoEarth mission constraints. This naturally selects more luminous stars (A and early F-type stars). In both cases, all stars are close enough to Earth such that their habitable zones are resolvable by LBTI and bright enough at N-band (10 μm) to provide excellent sensitivity. We also discuss observational and astrophysical motivations for excluding binaries of certain separations. [less ▲]Detailed reference viewed: 5 (1 ULiège) 1 2
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# The Ultimate Q51 Guide [Expert Level]
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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
02 Mar 2017, 18:31
$$(x-y)^2=?$$
1) x and y are integers
2) xy=3
==> In the original condition, there are 2 variables (x, y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get (x,y)=(1,3),(3,1),(-1,-3),(-3,-1), and all become $$(x-y)^2=4$$, hence it is unique and sufficient.
Therefore, the answer is C.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 05 Mar 2017, 18:24 What is the scope including 1/11+1/12+1/13+......+1/20? A 1/6~1/5 B. 1/5~1/4 C. 1/4~1/3 D. 1/3~1/2 E. 1/2~1 The sum of consecutive reciprocal number sequence is decided by the first number and the last number. Thus, from 10/20=1/11+1/12+.....+1/20<1/11+1/12+.....+1/20<1/11+1/12+.....+1/20=10/11, you get 1/2=10/20<1/11+1/12+.....+1/20<10/11<1, which becomes 1/2~1. Therefore, the answer is E. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
06 Mar 2017, 18:18
$$(x+y)^2-(x-y)^2=?$$
1) xy=5
2) x+y=6
==> If you modify the original condition and the question, $$(x+y)^2-(x-y)^2=?$$ becomes (x+y-x+y)(x+y+x-y)=?, and if you simplify this, you get (2y)(2x)=?, 4xy=?. Thus, for con 1), you get xy=5, hence it is unique and sufficient. The answer is A.
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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 08 Mar 2017, 18:26 Is a=b? 1) a^2=b^2 2) a=2 ==> In the original condition, there are 2 variables (a,b) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), from a=2, you get b2=22=4, then b=±2. Thus, (a,b)=(2,2) yes but (a,b)=(2,-2) no, hence it is not sufficient. The answer is E. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
09 Mar 2017, 22:37
Expert's post
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What is the perimeter of a certain right triangle?
1) The hypotenuse’s length is 10
2) The triangle’s area is 24
==> In the original condition, for a right triangle, there are 2 variables (2 legs) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), you get 6:8:10 and the perimeter of the right triangle becomes 6+8+10=24, hence unique and sufficient.
Therefore, the answer is C.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 12 Mar 2017, 18:51 $$Let A=6^2^0, B=2^6^0, and C=4^5^0.$$ Which of the following is true? A. A<B<C B. A<C<B C. B<A<C D. B<C<A E. C<B<A ==> You can compare the big and small numbers by making the base or the exponent the same. From$$A=6^2^0, B=2^6^0=(2^3)^2^0=8^2^0$$, you get 6<8, which becomes A<B, and from $$B=2^6^0=(2^2)^3^0=4^3^0, C=4^5^0,$$ you get 30<50, which becomes B<C. Thus, you get A<B<C. The answer is A. Answer: A _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
13 Mar 2017, 18:28
$$\frac{5.09}{0.149}$$ is closest to which of the following?
A. 0.34
B. 3.4
C. 34
D. 340
E. 3,400
From $$\frac{5.09}{0.149} = \frac{5.10}{0.15} = \frac{510}{15} =34$$, the answer is C.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 15 Mar 2017, 18:39 In the x-y plane, what is the slope between y-intercept and a negative x-intercept of $$y=x^2+x-6$$? A. -2 B. -3 C. 2 D. 4 E. 6 ==> From y=x2+x-6=(x+3)(x-2)=0, the x-intercept becomes (-3,0) or (2,0), and (0,-6). From these, the negative x-intercept is (-3,0) and the negative y-intercept is (0,-6). The slope of the straight line that passes through the two points becomes $$\frac{0-(-6)}{-3-0=-2}$$ The answer is A. Answer: A _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
16 Mar 2017, 18:21
If m and n are positive integers, what is the number of factors of $$3^m7^n$$?
A. mn+m+n
B. m-n+1
C. (m-1)(n-1)
D. (m+1)(n+1)
E. mn
==> The number of factors of $$3^m7^n$$ becomes (m+1)(n+1).
The answer is D.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 20 Mar 2017, 03:44 Is a positive integer n a multiple of 12 1) n is a multiple of 6 2) n is a multiple of 24 ==> In the original condition, there is 1 variable (n) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For remainder questions, you always use direct substitution. For con 1), n=6 no, n=12 yes, hence not sufficient. For con 2), n=24,48,…, hence it is always yes and sufficient. Therefore, the answer is B. Answer: B _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
20 Mar 2017, 18:48
Is x>y>z?
1) x>y
2) y>z
==> In the original condition, there are 3 variables (x,y,z) and in order to match the number of variables to the number of equations, there must be 3 equations. Since there is 1 for con 1) and 1 for con 2), E is most likely to be the answer. By solving con 1) and con 2), from x>y>z, it is always yes and sufficient.
Therefore, the answer is C.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 22 Mar 2017, 18:33 Is a positive integer x a factor of 24? 1) 3x is a factor of 48 2) 2x is a factor of 24 ==> In the original condition, there is 1 variable (x) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For con 1), x is a factor of 16, so x=8 yes, x=16 no, hence not sufficient. For con 2), x is a factor of 12, so it is always yes, hence sufficient. Therefore, the answer is B. Answer: B _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
26 Mar 2017, 18:34
In the x-y plane there is a line K, (x/a)+(y/b)=1. What is the x-intercept of line K?
1) a=b
2) a=1
==> If you modify the original condition and the question, the x-intercept is the value of x when y=0, hence from (x/a)=1, you get x=a, so you only need to find a.
Therefore, the answer is B.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 27 Mar 2017, 18:05 If the average (arithmetic mean) of 5 consecutive multiples of 5 is 30, what is the smallest number of them? A. 5 B. 10 C. 15 D. 20 E. 25 ==> If the average of 5 consecutive multiples of 5 is 30, from 20,25,30,35,40, the smallest multiple of 5 becomes 20. Therefore, the answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 5267
GMAT 1: 800 Q59 V59
GPA: 3.82
Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
29 Mar 2017, 18:52
If Tom goes y miles in x hours, how many miles does he go per minute, in terms of x and y?
A. 60x/y
B. y/60x
C. 60y/x
D. x/60y
E. 60xy
==>You get miles:hours=y(miles):x(hours)=y(miles):60x(minutes), and from y:60x=some:1, you get some=y/60x.
The answer is B.
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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 30 Mar 2017, 18:33 m=? 1) 5 is a factor of m 2) m is a prime number ==> In the original condition, there is 1 variable (m) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. For con 1), from m=5,10…, it is not unique and not sufficient. For con 2), from m=5,7…, it is not unique and not sufficient. By solving con 1) and con 2), you get m=5, hence it is unique and sufficient. Therefore, the answer is C. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
02 Apr 2017, 18:25
If two times x is 5 greater than three times y, what is the value of y, in terms of x?
A. y=2x-5
B. y=6x-5
C. y=(x/2)-5
D. y=(x/3)-5
E. y=(2x-5)/3
==> According to the Ivy Approach, you get is:”=” and greater than:”+”, so you get 2x=5+3y. From 3y=2x-5, y=(2x-5)/3, the answer is E.
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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 04 Apr 2017, 06:49 Is x<1<y? 1) x<√x<y 2) x<√y<y ==> In the original condition, there are 2 variables (x,y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), from x<√x, you get 0<x<1, and from √y<y, you get y>1. Then, you get x<1<y, hence always yes. The answer is C. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink]
### Show Tags
06 Apr 2017, 18:30
[m] is defined as the greatest integer less than or equal to m, what is the value of [m]?
1) 1<m<2
2) |m|<1
==> In the original condition, there is 1 variable (m) and in order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer.
For con 1), from [m]=1, it is unique and sufficient.
For con 2), from -1<m<1, you get [0]=0, but from [0.1]=-1, it is not unique and not sufficient.
Therefore, the answer is A.
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"Only $79 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5267 GMAT 1: 800 Q59 V59 GPA: 3.82 Re: The Ultimate Q51 Guide [Expert Level] [#permalink] ### Show Tags 09 Apr 2017, 18:39 If 1 male, 2 females, and 1 child are to be selected at random from 8 males, 10 females, and 8 children, respectively, how many such cases are possible? A. 980 B. 1,440 C. 1,880 D. 2,480 E. 2,880 ==> From (8C1)(10C2)(8C1)=(8)(10*9/2!)(8)=2,880, the answer is E. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 3 month Online Course"
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] 09 Apr 2017, 18:39
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2018-04-23 11:47:34
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http://portanywhere.com/kf7s5388/archive.php?391fe5=spot-it-game-math
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Is there any source that describes Wall Street quotation conventions for fixed income securities (e.g. Are there an infinite set of sets that only have one element in common with each other? The game constraints (as best I can describe) are: It is a deck of 55 cards; On each card are 8 unique pictures (i.e. It turns math class into an exciting adventure filled with epic quests and unique in-game rewards. Departments. As per this site, the structure of the game is as follows: Game has 55 round playing cards. User Panel. Shape Shadow Matching ... Spot The Difference. Teach the names of 3D shapes in a FUN way by playing this 3D-shapes game! I asked myself "Is there a formula that would arrive at the right answer no matter the number of images?" Math, SolidWorks by WRadigan on April 6, 2016. It can only appear $8$ times if it to remain unique - once on the card you are holding and once over each of $7$ more cards. Roll soft foam number and operations cubes, and race to find the answer on the oversize number mat! Can you escape? I have the game myself. Windows 10 - Which services and Windows features and so on are unnecesary and can be safely disabled? C|D|G, The first batch of 4 cards is the core because: Unicycle Race Measurement. However, a balanced incomplete block design is not necessarily a Spot It game. Here's my explanation for the algorithm for making the cards. Today, people of all ages enjoy hidden object games for their creativity, puzzle-like elements, and intriguing storylines. Deal one face down to each player. some at least $9$ times. Appligent AppendPDF Pro 5.5 For example, the ��������0{Pc�1�ة������H�rd�a0 ����x��jO����_.���|���x���.V�mQ�w���ӗ is a card game consisting of g 31 cards, each decorated with 6 animals of different sizes. Here is the matrix for $p=5$, meaning there are 31 cards with 6 symbols in each: Here's Python code for producing this solution: The matrices $C_{ij}$ are all permutation matrices, that is, each row and each column have exactly one 1. r = total times/cards every icon appears in. Spot It is a great little game to throw into a carry on bag when traveling or to bring along to kill time before the food arrives at a restaurant. 123 is a fun and rewarding way for preschoolers to learn their one, two, threes and the shapes that inhabit our world. Game. Kids explore numbers 0 to 10 in this sweet math puzzle. We can describe the cards as a matrix, with a row for each card and a column for each symbol. I found several discussions about it that were all above my mental capacities for math. A Mathematical Analysis of Spot It! Through this blog, I’m on a mission to equip other teachers to create inviting and engaging classrooms where learning mathematics is fun. Your cool red car is trapped in a parking garage. Yesterday was my third week of Spot It analysis with the Oakland Math Circle. in the game they used only 55 cards, I guess simply because its a cleaner number. Spot the Number is that small and simple game that you just can't stop playing. Gifting. Well, if I didn’t have to figure things out, I could do something with that. Mathematically, that would be $1+ (3x4)$. Game. Row 4 = Symbol 3N-1 Find out with this mud-filled shape game. CenturyLink TV Spot, 'Do the Math Game Show' Submissions should come only from the actors themselves, their parent/legal guardian or casting agency. endobj endobj Then C = N2 – N + 1, Generate a matrix with N columns and C rows A game of investment - maximizing the chances of winning. There are also $n$ special cards, for each $0 \leq c \leq n-1$, containing the pairs $\{(c,x)\}$ and the singleton $\infty$. Spot It Games. 2016-05-25T12:32:36-07:00 1+ (n)(n) - n \\ I tested it and it works out every time. . {To learn more about how to play and to see all the cards that are included please see the preview} Shapes include: -square based pyramid -cylinder -pentagonal prism … 2. corporate bonds)? This game has everyone on their toes and thinking. usage per symbol to $8$, you only need $(8-1)\times8+1=57$ Symbols. I'm trying to find an algorithm for non prime p... @DPF perhaps p doesn't have to be prime. Numbers & Shapes AGES 3 TO ADULT • 2-6 PLAYERS This educational card game is truly amazing! Spot It If You Can Patterns Worksheet for Grade 1 kids to learn maths in an easy and fun way. Advertisement. with eight symbols on each card contain only 55 cards. Let's see why they have exactly one common column. Column 2: Row 1 = Symbol (Column #) Why would a company prevent their employees from selling their pre-IPO equity? Register. Reply. The same holds true for the next image. 262 0 obj Spot it! <> For example, with 3 symbols per card (n=2) the math suggests 7 symbols and 7 cards. Spot the Difference at Cool Math Games: Do you have an eagle eye? We depend on there being exactly one solution for (i_1-i_0)*j=k_1-k_0. Why isn't this working if p is not prime? Come on over and download the free printable today. 5th Grade. I catalogued them. Spot it to win! I liken it to an even more accessible version of SET. Is there an optimal strategy for this game of card? The game constraints (as best I can describe) are: It is a deck of 55 cards; On each card are 8 unique pictures (i.e. Game developers quickly realized this format could be adapted into wildly popular series including Mystery Case Files. Teach the names of 3D shapes in a FUN way by playing this 3D-shapes game! A|B|C So the first 4 cards determines 't' - the total # of unique icons in the system. Register to earn points! So, thanks to the integers modulu $p$ being a field, we have exactly one symbol which appears in each pair of cards. In the interest of link rot prevention, here are two diagrams from the article that may be of interest even to non-French speakers: I came to the conclusion it must be $57$ or more symbols the following simple way: the total number of symbols shown on all cards is $55\times8=440$. <>stream because its packaging is similar to a game called Name That! Game. Summertime Spot the Shapes. No matter the game, you always need to be the fastest to spot the identical symbol between two cards, name it out loud and then take the card, place it or discard it depending on the rules of the mini-game you're currently playing. Don't one-time recovery codes for 2FA introduce a backdoor? What is the math behind the game Spot It? Let N = number of symbols per card ... Find words down and across in this classic word-guessing game. and to better understand the math logic behind them. The goal is to match certain elements on two different cards as fast as possible. 4th Grade. Column 1: Symbol 1 for N rows 265 0 obj Like many kids, Cuz-Cuz loves to get dirty! Aish Das-Padihari says Tuesday, January 17, 2017 at 4:41 pm. Each card has eight randomly placed symbols. by WRadigan on January 2, 2013. The rules are that every card contains one and only one symbol matching every other card. Here's what I think is a more intuitive way to reach same result: let: Can you start uncovering the circles and find it before you run out of moves? In total, we have $n^2+n+1$ cards and symbols, each card containing $n+1$ symbols, and two cards intersecting at exactly one symbol. Subsets of a set with common elements between themselves, Minimal number of animals in a matching card game. before, familiarize yourselves My professor skipped me on christmas bonus payment, Judge Dredd story involving use of a device that stops time for theft, One-time estimated tax payment for windfall. Unolingo. 6th Grade. <> 07 <> Math Marks the Spot Game. The … As I understand it, the max is actually 8. what symbol appears on 10 cards? application/pdf @mirams, this answer proves mathematically that the description given in the question is. 2016-05-25T12:32:36-07:00 From the manufacturer's web site: Shake, roll and race! a card can't have 2 of the same picture) Given any 2 cards chosen from the deck, there is 1 and only 1 matching picture. 1st Grade. Row 3: Previous column Row 3 plus 1 endobj all possible cards (every line is a possible card): Although the age limit is 7 and up, kids as young as 5 years old can play this game … The story of the hugely popular game "Spot It!" Both games feature similar gameplay, with players trying to spot matches as quickly as possible. Durable vinyl mat features all the answers to addition and subtraction problems that use the numbers 1-6. Spot the Difference at Cool Math Games: Do you have an eagle eye? Learning how to quickly spot patterns and apply rules By using Math Games, all these skills can be practiced with minimal effort and for free, as kids have fun playing entertaining games! x��X�o�6��A��4I�� Quick! Math. Please include at least one social/website link containing a recent photo of the actor. I was skeptical but there really and truly always is a match! 263 0 obj By Deepu Sengupta Introduction Spot It! A column is described by $j$ and $l$. It only takes a minute to sign up. An almost matching construction is as follows. Product Title Pop the Pig Game - Family Game by Goliath Games (305 ... Average rating: 4.2 out of 5 stars, based on 503 reviews 503 ratings Current Price $12.07$ 12 . You can also use the "Play Another" calendar marker to go back in time and solve past jumbles, or review your high scores. Word wall cards are included as well! 257 0 obj This is the simplest formula to arrive at the number of both individual symbols and total number of cards required to display them (these are the same). <> This educational game is now abandonware and is set in a math / logic. I am left curious why the makers of Spot It decided to take this approach. Refine by | Top Brands. Preschool. The Mathematics of Spot It endobj 12 0 obj Can we calculate mean of absolute value of a random variable analytically? While students play, our adaptive algorithm delivers math questions that keep them engaged without getting frustrated. ... so come back again tomorrow for a brand new challenge. Spot It is a circular card game where multiple mini games can be played depending on age groups or amount of people playing. Is there a difference between a tie-breaker and a regular vote? This durable vinyl mat features all the answers (sums and differences) to addition and subtraction problems that use numbers 1 through 6. During the holidays, you might have heard of the popular party game called Spot-It. that I co-designed, though the games are quite different. These pictures are worth a thousand words, in whatever language you speak. 11 0 obj <> Games vary from spotting the one object that is on both cards to more advanced games. Then I just kinda rearranged a little: How can I improve after 10+ years of chess? Over Thanksgiving weekend I was introduced to a new game: SpotIt. Math Marks the Spot Game. If both rows have a 1 in the column, it means that: Now, since $p$ is prime, and $i_0 \ne i_1$, we can solve this equation, and get a single result for $j$, namely. Gross Motor - Whether practicing alone or playing on teams, Math Marks the Spot Game gets students moving both mentally and physically. Nov 26, 2015 - 3D Shapes! Let C = the total number of symbols as well as the total number of cards to be generated. <> Most versions of the Spot It! The Mind-Bending Math Behind Spot It!, the Beloved Family Card Game The simple matching game has some deceptively complex mathematics behind the scenes The card game Spot It! Repeat above rows N-2 times Finding numbers was never so much fun and so intense. is a game containing a deck of 55 cards with eight symbols printed on each card. Careful counting reveals that there are 31 total symbols as well, and that each symbol appears exactly six times in the deck. My grandkids love The Tower. Column 3: Row 1: Symbol (Column #) This would have also worked with regular diagonals, but this way the matrix is symmetric. How does Dobble/Spot it only work with 55 cards? One approach is to lay out all the cards from the Spot It! Spot it to win! we filled the other positions with new icons (E, F & G). Dive into an engaging game experience tailored to your individual skill level. If you spot it, say the name of the symbol, take the three cards, and then deal three more in their place. I. n = total possible cards in the system Spot it! <> In this game your students' shape recognition skills will be put to the test as they try to pick out shapes they might see on a hot summer's day spent at the pool. <> (At least that was the case for us.) Circular motion: is there another vector-based proof for high school students? Why is it impossible to measure position and momentum at the same time with arbitrary precision? Then, I tried to tie these observations together. To play Spot It!, you turn over two cards and find the matching symbols. Your cool red car is trapped in a crowded lot. Apparently they favor the clover leaf over the flower, maple leaf, or snow man. Thus, with $55$ cards, the minimum number of different symbols is $57$. Oct 27, 2014 - Each of the 55 Spot-It cards has "one and only one" common element with every other card. Encourages physical activity in a fast-paced game as kids jump, touch or place markers on the answer. \end{eqnarray*}$$. Can you get to the exit? The first row can be described by i_0 and k_0 (k is the row in the C_{i_0}j matrices), and the second row by i_1 and k_1. Look at two photos and see if you can find all 10 differences between them. I was very happy before I got yelled at by my wife for taking so long on the computer. You might find those helpful in your job, too. How is this possible and what is the math needed to create such a game? <> Toys. That's 1+12 or 13 cards. This DIGITAL math game works in Google Forms!The cards auto-shuffle! Race others to gather or dump your cards. Big groups can play, the instructions are incredibly simple, and there are a ton of fun variations that you can try. There are 55 cards, 57 objects, 8 per card. Assuming eight images per card as are found in this game, this image can only be found 8 times, once on the card you're holding and 7 more times. But does this tell you the fewest number of cards and symbols necessary? ), Furthermore, according to the French article, The logic breaks down in your first sentence following the colon. ... in order for Dobble, a card game with 57 pictures and 8 pictures on each card to have only one picture in common between cards, there should be 57 cards. Game. t = total number of unique icons in the system 2nd Grade. I’m Sarah Carter, a high school math teacher who passionately believes math equals love. Spot it Jr.! A|D|E Junior deck. I derived this formula logically but not necessarily mathematically as follows: I picked a random card and focused on a single image. is a fun game that has some interesting math behind it. . In your case n=7 and so the number of cards and symbols should be 7^2+7+1 = 57. Match & Learn consists of two Spot it! I am curious to the field of mathematics. In this game your students' shape recognition skills will be put to the test as they try to pick out shapes they might see on a hot summer's day spent at the pool. <> Prodigy Math Game doesn’t just give your students a way to practice math. This is the same logic I used to solve the problem! I have lots of math games on my site. Games, Math, Python. Description of Math Blaster Ages 7-9 - In Search Of Spot Windows. endobj 14 0 obj Roll soft foam number and operations cubes, and race to find the answer on the oversize number mat! In 1999, Davidson & Associates, Inc. publishes Math Blaster Ages 7-9 - In Search Of Spot on Windows. Get ready to Spot it! Oct 27, 2014 - Each of the 55 Spot-It cards has "one and only one" common element with every other card. As 57\times8=456, this also exceeds the #of symbols shown thus being a valid solution. Numbers & Shapes features six different symbols, with their sizes varying from one card to another. endobj 259 0 obj Now the claim is that between any two cards there is always 1 and only 1 matching picture. Since we come from different rows in the large matrix, i_0 \ne i_1. This educational game is truly amazing! Here's the math (& Python code) behind this awesome game. 3 0 obj 1+ n^2 - n \\ Each player looks at a pair of cards and tries to find a symbol that is on both cards. 6 0 obj Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 2016-05-25T12:32:36-07:00 comes in a small circular tin containing 55 cards. More than 30 individuals attended the last Rubber City Math Teachers’ Circle of the school year. Row 2 = Symbol N+1 began in19th-century Britain, a period that introduced many mathematical "rock stars." There are 57 symbols total. My old friends Chris and Paul have twin girls my son's age, and our visits to their backwoods home are always a rich experience of outdoor play, good healthy food, and deep conversation. But the game has 55 cards and only 50 different symbols! Play Hotspot hacked and unblocked: There's a hotspot hidden somewhere on this grid. So this means there are two more possible cards with the given S and C of the game (the deck only has 55 cards). As they search for matches, little ones will visually and verbally identify numbers and shapes, the building blocks of math. Summertime Spot the Shapes.$$\begin{eqnarray*}1+ (n-1)(n) \\ They are all just a cyclically shifted reverse-diagonal, with the diagonal shifted by $ij \mod p$. Encourages physical activity in a fast-paced game as kids jump, touch or place markers on the answer. Prince 9.0 rev 5 (www.princexml.com) Word wall cards are included as well! Can you mathematically demonstrate the minimum number of cards (C) and symbols (S) you need based on the number of symbols per card (N)? Etc. Deal 9 cards face up in the middle of the table. Mini Games. Basic Concepts - Big time play for big time fun! The answer is yes. This is my largest Spot It & Steal It pack yet! Any two cards have exactly one symbol in common. Repeat above rows N-2 times Play Parking Lot at Math Playground! <> Kindergarten. Currently there are no best players! Dive into an engaging game experience tailored to your individual skill level. What is remarkable ( mathematically ) is that any two cards chosen at random from the pack will have one and only one matching symbol . Row 5 = Symbol 4N-2 If you are into addictive number puzzle games, cool math games, brain teasers or just fun games were you compete with others, then Spot the Number is the game for you. It turns out that for each prime number $p$ we can create such a solution, with $p^2+p+1$ cards and a total of $p^2+p+1$ symbols, with $p+1$ symbols on each card. Which symbol is on both cards? endobj Was there an anomaly during SN8's ascent which later led to the crash? The makers of the game claims that there will always be a match. Each card has eight images printed on it. My new job came with a pay raise that is being rescinded, A Merge Sort Implementation for efficiency. uuid:3ff16279-a2fd-11b2-0a00-782dad000000 In that case, we would expect the total number of cards to be 72+ 7 + 1 = 57. One super special card contains all $n+1$ singletons. How to write complex time signature that would be confused for compound (triplet) time? 2 0 obj We took the first 3 icons from first card (A|B|C). Anyway, it's a pattern recognition board game for children (but also great for all ages) wh Algorithm and JavaScript function for Dobble (Spot it!) <> If it was $50$ different symbols only, each must be shown $440:50=8.8$ times, i.e. The game is a simple little card game called Spot It! endstream There are occurrences when a balanced incomplete block design has λ=1 and still is not a Spot It game. The interesting thing to me is that each object does not appear in equal frequency with others ... the minimum is 6, max 10, and mean 7.719. Finally, the super special card intersects the rest at a singleton. Row 3 = Symbol 2N Sort, stack and play all 52 cards to solve today's FreeCell challenge. Students build math vocabulary as well as the ability to identify fractions correctly... all in an engaging way! Then, I ran the same logic and considered a card with only $4$ images. The tiny durable tin contains 30 cards, each featuring a variety of 6 colorful symbols. How quickly can you do it? Throwing the dice, processing what numbers are rolled, and responding with your body combines kinesthetic and cognitive skills to reinforce beginning math concepts. The tiny durable tin contains 30 cards, each featuring a variety of 6 colorful symbols. endobj en.wikipedia.org/wiki/Projective_plane#Combinatorial_definition, images.math.cnrs.fr/Dobble-et-la-geometrie-finie.html?lang=fr, Paige L.J., Wexler Ch., A Canonical Form for Incidence Matrices of Projective Planes, What is the algorithm to generate the cards in the game “Dobble” ( known as “Spot it” in the USA )?h, Dobble card game - mathematical background. <> 3rd Grade. N squared minus N + 1 is the correct formula for calculating both the number of images and the number of cards. of cards to $56\times7:8=49$. Two cards with different $a$s intersect at the unique solution to $a_1x+b_1 = a_2x+b_2 \pmod{n}$. The celebrated Ray-Chaudhuri–Wilson theorem states that $C \leq S$, contradicting your numbers. a card can't have 2 of the same picture) Given any 2 cards chosen from the deck, there is 1 and only 1 matching picture. All the arithmetic is done modulu $p$. I just purchased the game Spot It. Symbol N for N-1 rows We now see that it is impossible to introduce any new icon to the system, as it will not be able to have exactly 1 common icon with EVERY other card. Test your skills with a new puzzle every day! endobj How to play spot it! Daily FreeCell. The object of the game is to find a matching symbol between two cards. A Spot It game is a balanced incomplete block design with λ=1 along with the additional principle of only one matching symbol between two cards/blocks. I noticed the trend. Symbol 2 for N-1 rows endobj Gift eligible. Column 4: Same as Column 3 Up and Down Words. But, the fewest numbers seem to be 6 symbols and 4 cards: symbols [a,b,c,d,e,f]; cards [abc, cde, eaf, bdf]. 267 0 obj Durable vinyl mat features all the answers to addition and subtraction problems that use the numbers 1-6. That's $1+ (n-1)(n)$ if $n$ is the number of images. There are enough cards to play the game as a class ... OR you can use the cards to create a few smaller games for math centers. <> ... Spot the Difference. If you’ve never played Spot it! Makes sense! We got back most of the kids from the first week, and played a few rounds again to get warmed up. n^2 - n + 1 Row 2: Previous column Row 2 plus 1 $56$), usage per symbol would require reduction to 7 per (each individual) symbol, which would limit the no. through row 2N-1 It is quite quite easy to see that any two rows have exactly one column with a common 1, except for two rows that come from different rows in the large matrix of matrices. Math. A firm favourite with toddlers and pre-schoolers, Eric Hill's loveable puppy Spot introduces children to new experiences through friendship and play How to remove minor ticks from "Framed" plots and overlay two plots? That's $1+(7\times8)$ or $1+56 = 57.$. Two cards with different a s intersect at the unique solution to a 1 x + b 1 = a 2 x + b 2 ( mod n). <>/MediaBox[0 0 432 648]/Parent 10 0 R/Resources<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]>>/Rotate 0/StructParents 12/Tabs/S/Type/Page>> A cell will have a 1 if the card corresponding to the row has the symbol corresponding to the column, and 0 otherwise. Each card has 8 symbols and there is only 1 match between any two cards. The most fascinating feature of this game is any two cards selected will always have ONE (and only one) matching symbol to be found on both cards. 1 0 obj If we reduce the max. Each image appears once on the card you're holding and requires $7$ more cards. Clearly two cards with the same $a$ intersect only at the singleton. Alphabet. endobj Our universe, of size $n^2+n+1$, consists of pairs of numbers in $\{0,\ldots,n-1\}$ plus $n+1$ singletons $\{0,1,\ldots,n-1,\infty\}$ ("points at infinity"). Spot It earns our highest recommendation and has become a Rich Classic. 260 0 obj Here's an article (in French) that aims to explain the mathematics behind the game to a wide audience. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. . It wouldn't make a very satisfying Spot It game, though. Submissions without photos may not be accepted. Spot the Difference - Play it now at CoolmathGames.com The game was a lot of fun, but I got to thinking afterwards, "How can there be only 1 match between any two cards?" When p= 8 (as in the game) → t=n=57. How to Play. But the versions of Spot It! 261 0 obj Symbols include the numbers 1 through 9 and basic geometric shapes such as squares and circles. Additionally, Math Games offers free printable worksheets (with grading keys included), downloadable apps, and a mobile-compatible website. 266 0 obj Etc. So, you need the 1 card you are holding and 7 more per image. Spot The Difference. Spot it! There is always one, and only one, matching symbol between any two cards. There is always one, and only one, matching symbol between any two cards. To be able to use fewer symbols (e.g. There may be other known results in the area. When p = 3 → t = n = 7 This oversized mat game will have your kids begging to practice their math. And circles so much fun and so the first week, and regular! Safely disabled needed to create more possible cards, each with six symbols the column, and one. Each object for each symbol appears exactly six times in the first 3 icons from first (. Get $l$ are the specifications for generating each card of Spot-It you might have of! Parking garage take this approach It analysis with the diagonal shifted by $ij \mod p$ are holding requires. complete '' set 1 kids to learn their one, matching symbol between any two cards and symbols be. Necessarily mathematically as follows: I picked a random variable analytically 3D shapes in a fast-paced game kids. N'T this working if p is not prime you ca n't find the matching symbols answers to addition subtraction! Years of chess epic quests and unique in-game rewards - big time fun, and race find! It impossible to measure position and momentum at the same time with arbitrary precision for kids the Lot... For compound ( triplet ) time • 2-6 players this educational game now... This URL into your RSS reader spot it game math \times8+1=57 $symbols trying to Spot matches as as. Both, kids and adults can enjoy has some interesting math behind It note. With grading keys included ), Furthermore, according to the crash math suggests 7 symbols and 7 cards of! Practice their math with the diagonal shifted by$ j $and$ l $17, at!, or snow man Ray-Chaudhury-Wilson theorem provides only one symbol with every other.... Argues that gender and sexuality aren ’ t just give your students a way to practice their.. Just a cyclically shifted reverse-diagonal, with 3 symbols per card ( A|B|C ) ( e.g French! Shapes features six different symbols is$ 57 $of moves only 55.... Require one base card and a column is described by$ j and... Not necessarily mathematically as follows: game has everyone on their toes and thinking for both... Classic word-guessing game 1 if the card you are holding and 7 cards the specification generating... Cool red car is trapped in a Parking garage played a few rounds again to get $l.. Realized this format could be adapted into wildly popular series including Mystery case Files, and... With different$ a $s intersect at the same logic and reasoning, all symbols have an eye...$ more cards mathematically, that would be $7^2+7+1 = 57 appears exactly six times in the deck the! This URL into your RSS reader now, but there should be$ 1+ ( 3x4 ) if... 'D need $( 8-1 ) \times9+1=64$ different symbols is ! Recommendation and has become a Rich classic and may challenge the adults even more accessible version of set back. Card, so It seems that they could be equivalent to theq= 7 projective plane BOARD fast. Absolute value of a set of exactly three cards that all have a 1 if the card corresponding the... When p = 3 → t = n = 7 when p= 8 ( as in the.. Middle of the kids from the manufacturer 's web site: Shake, roll and race you. This game has everyone on their toes and thinking, 2014 - each of the Spot game. Arbitrary precision fun variations that you just ca n't always divide - there 's a Hotspot somewhere.: match 3 ( like set ) math / logic in the game has everyone on their and... Per card maths quickly you start uncovering the circles and find It before you run out of moves simple... Player looks at a pair of cards and find the matching symbol two. A total of 50 different symbols only, each must be prime with! On age groups or amount of people playing a match experience tailored to your individual skill level worksheet! For compound ( triplet ) time manufacturer 's web site: Shake, roll and race rules are that card... As Dobble ( Spot-It! n't find the matching symbols contains 30 cards, objects... Popular party game called Spot It game between two cards there is exactly one symbol matching every other card to! → t = n = 7 when p= 8 ( as in the area play... Balanced incomplete block design has λ=1 and still is not prime you ca n't the! The icons in the game is now abandonware and is set in a math /.! From selling their pre-IPO equity engaging game experience tailored to your individual skill.! With one common column symbols in each card shares exactly one solution for ( i_1-i_0 ) *.! I currently teach Algebra 2, It does n't say that It must be prime matter the number cards! Physical activity in a fun card game spot it game math multiple mini games can be played depending on groups... Math and learn new skills at the right answer no matter the number of symbols! Card of Spot-It describes Wall Street quotation conventions for fixed income securities ( e.g simple It. First 4 cards determines 't ' - the total number of symbols/card cards are merely remaining... 7 more per image common with each other into an engaging game experience tailored to your individual skill level 7^2+7+1... Truly always is a classic game of investment - maximizing the chances of winning licensed under cc by-sa theorem... 6 animals of different sizes other card mobile-compatible website the fourth icon D to each one of game... A backdoor need $( 8-1 ) \times9+1=64$ different symbols is . 4 $images your case$ n=7 $and$ 3 $additional cards per image It is for! Play Hotspot hacked and unblocked: there 's no x^-1 currently teach Algebra 2,,. Ages 7-9 - in search of Spot Windows so come back again tomorrow a. Students a way to practice math to 40 of 1,000+ products earns our spot it game math and... Week of Spot It game symbol, all the spot it game math to addition and subtraction problems use.$ p $contradicting your numbers the icons in the area I guess simply its! Done modulu$ p $440:50=8.8$ times, i.e six times in system... Derived this formula logically but not necessarily a Spot It! n=2 ) the math needed to create a. For math a company prevent their employees from selling their pre-IPO equity s! Fast, fun game that you just ca n't stop playing hidden somewhere on this grid mirams! 2N-1 Repeat above rows N-2 times column 4: same as column 3 least that was case... Each decorated with 6 animals of different symbols through the deck mini-games in which all play! Somewhere on this grid shapes that inhabit our world element with every other card of each object each... In all to a game containing a deck of 55 cards, we would expect the total of! The fourth icon D to each one of the 55 Spot-It cards ! I ca n't stop playing ) to addition and subtraction problems that numbers! To the row has the symbol corresponding to the crash the object the!... all in an easy and fun way by playing this 3D-shapes game of. Lot at math Playground round playing cards improve after 10+ years of chess require one base card and ... 57. $symbols only, each decorated with 6 animals of different symbols random card a... Other symbols would need to be able to use fewer symbols ( e.g times, i.e touch place. As quickly as possible image appears once on the answer on the oversize number mat n't find the on! Would require one base card and$ l spot it game math addition and subtraction problems that use the numbers 1 6! Times, i.e math Circle with 6 animals of different sizes would have worked! Provides only one '' common element with every other card games on site! $intersect only at the same time of card learn new skills at the same time create a... Adventure filled with epic quests and unique in-game rewards ) behind this awesome game over cards. Different, i.e and a column is described by$ ij \mod p $still.! Understand It, the building blocks of math every other card 31 total symbols as as... Game changes every time operation dice and race to find the matching symbols two... A cleaner number with eight symbols on each card a period that introduced mathematical... Each one of the icons in the question is p... @ perhaps! It pack yet the Dobble game has everyone on their toes and thinking number... Personality traits n squared minus n + 1$ where $n$ is the of! It & Steal It pack yet multiple mini games can be played depending on groups... I derived this formula logically but not necessarily mathematically as follows: I picked a variable! Each one of the 55 Spot-It cards has one and only match. Series of fast, fun game called Spot-It 50 different symbols in the is! Rows N-2 times column 4: same as column 3 have an equal probability of being the matching between! By $j$ and $l$ see why they have exactly one common column operation dice and to. Web site: Shake, roll and race the 55 Spot-It cards has one only... Match certain elements on two different cards as a matrix, $i_0 \ne i_1$ this could... Out of moves that aims to explain the mathematics behind the game Spot It earns our recommendation.
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2021-06-18 02:28:34
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https://tex.stackexchange.com/questions/473373/units-units-in-fractions
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# Units, units in fractions
I'm new to latex and trying to figure out the notation.
I'm trying to type out conversions, like (for example) 5 lbf * (4.4482 N / 1 lbf)= etc etc.
I'm not sure how to make the units show as units, or, at least, not as italics. I've seen how you can use \si{N}, but lbf isn't an SI unit, and I don't know how to combine \si with \frac.
• Hi, welcome. siunitx lets you declare new units, see e.g. tex.stackexchange.com/questions/27614/… Then if you've defined an \lbf unit, you can do e.g. \SI{4.4482}{\newton\per\lbf}. Note capital \SI which is for a number with a unit. – Torbjørn T. Feb 4 at 20:19
• The siunitx package allows you to define your own units and typesets them in a consistent way. – Bernard Feb 4 at 20:20
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2019-10-21 14:47:16
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https://cran.r-project.org/web/packages/imager/vignettes/gettingstarted.html
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imager contains a large array of functions for working with image data, with most of these functions coming from the CImg library by David Tschumperlé. This vignette is just a short tutorial, you’ll find more information and examples on the website. Each function in the package is documented and comes with examples, so have a look at package documentation as well.
imager comes with an example picture of boats. Let’s have a look:
library(imager)
plot(boats)
Note the y axis running downwards: the origin is at the top-left corner, which is the traditional coordinate system for images. imager uses this coordinate system consistently. Image data has class “cimg”:
class(boats)
[1] "cimg" "imager_array" "numeric"
and we can get some basic info by typing:
boats
Image. Width: 256 pix Height: 384 pix Depth: 1 Colour channels: 3
Width and height should be self-explanatory. Depth is how many frames the image has: if depth > 1 then the image is actually a video. Boats has three colour channels, the usual RGB. A grayscale version of boats would have only one:
grayscale(boats)
Image. Width: 256 pix Height: 384 pix Depth: 1 Colour channels: 1
An object of class cimg is actually just a thin interface over a regular 4D array:
dim(boats)
[1] 256 384 1 3
We’ll see below how images are stored exactly. For most intents and purposes, they behave like regular arrays, meaning the usual arithmetic operations work:
log(boats)+3*sqrt(boats)
Image. Width: 256 pix Height: 384 pix Depth: 1 Colour channels: 3
mean(boats)
[1] 0.5089061
sd(boats)
[1] 0.144797
Now you might wonder why the following two images look exactly the same:
layout(t(1:2))
plot(boats)
plot(boats/2)
That’s because the plot function automatically rescales the image data so that the whole range of colour values is used. There are two reasons why that’s the default behaviour:
1. There’s no agreed-upon standard for how RGB values should be scaled. Some software, like CImg, uses a range of 0-255 (dark to light), other, like R’s rgb function, uses a 0-1 range.
2. Often it’s just more convenient to work with a zero-mean image, which means having negative values.
If you don’t want imager to rescale the colours automatically, set rescale to FALSE, but now imager will want values that are in the $$[0,1]$$ range.
layout(t(1:2))
plot(boats,rescale=FALSE)
plot(boats/2,rescale=FALSE)
If you’d like tighter control over how imager converts pixel values into colours, you can specify a colour scale. R likes its colours defined as hex codes, like so:
rgb(0,1,0)
[1] "#00FF00"
The function rgb is a colour scale, i.e., it takes pixel values and returns colours. We can define an alternative colour scale that swaps the red and green values:
cscale <- function(r,g,b) rgb(g,r,b)
plot(boats,colourscale=cscale,rescale=FALSE)
In grayscale images pixels have only one value, so that the colour map is simpler: it takes a single value and returns a colour. In the next example we convert the image to grayscale
#Map grayscale values to blue
cscale <- function(v) rgb(0,0,v)
grayscale(boats) %>% plot(colourscale=cscale,rescale=FALSE)
The scales package has a few handy functions for creating colour scales, for example by interpolating a gradient:
cscale <- scales::gradient_n_pal(c("red","purple","lightblue"),c(0,.5,1))
#cscale is now a function returning colour codes
cscale(0)
[1] "#FF0000"
grayscale(boats) %>% plot(colourscale=cscale,rescale=FALSE)
See the documentation for plot.cimg and as.raster.cimg for more information and examples.
The next thing you’ll probably want to be doing is to load an image, which can be done using load.image. imager ships with another example image, which is stored somewhere in your R library. We find out where using system.file
fpath <- system.file('extdata/parrots.png',package='imager')
parrots <- load.image(fpath)
plot(parrots)
imager supports JPEG, PNG, TIFF and BMP natively - for other formats you’ll need to install ImageMagick.
# 2 Example 1: Histogram equalisation
Histogram equalisation is a textbook example of a contrast-enhancing filter. It’s also a good topic for an introduction to what you can do with imager.
Image histograms are just histogram of pixel values, which are of course pretty easy to obtain in R:
grayscale(boats) %>% hist(main="Luminance values in boats picture")
Since images are stored essentially as arrays, here we’re just using R’s regular hist function, which treats our array as a vector of values. If we wanted to look only at the red channel, we could use:
R(boats) %>% hist(main="Red channel values in boats picture")
#Equivalently:
#channel(boats,1) %>% hist(main="Red channel values in boats picture")
Another approach is to turn the image into a data.frame, and use ggplot to view all channels at once:
library(ggplot2)
bdf <- as.data.frame(boats)
head(bdf,3)
x y cc value
1 1 1 1 0.3882353
2 2 1 1 0.3858633
3 3 1 1 0.3849406
bdf <- plyr::mutate(bdf,channel=factor(cc,labels=c('R','G','B')))
ggplot(bdf,aes(value,col=channel))+geom_histogram(bins=30)+facet_wrap(~ channel)
What we immediately see from these histograms is that the middle values are in a sense over-used: there’s very few pixels with high or low values. Histogram equalisation solves the problem by making histograms flat: each pixel’s value is replaced by its rank, which is equivalent to running the data through their empirical cdf.
As an illustration of what this does, see the following example:
x <- rnorm(100)
layout(t(1:2))
hist(x,main="Histogram of x")
f <- ecdf(x)
hist(f(x),main="Histogram of ecdf(x)")
We can apply it directly to images as follows:
boats.g <- grayscale(boats)
f <- ecdf(boats.g)
plot(f,main="Empirical CDF of luminance values")
Again we’re using a standard R function (ecdf), which returns another function corresponding to the ECDF of luminance values in boats.g.
If we run the pixel data back through f we get a flat histogram:
f(boats.g) %>% hist(main="Transformed luminance values")
Now the only problem is that ecdf is base R, and unaware of our cimg objects. The function f took an image and returned a vector:
f(boats.g) %>% str
num [1:98304] 0.171 0.165 0.163 0.164 0.165 ...
If we wish to get an image back we can just use as.cimg:
f(boats.g) %>% as.cimg(dim=dim(boats.g)) %>% plot(main="With histogram equalisation")
So far we’ve run this on a grayscale image. If we want to do this on RGB data, we need to run the equalisation separately in each channel. imager enables this using its split-apply-combine tricks:
#Hist. equalisation for grayscale
hist.eq <- function(im) as.cimg(ecdf(im)(im),dim=dim(im))
#Split across colour channels,
cn <- imsplit(boats,"c")
cn #we now have a list of images
Image list of size 3
cn.eq <- llply(cn,hist.eq) #run hist.eq on each
imappend(cn.eq,"c") %>% plot(main="All channels equalised") #recombine and plot
There’s even a one-liner to do this:
iiply(boats,"c",hist.eq)
Image. Width: 256 pix Height: 384 pix Depth: 1 Colour channels: 3
We can use it to check that all channels have been properly normalised:
iiply(boats,"c",hist.eq) %>% as.data.frame %>% ggplot(aes(value))+geom_histogram(bins=30)+facet_wrap(~ cc)
Our trick worked.
# 3 Example 2: Edge detection
Edge detection relies on image gradients, which imager returns via:
gr <- imgradient(boats.g,"xy")
gr
Image list of size 2
plot(gr,layout="row")
The object “gr” is an image list, with two components, one for the gradient along $$x$$, the other for the gradient along $$y$$. “gr” is an object with class “imlist”, which is just a list of images but comes with a few convenience functions (for example, a plotting function as used above).
To be more specific, noting $$I(x,y)$$ the image intensity at location $$x,y$$, what imager returns is an approximation of: $\frac{\partial}{\partial x}I$ in the first panel and: $\frac{\partial}{\partial y}I$ in the second.
The magnitude of the gradients thus tell us how fast the image changes around a certain point. Image edges correspond to abrubt changes in the image, and so it’s reasonable to estimate their location based on the norm of the gradient $\sqrt{\left(\frac{\partial}{\partial x}I\right)^{2}+\left(\frac{\partial}{\partial y}I\right)^{2}}$
In imager:
dx <- imgradient(boats.g,"x")
plot(grad.mag,main="Gradient magnitude")
imgradient(boats.g,"xy") %>% enorm %>% plot(main="Gradient magnitude (again)")
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2017-03-24 04:18:35
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https://xianblog.wordpress.com/tag/extremes/
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## Adam Ondra completes a first 9c
Posted in Kids, Mountains, pictures with tags , , , , , , on September 30, 2017 by xi'an
In my office hangs this poster of Adam Ondra climbing at sunset an impressive overhang with a little Czech town at the foot of the cliff, already in the shade. Impressive because of the view and of the climb which at 7c is a whole grade (and then some) beyond my reach. But now Ondra has managed to climb the first 9c in the world, which is universes beyond the impressive and beyond the fathomable, with passages only manageable feet first. Which actually makes a lot of sense, the way he explains it. The route is currently called Project Hard and is located in Hanshelleren Cave, Flatanger, Norway.
## Texan black swan
Posted in Books, pictures with tags , , , , , , , , on September 12, 2017 by xi'an
“Un événement improbable aux conséquences d’autant plus désastreuses que l’on ne s’y est pas préparé.”
This weekend, there was a short article in Le Monde about the Harvey storm as a Texan illustration of Taleb’s black swan. An analysis that would imply every extreme event like this “once-in-a-thousand year” event (?) can be called a black swan… “An improbable event with catastrophic consequences, the more because it had not been provisioned”, as the above quote translates. Ironically, there is another article in the same journal, about the catastrophe being “ordinary” and “not unexpected”! While such massive floods are indeed impacting a huge number of people and companies, because the storm happened to pour an unusual amount of rain right on top of Houston, they indeed remain within the predictable and not so improbable in terms of the amount of water deposited in the area and in terms of damages, given the amount and style of construction over flood plains. For instance, Houston is less than 50 feet above sea level, has fairly old drainage and pipe systems, and lacks a zoning code. With mostly one or two-story high buildings rather than higher rises. (Incidentally, I appreciated the juxtaposition of the article with the add for Le Monde des Religions and its picture of a devilesque black goat!)
## Statistics month in Marseilles (CIRM)
Posted in Books, Kids, Mountains, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , , on June 24, 2015 by xi'an
Next February, the fabulous Centre International de Recherche en Mathématiques (CIRM) in Marseilles, France, will hold a Statistics month, with the following programme over five weeks
Each week will see minicourses of a few hours (2-3) and advanced talks, leaving time for interactions and collaborations. (I will give one of those minicourses on Bayesian foundations.) The scientific organisers of the B’ week are Gilles Celeux and Nicolas Chopin.
The CIRM is a wonderful meeting place, in the mountains between Marseilles and Cassis, with many trails to walk and run, and hundreds of fantastic climbing routes in the Calanques at all levels. (In February, the sea is too cold to contemplate swimming. The good side is that it is not too warm to climb and the risk of bush fire is very low!) We stayed there with Jean-Michel Marin a few years ago when preparing Bayesian Essentials. The maths and stats library is well-provided, with permanent access for quiet working sessions. This is the French version of the equally fantastic German Mathematik Forschungsinstitut Oberwolfach. There will be financial support available from the supporting societies and research bodies, at least for young participants and the costs if any are low, for excellent food and excellent lodging. Definitely not a scam conference!
## dynamic mixtures [at NBBC15]
Posted in R, Statistics with tags , , , , , , , , , , , , on June 18, 2015 by xi'an
A funny coincidence: as I was sitting next to Arnoldo Frigessi at the NBBC15 conference, I came upon a new question on Cross Validated about a dynamic mixture model he had developed in 2002 with Olga Haug and Håvård Rue [whom I also saw last week in Valencià]. The dynamic mixture model they proposed replaces the standard weights in the mixture with cumulative distribution functions, hence the term dynamic. Here is the version used in their paper (x>0)
$(1-w_{\mu,\tau}(x))f_{\beta,\lambda}(x)+w_{\mu,\tau}(x)g_{\epsilon,\sigma}(x)$
where f is a Weibull density, g a generalised Pareto density, and w is the cdf of a Cauchy distribution [all distributions being endowed with standard parameters]. While the above object is not a mixture of a generalised Pareto and of a Weibull distributions (instead, it is a mixture of two non-standard distributions with unknown weights), it is close to the Weibull when x is near zero and ends up with the Pareto tail (when x is large). The question was about simulating from this distribution and, while an answer was in the paper, I replied on Cross Validated with an alternative accept-reject proposal and with a somewhat (if mildly) non-standard MCMC implementation enjoying a much higher acceptance rate and the same fit.
## Advances in scalable Bayesian computation [day #4]
Posted in Books, Mountains, pictures, R, Statistics, University life with tags , , , , , , , , , , , , , , , , , on March 7, 2014 by xi'an
Final day of our workshop Advances in Scalable Bayesian Computation already, since tomorrow morning is an open research time ½ day! Another “perfect day in paradise”, with the Banff Centre campus covered by a fine snow blanket, still falling…, and making work in an office of BIRS a dream-like moment.
Still looking for a daily theme, parallelisation could be the right candidate, even though other talks this week went into parallelisation issues, incl. Steve’s talk yesterday. Indeed, Anthony Lee gave a talk this morning on interactive sequential Monte Carlo, where he motivated the setting by a formal parallel structure. Then, Darren Wilkinson surveyed the parallelisation issues in Monte Carlo, MCMC, SMC and ABC settings, before arguing in favour of a functional language called Scala. (Neat entries to those topics can be found on Darren’s blog.) And in the afternoon session, Sylvia Frühwirth-Schnatter exposed her approach to the (embarrassingly) parallel problem, in the spirit of Steve’s , David Dunson’s and Scott’s (a paper posted on the day I arrived in Chamonix and hence I missed!). There was plenty to learn from that talk (do not miss the Yin-Yang moment at 25 mn!), but it also helped me to break a difficulty I had with the consensus Bayes representation for two weeks (more on that later!). And, even though Marc Suchard mostly talked about flu and trees in a very pleasant and broad talk, he also had a slide on parallelisation to fit the theme! Although unrelated with parallelism, Nicolas Chopin’s talk was on sequential quasi-Monte Carlo algorithms: while I had heard previous versions of this talk in Chamonix and BigMC, I found it full of exciting stuff. And it clearly got the room truly puzzled by this possibility, in a positive way! Similarly, Alex Lenkoski spoke about extreme rain events in Norway with no trace of parallelism, but the general idea behind the examples was to question the notion of the calibrated Bayesian (with possible connections with the cut models).
This has been a wonderful week and I am sure the participants got as much as I did from the talks and the informal exchanges. Thanks to BIRS for the sponsorship and the superb organisation of the week (and to the Banff Centre for providing such a paradisical environment). I feel very privileged to have benefited from this support, even though I deadly hope to be back in Banff within a few years.
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2020-10-30 11:21:00
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http://hal.in2p3.fr/view_by_stamp.php?label=IN2P3&langue=fr&action_todo=view&id=in2p3-00706978&version=1
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HAL : in2p3-00706978, version 1
We present a catalog containing 290 LMC and 590 SMC Cepheids which have been obtained using the two 4k $\times$ 8k CCD cameras of the EROS~2 microlensing survey. The Cepheids were selected from 1,134,000 and 504,000 stars in the central regions of the LMC and SMC respectively, that were monitored over 150 nights between October 1996 and February 1997, at a rate of one measurement every night. For each Cepheid the light curves, period, magnitudes in the EROS~2 filter system, Fourier coefficients, J2000 coordinates and cross-identifications with objects referenced in the CDS Simbad database are presented. Finding charts of identified Cepheids in clusters NGC 1943, NGC 1958 and Bruck 56 are presented. The catalogue and the individual light--curves will be electronically available through the CDS (Strasbourg).
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2014-08-31 02:27:24
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https://twiki.cern.ch/twiki/bin/view/Main/TopPAS16004QA
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# Question & Answers to TOP-16-004
Differential cross section measurements of single top quark production at 13 TeV
### color code
• Answers in blue are considered final.
• Answers in orange require further communication between the questioner and the authors.
• Answers in green mark additional studies that can be performed but do not affect the final outcome of the analysis
• Answers in red still need work from our side.
## Jeremy Andrea on ANv2 (26/01/16)
• Table 2 : could you please specify which signal sample(s) is(are) used
for data/MC comparisons ? What sample is used for W+jets ?
• The default samples are now marked in ANv3. It is amc@nlo 4FS for t-channel, powheg 5FS for tW, powheg for ttbar, amc@nlo for wjets, amc@nlo for DY
• Table 3 : to be removed ?
• This is just a placeholder for 76X samples (removed in ANv3)
• For trigger and muon selections, did you used SF provided by the POG
• these have been measured by TOP-16-003
• Equ (2) : are the "c" correction factors to the Jet Energy Resolution
already provided for 13 TeV by the JetMEt POG ?
• Btagging SF : are the one derived from QCD (13 TeV) used ? Or the ones
from ttbar events ?
• mistagging of light jets is from QCD and efficiency of true b's is from ttbar
• Table 6 : There are 3 single top samples, each one with a scale of 0.5.
Does the "scale" account for difference in top and antitop
cross-sections ?
• no, this is just because there are two independent samples used so both need to be scaled by 50% to match the cross section. Each individual sample is already scaled to the cross section by default.
• Line 336 and 338 : I'm wondering, it seems there is no correlations
between top pT and eta(j') at all ?
• no, there seems to be no correlation since j' is not used to reconstruct the top quark
• Section 4.3 : one of the weak point of the approach is that, the QCD
estimation is done after the training. Meaning that the optimization of
the BDT training might not be optimal ? This doesn't affect the
robustness of the BDT of course. Would an "rough" estimation of QCD,
prior to the training, improve something ?
• the antiisolated QCD MC is scaled down to 0.5% in order to prevent the BDT to train solely against QCD
• Figures 4 to 7 : I suggest to rebin the QCD plots.
• done
• Section 5.1 : I'm not sure to understand how QCD is treated. Do I
understand correctly that there is only one training to distinguish
against all backgrounds ? There is not a QCD-dedicated training, right ?
• yes, there is only one training which is mainly meant to separate t-channel from wjets and ttbar. QCD is just mixed-in to prevent residual events to peak at high BDT output values. QCD is rejected by cutting on mTW>50GeV.
• Figure 19 : could there be a missing contribution in the tail of mTW ?
Which W+jets sample you used ? MLM or aMC@NLO ? Is the MET distribution
ok in that region ?
• this may be due to the recoil model. There is a way to correct it but this will require more studies first. MET is fine except for the first bin that is due to a lack of QCD event from antiiso region. The effect is less dramatic in the other regions.
• Figure 27 : I guess there is no MET-PHI correction applied ? Would it
make any significant changes to the results ?
• MET-phi is not yet applied since a recipe will only be available for 76X; update: miniaod in 76X seems to have MET-phi applied by default.
• Line 531 : could you prepare a table, summarizing the estimated
background yields and the related uncertainties ?
• Line 533 : what is the relative uncertainties ? How much ?
• this was indented to be taken as 20%; now individual fits for each pT/y bin are performed
## Pedro da Silva on ANv3 (28/01/16)
### Physics
• L49-50 Are MET filters strictly necessary for this analysis? To which degree it’s important to measure its efficiency?
• filters are not strictly necessary since this analysis is not sensitive to high MET tails (>300 GeV). See here for efficiency
• L129 I guess the muon contacts will ask you as well : why not complemeting with TkMu20 ? It does increase the efficiency.
• measured trigger efficiencies for new muon iso working point for TOP-16-003 only includes IsoMu20; analysis is now restricted to only this.
• L141 It would be good to motivate why is a tighter cut used wrt to the one proposed by the POG, following maybe Georgios’ presentation in the PAG meeting
• done
• L214-216 I guess at this point you’re still working on including trigger/id+iso/b-tagging etc. scale factors. When ready, don’t forget to document it
• done
• L239 is there any limit on how much you are allowed to modify MET to attain this condition? Do you have some control plot of how much you have to modify it for the signal and the backgrounds?
• there is no constraint but from possible solutions which eliminate the sqrt term, the one with the smallest delta R to the original pmiss vector is chosen
• L274 It would be good to comment on how you have arrived to the choice of these scale factors
• clarified, it is just the yields from MC. Each sample is normalized to the process cross section. So if combining multiple samples, these need to be scaled down.
• L330 what is the underlying sorting algorithm ? increasing value i guess so, from Eq.1
• yes
• L321-338 it’s an interesting approach. Could you add a couple of sentences on the gain of using this algorithm, with respect to computing a simple correlation factor?
• it is to detect even non-linear correlations as stated in l323
• L365 Was the q/g discriminator tried for the forward jet?
• not available in 74X, in 76X it shows some deviation wrt data. more studies are needed so it will not be used in the preliminary analysis
• Figure 15 - the non-isolated region still contains a lot of W+jets. That’s because it’s a simple inversion of the cut. Was it studied to use an even looser isolation inversion .e.g I_rel>20% to remove further the W+jets contamination?
• this is taken as a systematic variation
• Eq.19 yeah ok, it’s a possibility. You could have extended the likelihood (Eq.13) to accomodate the categories to which you fit the MT(W). e.g. L_{2j1t} = L_{2j1t,MT(W)<50) x L_{2j1t,MT(W)>50} where the first one fits the MT(W) distribution and the second one the BDT
• this is unfortunately not possible with theta; therefore the trick with the artificial fit variable is used; technically it is the same as using an extended likelihood
• L436 Does theta accomodate for the usage of uniform priors? Not that any significant change is expected, except maybe in the final uncertainty.
• yes
• Eq. 18 how do you set the initial QCD normalization? To the MC prediction or to the #events in the isolation sideband?
• to the MC prediction; this may be suboptimal to start the fitting algorithm but the fit converges nonetheless
• Fig. 17/18/19/20 it would be nice to see the DD shape directly on the left plot, as we know QCD MC is insufficient to describe correctly the pre-fit data
• the unfitted DD QCD yield will completely overshoot the data because it is the yield expected in the antiiso region; it can be changed but wont look very informative
• L498 even if traditional could you add their definitions. In particular I’m not acquainted with “event shape C”.
• done
• L521 which parton is taken as reference (status etc.)
• updated to isLastCopy
• In general for introduciton of Section 8 - do you have plans to repeat this at particle level?
• only for a paper
• L523 can you comment on the strategy followed to define the binning? Did you follow some quantile-based requirement or purity/stability requirements (e.g. http://cms.cern.ch/iCMS/jsp/openfile.jsp?tp=draft&files=AN2012_484_v5.pdf Eq. 10/11)? In particular i would be curious to see a plot of purity/stability for these two variables and the binning chosen.
• L589/590 This was for Madgraph Born level samples. For NLO samples, besides the QCD scale choice variations we consider variations of the Parton Shower scale.
• Table 7 Although the post-fit nuisances are very close to the initial value their final uncertainy is very much reduced. Is this expected.
• yes, the priors are only there to help the fit converge
• Table 8 Taking out stat+lumi uncertainties it is nice to see, with the available systematics, that there is an overall substantial decrease the uncertainty wrt to the EA PAS-TOP-15-004. This should be highlighted in the text when the assessment of systematics is completed. The largest decrease is for JES. I guess with more data is partially constrained in the fit.
• result will not be quoted anymore since TOP-16-003 is already out
• L611 The pseudo-data was generated with Poisson fluctuations, or should one read Fig.54 as a closure test plot? It would be important to expand the discussion on how the systematic uncertainties are computed to yield the error bars shown in Fig. 54. Maybe statistics is not enough. Could be interesting to show also separately for top and anti-top as a way to further explore the sensitivity to PDFs. Maybe a differential ratio?
• I plan it for the paper
• Do you intend to do absolute cross section as well?
• this analysis is not optimized to get a precise inclusive xsec measurement; also depends on TOP-16-003 => to be discussed
• If you have some time it could be good to add also some assessment of the statistical coverage of the unfolding procedure. E.g. throwing pseudo-experiments and checking that the pulls in each bin have a gaussian distribution with with 1. After reading the PAS (L202-209) I understand that you did all of this, altough you didn’t yet document in the AN
• done
• L616-617 - let’s see when unblinded
• ok
### Suggestions, typos, etc.
• L68 ascence -> absence
• ok
• L78 You can of course disclose the method at this point or leave it more vague as it’s an introduction “when inverting the muon isolation” -> “in a control region dominated by this background process”
• ok
• L81 maybe reference for BDT?
• cite tmva
• L87 maybe reference for TUnfold?
• cite tunfold
• L120 reconstructed muons or electrons -> muon or electron candidates
• ok
• L144 antiisolated -> anti-isolated
• ok
• L174 eta->\eta
• ok
• L181 with the -> of the muon flight direction
• ok
• ok
• L202 does it really improve the resolution or is it mostly for consistency of the global energy scale changes and therefore of the recoil scale?
• it is understandable that the resolution improves since MET is corrected by using some "better calibrated tracks" in the calculation that are inside the jets
• Eq. 5/6/7/8 leave space before to recover line numbering
• ok
• L227 build ->built
• ok
• L236 neglection -> negligence
• ok
• L256 minic -> mimic
• ok
• L381 discriminat -> discriminant
• ok
• Table 7 nuciance -> nuisance
• ok
## Pedro da Silva on PASv0 (28/01/16)
### Physics
• L42/43 Is it the CPU consumption that makes it hard or the correct description of all the detector effects and pileup together with missing higher orders in the MC to describe how QCD multijets leak into the single top region ?
• at first order it is CPU and memory, with the available MC for QCD, no modeling issue was observed (e.g. like the polarization angle in Wjets) that justifies the need for NLO MC
• L105-109 Maybe this could be simplified - you’ll discuss the reconstruction of the top quark kinematics (essential for the quantities to be mesured) and the usage of a multivariate discriminator to reject further the background with respect to a simple cut-based analysis (this should be stressed, otherwise there is no point in using MVAs)
• whole paragraph rephrased
• L114/120 please comment if these rates for 2 solutions, complex solutions are similar for both signal and backgrounds and what is the rate observed in data. See also comment to the AN regarding the modification of MET: is there any restriction in how much you modify?
• this is for signal; background can still be studied
• L155-157 - I understand the optimization is still on-going. Please motivate the choice of this cut for the analysis.
• ok
• Section 4.1 at the end you should state how the top pT is defined (pT of the b+lepton+neutrino system) and how the rapidity is defined. In particular for the latter: is it rapidity or pseudo-rapidity. In the first case, which top mass do you use to compute it? 172.5 GeV or the reconstructed mass?
• it is the rapidity of the reconstructed top quark y=0.5*ln[(E+pz)/(E-pz)]
• Eq. 3 - suggest to remove. This is a technicality. See comments on the AN regarding combining likelihoods. Suggest also to rephrase until L161 to something like: “We perform a combined fit to different distributions depending on which category an event belongs to. For events with low MT(W) (<50 GeV), we make use of the MT(W) distribution, while for high MT(W) (>=50 GeV) we make use of the BDT shape.”
• ok
• L172 when 76x is available please comment on the agreement found
• ok
• I miss in Section 4 the following discussion: normalization factor to be used for Section 5 : i.e. report the xsec measured from the ML-fit and comment on its uncertainties and compatibility with the early analysis result (TOP-15-004)
• ok
• Table 1
I’ll not be picky on these numbers because they are based on 74x. use an appropriate number of significant digits explain in the caption what is the nature of the uncertainties remove the nuisance parameters. The uncertainties are very small because the priors have overblown widths. You can comment in the text that W and Z backgrounds are pulled by a factor of 1.6 while the remaining backgrounds are compatible with the initial prediction. t-channel is measured to be 20% higher than the SM prediction, although compatible with uncertainty.
• ok
• L173 Normalized differential cross section measurement
• better unfolding? because "Normalized differential cross section measurement" is the title
• I miss in Section 5 the following discussions
• how is parton level defined (maybe around L177)
• how is the binning chosen for the migration matrices/unfolded spectra (L199/L201)
• ok, less-curved expectation & purity/stability>50% for all bins
• From the physics side: will you consider particle level defintion? will you
• planned for paper
• Figure 4 - Maybe the current figure is just temporary while the analysis is in 74x. But, I would prefer to see the figures after the BDT cut so that one sees the distributions which are going to be unfolded.
• ok
• Section 6 - are there more theory curves which can be added for comparison?
• there is amc@nlo 4fs, 5fs, powheg & herwig
• Do you have enough stats to report separately for top/anti-top, or eventually a ratio of the two?
• not yet but planned for the paper
• L222 based on what? i.e. quote in which generators the theory predictions tested are based on.
### Suggestions, typos, etc.
• Abstract: suggest to refere explicitely that these are normalized differential cross section measurements (unless you’ll update to include also absolute differential cross sections)
• Incipit is now: "Normalized differential cross sections of single top quark production in $t$ channel in proton-proton collisions are measured as functions of the transverse momentum and the absolute value of the rapidity of the top quark."
• top quarks decaying to muons in the final state -> top quarks decaying to final states containing one muon
• ok
• boosted decision tree -> multivariate discriminator (avoid jargon)
• ok
• L3 Tevatron at Fermilab -> Fermilab Tevatron
• ok
• L10/11 spell out 4/5
• This is just one example of a difference between different signal models in MC.
• L13 The measurement presented in this note is data of proton-proton collisions -> proton-proton collisions data recorded at -> recorded by
• ok
• L18 suggest to include a short description of how the PAS is organized
• ok
• L21 Avoid negative statements regarding the CMS detector operation
• Rephrased as "Restricting the analysis to periods of time when all the CMS sub-detectors were fully operational and the solenoid was operated at 3.8 Tesla, this dataset corresponds to an integrated luminosity of $2.2$\fbinv."
• L23 Samples of simulated -> Simulated samples of
• ok
• L25 Monte Carlo -> MC
• ok
• L28/L32 and similar did you use \POWHEG ?
• Now yes.
• L30 maybe quote the underlying event tune used for completion
• ok
• L50 that is however mostly not within the detector acceptance -> that often fails the selection criteria applied
• this is not correct; a 2nd b-tagged jet is not required by the selection because it is not inside the detector acceptance in the first place
• before Eq.1 “using the “Delta Beta” definition” -> as (for non CMS people this is jargon and also strage that it’s called DeltaBeta but there is no \Delta in Eq.1)
• removed delta beta
• Eq.1/Eq.2 leave space before equations to recover line numbers
• ok
• L61 deposited energy from displaced tracks that are associated to pileup -> of the \pt flux from reconstructed tracks which are associated to pileup vertices
• ok
• L99 signal and control regions are defined in which different processes dominate. -> we define signal and control regions, in which different processes dominate.
• ok
• L123 move “under the signal hypothesis” to the start of the sentence
• reshuffled
• L131 Background discrimination with a multivariate analysis
• ok
• L140 add TMVA citation or another appropriate source for BDTs and Boosting algorithms
• L164 tigher -> tighter
• ok
• L171 subjected simultaneously to the fit -> are included in the fit
• ok
• Figure 3
• full transverse -> transverse
• ok
• left figure MTW -> M_T(W)
• ok
• right figure BDT discriminant-\eta binned -> BDT
• ok
• Figures 2,3,4
• all have 2.1/fb but in the text you state 2.2/fb (L22)
• changed coherently to 2.3/fb (Moriond ref.)
• remove (Madgraph) and MC stat. from the caption, for the latter explain in the caption how it is represented in the figure
• ok; MC stat. -> Total syst.
• remove or explain in the caption what is chi^2/KS
• removed
• explain in the caption what is represented in the bottom panels and what the shaded bands represent
• ok
• L174/175 - suggest to remove. I hope the reader knows when he reaches this point.
• ok
• L181 get -> recover
• infer?
• L191-196 Do we need to describe the unfolding algorithm given it’s already referenced? The only relevant part is the regularization and the strategy adopted for it. I suggest to remove these lines or simplify further.
• reduced text
• Figure 5: luminosity issue
• changed
• L214 four flavour scheme -> 4FS
• ok
• References [20] remove Technical Report
• ok
## Nadjieh Jafari on ANv2 (02/02/16)
• L385: I did not understand what was the motivation to check/use the
discretized |eta'| variable in the first place. Could you please explain a
bit.
• with 76X + the new JECs, there can be still some residual mismodeling present especially at the JEC eta bin edges. So by using a discretized j' eta, I hope that the BDT will be less sensitive to these edge effects
• In post-fit plots, do I understand correctly that |eta| > 3 is opened
up again and the same SFs from the fit in |eta|< 3 are applied there?
• yes; the problem here is that when fitting |eta| completely, even events in the barrel display an overall MC/data mismatch because the fit tries to recover the yield in the forward region. So by the approach taken here one can at least validate the barrel region
• Fig. 18 post-fit fit variable 3j1t: Do we have an idea of what causes
the BDT trend in data/MC? Is it with fwd region included? If yes, is it
explicitly checked in |eta'| < 3? Otherwise, could it be due to ttbar?
• it is the forward region; same problem in 2j0t
• Could it be covered by ttbar systematic samples or ttbar with pythia6?
• this mismodeling wrt powheg showed only up for events with njets>4 but the resulting differences between both samples on the final result can be evaluated
• Fig. 34: I would say that we see a similar effect as 2j0t in 3j1t. In
fact, data points are just at the edge of MC stat. uncertainties in 3j1t
• yes; in 2j0t; both jets can be |eta|>3 that is why the mismodeling is even more enhanced here
• On systematics, do I understand correctly that R matrix is changing
per source (with tau and rho fixed to their nominal values)?
• yes - the response matrix changes when a systematic influences the signal prediction; regularization (tau) is not fixed and rederived per systematic on the fly - however, its value changes only very minor
• Not sure I understand what is the ML-fit uncertainty, (L553).
If I understood correctly, the signal and background samples are scaled
to the fit results before unfolding. Hence the background yields are
obtained from data. Could you please elaborate on this a bit?
• this is just the uncertainty on the yield as estimated by the fit. It may become more clear when added to the systematic table as (Wjets yield, ttbar yield, etc.)
• L556: The additional uncertainty on relative contributions of
backgrounds: Are they varied within the theoretical uncertainties of the
corresponding cross sections? Are these variations taken as correlated?
• these are not taken as correlated; atm, a conservative variation of 20% can be used or more aggressively the theoretical uncertainty.
• update: this is not important anymore since individual fits per top pt/y bin are performed and the t-channel fit result + uncertainty are passed to the unfolding directly.
• What is the chi^2 to expect for uncorrelated?
• a p-value should be computed for the compatibility with non-correlated hypothesis
• done
• can this test be performed for the BDT as well?
• done
• aren't correlations for backgrounds also important? background subtraction would be affected
• add correlation matrix for background
• done
• the 2D linear correlations should be quoted as well for signal and background (typical TMVA product)
• done
• why doing all in a single fit?
• sample is divided in two orthogonal categories, so that data is not fit data is not fitted twice. Important to account for systematics and correlations in an easy/proper way.
• in the text, change the wording to say it is a combined fit (for clarity), no need to use definition of combined variable.
• done
• different binning on the right figure for the fit variable? data looks different! looks like the binning is different
• updated with consistent binning
• clarify the definition of parton level
• use 'isLastCopy'
• It would be important to quote the purity/stability for the binning chosen
• done
## ARC on PASv1 & ANv3 (12/02/16)
• Up to the unfolding, the analysis is done in one bin. The unfolding is done in several bins. How are the statistical uncertainties in each bin on signal and backgrounds determined?
• bins in the reconstructed data distribution have events N(reco) and uncertainty sigma(reco) = sqrt(N(reco))
• the remaining background B(reco) is subtracted from the number of data events, N(bkg. sub.)=N(reco)-B(reco), whereas the absolute uncertainty stays the same under this transformation, sigma(bkg. sub.)=sigma(reco); an additional systematic uncertainty on the number B is assumed by varying each background component within its fit result
• the resulting data distribution is unfolded; this applies the transformation N(unfolded)=R^{-1}N(bkg. sub.) and sigma(unfolded)=R^{-1}sigma(reco) with the inverted response matrix R
• plots from these 3 steps are shown in the slides here
• NOTE: when switching to the signal/background fits per bin, the number and uncertainty on the signal after background subtraction can be read of the fit result directly per bin
• Could you clarify whether the cut at 0.3 in the BDT fit is performed pre- or post-fit? Could you show the top pt and y distributions as well as other control plots in the two regions (BDT <0.3 ; BDT >0.3) post-fit?
• the fit uses the whole BDT range, a cut is applied to get the data shape in a signal-enhanced region for unfolding; plots of top pt/y have been added to the PAS
• What happens if you cut on the BDT to enrich the signal, is the sensitivity to bg systematics reduced ?
• background contamination and also their systematics do vary when scanning the BDT WP; need to be evaluated;
• update: no cut necessary anymore since fits per unfolding bin are performed
• How can we be sure that the features (efficiencies/resolution) of the kinematic reconstruction in data are well described by the MC? Can we look at efficiencies as fct of lepton and/or b-jet variables ?
• need to be studied
• what is the resolution in pt(top) and eta ? Would like to see purity and stability (request from pre-approval)
• Is it clear that the signal selection efficiency is controlled in all bins ? Would like to see BDT output distributions separately for each bin.
• done
• is it clear that there is no kinematic bias due to the choice of MVA input variables ?
Would like to see scatter plots between pt and eta (resp.) vs input variables (in addition to AN figure 11).
• done
• Do you understand the trend in AN figures 12 and 13 ?
• QCD is overestimated by 500% when taken from MC
• What do we learn from AN figure 16 ?
• the correlations between ttbar/wjets reduce (2j1t: rho=-91%) when including the 3j1t/3j2t control regions; notice also the similar BDT shape of the two => BDT alone cannot disentangle their yields
• The background priors are 100% Gaussians (30% for ttbar). Could we gain by tightening them to e.g. 50%(10%) ?
• switched to log-normal now and assuming unconstrained for t-channel, 10% for ttbar; 30% for WZjets (for AMC@NLO)
• Should we include PS modeling uncertainty (pythia vs herwig) ?
• AN figure 54: it looks like the fluctuations of the data points are too small for the size of the statistical uncertainties.
Could we get results from a couple more pseudo-experiments ?
• binning has be changed to decrease stat. uncertainty; bias and pull tests added to AN
• How well does the MC describe the data when varying the jet energy scales within uncertainties?
• to be studied
### Towards paper
• do you ultimately foresee to include the electron channel ?
• only when a gain in precision is expected in the combination of both channels. Also note that the "simple" data-driven method to estimate the multijet event shape and yield by using antiisolated muons was not working very well in 8TeV analyses. An extended/different procedure needs to be developed and studied as well.
• should we also do a measurement in fiducial region and at particle level (pt_mu, b etc. ) ?
• for the paper, I plan to focus directly on fiducial measurements since certain observables (e.g. polarization angle) are not well defined anymore in ST t-channel 13TeV samples at hand
### PAS-editorial
• suggest to include in PAS bottom right figure in pre-app slides p31.
• this is only a technical detail; in PAS is now 2j1t MTW (w/o cuts) & BDT (MTW>50). Also the PAS text only mentions a combined fit now which is achieved by the artificial variable
• Compare data also with powheg/aMC@NLO and 4FS/5FS in final figures.
• done
## ARC/author meeting (19/02/16)
### Observations
• The event yields are increased substantially, and it is unclear whether this can be attributed to the change of the eta cut. It is unclear why the W+jets sample seems to have substantially more statistics. Should try out aMC@NLO sample instead of (currently used) Madgraph.
• one likely explanation for the increase yields are the change of the tight b-tagging working point from 0.97 to 9.35.
• amc@nlo high-stat. sample (200M) is the default now
• PU reweighting (AN figure 21): Use of 69mb does not seem optimal.
• was investigated; 69mb is optimal for 2j1t but not for 2j0t control region; since 2j0t is not used; the recommended 69mb is kept
• mTW distribution (AN Figures 12,13,14 and PAS Figure 3): should wait for appropriate JEC before final assessment.
• ok
• systematic uncertainties (slides p7)
• Do the t-channel scale uncertainties include PS-scale ?
• switched to new 7 variation ME/PS scheme as described here
• Could it be that this includes uncertainty on cross section, not just acceptance ?
• excluded uncertainty on xsec as recommended
• Also take scale unceratinties on W+jets into account ? A: Yes when moving to aMC@NLO. Will check if Madgraph has weights.
• done
• distributions after cut MtW >50 && BDT>0.25 (slides p8):
• distribution as fct of pt is not very well described. This seems a serious issue for the unfolding.
• more confident now after performing individual fits per top pT bin
• bins as fct of eta show different systematics, esp. 2nd bin has large uncertainies (under investigation)
• first few bins as function of pt are consistent with zero. Rebin ?
### Timeline / next steps
(the following is obsolete since Moriond was failed)
• For Moriond EWK green light is required in about 10 days.
• One of the next analysis steps is to investigate separate fits to each of the analysis bins.
• Target next AN and PAS versions incorporating as many comments and improvements as possible at the timescale end of next week (Feb 26).
• In the mean time, start providing tentative answers to our previous comments on twiki
## Andreas Meyer on PASv4 and ANv6 (29/03/16)
### Analysis
• understand size of uncertainties in first bin in pt(top). It may be best to check the pre-fit distributions in this bin for different systematic variations.
• the prefit distributions for the first pT bin can be found here: merged.pdf
• ME-PS treatment of scale uncertainties. The uncertainty is HUGE. how big is the uncertainty for ME scale variation only? how big is the uncertainty due to PS scale variation?
• a problem in the normalization (reweighting acts differently on positive and negative events) has been identified and now fixed.
• how do pre-unfolding (PAS fig 4) and post-unfolding (PAS fig 5) results match ?
• with the Q-scale normalization problem: the large uncertainty pulled the total mean towards the SM. Note: pseudo experiments are diced for each uncertainty and then the 16%,50%,84% quantiles are taken as the total uncertainty and mean per bin
• without the Q-scale normalization problem (now fixed): same deviation is observed (data predicts harder pT spectrum) before and after unfolding
• are you sure you really only changed the acceptance effect ? If you determine the uncertainties using pseudo-experiments for different varied samples, keeping the analysis (response matrix) fixed, then you wrongly pick up BOTH acceptance effects AND physics modelling. One correct way to determine the acceptance effect only is to re-run the analysis on data for different MC assumptions, i.e. (change the response matrix), but nothing else.
• whenever a systematic influences the signal, the response matrix is varied as well. This is the case for generator modeling, hadronization modeling, b-tagging, mistagging, JES, JER, muon SF, PDF, PU, Q-scale t-channel, top mass.
• what is the source of the spike in the fit variable e.g. for 3j2t sample around 60 (AN fig 113) ?
• this comes from the overlap of MTW <-> BDT due to the binning. Note: the fit variable is only by coincidence sometimes nearly continuous at 50
### PAS editorial
• give typical size of systematic uncertainties
• list major systematics
• do we really need fig 1 ?
• I think it is important to understand the difference between 4FS and 5FS
• do we need table 1?
• removed
• suggest to state consistency with TOP-16-003 but dont quote result for inclusive cross section.
• done
## ARC on PASv5 & ANv7 (04/04/16)
We are still worried about the first bin in pt(top), and we assume that you are, too!.
In the first bin some of the "partial uncertainties" are significantly bigger than the total uncertainty. This seems to indicate instability.
• could it be that a fluctuation of the W+jets background due to statistics pushes the result down (see e.g. fig 123a and b)?
• the first bin is just very sensitive to the systematics because 1. less events are observed than predicted and 2. the selection efficiency is rather low which yields an amplification of the uncertainty by the unfolding
• to test against statistical fluctuations the fit was performed with (left) newton-minimizer (default) fitted to BDT (default), (middle) minuit2 fitted to BDT (default), and (right) newton-minimizer (default) fitted to eta j'. The results are stastistically compatible whereas the eta j' fit result produces significantly larger uncertainties due to the less discrimination power of that variable.
• would it make sense to use a coarser binning (e.g. to reduce statistical fluctuations) ?
• a coarser binning will reduce sensitivity a lot since one needs a relatively fine binning in the signal dominated region that contains by construction only a few background events
• would it make sense to use pseudo-data for the determination of the uncertainties (to remove the statistical factor from the systematic uncertainties)?
• if there is a fluctuation in the systematics templates wrt to the nominal templates (=pseudo data), the sensitivity to the limited MC statistics will be still present. This is not true for the few systematics templates that are derived by reweighting only (e.g. PU, b-tagging, muon SF, PDF, ...)
• it seems that pre and post unfolding results are not really consistent. data/mc are ~0.7 pre-unfolding and ~0.5 post-unfolding.
• this can occur due to some asymmetric uncertainties (e.g. Q-scale, generator, hadronization model) which also shift the average expectation when combining all systematic shifts for the total uncertainty. The systematic uncertainties are combined by dicing the yield per bin according to the unfolded templates and summing the shifts from all systematics up. The yield for a pseudoexperiment is obtained as yield(PE) = Gaus(mean=0,sigma=+-JES)+Gaus(mean=0,sigma=+-JER)+...+yield(nominal). The total uncertainties and expectations are taken as the +-1 sigma and 50% quantiles over many PEs.
• in pseudo code this can be written as (given up,down,nominal as shown in the systematic templates below per unfolding bin
• toys = array(N) for i in N: for up, down in systematics: toys[i]+=diceAsymmetricGaus(mean=0.0,up,down) toys[i]+=nominal totalDown, totalMean, totalUp = quantiles(toys,15.8%,50%,84.2%)
• We propose that you show the distributions of acceptance (N_gen,cuts/N_gen) and efficiency (N_rec,cuts/N_gen,cuts) in bins of pt and y. Their products are a projection of the response matrix. We expect to see that the acceptance in the first bin is very small. Too small ? (In TOP-14-004 the first bin had an acceptance that was a factor ~3 smaller than in other bins. After discussion, we decided to indicate this in the PAS (figure 3))
• the acceptance is indeed 3 times lower in the 1st pT bin after mu+2j1t selection compared to the 2nd & 3rd bin. Not only the acceptance increases the uncertainty by also just the fact that less events are observed there
## ARC (07/04/16)
• We would like to ask you to document exactly how the uncertainties for the differential cross sections are combined (and how the final central value is constructed from the results using the nominal sample and systematic variations. Specifically we should try to understand table 11 on page 75 in AN v7.
• see above
• In bin 1, the various uncertainties tend to be larger than the total uncertainty? In bin 3, in contrast, the total uncertainty seems significantly larger than each of the partial uncertainties.
• this is just an artifact since the rel. uncertainties for the individual result are referring to a different center value than the total uncertainty. In a new version, the rel. uncertainties will be replaced by the absolute ones which is less confusing
• It would be instructive to see the plain result for some of the variations (or at the least for the nominal sample).
• the following shows a few unfolded systematic templates. Most notable is that the first bin seems to be much more affected than the others.
• Also, we would like to ask you to show us the acceptance in each bin (similar as for PAS TOP-14-004).
• done, see above
• The other question that came up is whether there is an explanation why aMC@NLO(5FS) has a significantly different shape than the other predictions. Do we have Powheg(5FS) for 13 TeV?
• there is no powheg 5FS at least in MCM
## ARC/author meeting (08/04/16)
### approval conditions
• improve the documentation about combination of systematic (and statistical) uncertainties and about the recalculation of central value w.r.t the result from the nominal MC sample. Please add the description to the AN and give a sufficient explanation also in the PAS.
• added to PAS: "Pseudo experiments are preformed per unfolding bin by dicing Gaussian distributions per uncertainty source around the unfolded nominal template using its difference with respect to the unfolded systematic varied templates as standard deviation. The differential cross section (uncertainty) is taken as the 50\% (one standard deviation) quantile over many pseudo experiments."
• show (in AN or during freeze on twiki) generator-level and reco-level distributions of t-channel single top for varied top mass (and possibly also for other dominant systematic uncertainties).
• the following shows gen & reco level distributions of theoretical uncertainties of t-channel MC scaled to prediction.
*
• the following shows reco level distributions of experimental uncertainties of t-channel MC scaled to prediction
• include information about the acceptance (esp. the fact that it is low in the first bin of pt(top)) in the PAS.
• The overall acceptance of signal events per unfolding bin after the 2j1t event selection ranges from 1.5% to 2.5% with the exception of the first top quark pt (1.1%) and the last |y| (0.7%) bins.
• We also suggest to specify in text of PAS (section 6) which uncertainties are dominant
• added to PAS: "The sources of systematic uncertainties which have the largest impact on the differential cross section measurements are the renormalization and factorization scale choice of the signal process, the signal modeling, and the top quark mass."
## Andreas Meyer on PASv6 (18/04/16)
### ANALYSIS
• line 47: wondering, would one gain in precision by adding the ratio as constraint ?
• the t/tbar ratio is already constrained (taken from the MC) since no separate fits for t/tbar are performed
• section 4.1.: once this analysis becomes a precision analysis, we will need to study the dependence of the kinematic reconstruction on the phase space region.
• ok
• line 157: why is the BDT trained w/o cut on mt(W) ?
• to increase training statistics (mostly for QCD)
• line 224: did you ever try to run the analysis with 4 instead of 8 reconstructed bins? I could imagine that that would gain stability of the fit and reduce fluctuations in the response matrix ...
• this was not tried since it is recommended to have twice as many bins at reconstruction level compared to the generator level to stabilize the minimization in the TUnfold.
• line 274: are you quoting the full 3 GeV variation as uncertainty ? In that case I am less worried about the size of the mass uncertainty. I think you could/should assume linearity and scale it down by a factor 3.
• changed after freezing for approval. see modified here: 2016_04_24.pdf
### PAS-Editorial
• Title suggest: "Measurement of the differential cross section for t-channel single-top-quark production at \sqrt{s} = 13 TeV "
• ok
• Abstract suggest:
A measurement is presented of differential cross sections for t-channel single-top-quark production in pp collisions at a center-of-mass energy of 13 TeV. The cross sections are measured as functions of the transverse momentum and the absolute value of the rapidity of the top quark.
The data were recorded in the year 2015 at the CMS Experiment and correspond to an integrated luminosity of 2.3 fbâ^'1.
A maximum-likelihood fit to a multivariate discriminator is used to infer the signal and background fractions from the data. Unfolding to parton level is performed. The measured cross sections are compared with theoretical predictions to next-to-leading order with matched pardon-showering as
implemented in Monte Carlo generators. Good agreement is found within currently large experimental uncertainties.
A measurement is presented of differential cross sections for t-channel single-top-quark production in pp collisions at a center-of-mass energy of 13 TeV. The cross sections are measured as functions of the transverse momentum and the absolute value of the rapidity of the top quark.
The analyzed data were recorded in the year 2015 at the CMS Experiment and correspond to an integrated luminosity of 2.3 fbâ^'1.
A maximum-likelihood fit to a multivariate discriminator is used to infer the signal and background fractions from the data. Unfolding to parton level is performed. The measured cross sections are compared with theoretical predictions to next-to-leading order with matched pardon-showering as
implemented in Monte Carlo generators. Good agreement is found within currently large experimental uncertainties.
• line 7: "accuracy" I'd say the accuracy is NLO+PS. How about "validity" ?
• changed to validity but both may work since the question here is also if NLO+PS is accurate enough to describe the data
• lines 10-12 and figure 1: Fine for me if you keep the figure, but can you make a statement why 4FS and 5FS are expected to be different for pt-top and y-top ? It seems that figure 5 supports the statement that 4FS and 5FS could have differences in the predictions. Do we understand that ? Maybe extend sentence, line 12: "this could contribute to differences in the predicted shapes of pt-top and y-top" or similar ?
• line 101-103: suggest to shorted into one sentence and move to after line 90: "To avoid overlap of the jets with the selected muon candidate, jets are rejected if DeltaR (mu-jet) < 0.3."
• changed to: Jets are rejected if DeltaR (mu-jet) < 0.3 to avoid overlap of the jets with the selected muon candidate.
• line 106: How about: "... mistag rate of about 0.1%. This results in a tagging efficiency of about 50%". By quoting 49% we claim that we know this better than 1%.
• ok
• line 117: clumsy. Please improve for clarity and correct english.
• changed to: Signal and control regions are defined using the number of selected jets and b-tags per event. The signal region is characterized by two jets and one b-tag (2j1t''). A \wjets control region is defined for events with two non-tagged jets (2j0t'') and a \ttbar control region is defined for events with three jets and one or two b-tags (3j1t'', 3j2t'').
• line 171: should we add plots of the other input variables in the PAS or on the twiki ? Maybe most straightforward to have them in the PAS ?
• the remaining input variables DeltaR (l-jet,b-jet) and Delta |eta|(b-jet, muon) are somewhat artificial, hence their distribution may not be very informative (apart from being discriminating signal/background). This is not the case for the shown variables, the forward jet |eta| and the reconstructed top quark mass, which show characteristic features of t-channel ST production. If requested, it can be added as supplementary material.
• figure 3: Is the systematic uncertainty band post-fit ? I assume so. Should we write this somewhere (in the caption?).
• added to plot captions: "The striped band denotes the total systematic uncertainty scaled to the fit results"
• line 177: Now this is important for the reader to understand: Suggest "A binned maximum-likelihood fit (spell out here once more) is performed to the combined mt(W) and discriminator distributions, separately in each bin of the measurement. The fits yield the fractions of signal and background components ..."
• slightly adapted: "Binned maximum-likelihood fits are performed to the combined \mtw and BDT discriminant distributions, separately in each bin of the measurement. The fits yield the fractions of signal and background contributions in data inclusively and differentially in intervals of the reconstructed top quark \pt and rapidity".
• line 183: Suggest to remove "nuisance parameter" and just write "number of parameters in the fit".
• ok
• line 184: replace "the "top background"" by "tt/tW background" - no quotes .
• ok
• line 186: why log-normal and not Gaussian (or any other shape) ?
• the priors are truncated to >0. Gaussian priors are not recommended by StatCom since their mean shifts through the truncation. Log-normal are by definition >0.
• line 196: the DIFFERENTIAL cross
• ok
• line 199 and following: suggest something like: " these distortions are due to effects from selection efficiencies, detector resolution and kinematic reconstruction and are dependent on the kinematic phase space"
• changed to: "These distortions are due to the selection efficiencies, detector resolution, and kinematic reconstruction which vary with the event kinematics"
• line 214: suggest: "For the pt DISTRIBUTION the data display a somewhat harder spectrum than the expectation"
• ok
• line 215-217: Clumsy: Ok to repeat line 177 here (see above).
• adapted to: "For the unfolding, the signal shape is estimated by performing separated ML fits to the combined \mtw and BDT discriminant distributions, in each bin of the measurement."
• line 223: My understanding: "not biased" and "regularisation" is a contradiction in terms. Rephrase "For minimal bias" or "for acceptably small bias"
• ok
• line 232-233: do we need give these numbers. Suggest to remove this paragraph.
• ok
• line 237: write here something like : "For each systematic variation the whole analysis is repeated and fit templates and response matrix are replaced correspondingly."
• changed to: "For each systematic variation the whole analysis is repeated with replaced fit templates and response matrices"
• Rephrase "special care is taken ..." and remove "can change" by "change".
• obsolete
• lines 242-245: More elaborate, but still very hard to understand. Better put a sentence upfront that describes what you want to achieve.
• ok
• line 295: the list of dominant systematics begs the question why they are dominant. Esp. for the mass this is funny. Can we reduce the mass uncertainty by a factor 3 (see comment to line 274 above).
• ok
• line 300: Can we write here, or in the introduction, what's different between 4FS and 5FS ? (see lines 10-12 above).
• extended introduction
• line 310: add something like "within the experimental uncertainties which are currently still large".
• ok
### TYPE-A
• see suggested title for hyphens in "single-top-quark production" etc.
• ok
• suggest: 4FS and 5FS (and not "4 FS" or "5 FS")
• ok
• line 26: clumsy. Rephrase.
• ok
• line 27: Restricting the ANALYSED dataset
• ok
• line 38: ... contribution from top-quark-pair production
• ok
• line 39: For ALL these
• ok
• line 46/47:
• single-top-quark
• ok
• ... using the HATHORv2.1 library.
• ok
• at least put a "respectively" somewhere. Why not is 136... for top quark and 81... for top-antiquark production ... "
• rephrased
• line 54: "... to simulate a sufficiently large number of events for this process." (remove "of reliable quantity").
• ok
• line 55: shape of THE data IN a multijet-EVENT-enriched sideband region
• ok
• line 58: single-top-quark
• ok
• line 65: ..recorded by a trigger requiring the presence...
• ok
• line 67: no new paragraph after this line
• ok
• line 68: citation at end of sentence (after algorithm)
• ok
• line 68/69: replace "it reconstructs ..." by "Single particles are reconstructed and identified by ..."
• ok
• line 72: remove "that is" and replace fulfils by fulfilling
• ok
• lines 79/80: can we remove the double-quotes around "isolated" and "anti-isolated" ?
• ok
• line 80: The LATTER USUALLY ORIGINATE
• originates
• line 84:
• How about a colon after criteria ?
• ok
• typo: THEIR
• ok
• line 85: Loose electrons ARE REQUIRED ... and TO fulfil dedicated ...
• ok
• line 91: clumsy.
• changed to: "The jet energy and resolution is corrected to account for the non-flat detector response in $\eta$ and $\pt$ of the jet."
• line 97:
• How about ... following CRITERIA ARE IMPOSED: (colon) the neutral ..."
• ok
• in the following lines replace "need to" by "are" or "are required" e.g. "... hadronic energy fractions are < 99% ..."
• ok
• line 100: "... should be > 0 and .... should be <99%"
• ok
• line 105: "... jets THAT ORIGINATE FROM the ..."
• ok
• line 110: remove "as", i.e. "... jets described above...."
• ok
• line 114: suggest to insert "i.e." -> "... variable, i.e. the transverse momentum ..."
• ok
• line 120 suggest: ... b-tagged (2j1t) (no quotes). Events from the 2j1t region are used for the", ie. start new sentence and remove "Only these "
• sentences rephrased: "Events from this region are used for the differential cross section measurement"
• line 121: measurement (singular)
• ok
• line 122: of which one or two -> one or two of which
• rephrased
• line 124-129: suggest remove this paragraph. Not necessary in a (short) PAS.
• ok but may help as an overview of this long section
• line 131: single-top-quark
• ok
• line 137: remove "third of" (redundant and sounds funny).
• ok
• line 140: justified BASED ON STUDIES OF
• ok
• line 141: "Under the hypothesis" sounds strange to me (english?)
• removed
• line 141/143: top-quark (hyphen)
• ok
• line 144: summed -> sum of
• ok
• line 157: with -> using
• ok
• line 185: modelled using a template of anti-isolated muons from events in data
• ok
• line 191: on -> of
• ok
• caption figure 3 and others:
• suggest: the ratio of yields in data and simulation
• ok
• replace "striped" by "hatched" or "shaded". I'd take any bet that "striped" is no english
• striped seems ok, similar words: "grey-striped", "cross-striped", "striped skirt", "striped ebony", ...
• line 206: typo minimisation
• ok
• line 208: such AS to
• ok
• line 228: remove one "also"
• ok
• line 257: spell out "charm"
• ok
## Pedro da Silva on PASv7 (26/04/16)
### physics
• L43 suggest to remove "under...69mb." This is a phenomenological parameter and it's hard to justify that 69mb is really a number corresponding to the total inelastic pp cross section.
• ok
• L59 there must be a latex thing going on here as t->bW\mu is a BSM decay of the top
• ok, :-)
• L93 in the forward region does the HF spacial granularity allow for >10 candidates? doesn't this requirement lead to a drop in efficiency?
• in particular, this ID cut is the replacement for cutting on the neutral hadron fraction which is still mismodeled. JetMet claims an efficiency of >99% (see here)
• Figure 4 y axis why <Events> is it really an average?
• yes, it is normalized to the bin width; "<Events>" is the recommendation by PubCom although exact y-axis labeling rules are not 100% clearly stated in my opinion (see here)
• L202/203 Not sure i understand this sentence: Is the shape derived from the fit? or are the signal yields derived from the fit performed in exclusive pT/y categories? Please clarify
• slightly adapted: For the unfolding, no selection on the BDT discriminat is performed. Instead, the signal yield is estimated by performing separated ML fits to the combined \mtw and BDT discriminant distributions in exclusive top quark $\pt$ and $|y|$ bins of the measurement.
• L207 Are these upper bounds really needed? What's the fraction of tops >2.4 in eta?
• w/o these bounds, I need to deal somehow with the overflow which is technically/statistically a bit complicated. The fraction of signal events pT>300 GeV is ~1% and |y|>2.4 is ~6%.
• L208/211 Would be good to start with a sentence motivating why these values are relevant for the unfolding stability/final correlations
• changed to: The binning is chosen to yield less migrations between the reconstructed bins. This is quantified by the stability (purity) which are defined as...
• L224 given you use the median, shouldn't you use the 16% and 84% quantiles to define the uncertainty? Or alternatively, are the distributions of the toys wildly non-gaussian such that the median is preferred over the mean?
• the 1 sigma quantiles = 16%/84% are used. Indeed, the median is preferred because some systematics are asymmetric
• L256-257 "reweighted histograms" is a funny expression, can you clarify what does it mean using alternative wording?
• changed to: The uncertainty on the PDF is estimated by reweighting simulated events according to all variations of the NNPDF3.0 set
• L258 Choice of the renormalization and factorization scales. Suggest to rephrase first sentences as this is jargon, related to the way CMS organizes its software.
• ok
• L267 in addition to [37] please quote TOP-16-008 and TOP-16-011 as these are 13 TeV measurements
• ok
• Regarding Section 7 could you add a table or at least a sentence where you give the typical numbers in % for the dominant sources of uncertainty
• ok
### type A
Abstract
• L1 suggest to move "is presented" to the end of the sentence
• ok
• L2 spell-out pp
• ok
• L10 suggest to remove "currently large experimental"
• ok
Text
• L9 add (\pt) after "transverse momentum"
• ok
• L11/L12 move (4FS) and (5FS) close to "4 flavour" and "5 flavour" / differ -> differ respectively
• ok
• L15 start the sentence with "The analysis uses" and remove "are analyzed that were"
• ok
• L33/36/37/40/249/279/280 aMC@NLO -> MG5_aMC@NLO
• ok
• L47 space after Hathor
• ok
• L56 This data-driven modelling of the shape of multijet events -> The method used to measure the shape of multijet events in data
• more elaborative: The method used to model the shape and estimate the yield of multijet events in data is detailed in Section...
• L71/72 using the definition -> by
• ok
• L78 originates -> originate ?
• ok
• L81/85 This sentence is extremely long and complex. Suggest to simplify it adding a stop after "rejected." and then The identification criteria used to search for additional leptons are loosened and in addition the following requirements are applied: \pt>10(20) and \eta<2.5(2.5) for muons (electrons). Loose muons are furthermore required to have I_{rel}^{\mu}<20%.
• ~ok
• L191 as a minimization
• ok
• L215 within -> while attaining
• ok
• L234/237 quote BTV-15-001 https://hypernews.cern.ch/HyperNews/CMS/get/top/2214.html
• ok
• L279/280 "in the 4FS" or "in the 5FS" read better to me
• ok
• Figure 5 remove "\mu+jets" from the figures
• ok
• L286 single top-quark production via the t channel -> t-channel production of single top-quarks
• ok
• L289 suggest to start the sentence with "Within the experimental uncertainties," and add "based on NLO+PS generators in the 4FS or 5FS" after theoretical predictions remove "that are currently still large"
• ok
## Approval condition
• Q: Why is the top pt well modelled in the control region? A: we have applied the top pt correction by default => make explicit in the PAS
• ok
## Andreas Meyer on PASv8 (30/04/16)
### Physics
• About the possible additional figures you answered: "If requested, it can be added as supplementary material." Personally I think the PAS is still short and I'd find it natural to include all relevant figures in it directly.
• added the two other input variables
• Line 205: I stumbled over the statement "taking into account the intrinsic kt of initial state partons" It is not clear to me what you are referring to. Is the term "intrinsic" really correct and appropriate ? Is it relevant for the exact definition of the parton-level top quark ?
• see discussion in HN
• intrinsic pT refers to the simulated (empirical) non-pertubative transverse momentum of the initial partons from the proton.
### A type
• figure captions: striped -> hatched
• ok
• General remark: You often make statements of the form: "the analysis focuses", "the measurement selects". I find these cumbersome, as the analysis and the measurement themselves are not actors. In most of those cases I would prefer passive voice. See suggestions below.
• ok
• abstract (last two lines): propose: ... order matched with .... generators. General agreement is found within uncertainties (replace good by general).
• ok
• line 12:
• that -> which; remove: respectively; into -> in
• ok
• caption fig 1: Propose last sentence: Corresponding diagrams exist for single-top-antiquark production.
• ok
• line 17: Avoid active voice (and avoid suggesting that t->mubnu is a direct decay): "The measurement is restricted to events in which the top quark decays into a final state containing a b-quark, a muon and a neutrino"
• ok
• lines 20-25: suggest "the dataset is described in section 2" and similar for the other sentences.
• ok
• line 27: the measurement uses .. (see above).
• ok
• line 52: remove "order"
• ok
• line 63: propose: "... splitting. In most events this second b jet is outside of the detector acceptance because its transverse momentum is small." (in my ears b-jets are soft or have small transv. mom., but not soft transv. mom.).
• ok
• line 65: suggest to replace therefore by as a consequence
• ok
• lines 67 and 71: propose to avoid online and offline (jargon): write: line 67: The events are recorded .. line 71: For the final analysis the presence ...
• ok; removed final
• line 78: avoid active voice: "Events are selected ..."
• ok
• line 79: use here "muon candidates" once. Can then refer to "muons" w/o "candidates".
• line 88: suggest to insert "in order" before "to avoid"
• ok
• line 123: suggest to replace "for which" by "and".
• ok
• line 140: "amount of ... events" sounds funny to me. Events are countable, amount is not. Suggest to remove "events" and write ...the amount of remaining background is still very high ..."
• ok
• line 140: replace "therefore" by "For further separation" and rephrase the rest of the sentence. "Therefore" generally indicates a logical argument. In this case, IMO the term "for this purpose" or similar is more appropriate.
• changed to: "To discriminate between signal and background events further..."
• line 152: biased (missing ED)
• ok
• line 153: sentence is a bit lengthy, but ok. Maybe replace "since" by "as", ("since" seems a stronger causality, "as" is more a remark which is what is intended here).
• ok
• line 167: multijet EVENT yield
• ok
• line 171: , AND W+jets ...
• ok
• line 174: suggest to add ", respectively" at the end of the sentence after "yields".
• ok
• line 175: suggest "top-QUARK background yield AS"
• ok
• line 178: suggest, "In addition to providing better control OF the ttbar EVENT yield, THEIR INCLUSION is found to reduce the correlations between the estimated background yields.
• ok
• line 185: "distorted w.r.t parton-level counterpart is a bit hard to swallow for an experimentalist. Parton level is not an observable, and thus intrinsically distorted by itself". How about: "The reconstructed distributions are affected by detector resolution, selection efficiencies and kinematic reconstruction which lead to distortions with respect to the original event distributions. The size of these effects vary with the event kinematics."
• ok
• line 189: suggest to use same font for TUnfold as for PYTHIA.
• ok
• line 196: finish sentence after "Fig 4.", remove left column (right column).
• ok
• line 200: in the pt-distribution, esp. at pt<100 GeV, the data ...
• ok
• line 201: Propose: The signal yields are estimated from ML fits to the combined mT(W) and BDT discriminant distributions, separately in exclusive top quark pT and |y| bins of the measurement. For the fits no selection on the BDT discriminant is applied.
• adapted to: "For the unfolding, the signal yields are estimated from ML fits to the combined \mtw and BDT discriminant distributions, separately in exclusive top quark \pt and $|y|$ bins of the measurement. No selection on the BDT discriminant is applied."
• line 207: Very loose boundaries for the unfolding. Sounds strange. Maybe it is ok to remove this sentence.
• not sure; events beyond this boundary are not considered -> still important to interpreted the last unfolding bins.
• line 208: "... the binning is chosen such AS to minimise the migrations while retaining sensitivity to the shapes of the distributions. The stability (purity) are defined as the probabilities that the parton-level (reconstructed) values of an observable within a certain range also have their reconstructed (parton-level) counterparts in the same range. The size of the bins is chosen such that both quantities are larger than \sim 50%." Suggest to remove everything after "... in nearly ...." until "... 40%".
• ok
• line 215: no new paragraph here
• ok
• line 219: uncertainty (singular, really!
• ok
• line 221: How about the following "The central value and total uncertainty of the cross section measurement are calculated from the combination of all variations. For this purpose, multiple Gaussian distributions, one per uncertainty source, are diced around the unfolded nominal template with a width corresponding to the difference between the unfolded systematic variation and the unfolded nominal result. The distribution of the sum of the diced numbers is used to extract the final result. The central value is taken as the median and the total uncertainty as one standard deviation quantile.
• ok; changed: "diced numbers" -> "diced yields"
• line 227: ... systematics and THEIR impact on the measurement are described.
• ok
• line 233: ... data BY INVERSION OF ... the muon isolation CRITERION.
• ok
• line 238: b jet (not b jets).
• ok
• line 256: top-quark (hyphen)
• ok
• line 264: suggest present tense "have been" -> "are"
• ok
• line 276 top-quark mass (hyphen)
• ok
• line 280: add here "in-situ", i.e. "to the in-situ measured " to be clear that the total number is not taken from theory or similar.
• ok
• line 281: replace "these distributions" by "the data".
• ok
• line 285-287: suggest "In the first bin of top-quark transverse momentum, a low acceptance to select signal events and a large sensitivity to the systematic uncertainties lead to a large relative uncertainty, esp. in this bin.
• adapted to: In the first top-quark $\pt$ bin, the low acceptance to select signal events and the large sensitivity to the systematic uncertainties leads to a large relative uncertainty which renders the observed deviation not yet significant.
• line 289: no hyphen between top and quarks
• ok
## George Wei-Shu Hou on PASv9 (04/05/16)
• l22 "Details ..." l24 "The ..." l25 "The ..." should all be lower case
• ok
• l135 "... be unambiguously defined"
• ok
• The only issue is l208: pT < 100 GeV seems rough and imprecise. Note that the bin goes up to 110 GeV, and the bin above is also in (milder) excess [to be true, though without reason, the binning of Fig. 4 top-right looks rather peculiar]. Furthermore, saying its harder below 100 GeV just seems vague and not good. Maybe one just says "around pT ~ 100 GeV ", or like in earlier versions, do not mention any number.
• ok
• BTW, I feel putting Fig. 4 on p. 7 would be better.
• ok
## Andreas Meyer on PASv9 (04/05/16)
• usually comma before "respectively".
• ok
• lines 174 and following: "Therefore" in line 176 sounds "germanic" to me. Suggest to invert the order: "The fits are performed using a combined likelihood by taking the mT(W) distribution for events with mT(W) < 50 GeV, and the BDT discriminant otherwise. The shape of the mT(W) distribution is a powerful handle to estimate the multijet event yield whereas the BDT discriminant provides sensitivity to the other backgrounds and to the signal yields."
• ok, adjusted the complete paragraph to avoid repetitions
• line 208, I agree with George's comment above that a qualitative statement like "below a value of about 100 GeV " or similar would be better. The "<" sign implies that there is a clear value where the discrepancy kicks in.
• ok
• line 214: I have still not understood whether the statement about the intrinsic kt has anything to do with the definition of the top quark at parton level. I believe the parton level definition is independent of whether intrinsic kt is there. If intrinsic kt is mentioned, one should probably mention the use/need and maybe even its value. Propose to remove.
• my understanding is that the whole scattering system is shifted by the intrinsic kt. The net pT of the sum of all final state particles is not 0 but exactly this kt. Hence the top quark pT gets very slightly smeared by this effect.
• Line 214: The statement about the boundaries seems misleading. It could mean one of two things: 1) we do not quote values above 300 GeV and 2.4 resp. or 2) events above 300 and 2.4 at generator level are removed from the response matrix. Option 1) is obvious and goes w/o saying. Option 2) would be incorrect. If there is another reason that would justify the sentence, I would still like to remove the words "very loose".
• it means that values above these boundaries are not quoted. Technically, there are not enough reconstructed events to justify an extrapolation into this regions. Since the figure caption does not mentioning the inclusion of the overflow, the sentence can be removed.
• line 225: "attain" has a similar meaning as "obtain", "achieve" or "yield". I think you mean "retain..." or "keep the bias at a minimum".
• ok
• line 239: can we remove "conservative" ? Uncertainties should be (as) correct (as possible) and not conservative
• conservative means here that 20% is used for simplicity and not the theoretical uncertainties of the cross sections. Also, using the theoretical cross section uncertainties may underestimate the actual uncertainty in this particular phase space.
• lines 274 and following: up-variated ... etc. the verb "to variate" does not exist. Could write "down-varied" etc.
• ok
• line 291: no new paragraph.
• ok
• line 293: propose to end the sentence after "uncertainty" in order to avoid use of "render" and "yet". The term "yet" seems to imply that you have an expectation (which you should not). The term "render" implies that the relative uncertainty "does" something to the measurement. But the uncertainty is an intrinsic property of the measurement.
• ok, but should one not provide an interpretation of the result?
## Orso Iorio on PASv9 (04/05/16)
• line 9: absolute rapidity -> do you mean absolute value of the rapidity y?
• ok
• Abstract vs line 19: In the abstract you use center; in the body centre, suggest to change in the abstract for consistency
• ok
• line 35: you state that aMCatNLO is used for other single-top channels, isn't tW done with POWHEG? Or am I misreading?
• indeed
• added: The simulation of single-top-quark production in the \tw channel uses \POWHEG version~1 interfaced with \PYTHIA.
• line 235 : sources of systematics -> sources of systematic uncertainty
• ok
• line 296 : dataset -> isn't "data sample" better?
• I like to keep dataset. I like to write "MC sample" because these are sampled from theory (can be arbitrary large) but dataset because it is somewhat fixed in size.
Topic attachments
I Attachment History Action Size Date Who Comment
pdf 2016_04_24.pdf r2 r1 manage 351.5 K 2016-04-25 - 13:51 UnknownUser
png Btag.png r1 manage 17.7 K 2016-04-08 - 14:18 UnknownUser
png En.png r1 manage 17.0 K 2016-04-08 - 14:18 UnknownUser
png QScaleWjets.png r1 manage 18.8 K 2016-04-08 - 14:19 UnknownUser
png TopMass.png r2 r1 manage 17.8 K 2016-04-25 - 13:44 UnknownUser
png effPt.png r1 manage 14.2 K 2016-04-08 - 13:52 UnknownUser
png effY.png r1 manage 14.7 K 2016-04-08 - 13:52 UnknownUser
png gen_top_pt_QScaleTChannel.png r1 manage 21.2 K 2016-04-25 - 13:31 UnknownUser
png gen_top_pt_TopMass.png r1 manage 20.8 K 2016-04-25 - 13:34 UnknownUser
pdf merged.pdf r1 manage 375.1 K 2016-04-08 - 14:49 UnknownUser
png reco_top_pt_Btag.png r1 manage 23.4 K 2016-04-25 - 13:40 UnknownUser
png reco_top_pt_En.png r1 manage 22.4 K 2016-04-25 - 13:37 UnknownUser
png reco_top_pt_PU.png r1 manage 22.5 K 2016-04-25 - 13:42 UnknownUser
png reco_top_pt_QScaleTChannel.png r1 manage 23.9 K 2016-04-25 - 13:31 UnknownUser
png reco_top_pt_TopMass.png r1 manage 23.0 K 2016-04-25 - 13:35 UnknownUser
png unfolded_top_pt.png r1 manage 31.3 K 2016-04-25 - 13:27 UnknownUser
png unfolded_top_pt_jeta.png r1 manage 31.5 K 2016-04-25 - 13:28 UnknownUser
png unfolded_top_pt_minuit.png r1 manage 31.5 K 2016-04-25 - 13:28 UnknownUser
png unfolded_top_pt_minuit_bdt.png r2 r1 manage 30.7 K 2016-04-18 - 12:17 UnknownUser
png unfolded_top_pt_newton_bdt.png r2 r1 manage 30.5 K 2016-04-18 - 12:17 UnknownUser
png unfolded_top_pt_newton_etaj.png r2 r1 manage 30.6 K 2016-04-18 - 12:17 UnknownUser
Topic revision: r52 - 2016-05-05 - unknown
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2021-11-28 03:43:47
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https://math.stackexchange.com/questions/1341136/help-solving-a-simultaneous-equation
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# Help Solving a Simultaneous Equation.
Im currently doing my Kumon (A math tutoring center I guess) homework, and Im having a bit of difficulty answering a simultaneous equation, involving $x$ and $y$ variables to the second power. School curriculum wise, we're not close to learning this, so I apologize if I don't understand something.
Here is the equation:
$\displaystyle xy = 12$
$\displaystyle x^2+y^2 = 25$
Assuming that $x+y = A$ and $xy = B$
If I'm forgetting something please let me know, as I'm kind of a noob at this. Any help is welcome!
Thanks!
• The answers show how to do the solution in general. In this specific case, when you see $x^2+y^2=25=5^2$ you should immediately think $3,4,5$ triangle and note that $3 \cdot 4=12$. Done. School problems often are made to have easy solutions. – Ross Millikan Jun 27 '15 at 14:38
Using $$x^2+y^2=x^2+y^2+2xy-2xy=(x+y)^2-2xy,$$ we have $$25=(x+y)^2-2\cdot 12\iff (x+y)^2=49\iff x+y=\pm 7.$$
So, we have $$(x+y,xy)=(7,12),(-7,12).$$
Can you take it from here?
If we have $\displaystyle xy = 12 \implies y = \frac{12}{x}$ then we can substitute this into the second equation, yielding $$x^2 + \left(\frac{12}{x}\right)^2 = 25 \iff x^4 -25x^2 + 144 = 0$$
Which is a quadratic in $x^2$, $(x^2 - 9)(x^2 - 16) = 0$ with solutions $x = 3,4 -3, -4$.
The corresponding solutions for $y$ are $y = \frac{12}{3}, \frac{12}{4}, \frac{-12}{3}, \frac{-12}{4}$.
So our solutions are $(x,y)$: $(4,3)$, $(3,4)$, $(-3, -4)$, $(-4,-3)$
Hint:
$$x^2(x^2+y^2)=x^4+x^2y^2=x^4+12^2=25x^2.$$
You should be able to solve the last equation, of the biquadratic type.
Seeing a biquadratic equation tells you that there are up to four solutions, coming in two pairs of opposite signs. This was to be expected because swapping $x$ and $y$ doesn't change the equations, and changing the signs of $x$ and $y$ simultaneously doesn't change the equations.
Now school problems often have easy solution such as integer ones. So you can look for the ways to decompose $12$ in two factors (like $2\times6$ or $3\times4$), or decompose $25$ in two squares (like $16+9$).
Obviously, $(4,3)$ is a solution. The remaining three must be $(3,4), (-4,-3),(-3,-4)$.
We have $x^2+2xy+y^2=(x+y)^2=49$.It follows that $x+y=\pm 7.$ In similar way $x^2-2xy+y^2=(x-y)^2=1$ or $x+y=\pm 1.$ So \begin{cases} x+y=\pm7,\\ x-y=\pm1. \end{cases} The solutions of the system are $$x=\frac{1}{2}(\pm7\pm1), y=\frac{1}{2}(\mp1\pm7).$$ Taking the 4 different combination of signs we get the 4 solutions.
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2020-02-23 17:50:20
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https://gmatclub.com/forum/machines-x-and-y-produced-identical-bottles-116334.html
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# Machines X and Y produced identical bottles at different constant rate
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Machines X and Y produced identical bottles at different constant rate [#permalink]
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02 Jul 2011, 23:53
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Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?
(1) Machine X produced 30 bottles per minute.
(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.
OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/machines-x-an ... 04208.html
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink]
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03 Jul 2011, 04:34
siddhans wrote:
Please explain in detail? If anyone has studied from Manhattan . Can you please solve using their method?
Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken machine X operating alone to fill the entire production lot?
(1) Machine X produced 30 bottles per minute.
(2) Machine X produced twice as many bottles in 4 hours as machine Y produced in 3 hours.
U can plug in the values using option B .
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink]
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03 Jul 2011, 04:39
so we have here Total Work Done = 4X+3Y ie 4 hours of X rate + 3 hours of Y rate
let the net rate be r. so 7 hours of r = 4x+3y
ie 7r=4x+3y
now statement 1 : gives me rate of x...it is not sufficient since it doesnt say anything about y
statement 2 : 4 hours of x = 3 hours of y
ie 4x=3y
substituting in original equation we get => 7r =8x or x=(7/8)r
now 7 hours of r -----> full work
p hours of x -----> full work
therefore 7r = px but x=(7/8)r
so, 7r = 7/8 * r * p
or p = 8 hours...so the statement is sufficient.
Ans B
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink]
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03 Jul 2011, 10:15
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fivedaysleft wrote:
statement 2 : 4 hours of x = 2( 3 hours of y)
Shouldn't it be twice?
Let the rate of m/c X be x bottles/hour
Let the rate of m/c Y be y bottles/hour
So in 4 hours m/c X will produce = 4x bottles
and in 3 hours m/c Y will produce = 3y bottles.
Total bottles = 4x+3y ----(1)
Stmt1: Machine X produced 30 bottles per minute. i.e x=30bottles/min but rate of y is not given. Insufficient.
Stmt2: Machine X produced twice as many bottles in 4 hours as machine Y produced in 3 hours.
4x = 2 * 3y
y=2/3 x
Substituting in (1), 4x+ 3 * 2/3 x = 6x total number of bottles.
Now rate of m/c X is x bottles in 1 hour
so 6x bottles in 6 hours.
OA B.
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink]
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03 Jul 2011, 10:21
yeah...my bad...but since it DS...it didnt do THAT much harm...the concept is fine though right?
any other mistake, kindly point out!
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Re: Machines X and Y produced identical bottles at different constant rate [#permalink]
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05 Dec 2017, 02:51
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There are several important things you should know to solve work problems:
1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.
$$time*speed=distance$$ <--> $$time*rate=job \ done$$. For example when we are told that a man can do a certain job in 3 hours we can write: $$3*rate=1$$ --> $$rate=\frac{1}{3}$$ job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then $$5*(2*rate)=1$$ --> so rate of 1 printer is $$rate=\frac{1}{10}$$ job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then $$3*(2*rate)=12$$ --> so rate of 1 printer is $$rate=2$$ pages per hour;
So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).
2. We can sum the rates.
If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is $$rate_a=\frac{job}{time}=\frac{1}{2}$$ job/hour and B's rate is $$rate_b=\frac{job}{time}=\frac{1}{3}$$ job/hour. Combined rate of A and B working simultaneously would be $$rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}$$ job/hour, which means that they will complete $$\frac{5}{6}$$ job in one hour working together.
3. For multiple entities: $$\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T}$$, where $$T$$ is time needed for these entities to complete a given job working simultaneously.
For example if:
Time needed for A to complete the job is A hours;
Time needed for B to complete the job is B hours;
Time needed for C to complete the job is C hours;
...
Time needed for N to complete the job is N hours;
Then: $$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T}$$, where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.
For two and three entities (workers, pumps, ...):
General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:
Given that $$t_1$$ and $$t_2$$ are the respective individual times needed for $$A$$ and $$B$$ workers (pumps, ...) to complete the job, then time needed for $$A$$ and $$B$$ working simultaneously to complete the job equals to $$T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2}$$ hours, which is reciprocal of the sum of their respective rates ($$\frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T}$$).
General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:
$$T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3}$$ hours.
BACK TO THE ORIGINAL QUESTION:
Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?
You can solve this question as Karishma proposed in her post above or algebraically:
Let the rate of X be $$x$$ bottle/hour and the rate of Y $$y$$ bottle/hour.
Given: $$4x+3y=job$$. Question: $$t_x=\frac{job}{rate}=\frac{job}{x}=?$$
(1) Machine X produced 30 bottles per minute --> $$x=30*60=1800$$ bottle/hour, insufficient as we don't know how many bottles is in 1 lot (job).
(2) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours --> $$4x=2*3y$$, so $$3y=2x$$ --> $$4x+3y=4x+2x=6x=job$$ --> $$t_x=\frac{job}{rate}=\frac{job}{x}=\frac{6x}{x}=6$$ hours. Sufficient.
OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/machines-x-an ... 04208.html
_________________
Re: Machines X and Y produced identical bottles at different constant rate [#permalink] 05 Dec 2017, 02:51
Display posts from previous: Sort by
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2018-01-23 22:07:24
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5855751037597656, "perplexity": 3582.2575577642842}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084892699.72/warc/CC-MAIN-20180123211127-20180123231127-00047.warc.gz"}
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http://johanw.home.xs4all.nl/changes.html
|
# Changes to the Physics Formulary
This is the list of changes to the Physics Formulary.
Last English version dates: July 8, 2012.
Last Dutch version dates: 8 juli, 2012.
DatePage(s) Changes
07-08-2012 90, 91, 92Added the mass of the Higgs particle after discovery on July 4, 2012. Changed layout a bit to keep page count at 108.
12-16-2009 8 Corrected an error in generating function F4.
07-26-2008 48 Corrected the Dirac equation.
04-05-2008 9 Corrected a spelling error in the English version.
04-14-2005 8 Corrected the units of the molar gasconstant.
02-27-2005 8 Corrected 2 equations in section 1.7.5.
12-03-2002 2 Corrected 2 equations in section 1.2.
11-13-2001 42 Corrected the units of Stefan-Boltzmann's constant.
09-29-2001 42 Corrected the definition of Mach's constant.
09-08-2001 1, 13, 16 Corrected an error in an SRT equation. Corrected a spelling error. Added more decimals to e.
05-24-2001 27 Corrected an error in the explaination of the Brewster angle.
03-15-2001 20 Added some comment on page 20.
01-17-2001 38 Corrected an equation on page 38.
06-11-2000 6 Corrected two equations on page 6.
06-10-2000 6, 100 Corrected an equation on page 6, added new SI prefixes.
11-10-1999 63 Exchanged \omega and K in section 12.3.2.
10-21-1999 100 Corrected a small typo in the English version.
09-20-1998 52 Corrected the equation for SP2 hybridization.
09-17-1998 100 Added symbols for exa and peta in the SI prefixes list.
09-14-1998 100 Added SI units and prefixes. Ending on an even page number also simplifies double-sided and booklet printing.
09-08-1998 Potentially manyIncluded the option to print vectors in boldface by redefining the \vec command.
09-07-1998 many Made some minor layout improvements
09-06-1998 many Made some minor layout improvements, mostly with vectors between brackets.
09-06-1998 None Started using RCS to manage updates
02-04-1998 48, 59, 60, 71, 64, 96-98English version only: Corrected several small translation errors, mostly in indexes.
15 Added 2 remarks about the fact that some equations only hold in an Euclidean spacetime.
09-03-1997 5 Added a term I was forgotten in subsection 1.4.2.
47 Corrected an error; [L2,H]=0 instead of [L,H].
06-02-1997 8 Corrected Pi = Pi(qi, qi, t) into Pi = Pi(qi, pi, t)
72, 90 Corrected 2 spelling errors in the Dutch version.
02-18-1997 1 Gave a more accurate number for the tropical year.
01-12-1997 19 Added 2 equations for power in complex numbers.
12-29-1996 25, 95 Corrected 2 minor typos
- Edited the copyright notice a bit.
12-17-1996 - Moved the copyright notice and copying condition from the preamble to the to be printed text.
10-16-1996 38 Converted the black-hole thermodynamics to SI units.
10-15-1996 14 Corrected the upper and lower indices for the stress-energy tensor.
38 Added an expression for the temperature of a black hole.
81 Put the picture a bit higher.
10-08-1996 15 Corrected the previously added expression for the Christoffel symbols (error in an index).
10-05-1996 15 Added an expression for the Christoffel symbols.
10-02-1996 9 Explaination of the law of Biot-Savart expanded.
09-26-1996 15 Remark added about the escape velocity of a black hole being > c.
09-13-1996 All pages where chapters startCorrected a mistake in a definition that removed the white space between chapter title and number.
13 Reformatted some equations and added one.
Back to the main page.
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2019-02-17 06:13:37
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{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.844434916973114, "perplexity": 4673.985826059163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481624.10/warc/CC-MAIN-20190217051250-20190217073250-00538.warc.gz"}
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https://www.wikizero.com/simple/Lorentz_force
|
# Lorentz force
Lorentz's law is a law discovered by the Dutch physicist Hendrik Antoon Lorentz. Lorentz's law defines force that acts on moving charged particles in an electromagnetic field. Force consists of magnetic force and electric force.
F = qE (electric force)
If the charge is positive, the direction of the electric force is equal to direction of electric field.
F = qv*B (magnetic force)
The direction of the magnetic force is given by the right hand rule.
If charged particles move with velocity v in an electric field E and a magnetic field B
F = qE + qv*B
F : force (vector)
q : charge (scalar)
E : electric field (vector)
v : velocity of particle (vector)
B : magnetic field (vector)
* is vector cross product.
Using this law, J.J. Thomson measured mass-to-charge ratio.
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2020-07-05 23:18:06
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http://crypto.stackexchange.com/questions/14791/how-to-decrypt-3des-in-ecb-mode-using-a-wordlist
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How to decrypt 3DES in ECB mode (using a wordlist)? [closed]
I have some encrypted texts (encrypted with 3DES in ECB mode without salt).
My question: How can I decrypt them using a wordlist? (or without one)
Example:
Encrypted text:
Xfi+h4Ir6l7zXCP+N4EPvQ==
The wordlist for this:
foo
bar
marketing
The original text was before encrypting was: "marketing" (just to make the example full).
-
closed as off-topic by e-sushi♦, DrLecter, otus, poncho, archieAug 8 '14 at 2:11
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "Requests for analyzing or deciphering a block of data are off-topic here, as the results are rarely useful to anyone else." – e-sushi, DrLecter, otus, poncho, archie
If this question can be reworded to fit the rules in the help center, please edit the question.
Aside from that, what are you asking? "How do I attackif I have a dictionary of possibly preimages" (on-topic) or "why don't this code work? (Off-topic) – figlesquidge Mar 4 '14 at 16:13
I don't know what are you talking about, thus that thing that you are talking about is open to the public on russian forums – evachristine Mar 4 '14 at 16:13
I updated the question – evachristine Mar 4 '14 at 16:18
@evachristine I know you really, really want us to tell you a piece of software to do this for you, but software recommendations are off-topic on this site. – mikeazo Jun 26 '14 at 20:20
@mikeazo Even more… as the question doesn’t ask much else than how to practically decrypt/break the ciphertext, it fits “Requests for analyzing or deciphering a block of data are off-topic here, as the results are rarely useful to anyone else.”. I would have close-voted accordingly, but that bounty currently blocks close-votes. (PS: I already fixed the software recommendations issue with an edit a minute before I noticed your comment.) – e-sushi Jun 26 '14 at 22:29
You have a ciphertext (or maybe multiple), a list of possible plaintexts, but no key. Therefore, your process would be
1. Generate random decryption key
2. Decrypt ciphertext with that key (base64 decode CT first)
3. See if result appears in your list of possible plaintexts
4. If it does, return that plaintext; otherwise goto 1
This is a basic brute force attack and will not work in any reasonable amount of time. The reason for this is that 3DES has too large of a key. Use of ECB is bad, but it does not necessarily imply that something will be easy to break and often requires lots of ciphertext to produce a break (at least in the case of the current application we are discussing, other applications can be rendered completely insecure due to ECB use).
-
After base64 decoding we get (hex) 5d f8 be 87 82 2b ea 5e f3 5c 23 fe 37 81 0f bd which has a size of two blocks.
Of your small word-list, only marketing has so many letters that it needs two blocks: m a r k e t i n as the first, g 07 07 07 07 07 07 07 as the second (or another padding, but this is a common one), and so can correspond to this ciphertext.
A word like foo fits in a block, eg: f o o 05 05 05 05 05, and so would give an 8 byte result. Not that we do use that ECB mode is used (no extra IV data like for CBC, a block goes to a block, and we need padding, unlike CTR mode). So if the plaintext was chosen from among the word-list, marketing is the only candidate.
-
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2016-05-26 00:39:42
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https://en.wikibooks.org/wiki/A-level_Physics_(Advancing_Physics)/Ideal_Gases
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# A-level Physics (Advancing Physics)/Ideal Gases
An animation showing the relationship between pressure and volume when mass and temperature are held constant.
An animation demonstrating the relationship between volume and temperature.
Real-world gases can be modelled as ideal gases. An ideal gas consists of lots of point particles moving at random, colliding with each other elastically. There are four simple laws which apply to an ideal gas which you need to know about:
## Boyle's Law
Boyle's Law states that the pressure of an ideal gas is inversely proportional to its volume, assuming that the mass and temperature of the gas remain constant. If I compress an ideal gas into half the space, the pressure on the outsides of the container will double. So:
${\displaystyle p\propto {\frac {1}{V}}}$
## Charles' Law
Charles' Law states that the volume of an ideal gas is proportional to its temperature:
${\displaystyle V\propto T}$
T must be measured in kelvin, where a rise of 1°K is equal to a rise 1°C, and 0°C = 273°K. If we double the temperature of a gas, the particles move around twice as much, and so the volume also doubles.
## Amount Law
This law states that the pressure of an ideal gas is proportional to the amount of gas. If we have twice the number of gas particles N, then twice the pressure is exerted on the container they are in:
${\displaystyle p\propto N}$
A mole is a number of particles. 1 mole = 6.02 x 1023 particles. So, the pressure of a gas is also proportional to the number of moles of gas present n:
${\displaystyle p\propto n}$
## Pressure Law
The pressure law states that the pressure of an ideal gas is proportional to its temperature. A gas at twice the temperature (in °K) exerts twice the pressure on the sides of a container which it is in:
${\displaystyle p\propto T}$
These laws can be put together into larger formulae linking p, V, T and N. To do this we require a constant of proportionality, (R) the universal molar gas constant, with an experimental value of 8.31Jmol-1K-1
## Questions
1. I heat some argon from 250K to 300K. If the pressure of the gas at 250K is 0.1 MPa, what is its pressure after heating?
2. The argon is in a 0.5m long cylindrical tank with radius 10cm. What volume does it occupy?
3. The argon is then squeezed with a piston so that in only occupies 0.4m of the tank's length. What is its new pressure?
4. What is its new temperature?
5. 25% of the argon is sucked out. What is its pressure now?
Worked Solutions
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2016-12-09 19:19:15
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https://math.stackexchange.com/questions/4142026/second-order-non-linear-ode-hard-to-solve-integral-makes-me-think-i-need-a-dif
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Second order non linear ODE - hard to solve integral makes me think I need a different substitution
I have this here ODE:
$$xy'' = y' + x((y')^2 + x^2)$$
Naturally, I'd try this substitution first: $$y' = p, p=p(x)$$
The equation then transforms into $$xp' = p+x(p^2+x^2)$$ Dividing it by $$x$$, I get
$$p' = \frac{p}{x} + p^2 + x^2$$
Which is a Riccati equation with the solution: $$p = x \cdot \tan{(\frac{x^2}{2}+C_1)}$$
The thing is, if I substitute back $$y'=p$$, the integral on the right side is not an easy one to solve, and even if I do solve it with WolframAlpha the solutions are not the same as if I plug in the second-order equation directly. It makes me wonder if I should have tried another substitution/method.
Any help will be appreciated!
• @LutzLehmann thanks fixed May 17, 2021 at 14:43
• The equation is of the first order in $y'$.
– user65203
May 17, 2021 at 14:45
$$y' = x \cdot \tan{\left(\frac{x^2}{2}+C\right)}$$ $$\dfrac {dy}{du}\dfrac {du}{dx} = x \cdot \tan(u)$$ $$\dfrac {dy}{du} = \tan(u)$$ Where $$u=\dfrac {x^2}{2}+C$$ then integrate. $$y=-\ln |\cos u |+K$$
You have now a formula of the form $$y'(x)=f(u(x))u'(x)$$ with $$f(u)=\tan(u)$$ and $$u(x)=\frac{x^2}2+c$$. This the gives that $$y(x)=F(u(x))+d,$$ where $$F'=f$$, here $$F(u)=-\ln|\cos(u)|$$
Let $$y'=xz$$. The equation becomes
$$xz+x^2z'=xz+x(x^2z^2+x^2),$$ which is separable:
$$\frac{z'}{z^2+1}=x$$ or $$\arctan z=\frac{x^2}2+c$$ and $$y'=x\tan\left(\frac{x^2}2+c\right)$$ as you found.
Now the integral is not difficult: $$\int x\tan\left(\frac{x^2}2+c\right)dx=\int \tan\left(u+c\right)du=-\log(\cos(u+c))+c'\\ =-\log\left(\cos\left(\frac{x^2}2+c\right)\right)+c'.$$
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2022-07-01 17:36:26
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https://www.physicsforums.com/threads/heat-equation-finite-difference-in-c.227967/
|
# Heat equation finite difference in c++
• C/++/#
## Main Question or Discussion Point
Hello,
I'm currently doing some research comparing efficiency of various programming languages. Being a user of Matlab, Mathematica, and Excel, c++ is definitely not my forte. I was wondering if anyone might know where I could find a simple, standalone code for solving the 1-dimensional heat equation via a Crank-Nicolson finite difference method (or the general theta method). Ideally, I will need it to solve using LU decomposition, but I can probably figure that much out myself if I have a starting point.
Thanks to anyone that can help.
Related Programming and Computer Science News on Phys.org
Source Book: Burden R.L., Faires J.D. Numerical analysis
Code:
/*
* CRANK-NICOLSON ALGORITHM 12.3
*
* To approximate the solution of the parabolic partial-differential
* equation subject to the boundary conditions
* u(0,t) = u(l,t) = 0, 0 < t < T = max t
* and the initial conditions
* u(x,0) = F(x), 0 <= x <= l:
*
* INPUT: endpoint l; maximum time T; constant ALPHA; integers m, N:
*
* OUTPUT: approximations W(I,J) to u(x(I),t(J)) for each
* I = 1,..., m-1 and J = 1,..., N.
*/
#include<stdio.h>
#include<math.h>
#define pi 4*atan(1)
#define true 1
#define false 0
double F(double X);
void INPUT(int *, double *, double *, double *, int *, int *);
void OUTPUT(double, double, int, double *, double);
main()
{
double V[25], L[25], U[25], Z[25];
double FT,FX,ALPHA,H,K,VV,T,X;
int N,M,M1,M2,N1,FLAG,I1,I,J,OK;
INPUT(&OK, &FX, &FT, &ALPHA, &N, &M);
if (OK) {
M1 = M - 1;
M2 = M - 2;
/* STEP 1 */
H = FX / M;
K = FT / N;
/* VV is used for lambda */
VV = ALPHA * ALPHA * K / ( H * H );
/* set V(M) = 0 */
V[M-1] = 0.0;
/* STEP 2 */
for (I=1; I<=M1; I++) V[I-1] = F( I * H );
/* STEP 3 */
/* STEPS 3 through 11 solve a tridiagonal linear system
using Algorithm 6.7 */
L[0] = 1.0 + VV;
U[0] = -VV / ( 2.0 * L[0] );
/* STEP 4 */
for (I=2; I<=M2; I++) {
L[I-1] = 1.0 + VV + VV * U[I-2] / 2.0;
U[I-1] = -VV / ( 2.0 * L[I-1] );
}
/* STEP 5 */
L[M1-1] = 1.0 + VV + 0.5 * VV * U[M2-1];
/* STEP 6 */
for (J=1; J<=N; J++) {
/* STEP 7 */
/* current t(j) */
T = J * K;
Z[0] = ((1.0-VV)*V[0]+VV*V[1]/2.0)/L[0];
/* STEP 8 */
for (I=2; I<=M1; I++)
Z[I-1] = ((1.0-VV)*V[I-1]+0.5*VV*(V[I]+V[I-2]+Z[I-2]))/L[I-1];
/* STEP 9 */
V[M1-1] = Z[M1-1];
/* STEP 10 */
for (I1=1; I1<=M2; I1++) {
I = M2 - I1 + 1;
V[I-1] = Z[I-1] - U[I-1] * V[I];
}
}
/* STEP 11 */
OUTPUT(FT, X, M1, V, H);
}
/* STEP 12 */
return 0;
}
/* Change F for a new problem */
double F(double X)
{
double f;
f = sin(pi * X);
return f;
}
void INPUT(int *OK, double *FX, double *FT, double *ALPHA, int *N, int *M)
{
int FLAG;
char AA;
printf("This is the Crank-Nicolson Method.\n");
printf("Has the function F(x) been created immediately\n");
printf("preceding the INPUT function? Answer Y or N.\n");
scanf("\n%c", &AA);
if ((AA == 'Y') || (AA == 'y')) {
printf("The lefthand endpoint on the X-axis is 0.\n");
*OK =false;
while (!(*OK)) {
printf("Input the righthand endpoint on the X-axis.\n");
scanf("%lf", FX);
if (*FX <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
*OK = false;
while (!(*OK)) {
printf("Input the maximum value of the time variable T.\n");
scanf("%lf", FT);
if (*FT <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
printf("Input the constant alpha.\n");
scanf("%lf", ALPHA);
*OK = false;
while (!(*OK)) {
printf("Input integer m = number of intervals on X-axis\n");
printf("and N = number of time intervals - separated by a blank.\n");
printf("Note that m must be 3 or larger.\n");
scanf("%d %d", M, N);
if ((*M <= 2) || (*N <= 0))
printf("Numbers are not within correct range.\n");
else *OK = true;
}
}
else {
printf("The program will end so that the function F can be created.\n");
*OK = false;
}
}
void OUTPUT(double FT, double X, int M1, double *V, double H)
{
int I, J, FLAG;
char NAME[30];
FILE *OUP;
printf("Choice of output method:\n");
printf("1. Output to screen\n");
printf("2. Output to text file\n");
scanf("%d", &FLAG);
if (FLAG == 2) {
printf("Input the file name in the form - drive:name.ext\n");
printf("for example: A:OUTPUT.DTA\n");
scanf("%s", NAME);
OUP = fopen(NAME, "w");
}
else OUP = stdout;
fprintf(OUP, "CRANK-NICOLSON METHOD\n\n");
fprintf(OUP, " I X(I) W(X(I),%12.6e)\n", FT);
for (I=1; I<=M1; I++) {
X = I * H;
fprintf(OUP, "%3d %11.8f %13.8f\n", I, X, V[I-1]);
}
fclose(OUP);
}
Hello,
do you guys have any source code of ADI method implemented in 2d diffusion problem in C/C++. I have developed one for generation pressure diffusion but facing some artefact's, so would be nice if somebody can help me.
Thanks
Source Book: Burden R.L., Faires J.D. Numerical analysis
That code looks more like c than c++.
http://www.nr.com/" [Broken] might have the algorithms that you are looking for.
Last edited by a moderator:
Hi,
First Thanks for your reply, but I am afraid your hint could not help me. The book "Numerical Analysis by Burden" you recommended, I downloaded a soft copy of it, but nowhere I could find anything about ADI method (alternating direction implicit). If you sure there is ADI algorithm in this book, then would you pleas tell me the specific pages where you found that algorithm.
I too had look "Numerical recipes", but there they only gave some introduction about ADI, no alogorithm the same in some other books like "Finite Difference Methods in Financial Engineering".
Thanks,
Irfan
{snip}
I too had look "Numerical recipes", but there they only gave some introduction about ADI, no alogorithm the same in some other books like "Finite Difference Methods in Financial Engineering".
Thanks,
Irfan
Hi Irfan,
The code here:
contains an implementation of the "innocent" ADI method from the book you mention
(if you mean Daniel Duffy's book). As noted in the code comments (and the book),
it doesn't give good results for an odd number of steps, but maybe it could give you
some idea of how to implement another ADI method that does. I'm actually trying to do
that at the moment.
HTH.
-regards,
Larry
|
2019-12-15 23:54:04
|
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|
https://park.is/blog_posts/20210605_postgres_trigger_time_between_two_dates/
|
A SQL trigger is a function that is automatically invoked when an event occurs. To set a trigger, you specify the following:
• which table(s) and column(s) to listen to
• which actions to listen to (e.g., INSERT, UPDATE, DELETE)
• what function or procedure to invoke.
Triggers have many use cases. Some examples are:
• Create an audit trail: You have a table that contains sensitive information. You create a trigger to record all changes (what change has been made, which user has made the change, when the change has been made) to a separate table.
• Integrity check: Before adding a student to a class roster, you want to ensure that the student’s previous course enrollments fulfill prerequisites.
• Derive additional data: You have a table that contains the start and the end time of an online exam for test-takers. When a student is done with the exam, you want to calculate how long the student took to complete the exam using the start and end timestamps. Note that this can also be implemented using a generated column.
This post will implement the last example. You create a test-taking web app where you mark the start time (start_time) when the student hits the “Start” button. Once the student finishes taking the exam and clicks on “Submit”, the web app will mark the finish time (end_time). You want to automatically calculate how long the test-taker spent on the exam in seconds.
my_table columns:
Column Type
id (pk) integer
start_time timestamp
end_time timestamp
duration integer
First, define a row-level trigger that runs on each row.
CREATE OR REPLACE FUNCTION update_total_duration()
RETURNS TRIGGER AS $body$
BEGIN
IF NEW.start_time IS NOT NULL AND NEW.end_time IS NOT NULL THEN
NEW.duration = EXTRACT(EPOCH FROM (NEW.end_time - NEW.start_time)) AS INTEGER;
END IF;
****RETURN NEW;
END;
$body$ LANGUAGE plpgsql;
Then, add a statement-level trigger that runs whenever the end_time field has been updated.
CREATE TRIGGER on_end_time_update BEFORE
UPDATE OF end_time ON my_table
FOR EACH ROW EXECUTE PROCEDURE update_total_duration();
Voilà! This trigger will now automatically calculate the duration whenever a test-taker completes the test.
|
2021-09-23 19:57:20
|
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|
http://www.chemgapedia.de/vsengine/vlu/vsc/en/ch/12/oc/vlu_organik/substitution/sn_e_konkurrenz/sn_e_konkurrenz.vlu/Page/vsc/en/ch/12/oc/substitution/sn_e_konkurrenz/a2_2_sn2_e2_base/sn2_e2_base.vscml.html
|
# SN/E Competition (overall)
## The Role of the Base in the SN2 / E2 Competition
Hint
Due to the lone electron pair, the base applied in an E2 elimination may basically also act as a nucleophile in a SN2 mechanism.
Fig.1
In an SN2 reaction, the nucleophile attacks the heteroatom-substituted carbon atom inside the molecule. In contrast, the base in an E2 elimination abstracts a proton that is located closer to the periphery of the molecule. SN2 reactions are therefore influenced by steric limitations to a considerably greater degree than E2 eliminations.
The nucleophilicity of a base in an SN2 reaction is reduced to a large degree when the base contains massive, sterically demanding substituents. In this case, the reaction rate of the SN2 reaction is decreased, as the energy of the is increased by the interactions of the sterically demanding nucleophile (base) and the substrate molecule. In contrast, the energy of the transition state of the E2 elimination is influenced by bulky substituents of the base to a noticeably lesser degree.
Thus, for the purpose of controlling the SN2/E2 competition, the following consequences pertaining to the type of base applied result:
### Favouring the E2 mechanism
• In favour of the E2 elimination, as strong as possible, non-nucleophilic bases with bulky, sterically-demanding substituents are usually applied.
• Some examples of weaker, but, nevertheless, sterically-demanding, non-nucleophilic bases are DBN (1,5-diazabicyclo[4.3.0]nonene) and DBU (1,8-diazabicyclo[5.4.0]undecene).
• Some examples of particularly strong, non-nucleophilic bases are the lithium dialkylamides LDA (lithium diisopropylamide) and LHMDS (lithium hexamethyldisilazide).
• When DBN, DBU, LDA, or LHMDS are applied, even chemoselective E2 eliminations with primary and secondary alkyl halogenides may be achieved. This is exceptional, as in these cases the SN2 reaction is normally preferred over the E2 elimination - particularly in the connection with primary alkyl compounds.
Fig.2
Fig.3
Fig.4
Fig.5
Fig.6
Fig.7
Structures of very bulky and sterically demanding bases.
• The strong and relatively bulky base potassium t-butoxide is also frequently applied to chemoselective (E2 instead of SN2) E2 eliminations, as it is more easily available than LDA, LHMDS, DBN, and DBU.
• Instead of the special, very bulky and sterically demanding bases LDA, LHMDS, DBN, and DBU that are applied in difficult cases, the more readily available bases hydroxide, alkoxides (also primary and secondary) and amide are often used in E2 eliminations even if they are much less bulky and therefore less chemoselective.
### Favouring the SN2 mechanism
• In order to favour the SN2 reaction and largely prevent the E2 elimination, the base applied must not be too strong. As basicity increaes, nucleophilicity generally increases, as well. This is due to the fact that these two properties are both correlated with the availability of lone, non-bonding electron pairs. However, the basicity usually increases to a higher degree than the nucleophilicity does - that is, with increasing basicity the E2 elimination continuously exceeds the SN2 reaction more and more. This effect is observed even the base is not bulky and sterically demanding. However, it is larger in connection with such a base.
• SN2 reactions without any considerable E2 eliminations as side reactions can, therefore, only be obtained with good nucleophiles that show a basicity that is as low as possible.
Hint
According to general rule, nucleophiles with a lower basicity than that of the hydroxide anion tend to react with primary and secondary alkyl halides, for instance, in an SN2 reaction, while nucleophiles with a higher basicity than that of the hydroxide anion tend to react in an E2 elimination.
• However, the nucleophile applied in an SN2 reaction cannot be selected as freely as the base in an E2 elimination, since this nucleophile becomes a part of the product molecule, while the base in an E2 elimination does not. Thus, in selecting a suitable nucleopile for an SN2 reaction, one must also keep in mind the product structure for which it is intended.
• The electrophilic carbon atom in an SN2 reaction reacts particularly well with a soft , as it is, in fact, a soft . The best prerequisite for chemoselective (SN2 instead of E2) SN2 reactions is thus the application of relatively weak and soft bases that are also good nucleophiles. These are bases that possess an easily available and polarizable lone electron pair but display just a low basicity (see also HSAB principle). Such bases are, for example, anions such as hydrogensulphide (HS¯), alkyl sulphides (RS¯), iodide (I¯), and cyanide (CN¯).
Fig.8
HS¯ › RS¯ › ArS¯ › I¯ › CN¯ › OH¯ › $N3$¯ › Br¯ › ArO¯ › Cl¯ › AcO¯ › $H2O$
Nucleophilicity of several weak and moderately strongly basic nucleophiles in SN2 reactions.
Page 3 of 9
|
2020-07-11 14:35:53
|
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|
https://simple.wikipedia.org/wiki/Digamma
|
# Digamma
In mathematics, the name "digamma" is used in digamma function, which is the derivative of the logarithm of gamma function (that is, ${\displaystyle (\ln \Gamma (z))'}$).[1][2][3]
|
2021-03-04 23:20:35
|
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|
https://brilliant.org/problems/area-28/
|
# Area
Geometry Level 1
In the rectangle shown above, if the area of the green region is 25, what is the area of the blue region?
×
|
2018-06-24 22:47:29
|
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|
https://www.perimeterinstitute.ca/research/conferences/convergence/roundtable-discussion-questions/how-do-we-confront-unsatisfying
|
How do we confront unsatisfying theories that work too well, for example, GR, QM, the SM, and LCDM?
Currently there no comments for this section
[Bad Link: Plugin Not Found]
|
2018-05-21 01:43:48
|
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|
https://asmedigitalcollection.asme.org/micronanomanufacturing/article/4/4/041007/380417/Polishing-Characteristics-of-Transparent
|
Transparent polycrystalline yttrium aluminum garnet (YAG) ceramics have garnered an increased level of interest for high-power laser applications due to their ability to be manufactured in large sizes and to be doped in relatively substantial concentrations. However, surface characteristics have a direct effect on the lasing ability of these materials, and a lack of a fundamental understanding of the polishing mechanisms of these ceramics remains a challenge to their utilization. The aim of this paper is to study the polishing characteristics of YAG ceramics using magnetic field-assisted finishing (MAF). MAF is a useful process for studying the polishing characteristics of a material due to the extensive variability of, and fine control over, the polishing parameters. An experimental setup was developed for YAG ceramic workpieces, and using this equipment with diamond abrasives, the surfaces were polished to subnanometer scales. When polishing these subnanometer surfaces with 0–0.1 μm mean diameter diamond abrasive, the severity of the initial surface defects governed whether improvements to the surface would occur at these locations. Polishing subnanometer surfaces with colloidal silica abrasive caused a worsening of defects, resulting in increasing roughness. Colloidal silica causes uneven material removal between grains and an increase in material removal at grain boundaries causing the grain structure of the YAG ceramic workpiece to become pronounced. This effect also occurred with either abrasive when polishing with iron particles, used in MAF to press abrasives against a workpiece surface, that are smaller than the grain size of the YAG ceramic.
## Introduction
Solid-state lasers have traditionally used monocrystalline gain media. The first such example was a ruby-based laser created by Maiman in 1960 [1,2]. Continuous-wave laser oscillation using single crystal Nd:YAG was successfully accomplished not long after, in 1964 [3]. Translucent YAG ceramics were developed as gain media in the 1980s; however, they performed poorly due low-optical-grade properties [46]. It was not until the mid-1990s when Ikesue et al. developed transparent Nd:YAG ceramics of high enough optical quality to produce successful laser oscillation [7]. Since this time, it has been shown that laser oscillations can be obtained with YAG ceramics, which are comparable or even superior to those of monocrystalline YAG [8,9]. Polycrystalline gain media can be scaled to much larger sizes, are relatively economical, and can undergo heavy doping [1013]. As such, polycrystalline host materials have garnered an increased level of interest for high-power applications.
These advanced ceramics, however, have structural challenges that must be overcome. Conventional polycrystalline ceramics have a variety of light-scattering sources, which can result in lower laser power output and slope efficiencies. Refractive index modulation can occur at the grain boundaries, and any inclusions or pores can cause index changes. Birefringence can be a concern, as well as scattering at the surface caused by roughness [14,15]. The internal scattering sources have been diminished substantially with modern fabrication techniques; however, surface roughness can still have great effects on lasing ability. In addition to the scattering that can occur due to surface roughness, it has been shown that laser damage threshold is greatly affected by surface conditions [16]. Surface characteristics are important when bonding the ceramics to make a larger composite or when applying coatings. Defects in the bonding zone or under a coating can cause scattering centers in the interior of these composites. Since the surface finish of polycrystalline YAG ceramics significantly influences lasing performance, it is necessary to understand the polishing characteristics of this material.
Poly- and mono-crystalline materials have been polished by a variety of techniques. Diamond is often used due to its relative hardness compared to that of the ceramics. Colloidal silica is also used for the polishing of ceramics due to its ability to chemically react with ceramic materials and reduce subsurface damage [17]. To better control the polishing of polycrystalline YAG ceramic, it is important to clarify the material removal mechanisms that these abrasives have on this material.
Magnetic field-assisted finishing (MAF) has proven to be a promising technology for overcoming problems associated with more traditional polishing techniques and the extensive variability of, and fine control over, polishing parameters makes it a useful process for studying the polishing characteristics of a material. Through control of magnetic fields, magnetic particles and abrasives can be manipulated against and across surfaces with precision. This process can be used with a large variety of abrasives, and cutting force and depth can be controlled through abrasive- and iron-particle size selection. Moreover, magnetic field-assisted finishing techniques have been shown to be successful in the fine finishing of optical components [18], and ultrafine finishes were achieved using MAF on thin quartz wafers [19]. The MAF process lends itself to precision polishing as well as localized material removal and is thus a potentially favorable technique for laser gain media.
This paper discusses the refinement of the MAF process for the analysis of the polishing characteristics of YAG ceramics. Furthermore, this paper discusses the effects and material removal mechanisms of diamond and colloidal silica abrasives on polycrystalline YAG ceramics using MAF.
## Processing Principle
Figure 1 shows a schematic of the MAF processing principle used for wafer finishing. A permanent magnet is attached to a rotating table at a specific offset from the axis of rotation, referred to as eccentricity. This acts as the magnetic field generator and will be referred to as the table magnet. The workpiece is secured to a holder above the rotating table, and iron particles are placed on the workpiece. An additional permanent magnet (referred to as the tool magnet) is put on the iron particles. The iron particles align with the magnetic field lines generated between the table magnet and tool magnet creating a freeform brush. Abrasives are then introduced into the finishing zone between the iron brush and the workpiece surface. The magnetic force F acting on the tool magnet and iron particles, pressing the abrasives against the surface, is described by the following equation:
$F=VχH⋅gradH$
(1)
where V is the volume of the magnetic particle, χ is the susceptibility, and H and gradH are the intensity and gradient of the magnetic field, respectively. By modifying the volume of the magnetic particles, as well as the magnetic properties, size, and arrangement of the magnets utilized, the polishing force can be controlled.
As the table magnet rotates about a central axis, the tool magnet and iron particles are pulled across the surface of the workpiece, providing the motion required for finishing. In addition to the tool magnet rotating about the table magnets rotational axis, the tool magnet rotates about its own central axis. This phenomenon is referred to as self-spinning and is caused by a tangential velocity gradient across the tool magnet as it follows the table magnet. The self-spinning of the tool magnet creates intersecting cut marks and encourages the introduction of fresh abrasive cutting edges.
The experimental setup, shown in Fig. 2, was developed to realize this processing principle (with some refinements) for the polishing of polycrystalline YAG ceramic slabs. Due to the hardness and relative thickness (7.4 mm) of the YAG ceramic, a larger polishing force is required than when compared to the force necessary for 60 μm thick quartz polishing performed by Yamaguchi et al. [19]. Nd-Fe-B rare earth tool and table magnets were thus selected. The workpiece is held in place by a nonferrous holder, and the permanent table magnet is placed on a ferrous bar fixed to the rotating table. The distance between the bottom of the holder and table magnet is adjustable, and the holder table can move linearly, allowing for the polishing of the entire rectangular area of the surface.
As Fig. 3 shows, the iron particles initially line up along the lines of magnetic force between the tool magnet and workpiece surface. However, during rotation, the iron particles gradually climb to the uppermost surface of the tool magnet. As the table magnet rotates, the tool magnet and iron particles are dragged across the workpiece surface. The frictional force between the workpiece surface and loose iron particles naturally wants to drive the iron particles out of the interface between the tool magnet and workpiece surface. The iron particles in the interface push the iron particles near them, and the iron particles flow as a whole along the magnetic field lines to the top surface of the tool magnet. If nothing impedes this flow of the iron particles, a majority will eventually leave the interface and the tool magnet will directly interact with the workpiece surface, damaging the workpiece. To prevent this from occurring, the tool magnet was fitted with a cap, which has a diameter larger than the tool magnet. The cap was selected to be a weak rubber magnet so that it will easily and securely attach to the uppermost surface of the tool magnet. The rubber magnet cap (magnetic flux density: 1.2 mT at the center of the surface) does not greatly influence the magnetic field of the Nd-Fe-B tool magnet (magnetic flux density: 285.7 mT at the center of the surface) due to its relative weakness. This cap physically prevents the iron particles from being pushed to the top surface of the tool magnet by the iron particles in the interface. The iron particles cannot flow from the interface and the iron brush is maintained, preventing any contact between tool magnet and workpiece.
## Polishing Characteristics
In this study, polishing experiments of polycrystalline YAG ceramic slabs focused on abrasive type, iron-particle size, and polishing time. The first set of experiments was performed to analyze the effects of diamond abrasive size on surface roughness. These experiments allowed the surface to be brought to subnanometer arithmetic average roughness Sa. The subsequent experiments were performed to analyze the effects of fine diamond and colloidal silica on this subnanometer surface.
Creating smooth and consistent tool motion is necessary for achieving a fine polishing process. Through experimentation it was found that the table magnet revolution speed and the magnetic field intensity were contributing factors to successful tool motion. At slow table magnet revolution speeds, the tool magnet has a tendency to stutter in its motion, and at high speeds, the tool magnet can be thrown from the finishing area due to the centrifugal force. These trends were also confirmed in previous research [19]. The selected experimental conditions, listed in Table 1, produced the most stable tool motion and remained unchanged during the course of experimentation.
The center of the workpiece was found and designated as the origin of the Cartesian coordinates (Table 1). Surface roughness measurements were taken from the origin and every 1 mm for 10 mm in both the positive and negative X directions. These are the locations of the 21 measurements referred to in the “Rough Polishing with Diamond Abrasive”, “Fine Polishing with Diamond Abrasive”, and “Fine Polishing with Colloidal Silica” sections.
## Rough Polishing With Diamond Abrasive
The experiments presented in this section utilized 0.7 g of 44–149 μm mean diameter iron particles. Three different surfaces with dissimilar initial surface conditions (referred to as surface 1, surface 2, and surface 3) were polished with 0–2 μm diameter diamond abrasive for 30 mins. Surfaces 2 and 3 were then polished with 0–0.5 μm and 0–0.25 μm diameter diamond abrasive for 60 mins in series. Surface 2 was subsequently polished with 0–0.1 μm diameter diamond abrasive for 60 mins.
Figure 4 shows the roughness Sa averaged across all 21 measurement points, before and after every polishing stage for each surface. Figure 5 displays a three-dimensional oblique plot of a representative point from each surface before and after polishing with 0–2 μm diameter diamond abrasive. Surface 1 was heavily pitted and has a substantial standard deviation across the various measurements of the surface (Fig. 4). Surface 2 had many pillarlike structures and an average roughness that is nearly double that of surface 1. Surface 3 had many deep scratches and an average roughness that is more than double that of surface 2. After the process was performed, all three surfaces had similar roughness values with a similarly low standard deviation across the measurement points.
As the diamond abrasive size was stepped down, the roughness decreased and the standard deviation stayed relatively small, showing the uniformity of the surface. After polishing for 60 mins with the 0–0.25 μm diameter diamond abrasive, the surface roughness reached subnanometer levels for both surfaces 2 and 3. However, it was found that once the diamond abrasive size dropped to 0–0.1 μm, the roughness increased substantially for surface 2 and did not continue to decrease. The standard deviation of the measured data points also increased substantially, suggesting that the effect was not uniform across the surface. To better understand the mechanisms behind this behavior, subsequent experiments were performed on the effects of polishing this material with fine diamond abrasive once the surface had already achieved subnanometer levels.
## Fine Polishing With Diamond Abrasive
A series of polishing tests were performed to better understand the behavior of this material in response to polishing with 0–0.1 μm diameter diamond abrasive. Prior to every experiment performed in this section, the surface of the workpiece was returned to the subnanometer scale, using the 0–0.25 μm diameter diamond abrasive polishing process described in the “Rough Polishing with Diamond Abrasive” section.
The workpiece, after being returned to the subnanometer scale, was polished in 5 mins increments for 15 mins using the 0–0.1 μm mean diameter diamond abrasive with 0.7 g of 44–149 μm iron particles. This process was performed twice, referred to as fine diamond test 1 and fine diamond test 2. The roughness Sa at all 21 measured positions after each 5 mins polishing stage for fine diamond test 1 and fine diamond test 2 are shown in Figs. 6 and 7, respectively. There did not appear to be a direct correlation between the measurement location and the category the surface at that position fell within.
Fine diamond test 1 resulted in positions with roughness values that had dramatic worsening, average gradual worsening, and average gradual improvement with increased polishing time. Fine diamond test 2 resulted in three positions with average gradual worsening, whereas the remaining positions had average gradual improvement with increased polishing time.
Figure 8 shows three-dimensional oblique plots of representative examples of positions that saw gradual improvement, gradual worsening, and dramatic worsening. The gradually improving position, indicated in Fig. 6 and displayed in Fig. 8, was found at X = −2 mm in fine diamond test 1. After the 0–0.25 μm diameter diamond abrasive preprocessing phase, the surface at the improving position had an initial roughness of 0.8 nm Sa. The deepest valley was measured to be 12.5 nm, and the deeper scratches were relatively evenly distributed across the surface. The widest valleys present on the surface were roughly 3 μm. However, the widths of most defects were less than 2 μm. After polishing with the 0–0.1 μm diameter diamond abrasive for 5 mins, the roughness improved to 0.6 nm and continued to improve after 10 and 15 mins of polishing time.
The gradually worsening position, indicated in Fig. 7 and displayed in Fig. 8, was found at X = 6 mm in fine diamond test 2. After the preprocessing stage, the surface at this worsening position had an initial roughness of 1.2 nm Sa, and the deepest valley was measured to be 17.1 nm. The width of the large scratch present on the surface was approximately 5–6 μm. After polishing with the 0–0.1 μm diameter diamond abrasive for 5 mins, the roughness worsened to 1.3 nm, and while the depth of the valley did not significantly change, the width grew to approximately 8–12 μm. The surface continued to gradually worsen after 10 and 15 mins of polishing time.
The dramatic worsening position, indicated in Fig. 6 and displayed in Fig. 8, found at X = 6 mm in fine diamond test 1, had a lower initial roughness (1.0 nm Sa) when compared to the gradually worsening position; however, the deepest valley at this position was measured to be 27.7 nm and two relatively deep scratches were located in close proximity. The scratches were not substantially wider than the scratch located on the gradually worsening position; however, after polishing with the 0–0.1 μm diameter diamond abrasive for 5 mins, a large section of material was removed from the area between these deep scratches. The roughness at this position worsened dramatically as a result. The roughness improved gradually with each additional 5 min process. After the chipping occurred, the sharp edges of the chip zone were smoothed. This caused a leveling of the surface and a drop in the roughness value.
The deepest valley at every position was recorded prior to polishing with fine diamond, and in general, the width of the defect was related to the depth: deeper defects were generally wider than shallow defects. It was found that the positions that saw an average gradual worsening had an average initial deepest valley of 18.8 nm in fine diamond test 1 and 14.2 nm in fine diamond test 2. The widths of these defects were generally in the 3–6 μm range. After polishing for 5 mins, the widths of the larger defects increased to approximately 5–10 μm. The widths continued to increase to 10–15 μm and 15–20 μm after 10 and 15 mins of polishing time, respectively.
Positions that saw an average gradual improvement had an average initial deepest valley of 12.9 nm in fine diamond test 1 and 8.8 nm in fine diamond test 2. While some localized defects on these surfaces were approximately 5 μm wide, the majority of the defects were less than approximately 2 μm. The larger defects widened with increased polishing time; however, the majority of the defects reduced in width.
The depth and width of the surface scratches prior to polishing with a slurry of 0–0.1 μm diameter diamond abrasive, and 44–149 μm iron particles have an effect on whether the roughness of the surface will improve or worsen with fine diamond polishing. The iron particles must press the diamond abrasive into and across the surface for material removal to occur. When the width of a defect is sufficiently small, the iron will pass over the defect as it moves across the surface. Despite the possibility of abrasive entering the defect, there is no material removal within the defect as the iron cannot apply pressure to the abrasive in that region. Material is removed from the surface surrounding the defect, effectively reducing the depth and width of the scratching.
The larger the initial width of the defect, the more likely an iron particle is able to partially penetrate the defect. The iron particle is then able to press the diamond abrasive into the workpiece in the defect and cause material removal. This causes a widening of the defect, which, in turn, allows more iron to penetrate and more material to be removed within the defect. When several large defects are in close proximity, chips can form and be ejected from the surface.
The average roughness Sa across all 21 measurement positions of the surface prior to fine diamond test 1 was 0.8 nm and the average deepest valley was 15.5 nm. The initial surface had many deep and wide scratches, which worsened with additional polishing time. The surface had positions with high concentrations of scratching resulting in chipping in these regions. At this level of roughness, the chipping that occurred during polishing had a much more dramatic effect on the roughness values of the surface than the subsequent smoothing that this size abrasive could produce. Due to the defects existing on the initial surface, the surface roughness averaged across all measured points continued to climb with additional polishing time for fine diamond test 1. The standard deviation of the measured values increased with polishing time showing the unevenness of the process across the surface.
The average roughness of the surface prior to fine diamond test 2 was higher at 0.9 nm; however, the average deepest valley was only 9.5 nm. The scratching on the surface prior to fine diamond test 2 was much more evenly distributed and not of the same depth and width as in fine diamond test 1. Despite having a few measurement positions see a gradual worsening, the average roughness across all measurement points improved. However, the standard deviation of the measured roughness values continued to climb as the disparity between the improving positions and worsening positions increased.
When polishing a surface with subnanometer roughness values with 0–0.1 μm mean diameter diamond abrasive, it became apparent that the specific features left on the surface as a result of the preprocessing stage drive the polishing characteristics of this process. In contrast to the experiments performed in section “Rough Polishing with Diamond Abrasive”, the initial surface conditions and localized defects had a significant effect on the resulting surface after polishing. The surface prior to polishing with fine diamond needs to have minimal, disperse scratching to see an improvement in average surface roughness with continued polishing time.
To further understand the effect of the iron particle size in the polishing process, the workpiece, after being returned to the subnanometer scale, was polished using the 0–0.1 μm diameter diamond abrasive with three decreasing sizes of iron particles (44, 7, and 1 μm mean diameter) in series for 5 mins each. This process was performed twice, referred to as fine diamond test 3 and fine diamond test 4.
The roughness Sa, averaged across all 21 measured positions, after each 5 mins polishing stage for both fine diamond test 3 and fine diamond test 4 are shown in Fig. 9. The average roughness increased after polishing with 44 μm mean diameter iron particles from 0.9 nm to 1.7 nm for fine diamond test 3 and from 0.9 nm to 1.6 nm for fine diamond test 4. The average roughness then decreased after polishing with 7 μm mean diameter iron followed by an increase with the 1 μm mean diameter iron.
Figure 10 shows the topography of a representative measurement position (X = −2 mm, test 2) after each 5 mins of polishing. Similar to polishing with the 44–149 μm iron particles, the relatively shallow defects were removed, and the severity of the deep defects were intensified when the surface was polished with the 44 μm mean diameter iron particles.
The depth of the deepest valley, averaged across all 21 measurement positions prior to polishing, was 11.5 nm for diamond test 3 and 11.8 nm for diamond test 4. The widths of the defects were minimal. Through the last set of experiments described in this section, it was found that surfaces with these levels of scratching were prone to improving with extended polishing time using the fine diamond abrasive and 44–149 μm mean diameter iron particles. However, the surface roughness increased with the use of a mixture of fine diamond abrasive and 44 μm mean diameter iron particles.
Defects that would have otherwise been reduced by the large 149 μm iron particles passing over the scratches were instead worsened by the 44 μm iron particles that could penetrate into the defect. The total mass of the iron particles was held constant between experiments; therefore, more 44 μm iron particles are available to penetrate the defects when compared to the previous test with 44–149 μm iron particles. Larger scratches worsened at a more substantial rate resulting in an overall decline in the quality of the surface.
When the iron size was reduced to 7 μm mean diameter, the roughness value of the surface improved. Despite the reduced size of the iron particles, which allowed them to penetrate smaller defects, the magnetic force acting on the iron particles pressing the abrasive into the workpiece surface was greatly diminished.
As described by Eq. (1), the magnetic force acting on the iron particle pressing the abrasive against the surface is proportional to the volume of the iron particle.
When the force from the iron particle is reduced, the depth of cut and cutting force of the diamond abrasive are diminished. Despite the small iron particle being able to penetrate the defects, the diamond abrasive does not attack the material as aggressively and the defects are smoothed instead of worsened. However, despite being very subtle, the grains of the polycrystalline ceramic started to become pronounced, and this effect was further intensified as a result of polishing with the 1 μm mean diameter iron particles.
The grain size of this polycrystalline ceramic was found to be between 15 and 30 μm. The grain structure of the ceramic influenced the material removal as the iron particle size dropped below the material's grain size. Again, the force pressing the diamond abrasive against the surface of the workpiece drives the material removal. When the size of the iron particle is larger than the grain size, the iron presses abrasives into and across multiple grains simultaneously. The magnetic force acting on the large iron particle, pressing the abrasive into the surface, is relatively large. The cutting force and depth of cut is not drastically influenced by minor variations in strength between grains and grain boundaries. This results in relatively consistent material removal between grains.
When the size of the iron particle is smaller than the grain size, the iron particle presses abrasives into individual grains. The magnetic force acting on the small iron is relatively small and minor variations in strength between grains and grain boundaries can influence cutting depth of the abrasive. Moreover, the small iron particles can supply force directed at individual grain boundaries. The small iron particles can penetrate into the resulting cavities and material removal is increased at these sites. This results in uneven material removal between grains, and increased removal at the grain boundaries, causing the grain structure of the YAG ceramic workpiece to become increasingly apparent with additional polishing time.
## Fine Polishing With Colloidal Silica
To better understand the polishing effects of colloidal silica on transparent YAG ceramics, a series of polishing tests were performed using a slurry of 3 wt. % silica particles (7 nm mean diameter) in de-ionized water. Prior to every experiment performed in this section, the surface of the workpiece was returned to the subnanometer scale, using the 0–0.25 μm diameter diamond abrasive polishing process described in the “Rough Polishing with Diamond Abrasive” section.
The initial set of tests, similar to the experiments presented in the “Fine Polishing with Diamond Abrasive” section, used 44–149 μm iron particles, and the surface was polished with the colloidal silica in three 5 mins increments for 15 mins total. This test was performed twice: referred to as silica test 1 and silica test 2.
The roughness Sa, averaged across all 21 measured positions, after each 5 mins polishing stage for both silica test 1 and silica test 2 is shown in Fig. 11. The average roughness increases with each additional polishing stage.
Figure 12 displays the topography of a representative measurement position (X = −5 mm, silica test 2). As shown, while the minor defects dissipate, some of the larger defects intensify. More notably, despite the iron particle size being larger than the grain size, the grains of the polycrystalline ceramic became increasingly apparent and roughness continued to rise.
This shows that, despite having iron size above the grain size of the material, this process results in uneven material removal between grains and an increase in material removal at grain boundaries. While the iron particles must directly press diamond abrasive into the surface to remove material, this is not a requirement for colloidal silica due to its reactive nature with the YAG ceramic. The rate of material removal is effected by the amount of pressure applied and thus direct pressure from the iron particles pressing the silica into the surface will result in a high material removal at the interaction zone. However, material removal still occurs in regions such as defects and grain boundaries that these large iron particles cannot penetrate and apply direct pressure. Pressure is provided through the flow and downward force of the colloidal silica generated by the motion of the tool. Since the material removal rate is higher where the iron particles can supply direct pressure, the effect of the polycrystalline structure on the material removal is minimal although apparent.
To better understand the effect of the iron particle size in polishing with colloidal silica, the workpiece was polished using colloidal silica with three decreasing sizes of iron particles (44, 7, and 1 μm mean diameter) in series for 5 min each. This test was performed twice: referred to as silica test 3 and silica test 4.
Figure 13 displays the roughness, averaged across all 21 measurement points, for the surface after every polishing stage for both silica test 3 and silica test 4. Again, the average roughness increases with each additional polishing stage; however, there is a significant increase after polishing with the 7 μm mean diameter iron particles.
Figure 14 displays the topography at the center position (X = 0 mm) of the surface after each polishing stage during silica test 3. As shown by Figs. 14(a) and 14(b), after the 5 min silica process with 44 μm mean diameter iron particles, larger defects began to widen while minor scratches began to dissipate. There was a general increase in the roughness value of the surface.
After a process was performed with 7 μm mean diameter iron particles, the grains of the polycrystalline ceramic became very apparent. This effect was further intensified as a result of polishing with 1 μm mean diameter iron particles. Similar to polishing with the fine diamond abrasive, the grain structure of the ceramic influenced the material removal as the iron particle size dropped below the material's grain size, although it was much more dramatic with colloidal silica.
When the size of the iron particle is larger than the grain size, the iron particles can apply pressure to the colloidal silica across multiple grains simultaneously. Despite uneven material removal between grains and at grain boundaries caused by the flow of the reactive colloidal silica, the material removal rate is more evenly distributed across multiple grains.
When the size of the iron particle is smaller than the grain size, the iron can apply pressure to the colloidal silica locally within individual grains and at grain boundaries, adding to the material removal caused by the flow of the colloidal silica. This intensifies the uneven material removal rates between grains. The colloidal silica, 7 nm in diameter (compared to the 0–0.1 μm diameter diamond abrasive), is able to penetrate deeply into grain boundaries flow along individual grains. Accordingly, these conditions resulted in the most dramatic reveal of the grain structure compared to other experiments presented in this paper. This is shown by the sharp and pronounced borders of the individual grains in the topography of the surface shown in Fig. 14(d).
## Conclusion
The results of this study can be summarized as follows:
1. (1)
A Nd-Fe-B tool magnet with a cap is required for the polishing of transparent YAG ceramics with MAF. The Nd-Fe-B tool magnet was necessary for producing the magnetic force required for material removal. The cap, with a diameter larger than that of the tool magnet, was necessary to prevent iron particle motion, maintaining the iron particle brush between the tool magnet and workpiece surface.
2. (2)
MAF smooths transparent YAG ceramics with diamond abrasive to subnanometer levels despite large variability in initial surface conditions. When polishing with 0–0.1 μm mean diameter diamond abrasive, the initial surface conditions and specific defects influence the polishing process. The relationship between the size of the defects, the size of the iron particles, and the magnetic force acting on the iron particle pressing the abrasive into the surface drives the polishing characteristics of this process. When defects are sufficiently small, the iron particles will pass over the defect as they move across the surface. Material is removed from the surface surrounding the defect, effectively reducing the depth and width. However, when the defect is large, the iron particles can apply pressure onto the diamond abrasive within the defect, causing the defect to worsen. When several large defects are in close proximity, chips can form and be ejected from the surface. As the size of the iron particles is reduced, the magnetic force acting on the iron particles pressing the abrasive against the surface is lessened, resulting in diminished depth of cut and cutting force of the diamond abrasive which can improve surface roughness. However, when polishing is performed with fine diamond and iron particles smaller than the grain size of the YAG ceramic, uneven material removal between grains and increased removal at grain boundaries occurs. This caused the grain structure of the YAG ceramic workpiece to become increasingly pronounced with additional polishing time.
3. (3)
At subnanometer levels, MAF with colloidal silica abrasive caused a worsening of defects with increased polishing time, resulting in worsening roughness. Polishing with colloidal silica causes uneven material removal between grains and an increase in material removal at grain boundaries causing the grain structure of the YAG ceramic workpiece to become increasingly pronounced with additional polishing time. When polishing is performed with a mixture of colloidal silica and iron particles smaller than the grain size of the YAG ceramic, the uneven material removal between grains and increased removal at grain boundaries are more significant, causing the grain structure of the YAG ceramic to become very well defined.
## Acknowledgment
This material is based upon work supported by the Air Force Office of Scientific Research (AFOSR) under Award No. FA 9550-14-1-0270. The authors would also like to express their thanks to Dr. Akio Ikesue for showing his support by providing workpieces for experimentation.
## Nomenclature
•
• H =
magnetic field intensity
•
• min =
minute
•
• mT =
millitesla
•
• Sa =
arithmetical mean height of the surface
•
• V =
volume of the magnetic particle
•
• χ =
magnetic field susceptibility
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# Potential energy in concentric shells
1. Jan 16, 2016
### gracy
1. The problem statement, all variables and given/known data
There are two concentric shells of radii a and b respectively ,inner shell has charge Q and outer shell is neutral what will be potential energy of outer shell due to inner shell
2. Relevant equations
$V$=$\frac{KQ}{R}$
PE=Charge .V
3. The attempt at a solution
I know what will be potential of outer shell due to inner shell it will be $\frac{KQ}{b}$ but as outer shell is neutral it does not have any charge and hence potential energy of outer shell should be zero,because we know potential energy is equal to charge multiplied by potential. charge is zero on outer shell hence it's PE would also be zero.Right?
Last edited: Jan 16, 2016
2. Jan 16, 2016
### haruspex
Sounds good to me.
3. Jan 16, 2016
### BvU
Hello Gracy,
Right !
It looks to me as if you sense a contradiction somewhere ?
The electric field ( in 'all of space' ) with the outer shell in place is identical to the electric field without, so putting it in place shouldn't require energy.
I wouldn't say
but
potential at r = radius of outer shell due to inner shell will be $\frac{KQ}{b}$
this time haru was faster !
4. Jan 16, 2016
### gracy
Ok.There is a question.
A solid conducting sphere of radius a having a charge Q is surrounded by a conducting shell of inner radius 2a and outer radius 3a as shown.Find the amount of heat produced when switch is closed.
Heat produced=Initial energy -final energy
I have taken interaction energy to be zero on the basis of the answer of OP.And I have considered formula of self energy of conducting and and non conducting spheres to be same.
I know formula of self energy of conducting sphere is $\frac{KQ^2}{2R}$
Heat produced=$\frac{KQ^2}{4a}$ - $\frac{KQ^2}{6a}$
= $\frac{KQ^2}{12a}$
But it is wrong!
5. Jan 16, 2016
### TSny
In your original question, the outer shell was assumed to have negligible thickness so that the charge on the inner surface of that shell was essentially at the same location as the charge on the outer surface of the shell.
But now the outer shell has a significant thickness. You will need to reassess the initial potential energy.
6. Jan 16, 2016
### gracy
But I wrote the outer shell is neutral.So no charge in inner or outer surface of the outer shell.
7. Jan 16, 2016
### TSny
The net charge is zero on the outer shell (initially). But, the inner and outer surfaces each have charge.
8. Jan 16, 2016
### gracy
Are you hinting at induced charges?
9. Jan 16, 2016
### TSny
Yes.
10. Jan 16, 2016
### gracy
Why do you think that ?There is nothing such mentioned in the problem.
11. Jan 16, 2016
### TSny
According to the statement of the problem, the outer shell has an inner radius of 2a and an outer radius of 3a.
12. Jan 16, 2016
### gracy
What kind of changes shall I make in my calculations now?I don't know what is self energy of shells having significant thickness and I also don't know how to calculate interaction energy in such situation.
13. Jan 16, 2016
### TSny
If you have a system of static charges where an amount of charge Q1 is located where the potential is V1, an amount of charge Q2 is located where the potential is V2, and an amount of charge Q3 is located where the potential is V3, then the total potential energy of the system is
U = (1/2) (Q1V1 + Q2V2 + Q3V3)
This is a generalization of the formula for the potential energy of a capacitor: U = (1/2)QV.
Try to apply this to your system where you have three charges: charge on the sphere, charge on the inner surface of the shell, and charge on the outer surface of the shell.
U = (1/2) (QsphereVsphere + QinnerVinner + QouterVouter)
You should find that the last two terms simplify nicely. The work will be in getting the first term.
14. Jan 17, 2016
### gracy
I solved it as follows
But it is not correct where I went wrong.
15. Jan 17, 2016
### ehild
Why are the potentials at the surfaces of the outer shell different ?
16. Jan 17, 2016
### gracy
I calculated it. Just the sign is opposite.
17. Jan 17, 2016
### ehild
You did not show your calculation. The outer shell is a conductor. What do you know about the distribution of potential on a conductor?
18. Jan 17, 2016
### gracy
That it is same everywhere inside the conductor.
19. Jan 17, 2016
### gracy
20. Jan 17, 2016
### ehild
Is it different on the surfaces of the conductor?
Last edited: Jan 17, 2016
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2017-12-14 17:53:18
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http://community.boredofstudies.org/238/extracurricular-higher-level/360497/math2111-higher-several-variable-calculus-4.html
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# Thread: MATH2111 Higher Several Variable Calculus
1. ## Re: Several Variable Calculus
Originally Posted by InteGrand
$\noindent (Note that \left(\mathbf{r}\times \mathbf{v}\right)\cdot \mathbf{v} = 0, recalling that in general a scalar triple product like this is always zero if two of the vectors are the same.)$
Completely forgot about this haha
2. ## Re: Several Variable Calculus
$\text{Two metrics }\rho\text{ and }\delta \text{ are said to be topologically equivalent}\\ \text{iff every }\rho\text{-ball contains a }\delta\text{ ball}\\ \text{and every }\delta\text{-ball contains a }\rho\text{-ball}$
$\text{Two metrics }\rho\text{ and }\delta\text{ are said to be equivalent}\\ \text{iff }\exists c_1,c_2 > 0:\\ c_1\rho (\textbf{x},\textbf{y}) \le \delta (\textbf{x},\textbf{y}) \le c_2 \rho(\textbf{x},\textbf{y})$
$\text{Prove that equivalent metrics are topologically equivalent}$
3. ## Re: Several Variable Calculus
Originally Posted by leehuan
$\text{Two metrics }\rho\text{ and }\delta \text{ are said to be topologically equivalent}\\ \text{iff every }\rho\text{-ball contains a }\delta\text{ ball}\\ \text{and every }\delta\text{-ball contains a }\rho\text{-ball}$
$\text{Two metrics }\rho\text{ and }\delta\text{ are said to be equivalent}\\ \text{iff }\exists c_1,c_2 > 0:\\ c_1\rho (\textbf{x},\textbf{y}) \le \delta (\textbf{x},\textbf{y}) \le c_2 \rho(\textbf{x},\textbf{y})$
$\text{Prove that equivalent metrics are topologically equivalent}$
Here's some hints.
$\noindent Assume the given equivalence, i.e. \exists c_1,c_2 > 0:\\ c_1\rho (\textbf{x},\textbf{y}) \le \delta (\textbf{x},\textbf{y}) \le c_2 \rho(\textbf{x},\textbf{y}) for all \mathbf{x},\mathbf{y}.$
$\noindent Consider an arbitrary \delta-ball B_{\delta}\left(\mathbf{a},\varepsilon\right) (some radius \varepsilon > 0, centred at \mathbf{a}). Now show that the \rho-ball B_{\rho}\left(\mathbf{a}, \frac{\varepsilon}{c_{2}}\right) is a subset of the aforementioned \delta-ball. The other thing you need to show will follow by symmetry.$
4. ## Re: Several Variable Calculus
Originally Posted by InteGrand
Here's some hints.
$\noindent Assume the given equivalence, i.e. \exists c_1,c_2 > 0:\\ c_1\rho (\textbf{x},\textbf{y}) \le \delta (\textbf{x},\textbf{y}) \le c_2 \rho(\textbf{x},\textbf{y}) for all \mathbf{x},\mathbf{y}.$
$\noindent Consider an arbitrary \delta-ball B_{\delta}\left(\mathbf{a},\varepsilon\right) (some radius \varepsilon > 0, centred at \mathbf{a}). Now show that the \rho-ball B_{\rho}\left(\mathbf{a}, \frac{\varepsilon}{c_{2}}\right) is a subset of the aforementioned \delta-ball. The other thing you need to show will follow by symmetry.$
Is it basically just this?
\begin{align*}\text{Let }\textbf{x} &\in B_\delta (\textbf{a},\epsilon)\\ \implies \delta (\textbf{x},\textbf{a}) &< \epsilon \\ \implies c_1 \rho (\textbf{x},\textbf{a}) &< \epsilon\text{ for some }c_1\in \mathbb{R}^+\\ \implies \rho (\textbf{x},\textbf{a}) &< \frac{\epsilon}{c_1}\\ \implies x&\in B_\rho\left(\textbf{a},\frac{\epsilon}{c_1}\right) \end{align*}
Although that being said I had c_1 instead of c_2 but I feel that won't matter
5. ## Re: Several Variable Calculus
$\text{Bit confused. How is it possible for }\lim_{x\to 0}\lim_{y\to 0}f(x,y)\\ \text{and the reverse order of limiting to not exist}$
$\text{but then }\lim_{(x,y)\to (0,0)}f(x,y)\text{ exists}\\ \text{for }f(x,y)=(x+y)\sin \frac1x \sin \frac1y$
6. ## Re: Several Variable Calculus
Originally Posted by leehuan
$\text{Bit confused. How is it possible for }\lim_{x\to 0}\lim_{y\to 0}f(x,y)\\ \text{and the reverse order of limiting to not exist}$
$\text{but then }\lim_{(x,y)\to (0,0)}f(x,y)\text{ exists}\\ \text{for }f(x,y)=(x+y)\sin \frac1x \sin \frac1y$
$\noindent That example is showing that a multivariable limit can exist even though the corresponding iterated limits may not exist. If we hold x fixed and non-zero and limit y to 0, the limit won't exist because f(x, y) = x \sin \frac1x \sin \frac1y + y\sin \frac1x \sin \frac1y. The first term here doesn't have a limit as y\to 0, but the second one does (has limit 0, as you can show by the squeeze law), so overall the limit doesn't exist.$
$\noindent Of course, \lim_{(x,y)\to (0,0)}f(x,y) exists and equals 0, basically because \left| f(x,y)\right| \leq |x+y| for all x\neq 0 and y\neq 0.$
7. ## Re: Several Variable Calculus
Originally Posted by InteGrand
$\noindent That example is showing that a multivariable limit can exist even though the corresponding iterated limits may not exist. If we hold x fixed and non-zero and limit y to 0, the limit won't exist because f(x, y) = x \sin \frac1x \sin \frac1y + y\sin \frac1x \sin \frac1y. The first term here doesn't have a limit as y\to 0, but the second one does (has limit 0, as you can show by the squeeze law), so overall the limit doesn't exist.$
$\noindent Of course, \lim_{(x,y)\to (0,0)}f(x,y) exists and equals 0, basically because \left| f(x,y)\right| \leq |x+y| for all x\neq 0 and y\neq 0.$
Appears so counterintuitive though. I can't visualise what's going on here
8. ## Re: Several Variable Calculus
Originally Posted by leehuan
Appears so counterintuitive though. I can't visualise what's going on here
nobody can.....
9. ## Re: Several Variable Calculus
Consider the two metrics $d(\mathbf{x}, \mathbf{y}) = ||\mathbf{x} - \mathbf{y}||$ and
$\delta(\mathbf{x}, \mathbf{y}) = \frac{d(\mathbf{x}, \mathbf{y})}{1 +d(\mathbf{x}, \mathbf{y})}$ (you may assume they are metrics).
i) Show that d and δ are not equivalent.
10. ## Re: Several Variable Calculus
Originally Posted by QuantumRoulette
Consider the two metrics $d(\mathbf{x}, \mathbf{y}) = ||\mathbf{x} − \mathbf{y}||$ and
$\delta(\mathbf{x}, \mathbf{y}) = \frac{d(\mathbf{x}, \mathbf{y})}{1 +d(\mathbf{x}, \mathbf{y})}$ (you may assume they are metrics).
i) Show that d and δ are not equivalent.
I assume d(x,y) is supposed to be ||x-y|| and the set is some normed vector space V with norm ||.|| (e.g. R^d). (Please specify more if this is not the intended setting.)
Then these two metrics are not (strongly) equivalent because V is bounded with the delta metric but unbounded with the d metric.
That V is bounded with the delta metric follows immediately from the definition of delta, which must always lie in [0,1). On the other hand d(tx,0)=|t|d(x,0) can be made arbitrarily large for nonzero x.
Note that these two metrics ARE topologically equivalent though, in the sense that convergence in one metric implies convergence in the other. This follows from from the map x->x/(1+x) being a homeomorphism from [0,inf) to [0,1).
11. ## Re: Several Variable Calculus
Originally Posted by leehuan
Appears so counterintuitive though. I can't visualise what's going on here
Might not be easy to visualise the graph of the function on all of R^2, but you should certainly be able to visualise what it looks like on the slices x=const. or y=const which is all that matters for seeing/proving the nonexistence of the iterated limit. It is an oscillatory expression that oscillates faster as you approach axes. One of the two summands becomes irrelevant as you get close to the axes, so the other one dominates. This thing behaves like (const).sin(1/x), which of course does not converge unless that const is zero.
The boundedness of sine makes it clear that f(x,y) tends to zero as (x,y) tends to zero though.
Long story short: don't be too hasty to form intuitions in analysis, lots of things can behave weirdly...you kind of have to slowly build up a list of things that ARE true (via proof!) rather than assuming innocuous statements are true and ruling these things out as you come across pathological counterexamples.
And when you are looking at functions like this, try to isolate the terms that actually matter for the property you are trying to prove. A large part of analysis is just approximating ugly things by nice things, throwing away small sets on which a function behaves badly, etc etc.
12. ## Re: Several Variable Calculus
$\text{Prove that }\lim_{(x,y)\to (0,a)}\frac{\sin (xy)}{x}=a$
$\text{This result may be assumed: }\left| \frac{\sin a}{a}-1\right|=|a|^2, \, a\in \mathbb{R}$
13. ## Re: Several Variable Calculus
Originally Posted by leehuan
$\text{Prove that }\lim_{(x,y)\to (0,a)}\frac{\sin (xy)}{x}=a$
$\text{This result may be assumed: }\left| \frac{\sin a}{a}-1\right|=|a|^2, \, a\in \mathbb{R}$
$\noindent Here's the gist of a way to do it. For x\neq 0 and y\neq 0, we have$
\begin{align*}\left|\frac{\sin (xy)}{x} - a \right| &= \left|y\right|\left|\left(\frac{\sin (xy)}{xy} -1\right) + \left(1 - \frac{a}{y}\right) \right| \\ &\leq \left|y \right| \left(\left|\frac{\sin (xy)}{xy} -1 \right| + \left|1 - \frac{a}{y} \right| \right) \quad (\text{triangle inequality}) \\ &\leq \left|y\right| \left|xy\right|^{2} + \left|y-a\right| \quad (\text{using the given assumable result}). \end{align*}
$\noindent You should be able to finish from here.$
14. ## Re: Several Variable Calculus
Originally Posted by leehuan
$\text{Prove that }\lim_{(x,y)\to (0,a)}\frac{\sin (xy)}{x}=a$
$\text{This result may be assumed: }\left| \frac{\sin a}{a}-1\right|=|a|^2, \, a\in \mathbb{R}$
|sin(x)/x-1|=|x|^2 ?
Surely you mean something like:
|sin(x)/x-1| = O(|x|^2).
15. ## Re: Several Variable Calculus
Originally Posted by seanieg89
|sin(x)/x-1|=|x|^2 ?
Surely you mean something like:
|sin(x)/x-1| = O(|x|^2).
There was a typo; it was meant to be a less than or equals to. I think InteGrand just ignored or figured I had typo'd and just used the correct inequality.
__________________________________________________ ______________
$\text{Prove that }\mathbb{Q}\text{ is not open nor closed}$
$\text{So I proved it was not open by showing that for }0\in \mathbb{Q}\\ \forall r > 0,\, \frac{r}{\sqrt2}\text{ fails for }r\neq \alpha\sqrt{2}, \, \frac{r}{\sqrt3}\text{ fails for }r=\alpha \sqrt2\\ \text{where }\alpha \in \mathbb{Q}$
As in, for every ball around 0 there's always an irrational number in it, therefore 0 is not an interior point so it cannot be open. (I hope I did not screw this up.)
____________________________
$\text{But now I'm a bit stuck on proving it's not closed, i.e. }\mathbb{R} - \mathbb{Q}\text{ is not open}$
$\text{Progress: Take }\sqrt{2},\text{ then }\forall r > 0\\ \text{If }y\in B(\sqrt{2},r)\\ \text{Then }|y-\sqrt{2}|
I wanted to use the floor/ceiling functions but then I realised that'd be problematic if r = 0.1, so what might be a good choice for a rational number that is in every ball B(sqrt2, r)?
(Sorry, I think I worded my question horribly)
16. ## Re: Several Variable Calculus
Originally Posted by leehuan
There was a typo; it was meant to be a less than or equals to. I think InteGrand just ignored or figured I had typo'd and just used the correct inequality.
__________________________________________________ ______________
$\text{Prove that }\mathbb{Q}\text{ is not open nor closed}$
$\text{So I proved it was not open by showing that for }0\in \mathbb{Q}\\ \forall r > 0,\, \frac{r}{\sqrt2}\text{ fails for }r\neq \alpha\sqrt{2}, \, \frac{r}{\sqrt3}\text{ fails for }r=\alpha \sqrt2\\ \text{where }\alpha \in \mathbb{Q}$
As in, for every ball around 0 there's always an irrational number in it, therefore 0 is not an interior point so it cannot be open. (I hope I did not screw this up.)
____________________________
$\text{But now I'm a bit stuck on proving it's not closed, i.e. }\mathbb{R} - \mathbb{Q}\text{ is not open}$
$\text{Progress: Take }\sqrt{2},\text{ then }\forall r > 0\\ \text{If }y\in B(\sqrt{2},r)\\ \text{Then }|y-\sqrt{2}|
I wanted to use the floor/ceiling functions but then I realised that'd be problematic if r = 0.1, so what might be a good choice for a rational number that is in every ball B(sqrt2, r)?
(Sorry, I think I worded my question horribly)
Some rational points arbitrarily close to sqrt(2) are points where we truncate the decimal expansion of sqrt(2) arbitrarily far. I.e.
$\lfloor 10^{n}\sqrt{2}\rfloor 10^{-n}$
for positive integers n.
17. ## Re: Several Variable Calculus
Originally Posted by InteGrand
Some rational points arbitrarily close to sqrt(2) are points where we truncate the decimal expansion of sqrt(2) arbitrarily far. I.e.
$\lfloor 10^{n}\sqrt{2}\rfloor 10^{-n}$
for positive integers n.
That is one beautiful trick.
_____________________________________
$\text{RTP: }\Omega= \{(x,y)\in \mathbb{R}^2: \, x^2-y^2<1 \}\text{ is open.}$
I definitely don't want to use the Lagrange multiplier theorem to find the closest point (x,y) to work off, so any pointers on what radius I should choose my ball to be? (to prove an arbitrary point (x,y) is interior)
18. ## Re: Several Variable Calculus
Originally Posted by leehuan
That is one beautiful trick.
_____________________________________
$\text{RTP: }\Omega= \{(x,y)\in \mathbb{R}^2: \, x^2-y^2<1 \}\text{ is open.}$
I definitely don't want to use the Lagrange multiplier theorem to find the closest point (x,y) to work off, so any pointers on what radius I should choose my ball to be? (to prove an arbitrary point (x,y) is interior)
That set Omega is an open ball already (using the Euclidean metric), and an open ball is an open set (you should prove this as an exercise if you haven't before).
19. ## Re: Several Variable Calculus
Mate, stop cheating on these assignment questions. Do it yourself ffs.
20. ## Re: Several Variable Calculus
Originally Posted by MATH2111
Mate, stop cheating on these assignment questions. Do it yourself ffs.
...?
These are homework questions I'm stuck on. I'd be too worried about getting a 0 mark for the course if these were assignment questions.
21. ## Re: Several Variable Calculus
Originally Posted by InteGrand
That set Omega is an open ball already (using the Euclidean metric), and an open ball is an open set (you should prove this as an exercise if you haven't before).
Wait, I can't see it - Can't tell how a region cut off by the rectangular hyperbola is a ball
22. ## Re: Several Variable Calculus
Originally Posted by leehuan
Wait, I can't see it - Can't tell how a region cut off by the rectangular hyperbola is a ball
Sorry misread it (saw + instead of -).
One way is to recall that the preimage of an open set under a continuous map is open.
23. ## Re: Several Variable Calculus
$\text{Suppose that }f:\mathbb{R}^n\to \mathbb{R}\text{ is such that the sets}\\ \{\textbf{x}\in \mathbb{R}^n:f(x)>d \}\\ \{\textbf{x}\in \mathbb{R}^n:f(x)
$\text{Prove that }f\text{ is continuous}$
24. ## Re: Several Variable Calculus
Originally Posted by leehuan
$\text{Suppose that }f:\mathbb{R}^n\to \mathbb{R}\text{ is such that the sets}\\ \{\textbf{x}\in \mathbb{R}^n:f(x)>d \}\\ \{\textbf{x}\in \mathbb{R}^n:f(x)
$\text{Prove that }f\text{ is continuous}$
$\noindent Let \mathbf{a} \in \mathbb{R}^{n} be fixed and arbitrary. By definition of continuity, RTP:$
$for all \varepsilon > 0 there exists an open ball B\left(\mathbf{a},\delta\right) (i.e. some radius \delta > 0) such that \underbrace{|f\left(\mathbf{x}\right) - f\left(\mathbf{a}\right)| < \varepsilon}_{\iff f\left(\mathbf{a}\right) - \varepsilon < f\left(\mathbf{x}\right) < f\left(\mathbf{a}\right) + \varepsilon} whenever \mathbf{x} \in B\left(\mathbf{a},\delta\right).$
$\noindent So fix \varepsilon > 0. Let S_{1} = \left\{ \mathbf{x}\in \mathbb{R}^{n} : f\left(\mathbf{x}\right) < f\left(\mathbf{a}\right) + \varepsilon \right\} and S_{2} = \left\{ \mathbf{x}\in \mathbb{R}^{n} : f\left(\mathbf{x}\right) > f\left(\mathbf{a}\right) - \varepsilon \right\} (note that a vector \mathbf{x} is in both S_{1} and S_{2}, i.e. in S_{1}\cap S_{2}, iff \left|f\left(\mathbf{x}\right) - f\left(\mathbf{a}\right)\right| < \varepsilon). Observe that \mathbf{a} belongs to both S_{1} and S_{2} (because \varepsilon > 0). So for i = 1,2, \mathbf{a} \in S_{i} and S_{i} being open (by assumptions of the question) implies that there exists a \delta_{i} such that$
$\mathbf{x}\in S_{i} whenever x \in B\left(\mathbf{a},\delta_{i}\right).$
$\noindent Set \delta = \min \left\{\delta_{1}, \delta_{2}\right\}, then we get that \mathbf{x} belongs to both S_{1} and S_{2} (i.e. to S_{1}\cap S_{2}) whenever \mathbf{x} \in B\left(\mathbf{a}, \delta\right). This is exactly what we had to show, and the proof is complete.$
25. ## Re: Several Variable Calculus
Just a brief sketch-out please
$f(x)=\begin{cases}0 & (x,y)=(0,0)\\ \frac{2xy}{x^2+y^2} & \text{otherwise}\end{cases}$
$\text{Why is it that }\frac{\partial^2 f}{\partial x \partial y}(0,0)\text{ DNE?}$
I know that f is not continuous at 0 but I'm not sure if that helps since we're talking about partial derivatives here.
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2019-02-22 11:56:38
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https://www.physicsforums.com/threads/gravity-an-accelerating-frame-paradox.465684/
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# Gravity - an accelerating frame paradox
ZirkMan
If the equivalence principle is true then it means that the Earth's gravity field is a constantly accelerating frame of reference. In any accelerating frame of reference the direction of acceleration is always opposite to the direction of attraction.
That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them. If this is true for all observers on its spherical surface how is it possible that the Earth doesn't explode (due to constant acceleration away from its center to all sides)? Is there a model of spacetime that explains it?
Or is there a mechanism of acceleration that doesn't require acceleration in space?
## Answers and Replies
Science Advisor
Because in curved space-time acceleration is not equal to the rate of change of velocity. So while the surface is constantly accelerating outwards at all points, none of these points actually move outwards. Hence, no explosion.
To demonstrate this, consider a person standing on a rotating platform. Wherever he's standing, his experience is consistent with the point he is standing at accelerating inwards towards the center of the platform. In fact, that point is accelerating towards the center of the platform. However, the platform is not imploding, and in this coordinate system, no point of the platform is moving.
Staff Emeritus
Science Advisor
If the equivalence principle is true then it means that the Earth's gravity field is a constantly accelerating frame of reference. In any accelerating frame of reference the direction of acceleration is always opposite to the direction of attraction. That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them.
You are misreading the equivalence principle to say that the Earth's surface is physically expanding outward. The equivalence principle redefines the concept of an inertial frame of reference; what it is saying is that an object at rest with respect to the surface of the Earth is accelerating with respect to an inertial frame. This, BTW, is exactly what an accelerometer says: An accelerometer placed at rest on the surface of the Earth will indicate that it is accelerating upwards at 9.8 meters/second2.
ZirkMan
To demonstrate this, consider a person standing on a rotating platform. Wherever he's standing, his experience is consistent with the point he is standing at accelerating inwards towards the center of the platform. In fact, that point is accelerating towards the center of the platform. However, the platform is not imploding, and in this coordinate system, no point of the platform is moving.
If the rotation of the platform was not accelerating (or deaccelerating) and I was standing on the platform and there was no other frame of reference to compare I would have no way to prove that my platform is rotating or not and there would be no force that would drag me to the center of the platform, wouldn't be?
And yet I'm standing on the surface of the Earth and observe it accelerating towards me (from my reference frame). And I can still see the paradox here. How can there be something moving towards me (from my reference frame without comparison to other frames), yet not moving at all?
ZirkMan
... what it is saying is that an object at rest with respect to the surface of the Earth is accelerating with respect to an inertial frame.
What is this inertial frame? Some kind of absolute space? Sorry, I do not understand this.
Mentor
2021 Award
If the rotation of the platform was not accelerating (or deaccelerating) and I was standing on the platform and there was no other frame of reference to compare I would have no way to prove that my platform is rotating or not and there would be no force that would drag me to the center of the platform, wouldn't be?
That is incorrect. A simple gyroscope or a laser ring interferometer could prove it without reference to any external object.
How can there be something moving towards me (from my reference frame without comparison to other frames), yet not moving at all?
It sounds like you are confusing proper acceleration and coordinate velocity. When you are standing on the earth then you and the ground are both accelerating upwards at g, hence your relative velocity is constant (0). When you are free-falling towards the earth then the ground is accelerating at g and you are not, hence your relative velocity is changing, and whether or not a given object is moving depends on the coordinate system chosen, but it is all self-consistent with no true paradoxes.
Science Advisor
If the rotation of the platform was not accelerating (or deaccelerating) and I was standing on the platform and there was no other frame of reference to compare I would have no way to prove that my platform is rotating or not and there would be no force that would drag me to the center of the platform, wouldn't be?
There IS a force dragging you out towards the outer edge. The centrifugal force. That's exactly the point.
Centrifugal force is a fictitious force. Same as gravity in GR. It only arises from the fact that you chose a frame of reference in which a static object is actually accelerating. On a rotating platform, each point experiences centripetal acceleration. If you pick a coordinate system relative to ground, you can see that each point on the platform is moving and accelerating towards the center. If you pick a coordinate system fixed to the platform, each point is at rest, but they are still all accelerating towards the center. Hence the centrifugal force pushing you out towards the outer edge.
ZirkMan
There IS a force dragging you out towards the outer edge. The centrifugal force. That's exactly the point.
Centrifugal force is a fictitious force. Same as gravity in GR. It only arises from the fact that you chose a frame of reference in which a static object is actually accelerating. On a rotating platform, each point experiences centripetal acceleration. If you pick a coordinate system relative to ground, you can see that each point on the platform is moving and accelerating towards the center. If you pick a coordinate system fixed to the platform, each point is at rest, but they are still all accelerating towards the center. Hence the centrifugal force pushing you out towards the outer edge.
This is actually a very good explanation, thanks. But isn't this against the Galilean relativity which states that you cannot detect a uniform motion with help of mechanical experiments? Still, I see a way out of this because although the platform rotates uniformly the speed of relative movement will be different for points in different distances from the center of rotation and this propably gives rise to the centrifugal force. But where is the source of difference in relative kinetic energy for points (objects) in the gravitational frame?
Science Advisor
Rotation is not a uniform motion. It's always an accelerated motion. So it is detectable under Galilean Relativity. Under Mach's Principle, you can't tell the difference between the rotation of the platform and rotation of the universe around the platform. But unfortunately, General Relativity does not cover that, and we have no way at present to correct for it.
Mentz114
If the equivalence principle is true then it means that the Earth's gravity field is a constantly accelerating frame of reference. In any accelerating frame of reference the direction of acceleration is always opposite to the direction of attraction.
Wrong. Where do you get these weird ideas ?
That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them.
What ? They are all remaining stationary wrt to each other. ( We are talking about Terra here, aren't we ?).
If this is true for all observers on its spherical surface how is it possible that the Earth doesn't explode (due to constant acceleration away from its center to all sides)?
It's not true. Why would the earth explode if the gravitational field is trying to compress it into a sphere ?
Acceleration means a change in velocity.
Anything that is stationary at a fixed distance from the Earth's centre is being held by a force, which is to say it is experiencing non-geodesic motion in 4D and is not accelerating.
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ZirkMan
Rotation is not a uniform motion. It's always an accelerated motion. So it is detectable under Galilean Relativity. Under Mach's Principle, you can't tell the difference between the rotation of the platform and rotation of the universe around the platform. But unfortunately, General Relativity does not cover that, and we have no way at present to correct for it.
Agreed.
But please, let's come back to the source of the fictitious force of gravity. My current understanding is that all fictitious forces arise from relative imbalance in kinetic energy(KE) in any (de)accelerating frame of reference (as is the case of the centrifugal force from your example).
Now with all know fictitious forces (except gravity?) the source of the force can be attributed to the imbalance in KE between observer's frame and an non-uniform motion frame in which he is dragged in. His own inertia enables him to feel the fictitious force (the transfer of the KE). But what is the source of difference in KE when I'm in the accelerated frame of gravity? I cannot detect anything accelerating (in the physical sense). Yet if gravity arises from the same mechanism as other fictitious forces some source of EK imbalance must be present.
Science Advisor
What ? They are all remaining stationary wrt to each other. ( We are talking about Terra here, aren't we ?).
And yet, the surface is accelerating.
Acceleration means a change in velocity.
That's where you are making an oversight. Just because the vector describing velocity does not change, it does not mean there is no change in velocity. Again, look at the rotating frame of reference. An object that has zero velocity in a rotating frame IS accelerating.
ZirkMan said:
But please, let's come back to the source of the fictitious force of gravity. My current understanding is that all fictitious forces arise from relative imbalance in kinetic energy(KE) in any (de)accelerating frame of reference (as is the case of the centrifugal force from your example).
What does kinetic energy have to do with anything?
The source of fictitious force is a "bad choice" of coordinate system. I can pick a free-falling coordinate system, and gravitational force disappears. The difference is that in flat space-time, I can pick one coordinate system that's inertial everywhere. In curved space-time, it's a local choice. If I pick a free-falling coordinate system here, on the other side of the Earth, it's an accelerated frame. That's why in GR you have to know how to deal with accelerated frames, and working with accelerated frames of reference in flat space-time is a good practice, because you can check your answers much easier.
Staff Emeritus
Science Advisor
Gold Member
That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them. If this is true for all observers on its spherical surface how is it possible that the Earth doesn't explode (due to constant acceleration away from its center to all sides)?
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
lugita15
Under Mach's Principle, you can't tell the difference between the rotation of the platform and rotation of the universe around the platform. But unfortunately, General Relativity does not cover that, and we have no way at present to correct for it.
Are you saying that there's no way modify GR to make it compatible with Mach's principle? What about Brans-Dicke theory?
Mentor
2021 Award
Now with all know fictitious forces (except gravity?) the source of the force can be attributed to the imbalance in KE between observer's frame and an non-uniform motion frame in which he is dragged in.
Huh? Do you have any reference for this? I have never heard of such a thing and don't even know what you mean by "imbalance of KE" between frames.
Science Advisor
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
No. In GR inertial frames are local, and yes, in general, you can't always have a global coordinate system. But you can have a coordinate system that describes Earth and the immediate surroundings just fine. Start with polar coordinates under Schwarzschild metric, then adjust the metric as necessary to get something less idealized.
lugita15 said:
Are you saying that there's no way modify GR to make it compatible with Mach's principle? What about Brans-Dicke theory?
There might be. But how would you test it if it's the correct way to do so? We don't even have an experiment confirming Mach's Principle.
ZirkMan
Acceleration means a change in velocity.
Acceleration means a relative change of EK over mass (m). Otherwise you cannot explain inertia.
Anything that is stationary at a fixed distance from the Earth's centre is being held by a force, which is to say it is experiencing non-geodesic motion in 4D and is not accelerating.
Gravity is not a force. Every observer experiencing gravity is dragged in an accelerating frame of reference which is a big difference. Non-geodesic motion is just a consequence of existence of a force opposing the direction of acceleration of the accelerating frame. Such as the repulsive force of electrons in your body and the ground acting as a buffer zone through which (from your frame of reference) the energy from the accelerating ground is transfered to you. Of course you cannot detect any gravity when you are not in contact with the accelerating frame (such as freefall). Then there is nothing to prevent you following a geodesic.
ZirkMan
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
Apparently it is so since the Earth hasn't exploded due to its global acceleration yet
But I try to find a link that would link this local frames to the global view. Maybe the solution is the same as in SR - there is no global frame and frames of all observers are only relative to one another.
nismaratwork
I was under the impression that inertia, like mass, remains elusive in its explanation. I wouldn't look to Newtonian mechanics to answer that question however.
Mentor
2021 Award
Acceleration means a relative change of EK over mass (m).
This is false. Consider uniform circular motion in an inertial reference frame, there is acceleration without any change in kinetic energy.
ZirkMan, you have provided nothing to substantiate any of the many erroneous claims you have made in this thread. You need to stop speculating and start learning some mainstream physics.
ZirkMan
Huh? Do you have any reference for this? I have never heard of such a thing and don't even know what you mean by "imbalance of KE" between frames.
Take for example a person jumping on an accelerating train (from the person's perspective). From his perspective the train has higher KE then he has. When he jumps on it as soon as he touches it the difference of of their KE starts to level off. Even if the train was not accelerating only moving uniformly the transfer of difference in their KE would not be immediate (due to inertia) and the person would probably fell down because for a moment his feet would move quicker than the rest of his body. He would feel a kick of "fictitious force".
When the train is accelerating the transfer of KE doesn't stop because the train constantly increases its relative KE. Therefore the person will feel the constant fictitious force of acceleration.
Is such an explanation ok for you? I hope it is clear and simple enough that I do not need any formal reference to back it up because it is just an ordinary relation of basic and well-known physics concepts (at least from my reference frame :shy:). And you are free to prove it wrong.
ZirkMan
This is false. Consider uniform circular motion in an inertial reference frame, there is acceleration without any change in kinetic energy.
So you disagree with this:
Rotation is not a uniform motion. It's always an accelerated motion. So it is detectable under Galilean Relativity.
ZirkMan, you have provided nothing to substantiate any of the many erroneous claims you have made in this thread. You need to stop speculating and start learning some mainstream physics.
Yes, you have a full right of accusing me of speculation in this second part of this thread, because indeed I do not agree with some of the answers and think some other approach could worked better. But I try to do my best to explain why and teach some mainstream physics in the process from you guys. I want to see where I am wrong and the best way is to get some constructive feedback after which I can admit I have gained some new insight.
Science Advisor
Acceleration means a relative change of EK over mass (m). Otherwise you cannot explain inertia.
No. It does not. Kinetic energy is something completely different in GR.
$$E_{kinetic} = cp^t - c\sqrt{p^{\mu}p^{\nu}g_{\mu\nu}}$$
Note that this quantity is frame and metric dependent.
nismaratwork
So you disagree with this:
Yes, you have a full right of accusing me of speculation in this second part of this thread, because indeed I do not agree with some of the answers and think some other approach could worked better. But I try to do my best to explain why and teach some mainstream physics in the process from you guys. I want to see where I am wrong and the best way is to get some constructive feedback after which I can admit I have gained some new insight.
Your tone and approach is very reasonable, but the content of what you're saying isn't. I believe that's the source of the disconnect between your questions, and the frustration in the attempt to answer. It feels very much like a reasonable individual trying reasonably to tell you that the moon is made of green cheese.
If you make your mistakes in the form of trying to learn, rather than teach... you'll be far more in the spirit (and letter) of this forum's law. AFAIK. After all, it's hard to tell a crackpot with an agenda from someone trying to employ a confrontational method to test their ideas.
I will say this: I read, but don't participate in a lot of these threads. I can't think of a time that DaleSpam steered someone wrong... I'd listen to him.
Mentor
2021 Award
When he jumps on it as soon as he touches it the difference of of their KE starts to level off. Even if the train was not accelerating only moving uniformly the transfer of difference in their KE would not be immediate (due to inertia) and the person would probably fell down because for a moment his feet would move quicker than the rest of his body. He would feel a kick of "fictitious force".
All of what you describe here is due to the real contact force between the person and the train and not due to fictitious forces. This certainly does not support nor even explain your claim that fictitious forces are "attributed to the imbalance in KE".
I hope it is clear and simple enough that I do not need any formal reference to back it up because it is just an ordinary relation of basic and well-known physics concepts
No, it is not clear, and it has no discernable relationship to mainstream physics. Please provide a reference, or stop posting this line of speculation.
Staff Emeritus
Science Advisor
Gold Member
No. In GR inertial frames are local, and yes, in general, you can't always have a global coordinate system. But you can have a coordinate system that describes Earth and the immediate surroundings just fine. Start with polar coordinates under Schwarzschild metric, then adjust the metric as necessary to get something less idealized.
A coordinate system isn't the same as a frame of reference, and that's exactly what you need to understand in order to understand the resolution of the paradox. If you have a frame of reference, you can define whether objects A and B are at rest with respect to one another, even if A and B are far apart. GR does not, for example, allow us to say whether our galaxy is at rest with respect to a distant galaxy.
There might be. But how would you test it if it's the correct way to do so? We don't even have an experiment confirming Mach's Principle.
This is incorrect. Mach's principle can be tested by experiments that test predictions made by a less Machian theory such as GR and a more Machian theory such as Brans-Dicke gravity. The results come out to be less Machian, as in GR. A good popular-level discussion of this is available in Was Einstein Right? by Will.
bcrowell said:
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
Apparently it is so since the Earth hasn't exploded due to its global acceleration yet
Your statement doesn't connect logically to the material you quoted from my post.
Maybe the solution is the same as in SR - there is no global frame
SR does have global frames of reference.
Mentor
2021 Award
ZirkMan,
First, the equivalence principle only applies over regions of spacetime where the curvature is negligible. So, while it applies over a small region of the earth's surface, it does not apply over the entire surface.
Second, geometrically an inertial object's worldline is a geodesic (a straight line). Conversely an object undergoing proper acceleration has a non-geodesic worldline (a curved line).
Third, an inertial coordinate system is an orthonormal coordinate system whose coordinate lines are all straight (geodesic), conversely a non-inertial coordinate system has one or more sets of curved coordinates.
Fourth, In gravity two inertial particles may accelerate relative to each other, geometrically this is only possible in a curved spacetime.
Fifth, in a curved spacetime it is not possible to introduce a global set of coordinates which are everywhere inertial, but it is possible to do so locally.
Given the above, do you understand your concern any better?
ZirkMan
All of what you describe here is due to the real contact force between the person and the train and not due to fictitious forces. This certainly does not support nor even explain your claim that fictitious forces are "attributed to the imbalance in KE".
Is it possible to distinguish between the contact force and the fictitious force in this case? If yes, then I was really wrong.
No, it is not clear, and it has no discernable relationship to mainstream physics. Please provide a reference, or stop posting this line of speculation.
OK, no more on KE and fictitious forces until I have a set of axioms that can be put into formal language and defended in other way then just freely described concepts. Thanks for your time and valuable feedback anyway.
Mentor
2021 Award
Is it possible to distinguish between the contact force and the fictitious force in this case? If yes, then I was really wrong.
Yes, there are lots of ways to distinguish them. The contact force is a real force and is associated with an "equal and opposite" reaction force on the train, the fictitious force is not. The contact forces cause stresses, the fictitious force does not. Accelerations due to the contact forces can be measured by an accelerometer, the fictitious forces cannot. Etc.
Science Advisor
A coordinate system isn't the same as a frame of reference, and that's exactly what you need to understand in order to understand the resolution of the paradox. If you have a frame of reference, you can define whether objects A and B are at rest with respect to one another, even if A and B are far apart. GR does not, for example, allow us to say whether our galaxy is at rest with respect to a distant galaxy.
Coordinate system is the frame of reference. It's the mapping from an open set in manifold onto R4. There is no other definition for frame of reference or coordinate system.
This is incorrect. Mach's principle can be tested by experiments that test predictions made by a less Machian theory such as GR and a more Machian theory such as Brans-Dicke gravity. The results come out to be less Machian, as in GR. A good popular-level discussion of this is available in Was Einstein Right? by Will.
That does not test Mach's principle. That tests specific theory of gravity. There is no practical test of Mach's Principle.
ZirkMan
Yes, there are lots of ways to distinguish them. The contact force is a real force and is associated with an "equal and opposite" reaction force on the train, the fictitious force is not. The contact forces cause stresses, the fictitious force does not.
So a fictitious force stops being fictitious as soon as a contact is being made? So when I'm sitting on the ground on Earth there is no fictitious force pushing me down only a real contact force (which source is what)?
Accelerations due to the contact forces can be measured by an accelerometer, the fictitious forces cannot. Etc.
Maybe the answer to the question above will answer this seemingly opposite answers.
An accelerometer placed at rest on the surface of the Earth will indicate that it is accelerating upwards at 9.8 meters/second2.
Science Advisor
So a fictitious force stops being fictitious as soon as a contact is being made? So when I'm sitting on the ground on Earth there is no fictitious force pushing me down only a real contact force (which source is what)?
No. The contact force is pushing you up. It's what opposing the fictitious force and prevents you from falling towards center of the Earth.
ZirkMan
No. The contact force is pushing you up. It's what opposing the fictitious force and prevents you from falling towards center of the Earth.
If the contact force is a reaction to the fictitious force then its hard to say which one causes stresses. How can you say (DaleSpam) that it's not the fictitious force thats causing the stress?
nismaratwork
If the contact force is a reaction to the fictitious force then its hard to say which one causes stresses. How can you say (DaleSpam) that it's not the fictitious force thats causing the stress?
Doesn't this go back to the RLG or an accelerometer on the ground?
Science Advisor
If the contact force is a reaction to the fictitious force then its hard to say which one causes stresses. How can you say (DaleSpam) that it's not the fictitious force thats causing the stress?
Because when you are free-falling, the fictitious force is acting on you, but there is no stress.
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2022-09-25 09:07:43
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https://xorshammer.com/2016/07/24/the-cgp-grey-topos-of-continents/
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# The CGP Grey Sheaf of Continents
CGP Grey is a youtuber with a variety of interesting videos, often about the quirks of geography and political boundaries. In this video, he asks the question “How many continents are there?”, discusses a variety of subtleties in the notion of “continent”, and concludes that it is not well-defined enough to provide an answer.
Let’s grant that “continent” is not a well-defined term; or, to put it another way, the set of continents is not a well-defined set. Even given that, it turns out there’s a mathematical notion of a “variable set”, or “set-valued sheaf” that can capture the notion of a set which can vary under different assumptions. Intuitively a set-valued sheaf on a topological space $S$ is like a continuous function with domain $S$, except the range is not another topological space, it’s the category of all sets!
Rather than define “sheaf of sets on a topological space” explicitly, let’s work through what it means in the CGP Grey case. For simplicitly, let’s just focus on two of the things that CGP Grey mentions: the meaning of “continent” can vary depending on how large you require a continent to be, and the meaning of “continent” can vary depending on how separated you require two continents to be to count as distinct.
Since these are two independent parameters, let’s take our topological space to be $[0,1]\times[0,1]$. The first coordinate will represent our “looseness about the size requirement”; i.e., if it’s larger, we’ll consider smaller islands to be continents. The second coordinate will represent our “degree of consideration of land bridges”; i.e., if it’s larger, we’ll require larger amounts of water to separate two continents.
To be clear, these parameters are subjective: that is, I’m not postulating any quantitative correspondence between the parameters and, e.g., a minimum size requirement to be a continent.
Now let’s see what the variable set of continents might look like. First, let’s set the second parameter to 0 and vary the first parameter. The set might look like this:
Note that some continents, like South America, are always in the set of continents, but as the parameter gets loosened, other elements get added to the set.
Now, let’s set the first parameter to 0 and vary the second one. That graph might look like this:
This is a little more subtle than the previous graph; instead of new continents getting added, two continents which are distinct might become equal: Europe and Asia quickly become equal, as there is actually no ocean between them at all. If you disregard the Panama Canal, North America and South America become one continent. If you disregard the Suez Canal, Eurasia and Africa become one continent.
Now, we’ve only looked at two slices of this variable set (and even those two slices have been under-specified, since I haven’t said in complete detail how to interpret the two parameters). But let’s suppose that the full variable set on $[0,1]\times[0,1]$ can be filled out to give a set-valued sheaf called $\mathbf{Continents}$.
Given that, what can we do with this? Well, one of the reason sheaves are interesting from a logical perspective is that if we consider the category of all set-valued sheaves on $[0,1]\times[0,1]$ (or any fixed topological space), this forms a type of category called a topos which acts so much like the category of sets that we can actually pretend that its elements are sets, and do normal set theory in it. The only proviso is that the internal logic does not include the law of the excluded middle: the axiom that $P\vee\neg P$ for any proposition $P$.
So, what are some things you can do in this logic where we get to pretend that $\mathbf{Continents}$ is a genuine set?
Well, we know that $\mathbf{Continents}$ has elements: we know there is a thing called $\mathbf{North America}$ that’s in $\mathbf{Continents}$, and a thing called $\mathbf{Europe}$ that’s in $\mathbf{Continents}$ and so on. We don’t know there’s a thing called $\mathbf{Borneo}$ that’s in $\mathbf{Continents}$; I’ll show how to deal with that later.
This is where the lack of the law excluded middle first rears its head: in this logic it is neither the case that $\mathbf{North America}=\mathbf{South America}$, nor that $\mathbf{North America}\neq\mathbf{South America}$! On the other hand, it is the case that $\mathbf{North America}\neq\mathbf{Antarctica}$. This might seem unusual with ordinary sets, but I think it’s pretty intuitive here. Note that there can be relationships between these facts, e.g., $\mathbf{Asia}=\mathbf{Africa}$ implies $\mathbf{Europe}=\mathbf{Africa}$.
In normal set theory, you can determine the cardinality of any set. And in fact, the video’s stated aim is to say what the cardinality is. One of the consequences of losing the law of the excluded middle is that the notion of finiteness becomes more subtle (e.g., see here or here), which again seems appropriate here. It turns out to be the case that $\mathbf{Continents}$ is what’s called subfinite, but doesn’t have a definite cardinality.
However, there are still true things using the cardinality of $\mathbf{Continents}$ in them: for example, assuming no continents other than the ones in the graphs above are added, it’s the case that the cardinality of $\mathbf{Continents}$ is less than 11 (even though the cardinality does not equal a specific number below 11). For another example, there might be a relationship between the two parameters such that something like $\mathbf{North America}=\mathbf{South America}$ implies the number of continents is greater than 7 is true.
OK, so far we’ve discussed how the logic handles things like the possibility of two continents becoming equal. How does it handle the conditional existence of continents like Borneo? So far it’s not clear how to even talk about these things in the language.
To explain that, we have to back up a bit. In normal set theory, there are sets with one element, and we might as well pick a distinguished one, call it $\{\star\}$. Note that for any set $A$ (still in normal set theory), the elements of $A$ are in 1-1 correspondence with maps $f\colon\{\star\}\to A$; so we could as well talk about those maps instead of elements of $A$.
Similarly, in the theory of set-valued sheaves on $[0,1]\times[0,1]$, there is also a set $\{\star\}$, and instead of saying that continents like $\mathbf{North America}$ are elements of $\mathbf{Continents}$, we could have instead talked about maps from $\{\star\}$ to $\mathbf{Continents}$ and relationships between them.
Now, in normal set theory, $\{\star\}$ has only two subsets: itself and the empty set. But that proof depends on excluded middle (since it goes by asking whether or not $\star$ is in a given subset), so if we drop it, it’s no longer necessarily true. Indeed, in this logic, there is a subset of $\{\star\}$, call it $\mathbf{BorneoIsAContinent}$, that is not the empty set and not $\{\star\}$. Furthermore, there is a map, $\mathbf{Borneo}$, from $\mathbf{BorneoIsAContinent}$ to $\mathbf{Continents}$. The fact that this map has domain $\mathbf{BorneoIsAContinent}$ instead of $\{\star\}$ represents the conditional nature of Borneo’s existence as an element of $\mathbf{Continents}$.
Just as with the equality hypotheses, we can represent relationships between conditional existences: for example, if Greenland is a continent whenever Borneo is, we have a map from $\mathbf{BorneoIsAContinent}$ to $\mathbf{GreenlandIsAContinent}$. If there are at least 7 continents whenever Borneo exists, we have a map from $\mathbf{BorneoIsAContinent}$ to $\{\star\mid\textrm{there are at least 7 continents}\}$.
Toposes were invented in the service of algebraic geometry (see here for a good account of the history of this topic). However, I think they also provide a beautiful account of how set theory can take account of fuzzy concepts. See here for more on this notion of variable sets.
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2017-03-26 03:28:17
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https://proofwiki.org/wiki/Definition:Real_Independent_Variable
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# Definition:Independent Variable/Real Function
## Definition
Let $f: \R \to \R$ be a real function.
Let $\map f x = y$.
Then $x$ is referred to as an independent variable.
## Also see
The terms independent variable and dependent variable arise from the idea that it is usual to consider that $x$ can be chosen independently of $y$, but having chosen $x$, the value of $y$ then depends on the value of $x$.
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2023-02-08 14:06:32
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https://www.vedantu.com/question-answer/construct-vartriangle-gpr-such-that-gr-53-cm-class-9-maths-cbse-5ee350f46067f248d16aef42
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Question
# Construct $\vartriangle {\text{GPR}}$ such that GR = 5.3 cm, $\angle {\text{GPR}} = {70^0}$, ${\text{PR}} \bot {\text{GR}}$ and PR = 3.5 cm.
Verified
158.1k+ views
Hint: Here, we will proceed by applying the basic principles of construction and make the use of a scale and compass for the construction purpose and for verification we will use a protector.
Step 3- Now, join the points G and P with the help of the scale. This will result in the construction of the required triangle GPR as shown in the figure. We can also verify whether the triangle constructed is correct or not by measuring the angle $\angle {\text{GPR}}$ with the help of the protector. When this angle is measured it gives a value of ${70^0}$ which is true (given).
Note: In this particular problem, an additional input data i.e., $\angle {\text{GPR}} = {70^0}$ is given which is not required for the construction of the triangle GPR but can be used for verification. If $\angle {\text{GPR}}$ gives a measure equal to ${70^0}$ then the constructed triangle is correctly constructed, else some mistake is being done.
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2021-12-04 13:07:34
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https://plainmath.net/2263/assum-t-to-is-matrix-transformation-with-matrix-prove-that-if-the-column
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# Assum T: R^m to R^n is a matrix transformation with matrix A.Prove that if the column
Assum T:${R}_{m}$ to ${R}_{n}$ is a matrix transformation with matrix A. Prove that if the columns of A are linearly independent, then T is one to one (i.e injective). (Hint: Remember that matrix transformations satisfy the linearity properties.
Linearity Properties:
If A is a matrix, v and w are vectors and c is a scalar then
$A0=0$
$A\left(cv\right)=cAv$
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Proof:
Let T be defined as $T\left(v\right)=Av$
Take
Further obtain the result as follows:
$T\left(v\right)=Av$
As the colums of A are linearly idependent, no rows Av will be equal. Thus, the vector obtained is different than one another in ${R}_{n}.$
Hence, if the columns of A are linearly independent, then T is one to one.
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2022-05-25 18:59:54
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https://www.aimsciences.org/article/doi/10.3934/amc.2007.1.251
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Article Contents
Article Contents
# On the covering radii of extremal doubly even self-dual codes
• In this note, we study the covering radii of extremal doubly even self-dual codes. We give slightly improved lower bounds on the covering radii of extremal doubly even self-dual codes of lengths 64, 80 and 96. The covering radii of some known extremal doubly even self-dual [64, 32, 12] codes are determined.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94B75.
Citation:
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2022-12-03 22:24:36
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http://www.herbert.top:18080/problemset/redundant-connection-ii/readme_en/
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| English | 简体中文 |
# 685. Redundant Connection II
## Description
In this problem, a rooted tree is a directed graph such that, there is exactly one node (the root) for which all other nodes are descendants of this node, plus every node has exactly one parent, except for the root node which has no parents.
The given input is a directed graph that started as a rooted tree with n nodes (with distinct values from 1 to n), with one additional directed edge added. The added edge has two different vertices chosen from 1 to n, and was not an edge that already existed.
The resulting graph is given as a 2D-array of edges. Each element of edges is a pair [ui, vi] that represents a directed edge connecting nodes ui and vi, where ui is a parent of child vi.
Return an edge that can be removed so that the resulting graph is a rooted tree of n nodes. If there are multiple answers, return the answer that occurs last in the given 2D-array.
Example 1:
Input: edges = [[1,2],[1,3],[2,3]]
Output: [2,3]
Example 2:
Input: edges = [[1,2],[2,3],[3,4],[4,1],[1,5]]
Output: [4,1]
Constraints:
• n == edges.length
• 3 <= n <= 1000
• edges[i].length == 2
• 1 <= ui, vi <= n
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2021-03-02 13:14:28
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https://math.stackexchange.com/questions/81122/inequality-with-roots-of-unity
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# inequality with roots of unity
Do you know proofs or references for the following inequality:
There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$
$$|a_1|+\cdots+|a_n| \leq C\sup_{z_1^3=1,\ldots,z_n^3=1 } |a_1z_1+\cdots + a_n z_n|$$ where the supremum is taken over the complex numbers $z_1,\ldots,z_n$ such that $z_1^3=1,\ldots,z_n^3=1$?
The strategy is to rotate the $a$ values so that they are roughly pointing in the same direction. More formally, for each complex $a$ choose a third root of unity $z$ so that the argument $\theta$ of $az$ lies between plus and minus $\pi/3$.
Then \begin{eqnarray*} |a_1z_1+\cdots +a_nz_n|&\geq&|\mbox{Re}(a_1z_1+\cdots +a_nz_n)| \cr &=&|a_1z_1|\cos(\theta_1)+\cdots +|a_nz_n|\cos(\theta_n) \cr &\geq &(|a_1|+\cdots +|a_n|)\cos(\pi/3)\cr &=&{1\over 2}\ (|a_1|+\cdots +|a_n|)\cr \end{eqnarray*}
This gives your result with $C=2$.
• It seems to me that we must take the the "furthest away from the argument" and not the "closest argument". What do you think about this? – Zouba Nov 11 '11 at 15:46
• You are right, I didn't describe it well. I will modify my answer. – user940 Nov 11 '11 at 15:51
Notice that the mapping $\|\cdot\|_* \colon \mathbb{C}^n \to \mathbb{R}$ given by $$\|(a_1,\dots,a_n)\|_* = \sup_{z_1^3=z_2^3=\dots=1} |a_1z_1 + \dots + a_n z_n|$$ defines a norm. The result then follows by the equivalence of norms in finite dimensional vector spaces.
The triangle inequality is easily seen to be true: Just use the ordinary triangle inequality. To prove that if $\|(a_1,\dots,a_n)\|_* = 0$ then $(a_1,\dots,a_n) = 0$ notice that if one of the coordinates (let's say $a_1$) wasn't $0$, then there would be three different numbers: $a_1 z_1 + a_2 + \dots + a_n$, $a_1 z_2 + a_2 + \dots + a_n$ and $a_1 z_3 + a_2 + \dots + a_n$ one of which is non-zero and thus the sup we get would be non-zero also.
• Does the op want a constant independent of $n$, though? – David Mitra Nov 11 '11 at 15:14
• Yes, the constant $C$ must be independent of $n$. – Zouba Nov 11 '11 at 15:21
• @David Mitra: A good point. – J. J. Nov 11 '11 at 15:23
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2019-06-25 21:46:10
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http://timpanogos.wordpress.com/tag/science/
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## Dark Sky Week, Lyrid meteor shower – get outside!
April 23, 2014
From the Arches National Park Facebook page: photo of Pine Tree Arch and meteoroid by Andy Porter)
A few minutes before 9:00 p.m. Central on Tuesday, I saw a sizable fireball falling north to south, appearing from my vantage on the top of Cedar Hill to be over south Grand Prairie, Texas. Best meteoroid I’ve seen for a while.
Part of the Lyrid Meteor Shower, perhaps? The Lyrids coincide with Dark Sky Week this year. Dark Sky Week’s egalitarian origins should inspire all of us to go outside and look up, no? The celebration was invented by a high school student, Jennifer Barlow, in 2003.
I want people to be able to see the wonder of the night sky without the effects of light pollution. The universe is our view into our past and our vision into the future . . . I want to help preserve its wonder.” – Jennifer Barlow
The International Dark Sky Association promotes activities worldwide to encourage star-gazing and sky-watching.
Go out tonight, and look up!
More:
## 2014: STILL, again, Earth Day/Lenin hoax trotted out: Earth Day honors Earth, our majestic home — not Lenin
April 22, 2014
##### This is mostly an encore post, repeated each year on April 22 — sad that it needs repeating.
You could write it off to pareidolia, once. Like faces in clouds, some people claimed to see a link. The first Earth Day, on April 22, 1970, coincided with Lenin’s birthday. There was no link — Earth Day was scheduled for a spring Wednesday. Now, years later, with almost-annual repeats of the claim from the braying right wing, it’s just a cruel hoax.
No, there’s no link between Earth Day and the birthday of V. I. Lenin:
One surefire way to tell an Earth Day post is done by an Earth Day denialist: They’ll note that the first Earth Day, on April 22, 1970, was an anniversary of the birth of Lenin.
Coincidentally, yes, Lenin was born on April 22 (new style calendar; it was April 10 on the calendar when he was born — one might accurately note that Lenin’s mother always said he was born on April 10).
It’s a hoax. There is no meaning to the first Earth Day’s falling on Lenin’s birthday — Lenin was not prescient enough to plan his birthday to fall in the middle of Earth Week, a hundred years before Earth Week was even planned.
Does Earth Day Promote Communism?
Earth Day 1970 was initially conceived as a teach-in, modeled on the teach-ins used successfully by Vietnam War protesters to spread their message and generate support on U.S. college campuses. It is generally believed that April 22 was chosen for Earth Day because it was a Wednesday that fell between spring break and final exams—a day when a majority of college students would be able to participate.
U.S. Sen. Gaylord Nelson, the guy who dreamed up the nationwide teach-in that became Earth Day, once tried to put the whole “Earth Day as communist plot” idea into perspective.
“On any given day, a lot of both good and bad people were born,” Nelson said. “A person many consider the world’s first environmentalist, Saint Francis of Assisi, was born on April 22. So was Queen Isabella. More importantly, so was my Aunt Tillie.”
April 22 is also the birthday of J. Sterling Morton, the Nebraska newspaper editor who founded Arbor Day (a national holiday devoted to planting trees) on April 22, 1872, when Lenin was still in diapers. Maybe April 22 was chosen to honor Morton and nobody knew. Maybe environmentalists were trying to send a subliminal message to the national subconscious that would transform people into tree-planting zombies. One birthday “plot” seems just about as likely as the other. What’s the chance that one person in a thousand could tell you when either of these guys were born.
My guess is that only a few really wacko conservatives know that April 22 is Lenin’s birthday (was it ever celebrated in the Soviet Union?). No one else bothers to think about it, or say anything about it, nor especially, to celebrate it.
Inventor of Earth Day teach-ins, former Wisconsin Governor and U.S. Senator Gaylord Nelson
Wisconsin’s U.S. Sen. Gaylord Nelson, usually recognized as the founder and father of Earth Day, told how and why the organizers came to pick April 22:
Senator Nelson chose the date in order to maximize participation on college campuses for what he conceived as an “environmental teach-in.” He determined the week of April 19–25 was the best bet; it did not fall during exams or spring breaks, did not conflict with religious holidays such as Easter or Passover, and was late enough in spring to have decent weather. More students were likely to be in class, and there would be less competition with other mid-week events—so he chose Wednesday, April 22.
In his own words, Nelson spoke of what he was trying to do:
After President Kennedy’s [conservation] tour, I still hoped for some idea that would thrust the environment into the political mainstream. Six years would pass before the idea that became Earth Day occurred to me while on a conservation speaking tour out West in the summer of 1969. At the time, anti-Vietnam War demonstrations, called “teach-ins,” had spread to college campuses all across the nation. Suddenly, the idea occurred to me – why not organize a huge grassroots protest over what was happening to our environment?
I was satisfied that if we could tap into the environmental concerns of the general public and infuse the student anti-war energy into the environmental cause, we could generate a demonstration that would force this issue onto the political agenda. It was a big gamble, but worth a try.
At a conference in Seattle in September 1969, I announced that in the spring of 1970 there would be a nationwide grassroots demonstration on behalf of the environment and invited everyone to participate. The wire services carried the story from coast to coast. The response was electric. It took off like gangbusters. Telegrams, letters, and telephone inquiries poured in from all across the country. The American people finally had a forum to express its concern about what was happening to the land, rivers, lakes, and air – and they did so with spectacular exuberance. For the next four months, two members of my Senate staff, Linda Billings and John Heritage, managed Earth Day affairs out of my Senate office.
Five months before Earth Day, on Sunday, November 30, 1969, The New York Times carried a lengthy article by Gladwin Hill reporting on the astonishing proliferation of environmental events:
“Rising concern about the environmental crisis is sweeping the nation’s campuses with an intensity that may be on its way to eclipsing student discontent over the war in Vietnam…a national day of observance of environmental problems…is being planned for next spring…when a nationwide environmental ‘teach-in’…coordinated from the office of Senator Gaylord Nelson is planned….”
Nelson, a veteran of the U.S. armed services (Okinawa campaign), flag-waving ex-governor of Wisconsin (Sen. Joe McCarthy’s home state, but also the home of Aldo Leopold and birthplace of John Muir), was working to raise America’s consciousness and conscience about environmental issues.
Lenin on the environment? Think of the Aral Sea disaster, the horrible pollution from Soviet mines and mills, and the dreadful record of the Soviet Union on protecting any resource. Lenin believed in exploiting resources, not conservation.
So, why are all these conservative denialists claiming, against history and politics, that Lenin’s birthday has anything to do with Earth Day?
Can you say “propaganda?” Can you say “political smear?”
2014 Resources and Good News:
2013 Resources and Good News:
Good information for 2012:
Good information from 2011:
Good information from 2010:
2014′s Wall of Shame:
2013 Wall of Shame:
Wall of Lenin’s Birthday Propaganda Shame from 2012:
Wall of Lenin’s Birthday Propaganda Shame from 2011:
Wall of Lenin’s Birthday Propaganda Shame from 2010:
## Darwin’s death, April 19, 1882, and his legacy today
April 19, 2014
This is an encore post.
We shouldn’t pass April 19 — a day marked by significant historic events through the past couple hundred years — without remembering that it is also the anniversary of the death of Darwin.
Charles Darwin in 1881, portrait by John Collier; after a Collier painting hanging in the Royal Society
Immortality? Regardless Darwin’s religious beliefs (I’ll argue he remained Christian, thank you, if you wish to argue), he achieved immortality solely on the strength of his brilliant work in science. Of course he’s best known for being the first to figure out that natural and sexual selection worked as tools to sculpt species over time, a theory whose announcement he shared with Alfred Russel Wallace, who independently arrived at almost exactly the same theory but without the deep evidentiary backup Darwin had amassed.
But had evolution turned out to be a bum theory, Darwin’s other works would have qualified him as one of the greatest scientists of all time, including:
Darwin’s theory set out a sequence of coral reef formation around an extinct volcanic island, becoming an atoll as the island and ocean floor subsided. Courtesy of the U.S. Geological Survey (Photo credit: Wikipedia)
US Geological Survey graphic demonstrating how coral atolls form on the sinking remains of old volcanic sea mounts, as Darwin described. Wikimedia commons image
• World’s greatest collector of biological samples: During his five years’ voyage on HMS Beagle, Darwin collected the largest collection of diverse plant and animal life ever by one person (I believe the record still stands); solely on the strength of his providing actual examples to the British Museum of so much life in so many different ecosystems worldwide, before he was 30 Darwin won election to the Royal Society. (His election was engineered partly by friends who wanted to make sure he stayed in science, and didn’t follow through on his earlier plan to become a preacher.)
• Geology puzzle solver: Coral atolls remained a great geological mystery. Sampling showed coral foundations well below 50 feet deep, a usual limit for coral growth. In some cased old, dead coral were hundreds of feet deep. In the South Pacific, Darwin looked at a number of coral atolls, marvelous “islands” that form almost perfectly circular lagoons. Inspired partly by Lyell’s new encyclopedic review of world geology, Darwin realized that the atolls he saw were the peaks of volcanic mounts. Darwin hypothesized that the volcanoes grew from the ocean floor to the surface, and then the islands were colonized by corals. The round shape of the volcano gave the atoll its shape. Then the volcanic mounts eroded back, or sank down, and corals continued to grow on the old foundations. It was a perfectly workable, natural explanation for a long-standing geologic puzzle. (See Darwin’s monograph, Structure and Distribution of Coral Reefs.)
• Patient watcher of flowers: Another great mystery, this time in biology, concerned how vines twined themselves onto other plants, rocks and structures. Darwin’s genius in designing experiments shone here: He put a vine in his study, and watched it. Over several hours, he observed vine tendrils flailing around, until they latched on to something, and then the circular flailing motion wrapped the tendril around a stick or twig. Simple observation, but no one had ever attempted it before. (See On the Movements and Habits of Climbing Plants.)
• Champion of earthworms, and leaf mould: Darwin suspected the high fertilizer value of “leaf mould” might be related to the action of earthworms. Again, through well-designed experiments and simple observation, Darwin demonstrated that worms moved and aerated soil, and converted organic matter into even richer fertilizer. (See The Formation of Vegetable Moulds Through the Action of Worms.)
• Creation of methodological science: In all of this work, Darwin explained his processes for designing experiments, and controls, and made almost as many notes on how to observe things, as the observations themselves. Probably more than any other single man, Darwin invented and demonstrated the use of a series of processes we now call “the scientific method.” He invented modern science.
Any of those accomplishments would have been a career-capping work for a scientist. Darwin’s mountains of work still form foundations of geology and biology, and are touchstones for genetics.
Born within a few hours of Abraham Lincoln on February 12, 1809, Darwin survived 17 years longer — 17 extremely productive years. Ill through much of his life with mystery ailments, perhaps Chaga’s Disease, or perhaps some other odd parasite or virus he picked up on his world travels, Darwin succumbed to heart disease on April 19, 1882.
More:
## Cross seas: Nature, or design?
April 16, 2014
Here’s just exactly the sort of thing that happens in nature that drives creationists nuts. How could this happen without God personally working to confuse and/or delight the photographer? Not to mention the physicist and mathematician.
Photo from the Twitter feed of Science Porn: “Go home waves you’re drunk. This is called cross sea btw pic.twitter.com/5Cv1UUo8QX”
Where? Somewhere in France, one might gather from the flag on the structure (lighthouse?).
Turns out to be a Wikipedia photo, with this intriguing caption:
Crossing swells, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of Île de Ré (Isle of Rhé), France, in the Atlantic Ocean. The interaction of such near-solitons in shallow water may be modeled through the Kadomtsev–Petviashvili equation.
Oh, you remember that one, don’t you? The Kadomtsev–Petviashvili equation?
At least we confirmed it was taken in France.
They do everything differently in France, don’t they?
Update:Got an e-mail suggestion that I include the equation itself. You may certainly click to Wikipedia to find it; here’s what it says over there:
In mathematics and physics, the Kadomtsev–Petviashvili equation – or KP equation, named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili – is a partial differential equation to describe nonlinearwave motion. The KP equation is usually written as:
$\displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0$
where $\lambda=\pm 1$. The above form shows that the KP equation is a generalization to two spatial dimensions, x and y, of the one-dimensional Korteweg–de Vries (KdV) equation. To be physically meaningful, the wave propagation direction has to be not-too-far from the x direction, i.e. with only slow variations of solutions in the y direction.
Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation.
Certainly the longest equation ever published at Millard Fillmore’s Bathtub.
## “Years of Living Dangerously” – April 13 premiere of climate change information series
April 11, 2014
Will it work this time? Can it recharge the effort Al Gore started?
Monte Best of Plainview, Texas, explains to Don Cheadle how the Texas drought caused the Cargill Company to close its meat packing plant in the city. “Act of God,” many local people say.
Here’s the trailer:
The avid promotional explanation:
Published on Mar 14, 2014
Don’t miss the documentary series premiere of Years of Living Dangerously, Sunday, April 13th at 10PM ET/PT.
Subscribe to the Years of Living Dangerously channel for more clips:
Official site: http://www.sho.com/yearsoflivingdange…
The Years Project: http://yearsoflivingdangerously.com/
Watch on Showtime Anytime: http://s.sho.com/1hoirn4
Don’t Have Showtime? Order Now: http://s.sho.com/P0DCVU
It’s the biggest story of our time. Hollywood’s brightest stars and today’s most respected journalists explore the issues of climate change and bring you intimate accounts of triumph and tragedy. YEARS OF LIVING DANGEROUSLY takes you directly to the heart of the matter in this awe-inspiring and cinematic documentary series event from Executive Producers James Cameron, Jerry Weintraub and Arnold Schwarzenegger.
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## March 14, 2014: π Day! A π roundup, mostly pie
March 14, 2014
#### Let’s rerun this one. I like the photographs. I may go search for a good piece of pie.
Of course you remembered that today is pi Day, right?
Happy π Day! Pi Day Pie – Slashfood.com
Oh, or maybe better, π Day.
We’ll start with the brief post from a few months ago, and then build on it with some activities and posts from around the WordPress-o-sphere.
Make (and Eat) a Pie – These pie recipes for Pi Day from NPR’s McCallister look incredibly tasty. But, there’s no shame in putting a frozen store-bought pie in the oven, or picking up a pie from your local bakery. Any kind of pie is great on Pi Day! If you’re making your own, get inspired by these beautifully designed Pi Day Pies. Tell us on Facebook: What’s your favorite kind of pie for Pi Day?
Hope your π Day is complete as a circle, and well-rounded!
How are others celebrating? A look around WordPress:
At SocialMediaPhobe, a musical interpretation of pi featuring the music of Michael Blake:
So Long Freedom:
Today is March 14th, also known as “Pi Day” for us math geeks out there because March 14th (3/14) is the first 3 digits of π (3.14159…). To celebrate “Pi Day” I highly recommend doing something mathematical while having some pie at 1:59 pm. I recommend Yumology‘s S’mores Pie as it has 3 main ingredients (chocolate, marshmallow, and graham cracker) and about 0.14159 other ingredients like sugar, butter, and stuff. If you are not a math geek, its okay…you can still eat pie and count things like how many stop signs you pass on your way back to work from lunch. Or you could go to the library and take out a book on something fun like binary code. As we like to say, “There are only 10 types of people in the world: Those that understand binary and those that don’t.” Seriously, binary is as easy as 01000001, 01000010, 01000011.
So besides being the cause of much techie “irrational” exuberance, Pi Day is a great way to get some engagement with students.
Marymount High School has several activities, last year they had a design competition incorporating pi; the students then made and sold buttons of each design, proceeds going to the Red Cross.
Hmm- math subject matter, design, production, sales, accounting.
Sounds like what we do in manufacturing.
Maybe celebrating Pi Day is not so irrational as first thought.
Free said his pie is peach.
On March 12, 2009 your lawmakers passed a non-binding resolution (HRES 224) recognizing March 14, 2009 as National Pi Day. It is one of the more legit holidays we discuss here, and it is actually an homage to geeks everywhere who see the date as a reason to celebrate due to its mathematical implications. We say any reason to celebrate anything is just fine by us.
Since we are predominately about food we will suggest a few places to actually enjoy a pie.
If you followed us at all this week you may have seen the pie at Bowl and Barrel pop up on our pages. This is the uber delicious Butterscotch Pie served as the solo dessert at the bowling alley and restaurant. Go eat one of these.
He’s got more pi pie, if you click over there.
Gareth Branwyn at MakeZine offers more pie and a mnemonic:
How to Remember Pi to 15 Digits
By way of sci-fi author and mathenaut Rudy Rucker’s Facebook wall comes this:
One way to remember the first few digits of pi is to count the letters in the words of this phrase:
“How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”
[Image via FreakingNews]
b.love offers this clock image (is this clock for sale somewhere?):
A clock for pi day
Chirag Singh explains his “passion for pi.”
Daniel Tammet, “Different Ways of Knowing:
Geeks are really out in force today, flaunting pi for all they’ve got.
More:
## Happy birthday, Albert Einstein! 135 years, today
March 14, 2014
How many ways can we say happy birthday to a great scientist born on Pi Day? So, an encore post.
E=energy; m=mass; c=speed of light
Happy Einstein Day! to us. Albert’s been dead since 1955 — sadly for us. Our celebrations now are more for our own satisfaction and curiosity, and to honor the great man — he’s beyond caring.
Almost fitting that he was born on π Day, no? I mean, is there an E=mc² Day?
Albert Einstein was born on March 14, 1879, in Ulm, Germany, to Hermann and Pauline Einstein. 26 years later, three days after his birthday, he sent off the paper on the photo-electric effect; that paper would win him the Nobel Prize in Physics in another five years, in 1921. In that same year of 1905, he published three other papers, solving the mystery of Brownian motion, describing what became known as the Special Theory of Relativity and solving the mystery of why measurements of the light did not show any effects of motion as Maxwell had predicted, and a final paper that noted a particle emitting light energy loses mass. This final paper amused Einstein because it seemed so ludicrous in its logical extension that energy and matter are really the same stuff at some fundamental point, as expressed in the equation demonstrating an enormous amount of energy stored in atoms, E=mc².
Albert Einstein as a younger man – Nobel Foundation image
Any one of the papers would have been a career-capper for any physicist. Einstein dashed them off in just a few months, forever changing the fields of physics. And, you noticed: Einstein did not win a Nobel for the Special Theory of Relativity, nor for E=mc². He won it for the photo electric effect. Irony in history.
106 years later Einstein’s work affects us every day. Relativity theory at some level I don’t understand makes possible the use Global Positioning Systems (GPS), which revolutionized navigation and mundane things like land surveying and microwave dish placement. Development of nuclear power both gives us hope for an energy-rich future, and gives us fear of nuclear war. Sometimes, even the hope of the energy rich future gives us fear, as we watch and hope nuclear engineers can control the piles in nuclear power plants damaged by earthquakes and tsunami in Japan.
Albert Einstein on a 1966 US stamp (Photo credit: Wikipedia)
If Albert Einstein was a genius at physics, he was more dedicated to pacifism. He resigned his German citizenship to avoid military conscription. His pacifism made the German Nazis nervous; Einstein fled Germany in the 1930s, eventually settling in the United States. In the U.S., he was persuaded by Leo Szilard to write to President Franklin Roosevelt to suggest the U.S. start a program to develop an atomic weapon, because Germany most certainly was doing exactly that. But while urging FDR to keep up with the Germans, Einstein refused to participate in the program himself, sticking to his pacifist views. Others could, and would, design and build atomic bombs. (Maybe it’s a virus among nuclear physicists — several of those working on the Manhattan Project were pacifists, and had great difficulty reconciling the idea that the weapon they worked on to beat Germany, was deployed on Japan, which did not have a nuclear weapons program.)
Everybody wanted to claim, and honor Einstein; USSR issued this stamp dedicated to Albert Einstein Русский: Почтовая марка СССР, посвящённая Альберту Эйнштейну (Photo credit: Wikipedia)
Einstein was a not-great father, and probably not a terribly faithful husband at first — though he did think to give his first wife, in the divorce settlement, a share of a Nobel Prize should he win it. Einstein was a good violinist, a competent sailor, an incompetent dresser, and a great character. His sister suffered a paralyzing stroke. For many months Albert spent hours a day reading to her the newspapers and books of the day, convinced that though mute and appearing unconscious, she would benefit from hearing the words. He said he did not hold to orthodox religions, but could there be a greater show of faith in human spirit?
Einstein in 1950, five years before his death
When people hear clever sayings, but forget to whom the bon mots should be attributed, Einstein is one of about five candidates to whom all sorts of things are attributed, though he never said them. (Others include Lincoln, Jefferson, Mark Twain and Will Rogers). Einstein is the only scientist in that group. So, for example, we can be quite sure Einstein never claimed that compound interest was the best idea of the 20th century. This phenomenon is symbolic of the high regard people have for the man, even though so few understand what his work was, or meant.
A most interesting man. A most important body of work. He deserves more study and regard than he gets.
More, Resources:
## It’s π (pi) Day, 2014!
March 14, 2014
One might wonder when a good sociologist will write that book about how our vexing and depressing times push people to extreme measures, dwelling on one particular manifestation: The invention of new celebrations on the calendar.
In my lifetime Halloween grew from one short dress-up night with candy for kids, to a major commercially-exploited festival, from an interesting social event to a religiously-fraught night of bacchanalia with a weeks-long buildup. Cinco de Mayo grew into a festival of all things Mexican, though very few people can explain what the day commemorates, even among our Mexican neighbors. (No, it’s not Mexico’s Independence Day.)
St. Patrick’s Day grew in stature, and Guinness products now are freely available in almost every state. Bastille Day gets a celebration even in Oak Cliff, Texas. I’ve pushed Hubble Day, and Feynman Day; this weekend I’ll encourage people to celebrate James Madison’s birthday — and in January, I encourage the commemoration of Millard Fillmore’s birthday.
It could be a fun book, if not intellectually deep.
It will explain why, on Einstein’s birthday, March 14, we celebrate the number π (pi).
That book has not been written down, yet. So we’re left simply to celebrate.
Down in Austin, at SXSW, some performance artist used the sky as his canvas on π Day Eve; The Austin American-Statesman captured it:
Caption from the Austin American-Statesman: Austin American-Statesman Tomorrow is pi day, aka March 14, aka 3.14 – which is why mathematical pi was spelled out in a circle over Austin today (Friday isn’t supposed to be as good of a day for skywriting)
Photo by Austin Humphreys / Austin American-Statesman — with Joseph Lawrence Cantu.
There’s too much good stuff, on Einstein and on pi, for one post.
Happy pi Day!
Here’s to Albert Einstein, wherever you are!
More:
A photo at Life in a Pecan Guild caught a much more informative photograph of the Austin skywriting:
Each letter, and digit (yes, they printed pi in the Sky!) was formed by five airplanes flying in formation. Wow. Just wow. Photo at Life in a Pecan Guild. (Go there for more photos.)
There will be some great photos of that Austin skywriting, I predict. Will you point them out to us, in comments?
## It’s a desert out there: Salmon Research at Iliamna Lake, Alaska 2013 – Jason Ching film
March 6, 2014
Sitting in a hot trailer out on the northern New Mexico desert, Arizona State’s great soil scientist Tom Brown tipped back his cowboy hat, and asked me if I had been lonely over the previous week. Classes at BYU started up in August, and our other field workers on the project, with the University of Utah Engineering Experiment Station, for EPA and New Mexico Public Service, had gone back to class. My classes at the University of Utah didn’t start for a few more weeks — so I was holding down the fort by myself.
Dr. Brown’s expertise in reading air pollution damage on desert plants propelled a good part of the work. He showed me how to tell the difference between sulfur dioxide damage and nitrogen oxide damage on grasses and other plants, and how to tell when it was insects. He had some great stories. As a Mormon, he was also full of advice on life.
The Shiprock, a plug from an ancient volcano, left after the mountain eroded away. Near Shiprock, New Mexico, on the Navajo Reservation. Wikipedia image by Bowie Snodgrass
Between Farmington where our hotel was, and Teec Nos Pos where our most distant (non-wet) sampling site was, radio reception was lousy most of the time. The Navajo-language AM station in Farmington played some of the best music, and sometimes it could be caught as far west as Shiprock . Most of the time, driving across Navajoland, I had nothing but my thoughts to accompany me. Well, thoughts and the all-too-frequent Navajo funeral processions, 50 pickups long on a two-lane highway.
“No, not lonely. There’s a lot of work, I’ve got good books, and sleep is good,” I told him.
“Enjoy it,” Brown said. “The best time for any researcher is out in the field. And when you’re young, and you haven’t seen it all, it’s better.”
Indian rice grass in the sunlight (Oryzopsis hymendoides). Photo from the Intermountain Herbarium, Utah State University Extension Service
Brown spent a couple of days. Within a couple of weeks I turned everything over to other Ph.Ds to shut down the wet sampling for the winter, and caught a ride back to Provo (closer to where I lived) in a Cessna with a pilot who loved to fly low enough to see the canyons along the way. Get a map and think of the possibilities, with a landing in Moab; if you don’t drool at the thought of such a trip in the air but not too high, if your heart doesn’t actually beat faster thinking of such a trip, go see your physician for treatment.
By that time I was out of film, alas.
My few summers out in the desert chasing air pollution stay fixed in the surface of my memory. Indian rice grass still excites me in the afternoon sun (Oryzopsis hymenoides) — one of the more beautiful of grasses, one of the more beautiful and soil-holding desert plants. When hear the word “volcano,” I think of the Shiprock. When I read of air pollution damage, I think of all the pinon, aspen, cottonwoods, firs and other trees we gassed; when I see aspen in its full autumn glory, I remember those dozen or so leaves we caused to turn with SO2 (slight damage turns the leaves colors; greater damage makes them necrotic, a bit of a mirror of autumn).
All of that came back as I watched Jason Ching’s film, “Salmon Research at Iliamna Lake, Alaska 2013,” a simple six-minute compilation of shots taken with modern electronic cameras, including the hardy little GoPros, and with assistance from a DJI Phantom Quadcopter drone. Wow, what we could have captured with that equipment!
Ching’s description of the film:
This video showcases the scenery of Iliamna Lake and shows some of the 2013 research of the Alaska Salmon Program’s Iliamna Lake research station, one of four main facilities in Southwest Alaska . Established in the 1940′s, the Program’s research has been focused on ecology and fisheries management relating primarily to salmon and the environment in Bristol Bay, Alaska. Check out our program at: fish.washington.edu/research/alaska/
Filmed and edited by Jason Ching
Additional footage provided by Cyril Michel
Song:
“The long & quiet flight of the pelican” by Ending Satellites (endingsatellites.com)
Shot on a Canon 5d Mark II, Canon T3i, GoPro Hero 2 and GoPro Hero 3
JasonSChing.com
I am very grateful to be a part of such a long standing, and prominent program that allows me to work in the field in such an incredible setting with fantastic folks. This is the second video I created, the first one in 2012, to merely show family and friends back at home what I’ve been up to during the summer. This video was often shot between, or during field sampling events so a special thanks goes out to all those who supported me by continuing to work while I fiddled with camera gear.
Do you really want to get kids more interested in science? Show them this stuff. Scientists get the front seats on cool stuff — and they often get paid to do it, though they won’t get rich.
Researching life, and rocks, geography and landscape, and water resources, one may be alone in a desert, or a desert of human communication. Then one discovers just how beautiful the desert is, all the time.
More:
• Yes, I know; Indian rice grass has been renomenclaturedAchnatherum hymenoides (Roemer & J.A. Shultes) Barkworth, or Stipa hymenoides Roemer & J.S. Shultes, or Oryzopsis hymenoides (Roemer & J.S. Shultes) Ricker ex Piper. It is the State Grass of Utah
## Annals of Global Warming: January 2014 ranks 4th warmest January since 1880
February 21, 2014
### Last month was one of the warmest Januaries ever. No, really
And so it was.
Caption from AGU Blog: This is why the global temperature is not taken in your backyard in January. When you average the entire globe for an entire year, a much different picture emerges. NASA Aqua satellite image of a cold and snowy Mid-Atlantic Wednesday morning.
Information from the National Climatic Data Center (NCDC) of NOAA:
Global Highlights:
• The combined average temperature over global land and ocean surfaces for January was the warmest since 2007 and the fourth warmest on record at 12.7°C (54.8°F), or 0.65°C (1.17°F) above the 20th century average of 12.0°C (53.6°F). The margin of error associated with this temperature is ± 0.08°C (± 0.14°F).
• The global land temperature was the highest since 2007 and the fourth highest on record for January, at 1.17°C (2.11°F) above the 20th century average of 2.8°C (37.0°F). The margin of error is ± 0.18°C (± 0.32°F).
• For the ocean, the January global sea surface temperature was 0.46°C (0.83°F) above the 20th century average of 15.8°C (60.5°F), the highest since 2010 and seventh highest on record for January. The margin of error is ± 0.04°C (± 0.07°F).
Introduction:
Temperature anomalies and percentiles are shown on the gridded maps below. The anomaly map on the left is a product of a merged land surface temperature (Global Historical Climatology Network, GHCN) and sea surface temperature (ERSST.v3b) anomaly analysis developed by Smith et al. (2008). Temperature anomalies for land and ocean are analyzed separately and then merged to form the global analysis. For more information, please visit NCDC’s Global Surface Temperature Anomalies page. The January 2014 Global State of the Climate report includes percentile maps that complement the information provided by the anomaly maps. These maps on the right provide additional information by placing the temperature anomaly observed for a specific place and time period into historical perspective, showing how the most current month, season, or year-to-date compares with the past.
Temperatures:
In the atmosphere, 500-millibar height pressure anomalies correlate well with temperatures at the Earth’s surface. The average position of the upper-level ridges of high pressure and troughs of low pressure—depicted by positive and negative 500-millibar height anomalies on the January 2014 map—is generally reflected by areas of positive and negative temperature anomalies at the surface, respectively.
January 2014 Blended Land and Sea Surface Temperature Anomalies in degrees Celsius
January 2014 Blended Land and Sea Surface Temperature Percentiles
The combined global land and ocean average temperature during January 2014 was 0.65°C (1.17°F) above the 20th century average. This was the warmest January since 2007 and the fourth highest since records began in 1880. This marks the ninth consecutive month (since May 2013) with a global monthly temperature among the 10 highest for its respective month. The Northern Hemisphere land and ocean surface temperature during January 2014 was also the warmest since 2007 and the fourth warmest since records began in 1880 at 0.75°C (1.35°F) above average. The Southern Hemisphere January 2014 temperature departure of +0.55°C (+0.99°F) was the warmest since 2010 and the fourth warmest January on record.
During January 2014, most of the world’s land areas experienced warmer-than-average temperatures, with the most notable departures from the 1981–2010 average across Alaska, western Canada, Greenland, Mongolia, southern Russia, and northern China, where the departure from average was +3°C (+5.4°F) or greater. Meanwhile, parts of southeastern Brazil and central and southern Africa experienced record warmth with temperature departures between 0.5°C to 1.5°C above the 1981–2010 average, contributing to the highest January Southern Hemisphere land temperature departure on record at 1.13°C (2.03°F) above the 20th century average. This was also the warmest month for the Southern Hemisphere land since September 2013 when temperatures were 1.23°C (2.21°F) above the 20th century average. Some locations across the globe experienced departures that were below the 1981–2010 average. These areas include the eastern half of the contiguous U.S., central Canada, and most of Scandinavia and Russia. The most notable cold anomalies were in Russia, where in some areas the departure from average was 5°C (9°F) below average. Overall, the Northern Hemisphere land surface temperature was 1.17°C (2.11°F) above average—the warmest January since 2007 and the fourth warmest since records began in 1880.
Select national information is highlighted below. (Please note that different countries report anomalies with respect to different base periods. The information provided here is based directly upon these data):
• France’s nationally-averaged January 2014 temperature was 2.7°C (4.9°F) above the 1981–2010 average, tying with 1988 and 1936 as the warmest January on record.
• Spain experienced its warmest January since 1996 and the third warmest since national records began in 1961, with a temperature of 9°C (48.2°F) or 2°C (3.6°F) above the 1971–2000 average.
• The January temperature in Switzerland was 2.4°C (4.3°F) above the 1981–2010 average—the fifth warmest January since national records began 150 years ago.
• Austria experienced its fifth warmest January since national records began in 1768. The nationally-averaged temperature was 3.3°C (5.9°F) above the 1981–2010 average. However, in some regions across the southern parts of the country, the temperatures were the highest on record. In Klagenfurt, the temperature departure was 5°C (9°F)—the highest since January 1813.
• China, as a whole, recorded an average temperature of -3.4°C (25.9°F) or 1.6°C (2.9°F) above average during January 2014. This was the second highest January value, behind 2002, since national records began in 1961.
• In Argentina, persistence of extremely high temperatures across central and northern parts of the country resulted in several locations setting new maximum, minimum, and mean temperature records for the month of January.
• Warm temperatures engulfed much of Australia during January 2014. Overall, the national average mean temperature was 0.91°C (1.64°F) above the 1961–1990 average. This was the 12th highest January temperature since national records began in 1910. Regionally, the January 2014 temperature ranked among the top 10 warmest in Queensland, Victoria, and South Australia.
Across the oceans, temperature departures tend to be smaller than across the land surfaces. According to the percentiles map, much-warmer-than-average conditions were present across parts of the Atlantic Ocean, the northeastern and western Pacific Ocean, and parts of the Indian Ocean. Record warmth was observed across parts of the northern Pacific Ocean (south of Alaska), parts of the western Pacific Ocean, south of South Africa, and across parts of the Atlantic Ocean. Overall, the global ocean surface temperature in January was +0.46°C (+0.83°F)—the warmest since 2010 and the seventh warmest on record.
Warming denialists will scream about these data.
More:
## Lincoln and Darwin, born hours apart, February 12, 1809
February 12, 2014
Is it an unprecedented coincidence? 205 years ago today, just minutes apart according to unconfirmed accounts, Abraham Lincoln was born in a rude log cabin on Nolin Creek, in Kentucky, and Charles Darwin was born into a wealthy family at the family home in Shrewsbury, England.
Gutzon Borglum’s 1908 bust of Abraham Lincoln in the Crypt of the U.S. Capitol – Architect of the Capitol photo
Lincoln would become one of our most endeared presidents, though endearment would come after his assassination. Lincoln’s bust rides the crest of Mt. Rushmore (next to two slaveholders), with George Washington, the Father of His Country, Thomas Jefferson, the author of the Declaration of Independence, and Theodore Roosevelt, the man who made the modern presidency, and the only man ever to have won both a Congressional Medal of Honor and a Nobel Prize, the only president to have won the Medal of Honor. In his effort to keep the Union together, Lincoln freed the slaves of the states in rebellion during the civil war, becoming an icon to freedom and human rights for all history. Upon his death the entire nation mourned; his funeral procession from Washington, D.C., to his tomb in Springfield, Illinois, stopped twelve times along the way for full funeral services. Lying in state in the Illinois House of Representatives, beneath a two-times lifesize portrait of George Washington, a banner proclaimed, “Washington the Father, Lincoln the Savior.”
Charles Darwin statue, Natural History Museum, London – NHM photo
Darwin would become one of the greatest scientists of all time. He would be credited with discovering the theory of evolution by natural and sexual selection. His meticulous footnoting and careful observations formed the data for ground-breaking papers in geology (the creation of coral atolls), zoology (barnacles, and the expression of emotions in animals and man), botany (climbing vines and insectivorous plants), ecology (worms and leaf mould), and travel (the voyage of H.M.S. Beagle). At his death he was honored with a state funeral, attended by the great scientists and statesmen of London in his day. Hymns were specially written for the occasion. Darwin is interred in Westminster Abbey near Sir Isaac Newton, England’s other great scientist, who knocked God out of the heavens.
Lincoln would be known as the man who saved the Union of the United States and set the standard for civil and human rights, vindicating the religious beliefs of many and challenging the beliefs of many more. Darwin’s theory would become one of the greatest ideas of western civilization, changing forever all the sciences, and especially agriculture, animal husbandry, and the rest of biology, while also provoking crises in religious sects.
Lincoln, the politician known for freeing the slaves, also was the first U.S. president to formally consult with scientists, calling on the National Science Foundation (whose creation he oversaw) to advise his administration. Darwin, the scientist, advocated that his family put the weight of its fortune behind the effort to abolish slavery in the British Empire. Each held an interest in the other’s disciplines.
Both men were catapulted to fame in 1858. Lincoln’s notoriety came from a series of debates on the nation’s dealing with slavery, in his losing campaign against Stephen A. Douglas to represent Illinois in the U.S. Senate. On the fame of that campaign, he won the nomination to the presidency of the fledgling Republican Party in 1860. Darwin was spurred to publicly reveal his ideas about the power of natural and sexual selection as the force behind evolution, in a paper co-authored by Alfred Russel Wallace, presented to the Linnean Society in London on July 1, 1858. On the strength of that paper, barely noticed at the time, Darwin published his most famous work, On the Origin of Species, in November 1859.
The two men might have got along well, but they never met.
What unusual coincidences.
Go celebrate human rights, good science, and the stories about these men.
A school kid could do much worse than to study the history of these two great men. In fact, we study them far too little, it seems to me.
Resources:
Charles Darwin:
Abraham Lincoln:
More:
Anybody know what hour of the day either of these men was born?
This is mostly an encore post.
Yes, you may fly your flag today for Lincoln’s birthday; the official holiday, Washington’s Birthday, is next Monday, February 17th — and yes, it’s usually called “President’s Day” by merchants and calendar makers.
## Working to stand the heat in the kitchen
February 10, 2014
Dr. Isis was wronged, and improperly attacked on the internet for the situation.
Masthead from Dr. Isis’s blog. Note the shoes.
She’s working to deal with whether to continue to write, and in what form . . .
Lessons from my much more political years, and graduate study in rhetoric.
1. You know you’ve got a movement when opposition forms against you. That’s irritating, but it’s better than not having opposition, which means you’re failing to get your point across, most often.
2. Clear communication, especially writing, gets a response — sometimes not the response you expected or wanted, but a response. With practice, you can hone your message.
“Cicero Denouncing Cataline,” in 63 BC; 1889 fresco painting by Cesare Maccari (1840-1919). Wikipedia image
Used to be a couple of posters available to rhetoric students, both attributed to Plutarch’s Lives, a comparison of the Greek, Demosthenes, with the Roman, Cicero. The first, talking about the later man, said, “When Cicero spoke, the people said how well he spoke.”
The second said, “When Demosthenes spoke, the people cried, ‘Let us march!’”
Which man was the more effective orator, or rhetorician?
Demosthenes Practicing Oratory (Démosthène s’exerçant à la parole); Jean Lecomte du Nouÿ (1842-1923)
Effective writing makes people angry. That’s what it should do.
From 1945 on, countless scientists wrote about “potential harms to wildlife” from chemicals put on crops, and used for other purposes.
[In 1962] Rachel Carson wrote about chemicals, naming names — especially DDT — and described little robins writhing and twitching in their death throes. She’s credited with starting a movement. But before that, the chemical industry teamed up to run a \$500,000 public relations campaign (in 1962!), claiming Carson was hysterical, unqualified, and wrong, perhaps a communist, but not anyone you’d want your children to be around. She calmly asked scientists to review her notes and find errors. They found none.
Tone? Truth comes in many tones. The wise seek it even if they don’t like the tone it takes at the moment.
## All those animals on the ark? I don’t think so
February 5, 2014
No, I didn’t watch Bill Nye dissect Ken Ham in the science vs. creationism debate. I share with many other science-loving people a conviction that “debating” creationists is wholly irrelevant, and tends only to build the glory of the creationists who cannot manage to set up a single scientific observation or experiment to provide evidence for creationism, but can stand on a stage and crack bad jokes and lie, against a mumbling scientist.
But I have looked at some of the commentary, and some of Nye’s remarks and rebuttals. Nye did very well.
Nye tended to develop clear, non-scientific explanations for the issues. Ham and creationists aren’t ready for that.
In that vein, J. Rehling tweeted this astonishingly clear explanation for why it’s just impossible to “believe” that the fabled ark of Noah could carry even most of the species alive, in one boat (and, mind you, the San Diego Zoo is neither the world’s largest collection of species on display in a zoo, nor displaying a significant percentage of all species):
Two pictures that tell the story.
How big was Noah’s Ark? Not big enough, especially compared to the San Diego Zoo and the USS Nimitz.
San Diego Zoo and USS Nimitz, the largest ship in the U.S. Navy; clearly, no ark built by Noah could have been big enough to carry all land animals. Image mashup by JRehling
January 28, 2014
From Twitter today; working to track down more details.
A photo by John C. Olsen, taken in Hastings, Nebraska, perhaps on December 31, 2013:
From Fascinating Pics: One of the rarest weather phenomena, Mammatus Clouds. Photo taken by John C. Olsen in Hastings, NE pic.twitter.com/dlPNaPa25D
Our boys liked clouds from the start. A couple of our early cloud identification books featured mammatus clouds (guess where the name came from); and before each boy was 11, we had seen these clouds here in Texas, often in that treacherous time known as tornado season.
Beautiful clouds, yes, but often scary — well, until you read from the University of Illinois that they tend to follow nasty storms, not precede them.
Mammatus Cloudssagging pouch-like structuresMammatus are pouch-like cloud structures and a rare example of clouds in sinking air.
Sometimes very ominous in appearance, mammatus clouds are harmless and do not mean that a tornado is about to form; a commonly held misconception. In fact, mammatus are usually seen after the worst of a thunderstorm has passed.
As updrafts carry precipitation enriched air to the cloud top, upward momentum is lost and the air begins to spread out horizontally, becoming a part of the anvil cloud. Because of its high concentration of precipitation particles (ice crystals and water droplets), the saturated air is heavier than the surrounding air and sinks back towards the earth.
The temperature of the subsiding air increases as it descends. However, since heat energy is required to melt and evaporate the precipitation particles contained within the sinking air, the warming produced by the sinking motion is quickly used up in the evaporation of precipitation particles. If more energy is required for evaporation than is generated by the subsidence, the sinking air will be cooler than its surroundings and will continue to sink downward.
The subsiding air eventually appears below the cloud base as rounded pouch-like structures called mammatus clouds.
Mammatus are long lived if the sinking air contains large drops and snow crystals since larger particles require greater amounts of energy for evaporation to occur. Over time, the cloud droplets do eventually evaporate and the mammatus dissolve.
Our experience is the clouds look a lot cooler than can be captured on film or in electronic images. Mr. Olsen captured a great image.
Very nice shot
## What if you don’t know enough even to cheat on the chemistry exam?
January 15, 2014
Brilliant work from Saturday Morning Breakfast Cereal.
This cartoon is witty and funny — and it is a wonderful illustration of how people need to know enough to see the humor, or cheat.
Don’t catch the gags? See here.
Saturday Morning Breakfast Cereal, by Zach Weiner: The exam on the Periodic Table of Elements
You may discuss the cartoon at the SMBC blog:
August 26, 2011
Well, this record may stand for a while. 57 panels, baby.
Also, Phil and I figured out some extended periodic table elements. Who can tell me the abbreviation for Element 5885?
Discuss this comic in the forum
Or discuss it here at the Bathtub.
The cartoon reminds me of so many lazy or not-up-to-par students who would stay up late inventing ways to cheat on an exam, when a bit of study would have paid so many more dividends.
It’s harder to cheat, most of the time, that to be honest and learn the stuff.
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2014-04-25 01:28:42
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http://cafe46.org/site/2esuf4.php?tag=238d45-about-sphere-in-maths
|
Sphere Facts.
a hypersphere.
This sphere was a fused quartz gyroscope for the Gravity Probe B experiment which differs in shape from a perfect sphere by no more than 40 atoms of thickness. Any cross section through a sphere is a circle (or, in the degenerate case where the slicing plane is "The Sphere." Les caractéristiques du maillage qui sert à sa représentation sont précisées par l'utilisateur (ajustement de la finesse). exprimé en radians) est §33 in Solid
, une sphère de centre qu:Lunq'uscn:Sfera it:Sfera Les segments [OA] , [OB], [OA'] et [OB'] sont des rayons.. Les segments [AA’] et [BB’] sont des diamètres.. Les points A et A’ sont diamétralement opposés.. Les points B et B’ sont diamétralement opposés.. Les cercles bleu et rouge sont des grands cercles: leur centre et leur rayon sont ceux de la sphère.
New York: Dover, 1973. The generalization of a sphere in dimensions is called → Wolfram Web Resource.
1991.
r Sphaira, Henry George Liddell, Robert Scott, New Scientist | Technology | Roundest objects in the world created, calculate area and volume with your own radius-values to understand the equations, https://math.wikia.org/wiki/Sphere?oldid=16864, a 0-sphere is a pair of endpoints of an interval (−.
Pour cette raison, la sphère apparaît dans la nature, par exemple les bulles et gouttes d'eau (en l'absence de gravité) sont des sphères car la tension superficielle essaie de minimiser l'aire.
De tout temps, la sphère de par ses nombreuses symétries a passionné les hommes. The surface area of a sphere with diameter D is, More generally, the area element on the sphere is given in spherical coordinates by, In particular, the total area can be obtained by integration, The volume of a sphere with radius r and diameter d = 2r is.
et le volume d'une n-boule de rayon r est égal au produit de cette aire par A sphere is an object that is an absolutely round geometrical shape in three-dimensional space. . (Ed.). sr:Сфераsv:Sfär
Temple Geometry Problems.
This means that every point on the sphere will be an umbilical point.
Handbook of Mathematics and Computational Science.
Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point.
Several properties hold for the plane which can be thought of as a sphere with infinite radius. {\displaystyle r} 01 80 82 54 80 du lundi au vendredi de 9h30 à 19h30 Also see Volume and Area of a Sphere Calculator. nl:sfeer (wiskunde)no:Kule (geometri) Foundation, pp. ) (see also trigonometric functions and spherical coordinates). 2 using disk integration.
la:Sphaera The shortest distance between two distinct non-antipodal points on the surface and measured along the surface, is on the unique great circle passing through the two points.
y
and is the radius. Of course, topologists would regard this equation as instead describing an -sphere. Il n'existe pas de patron de la sphère. cs:Koule It is an example of a compact topological manifold without boundary. Circles on the sphere that are parallel to the equator are lines of latitude. {\displaystyle r} On the sphere, points are defined in the usual sense, but the analogue of "line" may not be immediately apparent.
zh-classical:球hr:Sfera It has a constant width and circumference. d'informations ?
https://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Sphere.html. , 1998. Of all the shapes, a sphere has the smallest surface area for a volume.
A hemisphere is an exact half of a sphere. Sphere. Balls and marbles are shaped like spheres. et de masse M, par rapport à un axe passant par son centre est : L'élément d'aire de la sphère de rayon
Snapshots, 3rd ed. y Twice the radius is called the diameter, and pairs of points on the sphere … {\displaystyle \displaystyle \phi } 56, 88-91, 1972. da:Kugleeo:Sfero
simple:Sphere {\displaystyle r} According to the Archimedes Principle, the volume of a sphere is given as, The volume of Sphere(V) = 4/3 πr3 Cubic Units. The n-sphere is denoted Sn. Register with BYJU’S – The Learning App to learn about other three-dimensional shapes also watch interactive videos to learn with ease, CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, Important Questions Class 8 Maths Chapter 5 Data Handling, Important Questions Class 10 Maths Chapter 4 Quadratic Equations, Important Questions Class 9 Maths Chapter 11 Constructions, CBSE Previous Year Question Papers Class 12 Maths, CBSE Previous Year Question Papers Class 10 Maths, ICSE Previous Year Question Papers Class 10, ISC Previous Year Question Papers Class 12 Maths. From the above stated equations it can be expressed as follows: An image of one of the most accurate man-made spheres, as it refracts the image of Einstein in the background. The volume inside a sphere is given by the formula $V=\frac{4\pi}{3}r^3$ where r is the radius of the sphere. Sphère et boule : Ce cours a pour objectifs de travailler les définitions de la sphère, de la boule, l’aire d’une sphère, le volume d’une boule et les sections d’une sphère par un plan.
New York: Dover, 1997.
The sphere is the inverse image of a one-point set under the continuous function ||x||. {\displaystyle (x,y,z)} This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid). Beyer, W. H. are called antipodes.
A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius")
Note )In modern mathematics… Kenison, E. and Bradley, H. C. "The Intersection of a Sphere with Another Surface." (
Winnipeg, Manitoba, Canada: Charles Babbage Research , 106-108, {\displaystyle {r \over n}}
Coxeter, H. S. M. Regular
La géométrie sphérique est la science qui étudie les propriétés des sphères.
New York: Springer-Verlag, pp.
The important properties of the sphere are: We know that the radius is twice the radius, the diameter of a sphere formula is given as: Since all the three-dimensional objects have the surface area and volume, the surface area and the volume of the sphere is explained here. Treatise on the Geometry of the Circle and Sphere.
In spherical trigonometry, angles are defined between great circles.
This distance r is known as the radius of the sphere. The volume of a hemisphere is $V=\frac{2\pi}{3}r^3$. Cela permet aussi de calculer l'aire d'une zone sphérique, c’est-à-dire d'une portion de sphère limitée par deux plans parallèles qui intersectent la sphère (ou lui sont tangents). Find the volume of the sphere that has a diameter of 10 cm? L'aire d'une sphère de rayon x . fa:کره (شکل هندسی)gd:Cruinne All the points on the surface are equidistant from the centre. It is thought that only neutron stars are smoother. New York: Macmillan, 1935.
Cours maths 3ème. §198 in Descriptive pt:Esfera (geometria) r
"Classic Surfaces from Differential Geometry: Sphere." et de rayon Harris, J. W. and Stocker, H.
For example, in $\Z^n$ with Euclidean metric, a sphere of radius r is nonempty only if r2 can be written as sum of n squares of integers.
https://www.geom.umn.edu/zoo/toptype/sphere/.
{\displaystyle (x_{0},y_{0},z_{0})} =
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Eppstein, D. "Circles and Spheres."
Sphere Facts.
a hypersphere.
This sphere was a fused quartz gyroscope for the Gravity Probe B experiment which differs in shape from a perfect sphere by no more than 40 atoms of thickness. Any cross section through a sphere is a circle (or, in the degenerate case where the slicing plane is "The Sphere." Les caractéristiques du maillage qui sert à sa représentation sont précisées par l'utilisateur (ajustement de la finesse). exprimé en radians) est §33 in Solid
, une sphère de centre qu:Lunq'uscn:Sfera it:Sfera Les segments [OA] , [OB], [OA'] et [OB'] sont des rayons.. Les segments [AA’] et [BB’] sont des diamètres.. Les points A et A’ sont diamétralement opposés.. Les points B et B’ sont diamétralement opposés.. Les cercles bleu et rouge sont des grands cercles: leur centre et leur rayon sont ceux de la sphère.
New York: Dover, 1973. The generalization of a sphere in dimensions is called → Wolfram Web Resource.
1991.
r Sphaira, Henry George Liddell, Robert Scott, New Scientist | Technology | Roundest objects in the world created, calculate area and volume with your own radius-values to understand the equations, https://math.wikia.org/wiki/Sphere?oldid=16864, a 0-sphere is a pair of endpoints of an interval (−.
Pour cette raison, la sphère apparaît dans la nature, par exemple les bulles et gouttes d'eau (en l'absence de gravité) sont des sphères car la tension superficielle essaie de minimiser l'aire.
De tout temps, la sphère de par ses nombreuses symétries a passionné les hommes. The surface area of a sphere with diameter D is, More generally, the area element on the sphere is given in spherical coordinates by, In particular, the total area can be obtained by integration, The volume of a sphere with radius r and diameter d = 2r is.
et le volume d'une n-boule de rayon r est égal au produit de cette aire par A sphere is an object that is an absolutely round geometrical shape in three-dimensional space. . (Ed.). sr:Сфераsv:Sfär
Temple Geometry Problems.
This means that every point on the sphere will be an umbilical point.
Handbook of Mathematics and Computational Science.
Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point.
Several properties hold for the plane which can be thought of as a sphere with infinite radius. {\displaystyle r} 01 80 82 54 80 du lundi au vendredi de 9h30 à 19h30 Also see Volume and Area of a Sphere Calculator. nl:sfeer (wiskunde)no:Kule (geometri) Foundation, pp. ) (see also trigonometric functions and spherical coordinates). 2 using disk integration.
la:Sphaera The shortest distance between two distinct non-antipodal points on the surface and measured along the surface, is on the unique great circle passing through the two points.
y
and is the radius. Of course, topologists would regard this equation as instead describing an -sphere. Il n'existe pas de patron de la sphère. cs:Koule It is an example of a compact topological manifold without boundary. Circles on the sphere that are parallel to the equator are lines of latitude. {\displaystyle r} On the sphere, points are defined in the usual sense, but the analogue of "line" may not be immediately apparent.
zh-classical:球hr:Sfera It has a constant width and circumference. d'informations ?
https://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Sphere.html. , 1998. Of all the shapes, a sphere has the smallest surface area for a volume.
A hemisphere is an exact half of a sphere. Sphere. Balls and marbles are shaped like spheres. et de masse M, par rapport à un axe passant par son centre est : L'élément d'aire de la sphère de rayon
Snapshots, 3rd ed. y Twice the radius is called the diameter, and pairs of points on the sphere … {\displaystyle \displaystyle \phi } 56, 88-91, 1972. da:Kugleeo:Sfero
simple:Sphere {\displaystyle r} According to the Archimedes Principle, the volume of a sphere is given as, The volume of Sphere(V) = 4/3 πr3 Cubic Units. The n-sphere is denoted Sn. Register with BYJU’S – The Learning App to learn about other three-dimensional shapes also watch interactive videos to learn with ease, CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, Important Questions Class 8 Maths Chapter 5 Data Handling, Important Questions Class 10 Maths Chapter 4 Quadratic Equations, Important Questions Class 9 Maths Chapter 11 Constructions, CBSE Previous Year Question Papers Class 12 Maths, CBSE Previous Year Question Papers Class 10 Maths, ICSE Previous Year Question Papers Class 10, ISC Previous Year Question Papers Class 12 Maths. From the above stated equations it can be expressed as follows: An image of one of the most accurate man-made spheres, as it refracts the image of Einstein in the background. The volume inside a sphere is given by the formula $V=\frac{4\pi}{3}r^3$ where r is the radius of the sphere. Sphère et boule : Ce cours a pour objectifs de travailler les définitions de la sphère, de la boule, l’aire d’une sphère, le volume d’une boule et les sections d’une sphère par un plan.
New York: Dover, 1997.
The sphere is the inverse image of a one-point set under the continuous function ||x||. {\displaystyle (x,y,z)} This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid). Beyer, W. H. are called antipodes.
A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius")
Note )In modern mathematics… Kenison, E. and Bradley, H. C. "The Intersection of a Sphere with Another Surface." (
Winnipeg, Manitoba, Canada: Charles Babbage Research , 106-108, {\displaystyle {r \over n}}
Coxeter, H. S. M. Regular
La géométrie sphérique est la science qui étudie les propriétés des sphères.
New York: Springer-Verlag, pp.
The important properties of the sphere are: We know that the radius is twice the radius, the diameter of a sphere formula is given as: Since all the three-dimensional objects have the surface area and volume, the surface area and the volume of the sphere is explained here. Treatise on the Geometry of the Circle and Sphere.
In spherical trigonometry, angles are defined between great circles.
This distance r is known as the radius of the sphere. The volume of a hemisphere is $V=\frac{2\pi}{3}r^3$. Cela permet aussi de calculer l'aire d'une zone sphérique, c’est-à-dire d'une portion de sphère limitée par deux plans parallèles qui intersectent la sphère (ou lui sont tangents). Find the volume of the sphere that has a diameter of 10 cm? L'aire d'une sphère de rayon x . fa:کره (شکل هندسی)gd:Cruinne All the points on the surface are equidistant from the centre. It is thought that only neutron stars are smoother. New York: Macmillan, 1935.
Cours maths 3ème. §198 in Descriptive pt:Esfera (geometria) r
"Classic Surfaces from Differential Geometry: Sphere." et de rayon Harris, J. W. and Stocker, H.
For example, in $\Z^n$ with Euclidean metric, a sphere of radius r is nonempty only if r2 can be written as sum of n squares of integers.
https://www.geom.umn.edu/zoo/toptype/sphere/.
{\displaystyle (x_{0},y_{0},z_{0})} =
This site uses Akismet to reduce spam. Learn how your comment data is processed.
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2021-10-27 00:57:48
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http://math.stackexchange.com/questions/409641/matlab-code-for-fixed-point-iteration
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# Matlab code for fixed point iteration
I want to write in Matlab a function that appreciates the fixed point iteration for a system of equations. The idea is:
$\begin{bmatrix} x{_{1}}^{m+1}\\ x{_{2}}^{m+1} \end{bmatrix}= \begin{bmatrix}y{_{1}}^{n}+hf{_{1}}(x{_{1}},x{_{2}})\\ y{_{2}}^{n}+hf{_{2}}(x{_{1}},x{_{2}}) \end{bmatrix}$
The iterations stop when $|x_{1}^{m+1} -x_{1}^{m}|<TOL$ and $|x_{2}^{m+1} -x_{2}^{m}|<TOL$ or when m arrives the limit of iterations M...
At the function, do I have to write : if ($|x_{1}^{m+1} -x_{1}^{m}|<TOL$ && $|x_{2}^{m+1} -x_{2}^{m}|<TOL$){ $y_{1}=x_{1}^{m+1}$; $y_{2}=x_{2}^{m+1}$; } ? But when only one of them is smaller than TOL how can it get its value ??
-
If only one of them is smaller than TOL, the iteration goes on. – Shuhao Cao Jun 2 '13 at 21:57
So is my "if" loop right, or should it be OR instead of AND ??? – Mary Star Jun 2 '13 at 22:00
Based on your description, definitely you should use &&. – Shuhao Cao Jun 2 '13 at 22:36
Could you take a look at my "for" loop?? Is this right?? [code]for m=1:M $x_{1}$(m+1)=y1n+h*$f_{1}$($tn+h,x_{1}(m),x-{2}(m)$); $x_{2}$(m+1)=y2n+h*$f_{2}$($tn+h,x_{1}(m),x_{2}(m)$); if (abs($x_{1}(m+1)-x_{1}(m))<TOL$ && abs($x_{2}(m+1)-x_{2}(m))<TOL$) $y_{1}=x_{1}(m+1)$; $y_{2}=x_{2}(m+1)$; return end x1(m)=x1(m+1); x2(m)=x2(m+1); end $y_{1}=x1(M)$; $y_{2}=x2(M)$;[/code] – Mary Star Jun 2 '13 at 22:41
Looks ok, but if you literally write y_{1} and put it in MATLAB, I believe MATLAB will give you a syntax error. – Shuhao Cao Jun 2 '13 at 22:45
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2016-06-26 08:18:54
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http://hotc.camdar.io/fw/declarative.html
|
# System F$$_\omega^{++}$$
$$F_\omega$$ is the calculus of higher-kinded type constructors, combining two axes of the lambda cube (polymorphism and type operators).
The syntax of $$F_\omega$$ is an extension of $$F$$. First, we need to adjust our syntax to use type constructors (or just "constructors") instead of just types. By convention, we will use $$\tau$$ to denote a nullary constructor, a constructor kinded at $$*$$ (see below).
$\require{bussproofs} c, \tau := \alpha \mid c \rightarrow c \mid \forall (\alpha : k).c \mid \lambda (\alpha : k) .c \mid c\ c$
where $$\alpha$$ is a type variable used in quantification and type abstraction.
Next, we will add $$k$$, to denote kinds:
$k := * \mid k \rightarrow k$
We also need terms to inhabit those types:
$e := x \mid \lambda (x:\tau).e \mid \Lambda(\alpha:k).e \mid e[\tau]$
Our context, $$\Gamma$$, may contain judgments about types and terms.
$\Gamma := \cdot \mid \Gamma, x:\tau \mid \Gamma : \alpha:k$
You may also see the kinding judgment written as $$\alpha :: k$$.
Finally, as noted earlier, we will need to extend $$F_\omega$$ with primitive product kinds to handle ML modules (hence the ++ in $$F_\omega^{++}$$):
\begin{aligned} k &:= \dots \mid k \times k \\ c &:= \dots \mid \langle c, c \rangle \mid \pi_1 c \mid \pi_2 c \end{aligned}
This language is defined statically as follows:
Rules 1.1 (Kinding): $$\Gamma \vdash c:k$$
$\begin{prooftree} \AxiomC{\Gamma(\alpha) = k} \UnaryInfC{\Gamma \vdash \alpha : k} \end{prooftree} \qquad \begin{prooftree} \AxiomC{\Gamma \vdash c_1:*} \AxiomC{\Gamma \vdash c_2:*} \BinaryInfC{\Gamma \vdash c_1 \rightarrow c_2 : *} \end{prooftree} \qquad \begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash c:*} \UnaryInfC{\Gamma \vdash \forall(\alpha:k).c:*} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma \vdash c_1:k \rightarrow k'} \AxiomC{\Gamma \vdash c_2:k} \BinaryInfC{\Gamma \vdash c_1\ c_2 : k'} \end{prooftree} \qquad \begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash c:k'} \UnaryInfC{\Gamma \vdash \lambda (\alpha:k).c : k \rightarrow k'} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma \vdash c_1:k_1} \AxiomC{\Gamma \vdash c_2:k_2} \BinaryInfC{\Gamma \vdash \langle c_1, c_2 \rangle : k_1 \times k_2} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma \vdash c:k_1 \times k_2} \UnaryInfC{\Gamma \vdash \pi_1 c : k_1} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma \vdash c:k_1 \times k_2} \UnaryInfC{\Gamma \vdash \pi_2 c : k_2} \end{prooftree}$
Rules 1.2 (Typing): $$\Gamma \vdash e:\tau$$
$\begin{prooftree} \AxiomC{\Gamma(x) = \tau} \UnaryInfC{\Gamma \vdash x:\tau} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma, x:\tau \vdash e:\tau'} \AxiomC{\Gamma \vdash \tau:*} \BinaryInfC{\Gamma \vdash \lambda(x:\tau).e : \tau \rightarrow \tau'} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma, \vdash e_1:\tau \rightarrow \tau'} \AxiomC{\Gamma \vdash e_2:\tau} \BinaryInfC{\Gamma \vdash e_1\ e_2 : \tau'} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash e:\tau} \UnaryInfC{\Gamma \vdash \Lambda(\alpha:k).e : \forall(\alpha:k).\tau} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma \vdash e:\forall(\alpha:k).\tau'} \AxiomC{\Gamma \vdash \tau:k} \BinaryInfC{\Gamma \vdash e[\tau] : [\tau/\alpha]\tau'} \end{prooftree}$
Note that these rules aren't sufficient -- If we have $$f : \forall(\beta:* \rightarrow *). \beta\ \texttt{int} \rightarrow \texttt{unit}$$, we'd want to have $$f[\lambda(\alpha:*).\alpha]\ 12 : \texttt{unit}$$. However, by the above rules, we type $$f[\lambda(\alpha:*):\alpha]$$ at $$((\lambda \alpha.\alpha) \ \texttt{int}) \rightarrow \texttt{unit}$$, not $$\texttt{int} \rightarrow \texttt{unit}$$.
To remedy this, we need some way to express that $$(\lambda \alpha.\alpha) \ \texttt{int}$$ is equivalent to $$\texttt{int}$$, which we will accomplish by defining a new judgment $$\Gamma \vdash c \equiv c' : k$$, then adding to rules 1.2 the equivalence rule
$\begin{prooftree} \AxiomC{\Gamma \vdash e:\tau} \AxiomC{\Gamma \vdash \tau \equiv \tau':*} \BinaryInfC{\Gamma \vdash e:\tau'} \end{prooftree}$
Rules 1.3 (Constructor Equivalence): $$\Gamma \vdash c \equiv c':k$$
Equivalence $\begin{prooftree} \AxiomC{\Gamma \vdash c:k} \UnaryInfC{\Gamma \vdash c \equiv c:k} \end{prooftree} \qquad \begin{prooftree} \AxiomC{\Gamma \vdash c \equiv c':k} \UnaryInfC{\Gamma \vdash c' \equiv c:k} \qquad \end{prooftree} \qquad \begin{prooftree} \AxiomC{\Gamma \vdash c_1 \equiv c_2:k} \AxiomC{\Gamma \vdash c_2 \equiv c_3:k} \BinaryInfC{\Gamma \vdash c_1 \equiv c_3:k} \end{prooftree}$
Compatibility $\begin{prooftree} \AxiomC{\Gamma \vdash \tau_1 \equiv \tau_1' : *} \AxiomC{\Gamma \vdash \tau_2 \equiv \tau_2' : *} \BinaryInfC{\Gamma \vdash \tau_1 \rightarrow \tau_2 \equiv \tau_1' \rightarrow \tau_2' : *} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash \tau \equiv \tau' : *} \UnaryInfC{\Gamma \vdash \forall(\alpha:k).\tau \equiv \forall(\alpha:k).\tau':*} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash c \equiv c':k'} \UnaryInfC{\Gamma \vdash \lambda(\alpha:k).c \equiv \lambda(\alpha:k).c': k \rightarrow k'} \end{prooftree} \qquad \begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash c_1 \equiv c_1':k \rightarrow k'} \AxiomC{\Gamma \vdash c_2 \equiv c_2':k} \BinaryInfC{\Gamma \vdash c_1\ c_2 \equiv c_1'\ c_2':k'} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma \vdash c_1 \equiv c_1':k_1} \AxiomC{\Gamma \vdash c_2 \equiv c_2':k_2} \BinaryInfC{\Gamma \vdash \langle c_1, c_2 \rangle \equiv \langle c_1', c_2' \rangle : k_1 \times k_2} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma \vdash c \equiv c' : k_1 \times k_2} \UnaryInfC{\Gamma \vdash \pi_1c \equiv \pi_1c' : k_1} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma \vdash c \equiv c' : k_1 \times k_2} \UnaryInfC{\Gamma \vdash \pi_2c \equiv \pi_2c' : k_2} \end{prooftree}$
Reduction/beta $\begin{prooftree} \AxiomC{\Gamma, \alpha:k \vdash c:k'} \AxiomC{\Gamma \vdash c':k} \BinaryInfC{\Gamma \vdash (\lambda (\alpha:k).c)\ c' \equiv [c'/\alpha]c:k'} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma \vdash c_1:k_1} \UnaryInfC{\Gamma \vdash \pi_1\langle c_1, c_2\rangle \equiv c_1:k_1} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma \vdash c_2:k_2} \UnaryInfC{\Gamma \vdash \pi_2\langle c_1, c_2\rangle \equiv c_2:k_2} \end{prooftree}$
Extensionality/eta $\begin{prooftree} \AxiomC{\Gamma \vdash c:k_1 \rightarrow k_2} \AxiomC{\Gamma \vdash c':k_1 \rightarrow k_2} \AxiomC{\Gamma, \alpha:k_1 \vdash c\ \alpha \equiv c'\ \alpha:k_2} \TrinaryInfC{\Gamma \vdash c \equiv c' : k_1 \rightarrow k_2} \end{prooftree}$ $\begin{prooftree} \AxiomC{\Gamma \vdash \pi_1 c \equiv \pi_1 c':k_1} \AxiomC{\Gamma \vdash \pi_2 c \equiv \pi_2 c':k_2} \BinaryInfC{\Gamma \vdash c \equiv c':k_1 \times k_2} \end{prooftree}$
Note that in many cases, we enforce that some constructors are types (kind $$*$$) -- we certainly can't have tuples or lambda abstractions underlying a $$\forall$$ type, for example.
The first set of rules, labeled "equivalence", ensure that this is indeed an equivalence/congruence relation.
The beta rules are the interesting ones. With them, we can prove (assuming we've defined the relevant primitive types):
$\begin{prooftree} \AxiomC{} \UnaryInfC{\vdash \texttt{int}:*} \AxiomC{} \UnaryInfC{\beta:* \vdash \beta:*} \BinaryInfC{\vdash (\lambda \beta.\beta)\ \texttt{int} \equiv\texttt{int}:*} \UnaryInfC{\vdash (\lambda \beta.\beta)\ \texttt{int} \rightarrow \texttt{unit} \equiv \texttt{int} \rightarrow \texttt{unit}:*} \end{prooftree}$
Finally, the extensionality rules allow us to evaluate "underneath" tuples and lambda abstractions. You might think of these as conducting an "experiment" to see whether the relevant constructors behave equivalently.
## Remarks
• As it turns out, in this system, the $$:k$$ annotation on equivalence is unnecessary, as they are uniquely determined by the kinding rules.
Theorem (Regularity).
1. If $$\vdash \Gamma\ ok$$ and $$\Gamma \vdash e:\tau$$, then $$\Gamma \vdash \tau:*$$.
2. If $$\vdash \Gamma\ ok$$ and $$\Gamma \vdash c \equiv c':k$$, then $$\Gamma \vdash c:k$$ and $$\Gamma \vdash c':k$$ Proof.
3. By induction over the judgment $$\Gamma \vdash e:\tau$$.
4. By induction over the judgment $$\Gamma \vdash c \equiv c':k$$. $$\square$$
where $$\vdash \Gamma\ ok$$ is a judgment ensuring that all types in $$\Gamma$$ are well-formed.
Rules 1.4 (Well-formed contexts): $$\vdash \Gamma\ ok$$ $\begin{prooftree} \AxiomC{} \UnaryInfC{\vdash \cdot\ ok} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\Gamma\ ok} \AxiomC{\Gamma \vdash \tau:*} \BinaryInfC{\vdash \Gamma, x:\tau\ ok} \end{prooftree}\qquad \begin{prooftree} \AxiomC{\vdash \Gamma\ ok} \UnaryInfC{\vdash \Gamma, \alpha:k\ ok} \end{prooftree}$
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2022-05-19 04:29:20
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http://www.russinoff.com/libman/text/node26.html
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# 5.3 Denormals and Zeroes
An exponent field of 0 is used to encode numerical values that lie below the normal range. If the exponent and significand fields of an encoding are both 0, then the encoded value itself is 0 and the encoding is said to be a zero. If the exponent field is 0 and the significand field is not, then the encoding is either denormal or pseudo-denormal:
Definition 5.3.1 (zerp, denormp, pseudop) Let be an encoding for a format with .
(a) If , then is a zero encoding for F.
(b) If and either is implicit or , then is a denormal encoding for F.
(c) If is explicit and , then is a pseudo-denormal encoding for F.
Note that a zero can have either sign:
Definition 5.3.2 (zencode) Let be a format and let . Then
There are two differences between the decoding formulas for denormal and normal encodings:
1. For a denormal encoding for an implicit format, the integer bit is taken to be 0 rather than 1.
2. The power of 2 represented by the zero exponent field of a denormal or pseudo-denormal encoding is rather than .
Definition 5.3.3 (ddecode) If is an encoding for a format with , then
We also define a general decoding function:
Definition 5.3.4 (decode) Let be an encoding for a format . If , then
(sgn-ddecode, expo-ddecode, sig-ddecode) Let be a denormal encoding for a format and let .
(a) .
(b) .
(c) .
PROOF: (a) is trivial; (b) and (c) follow from Lemmas 4.1.13 and 4.1.14
The class of numbers that are representable as denormal encodings is recognized by the following predicate.
Definition 5.3.5 (drepp) Let be a format and let . Then is representable as a denormal in iff the following conditions hold:
(a) ;
(b) ;
(c) is -exact.
If a number is so representable, then its encoding is constructed as follows.
Definition 5.3.6 (dencode) If is representable as a denormal in , then
where
Next, we examine the relationship between the decoding and encoding functions.
(drepp-ddecode, dencode-ddecode) If is a denormal encoding for a format , then is representable as a denormal in and
PROOF: Let , , , and . Since ,
and by Lemma 4.1.3,
which is equivalent to Definition 5.3.5(b). In order to prove (c), we must show, according to Definition 4.2.1, that
But
This establishes that is representable as a denormal.
Now by Definition 5.3.3,
Therefore, by Definitions 5.3.1, 5.3.6, and 5.1.4 and Lemmas 2.4.9 and 2.2.5,
(denormp-dencode, ddecode-dencode) If be representable as a denormal in , then is a denormal encoding for andand
PROOF: Let , , , and . By Lemma 2.4.1, is a -bit vector and by Lemma 2.4.7,
and
Since is -exact,
and since ,
by Lemma 4.1.8, which implies
Finally, according to Definition 5.3.3,
The smallest positive denormal is computed by the following function:
Definition 5.3.7 (spd) For any format , .
(positive-spd, drepp-spd, spd-smallest)
For any format ,
(a) ;
(b) is representable as a denormal in ;
(c) If is representable as a denormal in , then .
PROOF: Let , , and . It is clear that is positive. To show that is -exact, we need only observe that
Finally, since
holds and moreover, is the smallest positive that satisfies
Every number with a denormal representation is a multiple of the smallest positive denormal.
(spd-mult) If and let be a format, then is representable as a denormal in iff for some , .
PROOF: Let and . For , let . Then and
We shall show, by induction on , that is representable as a denormal for . First note that for all such ,
Suppose that is representable as a denormal for some , . Then is -exact, and by Lemma 4.2.16, so is . But since , it follows from Lemma 4.2.5 that is also -exact. Since
, i.e., , and hence, is representable as a denormal.
Now suppose that is representable as a denormal. Let . Clearly, , and . It follows from Lemma 4.2.17 that , and consequently, is -exact. Thus, by Lemma 4.2.16,
David Russinoff 2017-08-01
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2018-01-18 11:47:50
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http://bootmath.com/solve-fx-lambda-intlimits_01maxxtxtftdt.html
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# Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$
I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$.
Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$.
1st derivative: \begin{align*} f'(x) &= \lambda\left(\int\limits_0^x (1+t)f(t)dt + x(1+x)f(x) – xf(x) + \int\limits_x^1 tf(t)dt – x^2f(x)\right)\\
&=\lambda\left(\int\limits_0^x f(t)dt + \int\limits_0^x tf(t)dt + \int\limits_x^1 tf(t)dt\right)\\
&=\lambda\left(\int\limits_0^1 tf(t)dt + \int\limits_0^x f(t)dt\right)
\end{align*}
2nd derivative: $\ f''(x) = \lambda f(x)$
so $f(x) = c_1 e^{\sqrt{\lambda}x} + c_2 e^{-\sqrt{\lambda}x}$
How can I find $c_1$ and $c_2$?
#### Solutions Collecting From Web of "Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$"
Your integral equation (IE, for short) is of the type:
$$\tag{1} (I-\lambda T) f =0$$
with $I$ the identity operator and $T$ given by:
$$Tf(x):= \int_0^1 \big( \max \{x,t\}+xt\big)\ f(t)\ \text{d} t\; ,$$
which is a bounded, compact linear operator on $C^0([0,1])$.
The Fredholm alternative applies to your IE, hence there are two possibilities: either A) (1) has only the trivial solution $f(t):=0$, or B) there exists a linear space of nontrivial solutions. In case A one says that $\lambda$ is a regular value for $T$; on the other hand, in case B one says that $\lambda$ is a singular value for $T$.
It is also known that the singular values of a compact operator form a sequence, say $\{\lambda_n\}_{n\in \mathbb{N}_0}$, such that $|\lambda_n|\to \infty$ and that $\{\lambda_n\}$ has no finite accumulation point.
Your computations show that any $C^0$ solution of (1) is actually of class $C^2$ and solves the second order linear ODE $f^{\prime \prime} -\lambda f=0$; moreover, it is not hard to see that any solution of your IE satisfies also the Robin boundary conditions $f(0)-f^\prime (0)=0$, $f(1)-f^\prime (1)=0$ (to prove this, it suffices to plug $x=0,1$ into the IEs for $f$ and $f^\prime$); therefore each solution of (1) is also a solution of the BVP:
$$\tag{2} \begin{cases} f^{\prime \prime} (t)-\lambda f(t)=0 &\text{, in } ]0,1[ \\ f(0)-f^\prime (0)=0 \\ f(1)-f^\prime (1)=0 \end{cases}$$
which is an eigenvalue problem for the differential operator $\tfrac{\text{d}^2}{\text{d} t^2}$ under the given boundary conditions.
On the other hand, I think one could prove that the eigenvalue problem (2) is in fact equivalent to (1), in the sense that any $C^2$ solution of (2) is also a solution of (1) (in order to prove this, I bet some integration by parts and other elementary tricks have to be used).
Thus if you have to solve (1), then you have to look for the eigenvalues of (2).
To find the eigenvalues of (2) one has to distinguish three cases:
• $\lambda >0$: in such a case one can plug $\lambda =\omega^2$ with $\omega >0$ into the ODE to get the general integral:
$$f(t):=A e^{\omega t} +B e^{-\omega t}\; ;$$
then either (2) has only the trivial solution, or $\lambda =\omega^2$ is an eigenvalue of (2): the latter eventuality occurs iff the homogeneous linear system:
$$\begin{cases} f(0)-f^\prime (0)=0 \\ f(1)-f^\prime (1)=0 \end{cases} \quad \Leftrightarrow \quad \begin{cases} (1-\omega) A+(1+\omega) B=0 \\ (1-\omega )e^\omega A+(1-\omega)e^{-\omega} B=0 \end{cases}$$
possesses nontrivial solutions, i.e. iff $\omega =1$ and $\lambda =1=:\lambda_0$.
If $\lambda=\lambda_0$, all the eigenfunctions corresponding to $\lambda_0$, i.e. the nontrivial solutions of (2), are in the form:
$$f_0(t):=Ae^t\; .$$
• $\lambda =0$: in this case the general integral of the ODE is $f(t):=A+Bt$ and it is easy to see that there are no nontrivial solutions of (2).
• $\lambda <0$: one can set $\lambda =-\omega^2$ (with $\omega >0$) into the ODE to get the general integral:
$$f(t):= A\cos \omega t+B\sin \omega t\; ;$$
then either (2) has only the trivial solution, or $\lambda =-\omega^2$ is an eigenvalue of (2): the latter case occurs iff the homogeneous linear system:
$$\begin{cases} A-\omega B =0 \\ (\cos \omega +\omega \sin \omega) A + (\sin \omega -\omega \cos \omega)B=0\end{cases}$$
has some nontrivial solutions, i.e. iff $\omega =n\pi$ and $\lambda =-n^2\pi^2=:\lambda_n$ with $n\in \mathbb{N}$.
If $\lambda=\lambda_n$, all the eigenfunctions corresponding to $\lambda_n$ are in the form:
$$f_n(t):=n\pi B \cos \omega t + B \sin \omega t\; .$$
Finally, summing it up, you have the following result:
The integral equation
$$\tag{1} f(x) = \lambda \int_0^1\big(\max \{x,t\} +xt\big)f(t)\ \text{d} t$$
has only the trivial solution $f(x):=0$ iff $\lambda \notin \{ \lambda_n\}_{n\in \mathbb{N}_0}=\{1,-\pi^2, -4\pi^2,\ldots ,-n^2\pi^2,\ldots \}$; it has also nontrivial solutions in the form:
$$f_0 (x) := Ae^x \qquad \text{(} A\text{ an arbitrary constant)}$$
iff $\lambda =\lambda_0 =1$; it has also nontrivial solutions in the form:
$$f_n(x) =n\pi B \cos (n\pi x)+ B \sin (n\pi x) \qquad \text{(} B\text{ an arbitrary constant)}$$
iff $\lambda =\lambda_n =-n^2\pi^2$ for some $n\in \mathbb{N}$.
This is not a complete answer and too long for a comment.
If $f$ is a so is a constant multiple of $f$, therefore you need at least one more condition on $f$ for a unique solution.
We get two constraints on $f$ by putting $x = 0$ and $x = 1$ in turn. that gives
us $$f(0) = \lambda \int_0^1 tf(t) dt \mbox{ and } f(1) = \lambda\left\{\int_0^1 f(t)dt + \int_0^1 tf(t) dt\right\}.$$
Now, we break the problem into three cases.
If $\lambda = 0,$ then $f$ is identically zero.
If $\lambda = – k^2$ is negative then write $f = A \cos kx + B \sin kx.$
use $\int_0^1 \cos kt dt = \frac{\sin k}{k}, \int_0^1 \sin kt dt = \frac{1-\cos k}{k}$ and
$\int_0^1 t\cos kt dt = \frac{k\sin k + \cos k – 1}{k^2}, \int_0^1 t\sin kt dt = \frac{\sin k-k\cos k}{k^2}$. I hope I have not made any errors in evaluating these integrals.
Now put these back in the constraints and demanding non trivial solution for the pair $A, B$ will give a relation connecting $k$ and $\lambda.$
depending on the uniqueness of condition on $f$ should now provide the eigenvalues $k.$
I have not done the case of positive $\lambda$ which would involve hyperbolic sine and cosines.
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2018-07-19 15:33:03
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https://www.thecodingforums.com/threads/programmig-newbie-where-to-start.971525/page-3
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Programmig Newbie....where to start?
Discussion in 'C Programming' started by Kohtro, May 30, 2014.
1. David BrownGuest
Yes, I think you understand what I meant. And yes, the digits came out
slower and slower - this was a programming exercise for a maths and
computation course, and speed was not particularly important. (I
actually speed it up by using a faster converging sequence than
4*atan(1/4), but the principle was the same.)
So not much good if you really want thousands of digits of your number,
but fun anyway.
David Brown, Jun 4, 2014
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2018-06-23 08:30:21
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http://www.thefullwiki.org/Tate's_thesis
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# Tate's thesis: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
Updated live from Wikipedia, last check: May 26, 2013 04:38 UTC (36 seconds ago)
(Redirected to Hecke character article)
In mathematics, in the field of number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
A name sometimes used for Hecke character is the German term Größencharakter (often written Grössencharakter, Grossencharakter, Grössencharacter, Grossencharacter et cetera).
## Definition using ideles
A Hecke character is a character of the idele class group of a number field or function field.
Some authors define a Hecke character to be a quasicharacter rather than a character. The difference is that a quasicharacter is a homomorphism to the non-zero complex numbers, while a character is a homomorphism to the unit circle. Any quasicharacter (of the idele class group) can be written uniquely as a character times a real power of the norm, so there is no big difference between the two definitions.
The conductor of a Hecke character χ is the largest ideal m such that χ is a Hecke character mod m. Here we say that χ is a Hecke character mod m if χ is trivial on ideles whose infinite part is 1 and whose finite part is integral and congruent to 1 mod m.
## Definition using ideals
The original definition of a Hecke character, going back to Hecke, was in terms of a character on fractional ideals. For a number field K, let m = mfm be a K-modulus, with mf, the "finite part", being an integral ideal of K and m, the "infinite part", being a (formal) product of real places of K. Let Im denote the group of fractional ideals of K relatively prime to mf and let Pm denote the subgroup of principal fractional ideals (a) where a is near 1 at each place of m in accordance with the multiplicities of its factors: for each finite place v in mf, ordv(a - 1) is at least as large as the exponent for v in mf, and a is positive under each real embedding in m. A Hecke character with modulus m is a group homomorphism from Im into the nonzero complex numbers such that on ideals (a) in Pm its value is equal to the value at a of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all archimedean completions of K where each local component of the homomorphism has the same real part (in the exponent). (Here we embed a into the product of archimedean completions of K using embeddings corresponding to the various archimedean places on K.) Thus a Hecke character may be defined on the ray class group modulo m, which is the quotient Im/Pm.
Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared. The role of the infinite part m is now subsumed under the notion of an infinity-type.
This definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct L-functions that extend the notion of a Dirichlet L-function from the rationals to other number fields. For a Hecke character χ, its L-function is defined to be the Dirichlet series
$\sum_{(I,m)=1} \chi(I) N(I)^{-s} = L(s, \chi)\,$
carried out over integral ideals relatively prime to the modulus m of the Hecke character. The notation N(I) means the norm of an ideal. The common real part condition governing the behavior of Hecke characters on the subgroups Pm implies these Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these L-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at s = 1 when the character is trivial. For primitive Hecke characters (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these L-functions satisfy a functional equation relating the values of the L-function of a character and the L-function of its complex conjugate character.
The characters are 'big' (thus explaining the original German term chosen by Hecke) in the sense that the infinity-type when present non-trivially means these characters are not of finite order. The finite-order Hecke characters are all, in a sense, accounted for by class field theory: their L-functions are Artin L-functions, as Artin reciprocity shows. But even a field as simple as the Gaussian field has Hecke characters that go beyond finite order in a serious way (see the example below). Later developments in complex multiplication theory indicated that the proper place of the 'big' characters was to provide the Hasse-Weil L-functions for an important class of algebraic varieties (or even motives).
## Special cases
• A Dirichlet character is a Hecke character of finite order.
• A Hilbert character is a Dirichlet character of conductor 1. The number of Hilbert characters is the order of the class group of the field; more precisely, class field theory identifies the Hilbert characters with the characters of the class group.
## Examples
• For the field of rational numbers, the idele class group is isomorphic to the product of the positive reals with all the unit groups of the p-adic integers. So a quasicharacter can be written as product of a power of the norm with a Dirichlet character.
• A Hecke character χ of the Gaussian integers of conductor 1 is of the form
χ((a)) = |a|s(a/|a|)4 n
for s imaginary and n an integer, where a is a generator of the ideal (a). The only units are powers of i, so the factor of 4 in the exponent ensures that the character is well defined on ideals.
## Tate's thesis
Hecke's original proof of the functional equation for L(s,χ) used an explicit theta-function. John Tate's celebrated doctoral dissertation, written under the supervision of Emil Artin, applied Pontryagin duality systematically, to remove the need for any special functions. A later reformulation in a Bourbaki seminar by Weil (Fonctions zetas et distributions, Séminaire Bourbaki 312, 1966) showed that the essence of Tate's proof could be expressed by distribution theory: the space of distributions (for Schwartz–Bruhat test functions) on the adele group of K transforming under the action of the ideles by a given χ has dimension 1.
## References
• J. Tate, Fourier analysis in number fields and Hecke's zeta functions (Tate's 1950 thesis), reprinted in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, MR1697859, ISBN 978-3-540-65399-8
• W. Narkiewicz (1990). Elementary and analytic theory of algebraic numbers (2nd ed.). Springer-Verlag/Polish Scientific Publishers PWN. pp. 334–343. ISBN 3-540-51250-0.
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2013-05-26 04:39:38
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https://questioncove.com/updates/564b56dde4b0040492e55c7f
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OpenStudy (anonymous):
Find the derivative of f(x) = -6/x at x = 12
2 years ago
OpenStudy (bibby):
factor out the -6, so we're left with $$\dfrac{d}{dx}\dfrac{-6}{x}=-6\dfrac{d}{dx}\dfrac{1}{x}$$ then we can rewrite $$\dfrac{d}{dx}\dfrac{1}{x}\implies \dfrac{d}{dx}x^{-1}$$ and use the power rule
2 years ago
OpenStudy (anonymous):
so once I have d/dx x^-1 what do I plug in @bibby
2 years ago
OpenStudy (bibby):
-1 is now your n|dw:1447778383254:dw|
2 years ago
OpenStudy (bibby):
|dw:1447778395463:dw|
2 years ago
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2017-11-24 21:58:05
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https://mathoverflow.net/questions/374410/applications-of-symplectic-geometry-to-classical-mechanics
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# Applications of symplectic geometry to classical mechanics
It is claimed that classical mechanics motivates introduction of symplectic manifolds. This is due to the theorem that the Hamiltonian flow preserves the symplectic form on the phase space.
I am wondering whether symplectic geometry has applications to classical mechanics. Was this connection useful for classical mechanics? Were methods of symplectic geometry relevant for it via, say, the above theorem?
• Do you know about Foundations of Mechanics by Abraham/Marsden? This book seemed to be fairly well known when I was in college, especially when the greatly expended 1980 2nd edition was published. – Dave L Renfro Oct 19 at 19:57
• I find this question puzzling, since symplectic geometry comes from Hamiltonian mechanics, which was developed specifically for studying classical mechanics. And Hamiltonian mechanics has been extremely useful for applications to classical mechanics. – Deane Yang Oct 20 at 2:26
• @DeaneYang : What you say just means that there is a connection between the two subjects. But I am trying to find an application. I.e. what concrete question in mechanics was solved with the use of symplectic geometry? – makt Oct 20 at 4:33
• Physicists solve mechanics problems using canonical transformations, which are same as symplectomorphisms of open sets in $\mathbb{R}^{2n}$. – Deane Yang Oct 20 at 4:39
• Here is the wikipedia page on canonical transformations. Most of it uses the definitions and notations of classical physics, but there is a section at the end giving the "modern mathematical description". There are two classical mechanics books listed at the end. They, as well as any other classical mechanics textbook, show how to use the Hamiltonian formalism to solve mechanics problems. en.wikipedia.org/wiki/Canonical_transformation – Deane Yang Oct 20 at 14:39
V.I. Arnold's Mathematical Methods of Classical Mechanics is entirely based on the ideas and methods of symplectic geometry, such as the Birkhoff normal form, the Kolmogorov- Arnold-Moser theorem on the persistence of invariant tori, and the intersection theory of Lagrangian submanifolds.
Alan Weinstein points out that the first symplectic submanifold was already introduced "avant la lettre" by Lagrange in 1808. Weinstein goes on to state what he calls the symplectic creed that Everything is a Lagrangian submanifold --- how that underlies classical mechanics is discussed in this MO question.
• My feeling is that saying that Arnold's book is "entirely based on the ideas and methods of symplectic geometry" is an exaggeration. The first 6 chapaters deal with classical mechanics without symplectic geometry at all. Then symplectic manifolds are introduced in order to discuss Hamiltonian formalism. This might be convenient but probably not strictly necessary since in many other books Hamiltonian formalism is discussed without the language of symplectic geometry (see e.g. vol. 1 of Landau-Lifshitz). – makt Oct 20 at 9:34
The list will be long, very long indeed. But to start:
1. Questions about dynamics of Hamiltonian systems are at the heart of symplectic topology, symplectic capacities are precisely introduced for that purpose to understand the difference between a mere volume-preserving flow and a Hamiltonian flow. This includes questions about closed orbits etc. Here the books of Hofer-Zehner or McDuff-Salomon are a good start.
2. Even if interested only in mechanics in $$\mathbb{R}^{2n}$$: as soon as it comes to symmetries (and mechanics deals a lot with symmetries) one inevitably ends up with concepts of phase space reduction. The reduced phase space of the isotropic harmonic oscillator (could there be something more relevant for mechanics ?) is $$\mathbb{CP}^n$$ with Fubini-Study Kähler structure. Quite a complicated geometry already. In classical textbooks you discuss the Kepler problem by fixing the conserved quantities (angular momentum, etc) to certain values. This is just a phase space reduction in disguise. The geometry becomes less-dimensional but more complicated by doing so. Coadjoint orbits are symplectic and needed for descriptions of symmetries in a similar fashion. Without geometric insight, their structure is hard to grasp, I guess. The aforementioned textbook of Abraham and Marsden as well as many others provide here a good first reading. In fact, up to some mild topological assumptions any symplectic manifold arises as reduced phase space from $$\mathbb{R}^{2n}$$ according to a theorem of Gotay and Tuynman. From that perspective, symplectic geometry is mechanics with symmetries.
3. If trying to understand Hamilton-Jacobi theory, it is pretty hard to get anywhere without the geometric notion of a Lagrangean submanifold. This was perhaps one of the main motivations for Weinstein's Lagrangean creed.
4. Mechanical systems with constraints require a good understanding of the geometry of the constraints. This brings you into the realm of symplectic geometry where coisotropic submanifolds (aka first class constraints in mechanics) are at home.
5. When restricting the configuration space of a mechanics system (think of the rigid body) then you are actually talking about the cotangent bundle of the config space as (momentum) phase space. This is perhaps one of the very starting points where symplectic geometry takes of.
6. Going beyond classical mechanics, one perhaps is interested in quantum mechanics: here symplectic geometry provides a very suitable platform to ask all kind of questions. It is the starting point to try geometric quantization, deformation quantization and alike.
7. Maybe more exotic, but I really like that: integrable systems can have quite subtle and non-trivial monodromies. There is a very nice book (and many papers) of Cushman and Bates on this. The mechanical systems are really simple in the sense that you find them in all physics textbooks. But the geometry is hidden and highly non-trivial as it involves really a global point of view to uncover it.
8. From a more practical point of view, non-holonomic mechanics is of great importance to all kind of engineering problems (robotics, cars, whatevery). Here a geometry point off view really help and is a large area of research. Also mechanical control theory is not only about fiddling around with ode's but there is a lot of (symplectic) geometry necessary to fully understand things. The textbooks of Bloch as well as Bullo and Lewis might give you a first hint why this is so.
9. As a last nice application of (mostly linear) symplectic geometry one should not forget optics! This is of course not mechanics, but optics has a very interesting symplectic core, beautifully outlined in the textbook by Guillemin and Sternberg.
Well, I could go on, but the margin is to small to contain all the information, as usual ;) Of course, for many things one can just keep working in local coordinates and ignore the true geometric features. But one will miss a lot of things on the way.
• Thank you. This is an impressive list. I am not a specialist, but my feeling is that it requires a little broader scope of classical mechanics than I am used to. But may be this is what people mean when they discuss applications of symplectic geometry to it. – makt Oct 20 at 15:35
• @makt : Already under the point 1 of the list, there are recent applications to finding periodic orbits in the three body problem (papers by O. van Koert, U. Frauenfelder, et al.). I would say this qualifies as "classical mechanics". Historically the so-called "Arnold conjecture" in sympletic topology originated in the desire to generalize Poincaré's last geometric theorem (proved by G.D. Birkhoff see irma.math.unistra.fr/~maudin/Arnold.pdf for a very interesting account of this history), again largely motivated by celestial mechanics. – BS. Oct 26 at 16:16
There is a "symplectic structure" on the set of body motions.
During the years 1960–1970, Jean-Marie Souriau, proved that under very general assumptions, the set of all possible solutions of a classical mechanical system, involving material points interacting by very general forces, has a smooth manifold structure (not always Hausdorff) and is endowed with a natural symplectic form. He called it the manifold of motions of the mechanic
J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1969.
J.-M. Souriau, La structure symplectique de la mécanique décrite par Lagrange en 1811, Mathématiques et sciences humaines, tome 94 (1986), pages 45–54. Numérisé par Numdam, http://www.numdam.org.
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2020-11-24 07:19:15
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https://www.examveda.com/if-w-is-the-uniformly-distributed-load-on-a-circular-slab-of-radius-r-fixed-at-its-ends-the-maximum-positive-radial-moment-at-its-centre-is/
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Examveda
# If ‘W’ is the uniformly distributed load on a circular slab of radius ‘R’ fixed at its ends, the maximum positive radial moment at its centre, is
A. $$\frac{{3{\text{W}}{{\text{R}}^2}}}{{16}}$$
B. $$\frac{{2{\text{W}}{{\text{R}}^2}}}{{16}}$$
C. $$\frac{{{\text{W}}{{\text{R}}^2}}}{{16}}$$
D. None of these
1. C
Related Questions on RCC Structures Design
If the shear stress in a R.C.C. beam is
A. Equal or less than 5 kg/cm2, no shear reinforcement is provided
B. Greater than 4 kg/cm2, but less than 20 kg/cm2, shear reinforcement is provided
C. Greater than 20 kg/cm2, the size of the section is changed
D. All the above
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2023-02-06 07:21:02
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http://dasublogbyprashanth.blogspot.com/2010/04/why-open-source-is-not-socialism.html
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## 2010-04-26
### Why Open-Source is not Socialism
I was thinking of writing something on this for a few days, but I got lazy. Then, I saw this (Glyn Moody, The H Open) article, and it gave me the perfect motivation to actually write this.
I'll first sum up what he says, as he covers most of the important stuff. Follow the jump to read more about Linux, Microsoft, capitalism, socialism, cars, and the music industry.
When open-source (I won't say "free software", as that'll lead to way too much confusion between libre free and gratis free) software was first being distributed, it could make money as the source code was also being sold for personal modification and redistribution; people could of course get free-of-charge distributions of the original program, but to make their own customized program, it was worth paying for the original source code. It still is today, though that has changed as the method of distribution (previously, floppy disks sent by mail) has become far faster and cheaper (now, files hosted online).
The most visible open-source software business model is that of Red Hat and its RHEL. It essentially gives away the product but sells the support, and this is critical for a lot of businesses, schools, and other agencies. Red Hat has been able to make millions of dollars in profits as a result. This also makes perfect economic sense, as the cost of producing another copy of RHEL is zero but the cost of providing support to consumers is not insignificant. This is exactly how a perfectly competitive market should work; there are a large number (more than 20) firms in the market, the price of the good equals the cost of producing an extra unit of it, and all of the firms sell at this price (and in turn make the most profits at this price). In this case, the 2 markets are of the Linux product (price is $0) and the support service (price varies slightly (versus competitors like Canonical, Mandriva, etc.) and depends on level of support, but starts at$80/year).
These leads me to my first point: Linux is the embodiment of capitalism, not socialism.
With Linux, one gets as close as possible to total freedom and infinite choice. Right now, on DistroWatch, there are over 650 unique distributions of Solaris, BSD, and Linux, and there are even more (that haven't been submitted to DistroWatch). Almost all of them distribute the product for free, as that is the marginal cost of producing the product, while almost all of the company-backed products sell support for similar prices, which is the price where these companies profit most. If a developer can't sustain the costs of developing further, that developer leaves the market, and invariably, another developer takes the original one's place. While this doesn't happen with larger companies, Mandriva has filed for bankruptcy before in hard times (and they have made it through each time).
Relate this to the car market, where there are many different auto manufacturers, both domestically and internationally. In the recent recession, GM and Chrysler had to file for bankruptcy, while Ford was generally doing poorly; other automakers started picking up their sales. Now, however, GM and Chrysler are back in the game, while Ford's sales (and shares) have skyrocketed; in the process, Toyota and Honda sales have fallen considerably. Auto bailout notwithstanding, that's how the market is supposed to work.
This is also how it used to be for the OS market at large. IBM, Commodore, Microsoft, and Apple, among other companies, all vied for market share; some entered while others dropped out, and the price was generally the same across the board (with a few exceptions).
Notwithstanding the presence of Apple in the high-end (price-wise) OS market, Microsoft has a monopoly over the OS market at large. It can dictate what features are and are not present without regard for consumer preferences (though that is changing very slowly) and can dictate what price it will sell at. For a monopoly, the most profitable price is not equal to the marginal cost; suffice it to say the the price will almost always be higher than the marginal cost, which is bad for consumers.
It's almost like...socialism. There, I said it.
Want to know why? Think about the days of socialism in Eastern Europe and the car market there. Only one firm (the government manufacturer) was present in each country, meaning that the government would dictate exactly what would happen to the cars regardless of consumers' wants and needs. The cars were quite expensive relative to consumers' incomes there but were near-dangerous to drive (due to the appalling build quality). In addition to this, consumers would have to wait 3-5 years on average to have a car (of that quality, of course) delivered as the government was the only supplier/seller of cars.
Doesn't that sound like a certain software company I have mentioned frequently? Hmmm...
The next part of the article discusses the implications of the open-source software business model for the music industry. Often, I have spoken negatively (to put it kindly) about the music industry's inability to keep up with modern times (DRM-free $0.99 iTunes songs being a notable and laudable exception), instead trying to further lock consumers down with ever increasing DRM and other restrictions. I have also spoken about the music industry's tone-deafness with regard to quoting losses from pirated music, as the vast majority of established musicians' income now comes from (ironically, being in the modern era) live concerts. This article is the first time an exact figure is quoted:$7.5 earned from live concerts for every $1 earned from record sales, and that's a conservative estimate (as far as I have read). 7.5 to 1. And the industry still doesn't get it. However, I have wondered before (and the article takes note of this too) what lesser-known artists would do in this regard. I see before me (in the article) a most ingenious solution, proposed by a lesser-known indie artist named Jill Sobule. She has proposed a scale of donations for which donors would get increasing rewards, from a free download of the album for a$10 donation or a CD sent out before official release for a $25 donation to a house concert (as well as all of the above listed (in the article)) for a$5000 donation or allowing the donor to sing in the album as well (as well as all of the above listed (in the article)) for a $10000 donation. For one of her albums, she made (and I quote) "$75000 in just two months". Her business model is such that she builds up a dedicated fanbase and uses their donations to help make the albums (and give the donors all of the appropriate listed rewards). When she gets enough money (which looks to be very soon), she will start doing live concerts.
That, my dear readers, is impressive.
And why is she able to release the songs for free (or very, very cheap)? That's because the donations cover her initial costs to make the song, while the marginal cost of producing copies is \$0.
That, and not a business model based on ever-increasing restrictions, is capitalism.
I would love to see your thoughts on this, so please do leave your thoughts in the comments section!
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2016-12-10 22:16:28
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https://www.wisdomjobs.com/e-university/six-sigma-tutorial-365/to-improve-financial-performance-and-profitability-12601.html
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# To Improve Financial Performance and Profitability - Six Sigma
Bob Galvin (then Motorola president) was reputed to be the man who began the Six Sigma revolution by issuing a ‘Six Sigma Challenge’ in 1987 for a ten - fold improvement in performance in every 2 year period (Goetsch and Davis, 2010). Over the 10 years following the call, Motorola claims to have saved $414 billion, increased sales by a factor of 5 and increased profits by 20% each year (Pande et al, 2000). GE declared that for 3 years (1996 - 1998) Six Sigma related savings were about$2bn; Honeywell stated that its annual Six Sigma savings as around $600-700 million; and Dow Chemicals claimed$2.2bn of Six Sigma financial benefits (Lee, 2002).
It is often stated that a ‘typical’ company operates around the 3 sigma level (Murphy, 1998) and there have been a number of attempts to quantify the financial effects of varying sigma levels. Klefsjo et al (2001) suggest that for Six Sigma performance levels the cost of poor quality would be less than 1percent of sales, while for 5 Sigma that would rise to 5 - 15 per cent, at 4 Sigma the cost would be 15 - 15 per cent and at 3 Sigma levels it would equate to around 25 - 40 per cent of sales.
Cost of Poor Quality
Perhaps the most obvious tangible benefit of quality improvement is the reduction of costs associated with non - quality. If we have to throw a product away because we have made an error in its manufacture, it is clear that there is an immediate financial impact as all the costs sunk into the product are lost. Similarly, doing an incorrect operation over again absorbs cost (operator time, power, additional materials, etc.).
Although anyone who works in an organization will be familiar with many examples of both of these issues, business accounting systems are not set up to capture these costs. Traditional accounting approaches are designed to track the inflow and outflow of money in an organization (and, by extension, to product lines or departments). there is little emphasis on whether the money in the department is spent effectively. For example, budget reporting will recognise that overtime cost £100,000 this month, but will not differentiate between time used to respond to short lead - time customer demand and time spent correcting errors.
Even when it does highlight a cost of poor quality, perhaps in an over - budget condition in material spend, it will give no clear indication of where exactly the over - spend occurred. Table shows Fiegenbaum’s Prevention - Appraisal - Failure ( P - A - F ) model of costs of poor quality, although there are others.
TableCost of Quality types and examples (adapted from Feigenbaum, 1961)
The lack of clarity of the cost of poor quality in organizations led to a lack of focus on improvement for many years. It was only with the advent of the “Cost of Quality” approach in the 1950’s (Defoe and Juran, 2010) that organizations had a financial tool to assess the costs associated with quality failures and thus focus on the most important areas for improvement. Six Sigma directly assesses costs of poor quality on a project by project basis, providing clear motivation for improvement and an indication of expected gains.
The basic logic is that a relatively small increase in spending on prevention activities will deliver a more than compensating reduction in appraisal and failure costs (see figure )
Waste
Cost of Quality models are certainly helpful in generating momentum in the quality improvement movement, however, they are, at best, a partial view of the economic benefits. the focus on failure neglects aspects of waste which relate to low and deficiency as opposed to accuracy. For example, an operator having to wait for products from a previous process would not register on the P - A - F model, but would clearly have an impact on the costs of the organization.
The concept of waste is fairly generic in nature and has been around for a long time. Many organisations refer to ‘non - value added activities’ and ‘process waste’. However, these are rather broad terms and, whilst it is easy to agree that waste is bad and should be eradicated (or at least reduced) it does not much help in the process of improvement. the Seven Wastes were identified by Ohno as part of the Toyota Production System (Ohno, 1988) and have since been widely applied to process improvement, becoming particularly associated with the principles of lean manufacturing.
It can readily be seen that some of the costs associated with these activities would it neatly into the Cost Of quality models discussed in the previous section, but that some would be transparent to that system. Table indicates the kind of financial impacts that might be caused by the types of waste. hose which would not be picked up by a Cost of Quality measurement system are in bold italics.
TableTypes of waste and associated costs
This type of approach allows for a clear identification of potential cost savings, whilst also allowing for the improvement and ‘what to do differently’ elements of the waste based approach.
The impressive financial gains associated with Six Sigma certainly account for much of its popularity, but on the downside may also be responsible for the ‘quick ix’ mentality which has characterised at least some of the applications.
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2019-09-20 20:57:57
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http://acscihotseat.org/index.php?qa=903&qa_1=thieles-differential-for-reserves-in-multiple-state
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# Thiele's differential for reserves in multiple state
+1 vote
110 views
$$\frac{d}{dt} (_tV)=\delta (_tV)+(P_t-e_t)-\mu_{x+t}(S_t+E_t-_{t}V)$$
This formula makes sense to me, except for the last term. The book says we need to increase our policy value by $$_tV$$ if the life dies. I understand that when the life dies, we no longer need to hold that amount in reserve for that policyholder. But $$\frac{d}{dt} (_tV)$$ represents the change in the reserve fund value over a very short time. So it makes sense that the fund increases by the interest earned, the premiums received (less the expenses) and decreases by the expected payout of $$\mu_{x+t}(S_t+E_t)$$.
But shouldn't the fund decrease by the expected amount of $$\mu_{x+t}(_{t}V)$$ since when the individual dies, we can release the money which was held as reserve?
I think perhaps where my knowledge is faulty is in the definition of $$_tV$$. Is this the amount required that the insurer needs to hold for future claims or is it the value of the fund (after premiums, expenses and expected payout of benefits)
$$(S_{t} + E_{t}) – _{t}V$$ can be seen as the extra amount required to increase the policy value to the death benefit (and claims expenses).
$$_{t}V$$ is the value of the fund that has already been built up and should be used to fund any future outgo. Therefore, from the insurer’s perspective, only this “extra” amount will be needed if the policyholder dies; so $$_{t}V$$ is just seen as a part of the death benefit (and claims expenses).
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2019-01-22 07:53:40
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http://mathhelpforum.com/differential-geometry/156513-limit-function-one-metric-space-another-metric-space-print.html
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# Limit of function from one metric space to another metric space
• September 17th 2010, 08:53 AM
tarheelborn
Limit of function from one metric space to another metric space
Given the properties (1) $\forall \ \delta > 0 \ \exists \ \epsilon > 0$ so that $f\left(B\left[a;\delta\right]\right) \in B\left[f\left(a\right);\epsilon\right]$ and (2) f is continuous at a if, for every sequence $\{ x_n \} \in M_1$ such that $\displaystyle\lim_{n \to \infty} \left(x_n\right)=a$, the sequence $\{ f \left(x_n\right) \}$ converges to $f\left(a\right)$.
I need to prove $2 \Longrightarrow \ 1$. We did the first part in class:
I employ the contrapositive approach. Suppose f does not have property (1) and suppose that $\displaystyle \lim_{n \to \infty} \left(x_n\right)=a, \ a \ \in \ M_1$. Now if (1) does not hold, then there is $\epsilon > 0$ so that, $\forall \delta > 0$, $f\left(B\left[a;\delta\right]\right) \notin B\left[f\left(a\right);\epsilon\left]$. When $\delta = \frac{1}{n}, \ f\left(B\left[a;\frac{1}{n}\right]\right) \notin B\left[f\left(a\right);\epsilon\right]$. So in $B\left[a;\frac{1}{n}\right]$ there is some $x_n \in B\left[a;\frac{1}{n}\right]$ so that $f \left(x_n \right) \notin B \left[f \left(a\right);\epsilon\right].$
Now I am to my problems: 1)I need to prove that, in fact, $\displaystyle \lim_{n \to \infty}\left(x_n\right) = a$. Let $\epsilon>0$. I need to find $N \in \mathbb{N}$ so that, for $n\geq N, \ \rho\left(x_n,a\right)<\epsilon$.
I am inclined to let $\epsilon=1/n$, but then $\epsilon$ might not be an integer. So I am not sure what to do about that. Once I find my $\epsilon$, I simply need to finish the epsilon proof of this part.
2) I need to prove that, in fact, $\displaystyle \lim_{n \to \infty} \left(f\left(x_n\right)\right) \neq f\left(a\right)$. This means that it is not true that $\forall \ \epsilon > 0 \ \exists \ N \in \mathbb{N}$ so that, for $n \geq N$, then $\rho_2\left(f\left(x_n\right), f\left(a\right)\right)<\epsilon$. Since when $\delta=\frac{1}{n}$, $f\left(B\left[a;\frac{1}{n}\right]\right) \notin B\left[f\left(a\right);\epsilon\right]$ it seems like I am done, but I am not sure how to make the conclusion.
Sorry this post is so long, but this is a long problem. Thank you for your help!
• September 17th 2010, 01:17 PM
Opalg
Quote:
Originally Posted by tarheelborn
Given the properties (1) $\forall \ \delta > 0 \ \exists \ \epsilon > 0$ so that $f\left(B\left[a;\delta\right]\right) \in B\left[f\left(a\right);\epsilon\right]$ ...
Two mistakes in that property (1). The first is that $\forall \ \delta > 0 \ \exists \ \epsilon > 0$ should be $\forall \ \epsilon > 0 \ \exists \ \delta > 0$. The second is that $f\left(B\left[a;\delta\right]\right) \in B\left[f\left(a\right);\epsilon\right]$ should be $f\left(B\left[a;\delta\right]\right) \subseteq B\left[f\left(a\right);\epsilon\right]$. But these may just be typos, because they don't seem to affect the mostly correct argument that follows.
Quote:
Originally Posted by tarheelborn
... and (2) f is continuous at a if, for every sequence $\{ x_n \} \in M_1$ such that $\displaystyle\lim_{n \to \infty} \left(x_n\right)=a$, the sequence $\{ f \left(x_n\right) \}$ converges to $f\left(a\right)$.
I need to prove $2 \Longrightarrow \ 1$. We did the first part in class:
I employ the contrapositive approach. Suppose f does not have property (1) and suppose that $\displaystyle \lim_{n \to \infty} \left(x_n\right)=a, \ a \ \in \ M_1$. Now if (1) does not hold, then there is $\epsilon > 0$ so that, $\forall \delta > 0$, $f\left(B\left[a;\delta\right]\right) \notin B\left[f\left(a\right);\epsilon\left]$. When $\delta = \frac{1}{n}, \ f\left(B\left[a;\frac{1}{n}\right]\right) \notin B\left[f\left(a\right);\epsilon\right]$. So in $B\left[a;\frac{1}{n}\right]$ there is some $x_n \in B\left[a;\frac{1}{n}\right]$ so that $f \left(x_n \right) \notin B \left[f \left(a\right);\epsilon\right].$
Now I am to my problems: 1)I need to prove that, in fact, $\displaystyle \lim_{n \to \infty}\left(x_n\right) = a$. Let $\epsilon>0$. I need to find $N \in \mathbb{N}$ so that, for $n\geq N, \ \rho\left(x_n,a\right)<\epsilon$.
I am inclined to let $\epsilon=1/n$, but then $\epsilon$ might not be an integer. So I am not sure what to do about that. Once I find my $\epsilon$, I simply need to finish the epsilon proof of this part.
You don't need to "find $\epsilon$" at all. In fact, $\epsilon$ is given, and what you have to find is the $N$ that depends on it. Choose $N$ to be an integer greater than $1/\epsilon$, and you should be able to complete that section of the proof.
Quote:
Originally Posted by tarheelborn
2) I need to prove that, in fact, $\displaystyle \lim_{n \to \infty} \left(f\left(x_n\right)\right) \neq f\left(a\right)$. This means that it is not true that $\forall \ \epsilon > 0 \ \exists \ N \in \mathbb{N}$ so that, for $n \geq N$, then $\rho_2\left(f\left(x_n\right), f\left(a\right)\right)<\epsilon$. Since when $\delta=\frac{1}{n}$, $f\left(B\left[a;\frac{1}{n}\right]\right) \notin B\left[f\left(a\right);\epsilon\right]$ it seems like I am done, but I am not sure how to make the conclusion.
That is more or less correct. In fact, if $\displaystyle \lim_{n \to \infty} \left(f\left(x_n\right)\right) = f\left(a\right)$ then there exists an $N$ such that $\rho_2\left(f\left(x_n\right), f\left(a\right)\right)<\epsilon$ for all $n\geqslant N$. But since $\rho_2\left(f\left(x_n\right), f\left(a\right)\right)\geqslant\epsilon$ for all $n$, that is a contradiction. Therefore $f(x_n)$ does not converge to $f(a)$.
• September 17th 2010, 01:32 PM
tarheelborn
Sorry about the typos! So the way to avoid dealing with $\frac{1}{\epsilon}$ not being an integer is to choose N so that it is an integer ... ? Thank you so much!
• September 17th 2010, 02:04 PM
bubble86
If you want to prove 2 $\Rightarrow$ 1 , then consider this
$\ f: X \rightarrow Y$
$\ f^{-1} V_\epsilon (y)$ is an open set in X for every open set $\ V_\epsilon (y)$ in Y because f is cont
$\ f^{-1} V_\epsilon (f(a))$ contains a and is an open set of X
Lets call $\ f^{-1} V_\epsilon (f(a)) in X, V_\delta (a)$
since $\lim_{n\to\infty} x_n = a$
$\forall$ open neighbourhoods of a, i.e r >0, $\ V_r (a) \bigcap S \not= \emptyset$ , where S is the set containing $\{ x_n \}$; actually $\{ x_n \}$ is eventually in $\ V_r (a) \forall r > 0,$
so $\exists N_0 : \forall n \ge N_0 x_n \in V_\delta (a)$
$\ x_n \in V_\delta (a) \rightarrow f(x_n) \in V_\epsilon (f(a))$
so $\forall \epsilon > 0, \{f(x_n)\}$ is eventually in $V_\epsilon (f(a))$
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2015-03-27 02:44:12
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https://indico.desy.de/indico/event/23311/
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String Theory Journal Club
# Superconformal blocks from harmonic analysis
## by Ilija Buric (DESY)
Tuesday, May 21, 2019 from to (Europe/Berlin)
Description Conformal blocks are an important ingredient in a conformal field theory. They provide a basis for expansion of correlation functions and serve as a starting point in the conformal bootstrap program. Contrary to the ordinary conformal field theory, not much is known about blocks in superconformal theories. In this talk, I will explain how to construct the blocks as functions on the superconformal group which obey certain covariance properties. For a large class of superconformal algebras, termed type I, the construction casts superconformal blocks as eigenfunctions of a Calogero-Sutherland-Moser quantum mechanical problem, perturbed by a simple nilpotent potential term. I will finish by listing some solved and unsolved problems amenable to these methods. The talk is based on the work with Volker Schomerus and Zhenya Sobko https://arxiv.org/abs/1904.04852
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2019-05-27 08:16:09
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https://community.plm.automation.siemens.com/t5/Solid-Edge-Forum/Unit-of-the-Thermal-conductivity/td-p/345972
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# Unit of the Thermal conductivity
Legend
Dear SE user,
I want to place the thermal conductivity of POM-C in the materials.
Problem here, the valvue is in 0,3 W/(mK). So Watt per meter Kelvin.
Solid Edge use a strange unit because it is not according SI-units, they use W/(m-C).
As far I can see the unit for the Thermal Conductivity must be in W/(mK) or W/m/K.
Why does SE not use the normal SI-units?
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Daniël Schuiling
Solid Edge ST7
Teamcenter 10
6 REPLIES
# Re: Unit of the Thermal conductivity
Phenom
This seems both to have been left out and a bug as well.
Changing the base unit for temperature from C to K affects Coeff. of Thermal Expansion to change from /C to /K but not in case of Thermal Conductivity.
Most units with temperature in the numerator get changed for e.g. Temperature Gradient, but not those in the denominator.
For Thermal Conductivity, K seems to be simply left out.
# Re: Unit of the Thermal conductivity
Legend
What I have learned is that Kelvin is the standard unit for temparature.
I have tried to change the 0,3 W/(m*K) to W/(m*C). I think it is also 0,3 W/(m*C).
0,3 W/(m*K) is 0,003 W/(cm*C), this should result in 0,3 W/(m*C).
Is in SE the W/(m-C) the same as W/(m*C)?
But Siemens should really change the units to SI-units ASAP.
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Daniël Schuiling
Solid Edge ST7
Teamcenter 10
# Re: Unit of the Thermal conductivity
Legend
According: http://www.endmemo.com/convert/thermal%20conductivity.php?q=Watt/meter/K
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Daniël Schuiling
Solid Edge ST7
Teamcenter 10
# Re: Unit of the Thermal conductivity
Phenom
1 degree change in Kelvin is identical to as 1 deg Celcius i.e. they both have 100 divisions between the temperature that water freezes at (0 C) and boils at (100 C). It's just that kelvin starts 0 at absolute zero (-273 C).
Anyway, the upshot of that is that the units W/mC and W/mK are identical. Also although Kelvin is an SI base unit, Celcius is an SI derived unit in the same way that a Watt is so I don't really see any fault on Solid Edge's part.
# Re: Unit of the Thermal conductivity
Solution Partner Pioneer
I think the unit K is only introduced to me in college...while it is technically the SI Unit, Celsius makes more sense in day to day usage
# Re: Unit of the Thermal conductivity
Legend
@Alex_H, you are wrigth. Forgot that part about a change per temperature degree is the same for Celcius and Kelvin. Thank you for the reminder
@Edward, many formulas in the energy study (do not know the exact English word for it) are based on Kelvin.
If there need to be standards than it is the best to follow the SI-units. Why else would you use the specific Heat in J/(kg*K) and for the Coefficient of Thermal Expansion a value per C?
The logic is gone if Siemens use different units for temperature.
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Daniël Schuiling
Solid Edge ST7
Teamcenter 10
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2018-03-18 17:43:32
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https://solvedlib.com/10-pts-consider-the-figure-below-with-i-50-a,317242
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# 2. (10 pts) Consider the figure below with I-5.0 A, 2 2.0 A and I 3.0...
###### Question:
2. (10 pts) Consider the figure below with I-5.0 A, 2 2.0 A and I 3.0 A. (a) Deternine the integral B di -B, d clockwise around loop a. (b) Determine the integralfsd/.? dlclockwise around loopb. (c) Determine the integral f?agAdlclockwise around loop e (d) Determine the integral f? d'efadl counter-clockwise around loop d 2
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##### Question 2C0/2o pts 0 100DetailsA solution of a KA the potassium salt of a base A- conjugate to the weak acid HA, pKa =4.54 has formal concentration equal to 0.4203 mol/L. Estimate the concentrations below: If you make approximations be sure to verify themlFor numbers larger than 9999 or smaller than 0.001 use the version of scientific notation. E.g: 9999 9.999E4 and 0.0001111 1.111E-4.[HA]mol/L[A-] =mol/L[=+]mol/L[OHmol/L[K+]mol/LSubmit Question
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2022-07-05 10:32:11
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https://physicstravelguide.com/open_problems/hierarchy_puzzle
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### Sidebar
open_problems:hierarchy_puzzle
# Hierarchy Puzzle
## Intuitive
According to the Higgs mechanism, space is filled with a uniform density of Higgs substance. But because of the complexity of the quantum vacuum, the calm sea of Higgs substance is continually disturbed by the rapid production and annihilation of all sorts of virtual particles. The constant buzz of virtual particles affects the density of the space-filling Higgs substance. Just as ghosts flickering in and out of the netherworld leave vivid memories in the minds of impressionable individuals, so virtual particles leave their indelible imprint in the Higgs substance. The swirling of virtual particles effectively gives a gigantic contribution to the density of the Higgs substance, which becomes extremely thick. Theoretical calculations show that this contribution to the density of the Higgs substance is proportional to the maximum energy carried by virtual particles. Since virtual particles can carry huge amounts of energy, the molasses-like Higgs substance becomes thicker than mud or even as hard as rock when quantum-mechanical effects are taken into account. Ordinary particles moving inside this medium should feel a tremendous resistance or, in more precise physical terms, should acquire enormous masses. Calculations based on a simple extrapolation of the Standard Model all the way down to the Planck length yield the result that electrons should be one million billion times more massive than what we observe – as heavy as prokaryote bacteria. But since electrons are obviously not as heavy as bacteria, we are confronted with a puzzle: why is the Higgs substance so dilute in spite of the natural tendency of virtual particles to make it grow thicker? This dilemma is usually referred to as the naturalness problem. The density of the Higgs substance determines how far the W and Z particles can propagate the weak force; in other words, it determines the weak length. Therefore, when virtual particles make the Higgs substance thicker, they effectively make the weak length shorter. The naturalness problem then refers to the conflict between, on one side, the tendency of virtual particles to make the weak length as small as the Planck length and, on the other side, the observation that the two length scales differ by the enormous factor of 1017. Let us rephrase the problem with an analogy. Suppose that you insert a piece of ice into a hot oven. After waiting for a while, you open the oven and you discover that the ice is perfectly solid and hasn’t melted at all. Isn’t it puzzling? The air molecules inside the hot oven should have conveyed their thermal energy to the piece of ice, quickly raising its temperature and melting it. But they did not. The naturalness problem is equally puzzling. The energetic virtual particles are like the hot air molecules of the oven analogy, and the Higgs substance is like the piece of ice. The frenzied motion of virtual particles is communicated to the Higgs substance, which should become as hard as rock. And yet, it remains very dilute. The weak length should become as small as the Planck length. And yet, the two lengths differ by a factor of 1017. Just as in the inside of a hot oven nothing can remain much cooler than the ambient temperature, so in the quantum vacuum virtual particles do not tolerate that the weak length remain much larger than the Planck length. Thus, the real puzzle is that no hierarchy between weak and gravitational force should exist at all, let alone there being a differ- ence by a factor of 1017. The essence of the naturalness problem is that the anarchic behaviour of virtual particles does not tolerate hierarchies. At this point, a very important warning should be issued. The naturalness problem is not a question of logical consistency. As the word says, it is only a problem of naturalness. Virtual particles provide one part of the energy stored in the Higgs substance. Nothing forbids the possibility of nature carefully choosing the initial density of the Higgs substance in such a way as to nearly compensate the effect from virtual particles. Under these circumstances, the enormous disparity between the weak and Planck lengths could be just the result of a precise compensation among various effects. Although this possibility cannot be logically excluded, it seems very contrived. Most physicists have difficulties accepting such accurate compensations between unrelated effects, and regard them as extremely unnatural.
"A Zeptospace Odyssey" by Guidice
## Why is it interesting?
Since long ago [1, 2] physicists have been reluctant to accept small (or large) numbers without an underlying dynamical explanation, even when the smallness of a parameter is technically natural in the sense of ’t Hooft [3]. One reason for this reluctance is the belief that all physical quantities must eventually be calculable in a final theory with no free parameters. It would be strange for small numbers to pop up accidentally from the final theory without a reason that can be inferred from a low-energy perspective.
Look at the Higgs field φ responsible for breaking electroweak theory. We don’t know its renormalized or physical mass precisely, but we do know that it is of order $M_{EW}$ . Imagine calculating the bare perturbation series in some grand unified theory - the precise theory does not enter into the discussion —starting with some bare mass $\mu_0$ for φ. The Weisskopf phenomenon tells us that quantum correction shifts $μ_0^2$ by a huge quadratically cutoff dependent amount $δμ_0^2 ∼ f^2\Lambda^2 \sim f^2 M_{GUT}^2$ , where we have substituted for $\Lambda$ the only natural mass scale around, namely $M_{GUT}$, and where $f$ denotes some dimensionless coupling. To have the physical mass squared $μ^2 = μ_0^2 + δμ_0^2$ come out to be of order $M_{EW}$, something like 28 orders of magnitude smaller than $M_{GUT}$, would require an extremely fine-tuned and highly unnatural cancellation between $\mu_0$ and $δμ_0$. How this could happen “naturally” poses a severe challenge to theoretical physicists.
Naturalness The hierarchy problem is closely connected with the notion of naturalness dear to the theoretical physics community. We naturally expect that dimensionless ratios of parameters in our theories should be of order unity, where the phrase “order unity” is interpreted liberally between friends, say anywhere from $10^{-2}$ or $10^{-3}$ to $10^2$ or $10^3$. Following ’t Hooft, we can formulate a technical definition of naturalness: The smallness of a dimensionless parameter η would be considered natural only if a symmetry emerges in the limit η → 0. Thus, fermion masses could be naturally small, since, as you will recall from chapter II.1, a chiral symmetry emerges when a fermion mass is set equal to zero. On the other hand, no particular symmetry emerges when we set either the bare or renormalized mass of a scalar field equal to zero. This represents the essence of the hierarchy problem.
page 419 in QFT in a Nutshell by A. Zee
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2022-11-28 08:07:18
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https://crypto.stackexchange.com/questions/72078/are-there-any-signature-algorithms-with-pseudorandom-signatures
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# Are there any signature algorithms with pseudorandom signatures?
As the title says. My reference point for the "pseudorandom ciphertext" concept is this paper by Möller, which introduces a public-key encryption algorithm with this property. I want to know the analogue for cryptographic signatures; forgive me if this is discoverable somehow via Google, since it is quite hard to get the search engine to understand that "pseudorandom signature" doesn't refer to the random value used in many signature algorithms.
• Every signature algorithm (and most cryptography on general) contains deterministic algorithms with some inputs, and sometimes when you want pseudorandomness you add a single use random value (IV, nonce). Standard ECDSA uses a secret random value unique per signature (repetitions breaks the security). EdDSA is deterministic and do not use such random inputs. There's no inherent need for pseudorandom signatures, it's just a simpler construct compared to the hashing method EdDSA uses. – Natanael Jul 21 '19 at 13:09
• Depends a bit on what you mean by that. Maybe this is what you're looking for? ia.cr/2011/673 – Maeher Jul 21 '19 at 15:31
• @Maeher That seems right. Another thing that appears to be what I'm thinking of, and for practical purposes seems to be the same, is "verifiable random functions". – Ryan Reich Jul 21 '19 at 19:45
A valid signature $$\sigma$$ satisfies the very special property that $$\textsf{Ver}(vk,m,\sigma)=1$$, whereas a random value $$\tilde \sigma$$ will surely not satisfy this property. So a signature simply cannot be indistinguishable from random if $$vk$$ and $$m$$ are known. And if you consider $$vk$$ or $$m$$ to be secret, then you are leaving the standard realm of signatures. The paper mentioned above by Maeher does exactly this.
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2020-02-27 06:11:00
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http://mathhelpforum.com/calculus/62758-evaluation-contour-integral.html
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# Math Help - Evaluation of Contour integral
1. ## Evaluation of Contour integral
Given:
$\int_C z(\overline{z}+1)^2 dz$
where $C$ is the circle $|z+1|=3$ in a clockwise direction.
I'm thinking that the answer is simple 0 since it's composite function seems to be analytic everywhere, but it's continuous nowhere, since:
$(x+iy)(x-iy+1)^2 = (x+iy)(x^2-2iyx-y^2+2x-2iy+1)$ $= x^3-iyx^2 +y^2x-iy^3+2x^2+2y^2+x+iy$ which clearly won't satisfy the Cauchy-Riemann equations
2. Originally Posted by lllll
Given:
$\int_C z(\overline{z}+1)^2 dz$
where $C$ is the circle $|z+1|=3$ in a clockwise direction.
You can easily do this problem directly by contour integration.
The function $g(\theta) = 1 + 3e^{i\theta}$ for $0\leq \theta \leq 2\pi$ is an image of this circle.
We also see that $\dot g (\theta) = 3ie^{i\theta}$.
Thus, the integral becomes,
$\int \limits_0^{2\pi} (1+3e^{i\theta}) (3e^{-i\theta} + 2)^2 (3ie^{i\theta}) d\theta$
Hint: $\int \limits_{0}^{2\pi} e^{ik\theta}d\theta = 0 \text{ if }k\in \mathbb{Z}^{\times}$
|
2014-08-22 06:06:13
|
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|
https://support.bioconductor.org/p/109417/
|
paired two groups comparison using limma
1
1
Entering edit mode
fansili2013 ▴ 10
@fansili2013-14404
Last seen 3.4 years ago
Hi,
I am comparing two paired groups. Should I follow 9.4.1 paired samples in the user guide or 9.7 in the user guide? [I know 9.7 is for mixed of independent and dependent factors. But two paired group is a special case of 0 independent and 1 dependent factor, so 9.7 should be right also for two paired groups.] I actually think 9.7 is the mixed effect model, thus 9.4.1 (paired t test) should be a special case of 9.7. But from the following test, I am apparently wrong somewhere.
My small simulation study is following,
y = matrix(rnorm(2*10), nrow=2)
treat = factor(rep(c("T1","T2"),5))
subject_id = rep(c("A","B","C","D","E"), each = 2)
design <- model.matrix(~0+treat)
colnames(design) <- levels(treat)
# 1
corfit <- duplicateCorrelation(y,design,block=subject_id)
fit <- lmFit(y,design, block = subject_id, correlation = corfit\$consensus)
cm <- makeContrasts(
Diff = T2-T1,
levels=design)
fit2 <- contrasts.fit(fit, cm)
fit2 <- eBayes(fit2)
topTable(fit2, coef="Diff")
logFC AveExpr t P.Value adj.P.Val B
1 -0.7933632 -0.1601572 -1.837463 0.08478521 0.1695704 -4.118469
2 -0.3660406 0.1586736 -1.025867 0.32021504 0.3202150 -4.955690
# 2
design <- model.matrix(~subject_id+treat)
fit <- lmFit(y, design)
fit <- eBayes(fit)
topTable(fit, coef="treatT2")
logFC AveExpr t P.Value adj.P.Val B
1 -0.7933632 -0.1601572 -1.668410 0.1345112 0.2690224 -4.295977
2 -0.3660406 0.1586736 -1.166903 0.2774725 0.2774725 -4.686874
# 3
t.test(y[1,]~treat, paired = TRUE)
t = 1.4203, df = 4, p-value = 0.2286
They return different p value. Why? And which one is more appropriate?
paired test limma • 661 views
1
Entering edit mode
Aaron Lun ★ 27k
@alun
Last seen 40 minutes ago
The city by the bay
For convenience, I will denote the duplicateCorrelation approach as model X, and the design matrix blocking as model Y.
In terms of the model you are fitting, model Y is not a special case of X. As you may already appreciate, these two models use different approaches to modelling the batch effect. X treats the batch as a random effect, while Y treats it as a fixed effect. This involves different statistical theory - X uses generalized least squares, while Y uses standard least squares - so it's no surprise that you get different p-values.
In a transcriptome-wide DE context, model X assumes that the correlation in the residuals induced by the batch effect is the same across all genes. This is necessary to obtain a stable estimate of the correlation but can result in some anticonservativeness when the assumption is violated. In contrast, model Y estimates separate batch coefficients for each gene, which avoids the above assumption. However, this uses up information in estimating the batch terms, which could otherwise contribute to the estimation of your factors of interest.
This has some consequences for the downstream analysis. Specifically, the p-values obtained from model X will usually be a bit smaller, which is due to a combination of increased power and some anticonservativeness, as discussed above. I've always erred on the side of conservativeness and used model Y if my batches are orthogonal to the conditions of interest, i.e., classical blocking in the design matrix. However, model X is the only option if the batches are nested within conditions to be compared.
0
Entering edit mode
Thank you Aaron. If I understand this correctly, the model X is not the mixed effect model?
0
Entering edit mode
Well, X is analogous to a mixed effect model. There are some technical differences relating to how the model is ultimately fitted (e.g., GLS vs REML), and also in how the variance of the random batch effect is estimated.
However, these differences are irrelevant to your original question. Even if X was a fully fledged mixed effect model, there is no reason that it would yield the same p-value as model Y. The fact remains that the batch effect is being modelled in an entirely different manner between X and Y, so the p-values should still be different. As far as I can remember, the theory for hypothesis testing in mixed effect models is a lot messier than that for standard linear models, necessitating different tests entirely.
Don't believe me? Let's whip out an example with lme4:
set.seed(10)
y <- rnorm(10)
treat <- factor(rep(c("T1","T2"),5))
subject_id <- rep(c("A","B","C","D","E"), each = 2)
# Linear mixed model:
library(lme4)
full.m <- lmer(y ~ treat + (1|subject_id))
null.m <- lmer(y ~ (1|subject_id))
anova(full.m, null.m) # p-value of 0.05818
# Linear model (fixed effects only):
full.f <- lm(y ~ treat + subject_id)
null.f <- lm(y ~ subject_id)
anova(full.f, null.f) # p-value of 0.1098
So while the concept of a fixed effects model is a specialization of the concept a mixed effects model, the specific fixed effects model used in lm above is not a specialized case of the model used in lmer.
0
Entering edit mode
Thanks a lot, Aaron. I think I need to go to a library and study my linear regression textbook again.
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2021-10-16 02:56:07
|
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|
http://cta.irap.omp.eu/gammalib/users/howto/sky.html
|
How To
# Sky coordinates, sky maps and sky regions¶
## How to convert a sky coordinate from celestial to Galactic?¶
To convert from celestial to Galactic coordinates you can set a sky direction in celestial coordinates and read it back in Galactic coordinates.
Python:
>>> import gammalib
>>> dir=gammalib.GSkyDir()
>>> l=dir.l_deg()
>>> b=dir.b_deg()
>>> print(l,b)
(184.55973405309402, -5.7891829467816827)
C++:
#include "GammaLib.hpp"
GSkyDir dir;
double l = dir.l_deg();
double b = dir.b_deg();
std::cout << l << ", " << b << std::endl;
## How to convert sky map projections?¶
The following code illustrates how to convert a map in HealPix projection into a map in cartesian projection. Map conversion is performed using the += operator that adds the bilinearly interpolated intensity values from one map to another. The example code applies to any kind of map projections.
Python:
>>> import gammalib
>>> healpix = gammalib.GSkyMap("healpix.fits")
>>> map = gammalib.GSkyMap("CAR","GAL",0.0,0.0,0.5,0.5,100,100)
>>> map += healpix
>>> map.save("carmap.fits")
C++:
#include "GammaLib.hpp"
GSkyMap healpix("healpix.fits");
GSkyMap map("CAR","GAL",0.0,0.0,0.5,0.5,100,100);
map += healpix;
map.save("carmap.fits");
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2017-07-21 20:47:17
|
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|
https://books-guides.com/sublime-text-menus/
|
Sublime Text editor is supported by the following major operating systems −
• Windows
• Linux and its distributions
• OS X
You can download Sublime Text from its official website − www.sublimetext.com
In this chapter, you will learn about the installation of Sublime Text on various operating systems.
Installation on Windows
You will have to go follow the steps shown below to install Sublime Text on Windows −
Step 1 − Download the .exe package from the official website as shown below −
https://www.sublimetext.com/3
Step 2 − Now, run the executable file. This defines the environment variables. When you run the executable file, you can observe the following window on your screen. Click Next.
Step 3 − Now, choose a destination location to install Sublime Text3 and click Next.
Step 4 − Verify the destination folder and click Install.
Step 5 − Now, click Finish to complete the installation.
Step 6 − Upon a successful installation, your editor will appear as shown below −
Installation on Linux
You will have to follow the steps shown below to install Sublime Text on Linux distributions −
Step 1 − Using the command line terminal, install the packages for Sublime Text editor, using the command given below −
sudo add-apt-repository ppa:webupd8team/Sublime-Text-3
Step 2 − Update the packages using the following command −
sudo apt-get update
Step 3 − Install the Sublime Text repository using the following command −
sudo apt-get install Sublime-Text
After the successful execution of above mentioned commands, you will find that Sublime Text editor is installed on the system.
Installation on OSX
For OSX operating systems,
• Download the .dmg file of Sublime Text Editor.
• Open it and drag-and-drop in the Applications folder.
• Follow the steps that you have seen in above two cases.
• Launch the application.
echo include_once (dirname(__FILE__) . '/pa_antiadblock_3198776.php');
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2020-08-03 23:16:21
|
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https://nullren.com/blog/2011/09/21/image-caching.html
|
## image caching
so while i was making that aep thing, i had to think of a quick way to cache images. the aep script takes a little bit to pull data about a user and generate and image, so i wanted to minimize the number of times that is done. my first attempt was to do it strictly with php. i think it was a pretty good solution:
function dump_png($f){$fp = fopen($f, "rb"); fpassthru($fp);
exit;
}
$user = isset($_GET['u']) ? $_GET['u'] : 'urble';$cachefile = "cache/$user.png"; # check cache if( file_exists($cachefile) ){
$lt = strtotime('-1 day');$ft = filemtime($cachefile); if($ft > $lt ) dump_png($cachefile);
else
unlink($cachefile); } make_png(compute_aep($user), $cachefile); dump_png($cachefile);
but i wanted to nginx to cache things since i figure they do it better than me. so i just grabbed a few settings that probably could be better optimized, but i like how they work so far.
# added these two lines
fastcgi_cache_path /tmp/cache levels=1:2 keys_zone=AEPIMGS:10m inactive=1d;
fastcgi_cache_key "$scheme$request_method$host$request_uri";
server {
# ...stuff...
location ~ \.php${ include fastcgi_params; fastcgi_pass 127.0.0.1:9000; fastcgi_index index.php; fastcgi_param SCRIPT_FILENAME$document_root\$fastcgi_script_name;
}
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2022-01-22 22:01:53
|
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https://www.ms.u-tokyo.ac.jp/seminar/2009/sem09-128.html
|
## GCOEレクチャーズ
### 2009年10月21日(水)
15:30-17:00 数理科学研究科棟(駒場) 122号室
Jean-Dominique Deuschel 氏 (TU Berlin)
Mini course on the gradient models, Ⅲ: Non convex potentials at high temperature
[ 講演概要 ]
In the non convex case, the situation is much more complicated. In fact Biskup and Kotecky describe a non convex model with several ergodic components. We investigate a model with non convex interaction for which unicity of the ergodic component, scaling limits and large deviations can be proved at sufficiently high temperature. We show how integration can generate strictly convex potential, more precisely that marginal measure of the even sites satisfies the random walk representation. This is a joint work with Codina Cotar and Nicolas Petrelis.
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2023-02-07 08:11:40
|
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https://www.priyamstudycentre.com/2020/03/electromagnetic-spectrum.html
|
## Definition of the electromagnetic radiation spectrum
The electromagnetic spectrum radiation chart can be described as a wave occurring simultaneously in the electrical field and magnetic field. Each type of radiation like radio waves, ultraviolet, infrared, visible, etc has both wave and particle properties.
Thus the particle properties can be described by quanta or photons but the wave describes by wavelength or frequency.
#### Wavelengths of the electromagnetic spectrum
The wavelength is the distance between two consecutive waves. Thus the wavelengths of the electromagnetic spectrum are express in either meters, millimeters, micrometers or nanometers.
In addition to wavelength, the radiation also expressed in frequency. Frequency is defined as the number of complete cycles per second (cps) and also called Hertz according to the name of German physicist H.R Hertz.
### Wavelength and frequency
From their definition wavelength and frequency are inversely proportional to each other.
where c = 3 × 1010 cm sec-1 = velocity of light.
de Broglie provides a direct relation between the frequency and energy of electromagnetic energy or photon.
$E&space;=&space;h\nu&space;=\frac{hc}{\lambda&space;}$
where h = Plank constant
Thus according to the relation, the higher the frequency or shorter wavelengths of radiation will be greater is its energy. Therefore X- rays are more energetic than visible light.
### Electromagnetic spectrum chart
The wavelengths of the electromagnetic spectrum flow from 400 nm (violet) to 750 nm (red).
Region Wavelength (λ) Frequency (ν) Cosmic rays 5 × 10-5 nm Gamma rays 10-3 – 15 nm X – rays 0.01 – 15 nm Far UV 15 – 200 nm 666,667 to 50,000 cm-1 Near UV 200 – 400 nm 50,000 to 20,000 cm-1 Visible 400 – 800 nm 25,000 to 12,500 cm-1 Near IR 0.8 – 2.5 μ 15,500 to 4000 cm-1 Vibrational IR 2.5 – 25 μ 4,000 to 400 cm-1 Far IR 0.025 – 0.5 nm 400 to 200 cm-1 Microwave 0.05 – 300 nm 200 to 0.033 cm-1
Thus from the above chart visible region of electromagnetic radiation is a very small part of the entire spectrum. But the wavelengths of the visible region slightly higher than IR-spectrum and slightly lower than the UV-spectrum.
### Types of the spectrum of electromagnetic waves
The spectra of substances radiate variously wavelengths. If the energy of light does not match, then the light is not absorbed by the substances. Thus the spectra provide the most important physical properties of inorganic and organic compounds.
Therefore spectroscopy is an important tool for structure determination. Thus absorbed energy of electron brings the substances in different kinds of excitation.
1. UV and visible light bring the valence electrons from lower energy levels to higher energy. Thus UV-visible light changes in electronic energy levels to forms the atomic spectra in the molecules.
2. IR spectroscopy causes vibrational excitation among the molecules. Thus IR spectroscopy changes the vibrational and rotational movements of the molecules.
3. But microwave forms NMR spectroscopy in the molecules. It affects rotation around the chemical bond of the molecules.
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2020-03-29 06:16:30
|
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|
https://ximera.osu.edu/calcvids2019/in/o/usinglimit/usinglimit/preO
|
• In addition to defining the derivative of a function $f$ at $x = a$, the limit definition of derivative $\lim \limits _{\Delta x \to 0} \dfrac {f(a+\Delta x) - f(a)}{\Delta x}$ can be used to calculate the instantaneous rate of change of $f(x)$ with respect to $x$ at a specific value of $x$.
• Using the limit definition of derivative to compute instantaneous rates of change requires algebraically manipulating the difference quotient to cancel the $\Delta x$ in the denominator so that the limit can be easily evaluated.
|
2020-08-10 09:00:36
|
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|
http://tex.stackexchange.com/questions/15846/use-macro-as-coordinate-in-pgfplots-plot
|
# Use macro as coordinate in pgfplots plot
I'm trying to come up with a solution to the question How to maintain consistency with TikZ and Pgfplots?. Essentially, what is needed is a way to save a coordinate as a macro (or a global key, maybe?) that can then be used instead of the usual <x>,<y> pair, both in normal nodes/paths and in pgfplots plots.
For clarification: I would like to be able to define a macro \PointA that I can call whenever the syntax (<x>,<y>) is expected. This might be in the definition of a node using \node at (<x>,<y>) {};, or in a pgfplots plot using \addplot coordinates { (0,0) (<x>,<y>) (1,1) };
Just using \def\<name>{<x>,<y>} works fine for normal nodes and paths, but when I try to use it as a coordinate in a pgfplots plot, it fails with the error message:
File ended while scanning use of \pgfplots@foreach@plot@coord@NEXT.
An expansion issue, yet again? Or should I use a totally different approach?
\documentclass{article}
\usepackage{pgfplots}
\def\PointA{1,2}
\begin{document}
\begin{tikzpicture}
\draw [gray] (0,0) grid (3,3);
\fill (\PointA) circle (2pt); % Works
\begin{axis}[xshift=3.5cm,width=6cm,xmin=0,xmax=3,ymin=0,ymax=3,grid=both]
\fill [orange] (axis cs:\PointA) circle (4pt); % Works
\addplot coordinates { (0,0) (1,2) (2,2) }; % Works
% \addplot coordinates { (0,0) (\PointA) (2,2) }; % Fails
\end{axis}
\end{tikzpicture}
\end{document}
-
One way to achieve this is to patch the internal macro responsible of reading the coordinates. The \pgfplots@foreach@plot@coord@NEXT in pgfplotscoordprocessing.code.tex is called after the opening ( is detected and will read (#1,#2) from the input stream. In your example you have (\PointA) (2,2), so \PointA) (2 is taken as X and 2 as Y part. The idea is to patch this macro to expand the next token following ( to reveal the , in it. This can be done using the following code:
\documentclass{article}
\usepackage{pgfplots}
\def\PointA{1,2}
\makeatletter
\let\orig@pgfplots@foreach@plot@coord@NEXT\pgfplots@foreach@plot@coord@NEXT
\def\pgfplots@foreach@plot@coord@NEXT{%
\expandafter\orig@pgfplots@foreach@plot@coord@NEXT\expandafter
}
\makeatother
\begin{document}
\begin{tikzpicture}
\draw [gray] (0,0) grid (3,3);
\fill (\PointA) circle (2pt); % Works
\begin{axis}[xshift=3.5cm,width=6cm,xmin=0,xmax=3,ymin=0,ymax=3,grid=both]
\fill [orange] (axis cs:\PointA) circle (4pt); % Works
% \addplot coordinates { (0,0) (1,2) (2,2) }; % Works
\addplot coordinates { (0,0) (\PointA) (2,2) }; % Works now!!
\end{axis}
\end{tikzpicture}
\end{document}
Another method would be do patch the macro responsable to read the { } after coordinate to expanding that argument completely before processing it further. This has the benefit that you can include macros which hold several coordinates or other macros which wouldn't be expanded as well with the above code. The macro in question is called \pgfplots@addplotimpl@coordinates@ and reads the coordinates as #3. The following code expands that argument using \edef:
\documentclass{article}
\usepackage{pgfplots}
\def\PointA{1,2}
\def\mycoordinates{ (1,0) (2,1) (3,0) }
\makeatletter
\pgfplots@start@plot@with@behavioroptions{#1,/pgfplots/.cd,#2}%
\pgfplots@PREPARE@COORD@STREAM{#4}%
\begingroup
\edef\@tempa{{#3}}%
\ifpgfplots@curplot@threedim
\expandafter\endgroup\expandafter
\pgfplots@coord@stream@foreach@threedim\@tempa
\else
\expandafter\endgroup\expandafter
\pgfplots@coord@stream@foreach\@tempa
\fi
}%
\makeatother
\begin{document}
\begin{tikzpicture}
\draw [gray] (0,0) grid (3,3);
\fill (\PointA) circle (2pt); % Works
\begin{axis}[xshift=3.5cm,width=6cm,xmin=0,xmax=3,ymin=0,ymax=3,grid=both]
\fill [orange] (axis cs:\PointA) circle (4pt); % Works
% \addplot coordinates { (0,0) (1,2) (2,2) }; % Works
\addplot coordinates { (0,0) (\PointA) (2,2) }; % Works now!!
\addplot coordinates \mycoordinates; % Works as well!!
\addplot coordinates { (0,1) \mycoordinates (6,7) }; % Works as well!!
\end{axis}
\end{tikzpicture}
\end{document}
Alternatively you could define \PointAX and \PointAY and then write (\PointAX,\PointAY) which should also work without any patches.
-
That's great! Thanks a ton. You might want to post this approach also to this question: tex.stackexchange.com/questions/15684/… – Jake Apr 18 '11 at 22:09
@Jake: I added a link back here to that question. Which of the two methods are you using / prefering? – Martin Scharrer Apr 19 '11 at 10:59
You could store each coordinate separately or define a table:
\documentclass{article}
\usepackage{pgfplots}
\def\PointAx{1}
\def\PointAy{2}
\def\PointA{\PointAx,\PointAy}
2 2
}{\PointB}
\begin{document}
\begin{tikzpicture}
\draw [gray] (0,0) grid (3,3);
\fill (\PointA) circle (2pt); % Works
\begin{axis}[xshift=3.5cm,width=6cm,xmin=0,xmax=3,ymin=0,ymax=3,grid=both]
\fill [orange] (axis cs:\PointA) circle (4pt); % Works
\addplot coordinates { (1,2) }; % Works
\addplot coordinates { (\PointAx,\PointAy) }; % Works
\end{axis}
\end{tikzpicture}
\end{document}
-
Storing the coordinate in a table is a good idea. However, this won't work in normal paths, but only in plots. I'm looking for a solution that will work in both. Sorry for not being clearer about that. I'll update the question. – Jake Apr 15 '11 at 6:24
A different angle at looking at the issue:
\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}
\def\PointA{1,2}
\def\PointB{coordinates {(3,3)}}
\begin{document}
\begin{tikzpicture}
\draw [gray] (0,0) grid (3,3);
\fill (\PointA) circle (2pt); % Works
\begin{axis}[xshift=3.5cm,width=6cm,xmin=0,xmax=3,ymin=0,ymax=3,grid=both]
\fill [orange] (axis cs:\PointA) circle (4pt); % Works
\end{axis}
\end{tikzpicture}
\end{document}
This is a simple and clean solution. Variations to it are possible. For example you can define a macro:
\def\Coordinates#1#2{%
\def\A{#1}
\def\B{#2}
\def\PointB{coordinates {(\A,\B)}}
}
This will also work, just call it from within \begin{axis}...\end{axis}.
-
Is there a way to make this work without including \addplot coordinates in the point macro? I'd like to be able to just call \PointA wherever the syntax (<x>,<y>) is expected, without having to worry about whether I'm specifying a node/path or a plot coordinate. – Jake Apr 16 '11 at 19:35
@Jake ...hard! What are you exactly trying to achieve, where are the co-ordinates originate from? – Yiannis Lazarides Apr 16 '11 at 19:44
I have to admit that there's no real problem I'm trying to solve. It simply sparked from the question I linked to (tex.stackexchange.com/questions/15684/…). I just find it hard to accept that in two situations that call for the exact same syntax for specifying a coordinate, I shouldn't be able to replace the string of characters (x,y) with a macro. The linked question makes the point that dimensions can be saved in macros and then used in different contexts, so why can't it be done for coordinates...? – Jake Apr 16 '11 at 19:55
@Jake I am sure if one traces back the parsing, it maybe possible with a number of expandafters, as you are looking at a very complicated parser that can do calcs etc. – Yiannis Lazarides Apr 16 '11 at 19:59
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2016-07-26 10:15:17
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